Equadiff 10: Czechoslovak international conference on differential equations and their applications, Prague, August 27-31, 2001 : papers

CZECHOSLOVAK INTERNATIONAL CONFERENCE ON DIFFERENTIAL EQUATIONS AND THEIR APPLICATIONS, PRAGUE, AUGUST 27 – 31, 2001 EQ...

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CZECHOSLOVAK INTERNATIONAL CONFERENCE ON DIFFERENTIAL EQUATIONS AND THEIR APPLICATIONS, PRAGUE, AUGUST 27 – 31, 2001

EQUADIFF 10 Papers edited by J. Kuben, J. Vosmanský

MASARYK UNIVERSITY PUBLISHING HOUSE

BRNO 2002

Published by the Masaryk University, Brno in co-edition with EQUADIFF 10, Mathematical Institute, Academy of Sciences of the Czech Republic, Prague e-mail: [email protected] website: http://www.math.cas.cz/~equadiff Edited by Jaromír Kuben and Jaromír Vosmanský Printed by the Masaryk University Press, Areál Kraví hora, 602 00 Brno First Edition 2002 55-972-2002 02/58 6/Př c Masaryk University, Brno, 2002 ISBN 80-210-2809-2

Preface This volume (PAPERS) is a part of the EQUADIFF 10 CD ROM and it was prepared in hypertext PDF format using LATEX 2ε . A limited number of copies is published also in the usual hard copy form. It contains 45 papers, accepted for presentation on the EQUADIFF 10 Conference in the form of communications, posters or extended abstracts and submitted for publication by the authors. A part of the articles is in the ﬁnal form and will not be published elsewhere, major part are preliminary versions. The expression “overview article” is used here in a rather broad sense and represents mostly the presentation of results already published or submitted for publication in one or more papers of the authors or a usual survey paper. This is mentioned in footnotes on the ﬁrst pages of papers on the base of the author’s opinion. The papers published in this volume (PAPERS) have not gone through a detailed referee process. The acceptance for the presentation at the Conference and a good response from the participants were considered as a proper evaluation, corresponding to this type of publication. We thank to Mrs. Milada Suchomelova from the Masaryk University, Brno, for the administrative work and e-mail correspondence connected with the preparation of this volume.

March 2002

Editors

3

Table of Contents

Diliberto’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ladislav Adamec (Masaryk University Brno)

9

Quasilinear Elliptic Dirichlet Problem in Nonregular Domains . . . . . . . . . . . . . 13 Darya E. Apushkinskaya (Saarland University), and Alexander I. Nazarov (St. Petersburg State University) Parallel realization of the Poisson-Boltzmann equation . . . . . . . . . . . . . . . . . . . . 17 Edik A. Ayrjan (LIT Dubna), Shura Hayryan and Chin-Kun Hu (Institute of Physics, Academia Sinica, Nankang), Imrich Pokorný (Technical University in Košice) and Igor V. Puzynin (LIT Dubna) Asymptotic behaviour of solutions of linear discrete equations . . . . . . . . . . . . . 21 Jaromír Baštinec (Brno University of Technology (VUT)), Josef Diblík (Brno University of Technology (VUT)) Geothermal Flow in Porous Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 Michal Beneš and Jiří Mikyška (Czech Technical University Prague) Nonlinear oscillators at resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 Denis Bonheure (Université Catholique de Louvain), Christian Fabry (Université Catholique de Louvain), Didier Smets (Université Catholique de Louvain) Landesman–Lazer Type Conditions and Quasilinear Elliptic Equations . . . . . . 45 Jiří Bouchala (VŠB-Technical University of Ostrava) Extremality results for diﬀusion equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 Alberto Cabada, José Angel Cid, and Rodrigo L. Pouso (Univ. of Santiago de Compostela) Attractors of nonautonomous inclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 Tomás Caraballo (Universidad de Sevilla), José Antonio Langa (Universidad de Sevilla), Valery S. Melnik (Institute of Applied System Analysis) and José Valero (Universidad Cardenal Herrera CEU) Approximation of attractors for MRDS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 Tomás Caraballo (Seville University), José A. Langa (Seville University), and José Valero (Cardenal Herrera CEU) Distributed Delayed Competing Predators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 Mario Cavani (Universidad de Oriente Cumaná)

Parabolic Diﬀerential Equations of Allen-Cahn Type in Image Processing . . . 83 Vladimír Chalupecký (Czech Technical University in Prague), and Michal Beneš (Czech Technical University in Prague) A general controllability theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 Veronika Chrastinová (Technical University Brno) Bifurcation of periodic points and normal form in reversible diﬀeomorphisms. 109 Maria-Cristina Ciocci (Universiteit Gent, Belgium.) Inﬁnitely Many Solitary Waves in Three Space Dimensions . . . . . . . . . . . . . . . . 119 Pietro d’Avenia and Lorenzo Pisani (Università degli Studi di Bari) Positive and oscillating solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 Josef Diblík (Brno University of Technology (VUT)) Inequalities for solutions of systems with “pure” delay . . . . . . . . . . . . . . . . . . . . 133 Josef Diblík (Brno University of Technology (VUT)), Denis Ja. Khusainov (Kiev University), and Violeta G. Mamedova (Kiev University) Solutions of a singular Cauchy problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 Josef Diblík (Brno University of Technology (VUT)), and Miroslava R˚ užičková (University of Žilina) An existence criterion of positive solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 Josef Diblík (Brno University of Technology (VUT)), Zdeněk Svoboda (Military Academy Brno(VA Brno)) Fieldless Methods for the Simulation of Induction Heating . . . . . . . . . . . . . . . . 143 Ivo Doležel, Institute of Electrical Engineering of the ASCR, Prague, Pavel Šolín, Institute of Electrical Engineering of the ASCR, Prague, and Bohuš Ulrych, Westbohemian University Pilsen Topological Properties of Nonlinear Evolution Equations . . . . . . . . . . . . . . . . . . 159 Vladimír Ďurikovič (Cyril and Methodius University) and Monika Ďurikovičová (Slovak Technical University) A doubly degenerate elliptic system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 María Teresa González Montesinos (University of Cádiz), and Francisco Ortegón Gallego (University of Cádiz) Thermo-elastic Hybrid Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 Marié Grobbelaar (University of Pretoria) Solvability of Some Higher Order Two-Point Boundary Value Problems . . . . . 183 Maria do Rosário Grossinho (ISEG, Universidade Técnica de Lisboa) and Feliz Manuel Minhós (Universidade de Évora) 6

Nonexistence of standing waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 Guzmán-Gómez Marisela (Autonomus Metropolitan University-Azc.) Heteroclinics for a class of fourth order conservative diﬀerential equations . . . 203 P. Habets (Institut de Mathématique Pure et Appliquée, Louvain la Neuve), L. Sanchez (Universidade de Lisboa), M. Tarallo (Università degli studi di Milano) and S. Terracini (Politecnico di Milano) About One Property of the Generalized Liénard Diﬀerential Equations . . . . . 217 Daniela Hricišáková Inertial Manifolds for Nonautonomous Dynamical Systems . . . . . . . . . . . . . . . . 221 N. Koksch (Technical University Dresden) and S. Siegmund (Georgia Institute of Technology, Atlanta) The Asymptotic Properties of Neutral Diﬀerential Equations . . . . . . . . . . . . . . 267 Dáša Lacková (Technical University Košice) A Class of Competing Models with Discrete Delay. . . . . . . . . . . . . . . . . . . . . . . . 275 Julio Marín (Universidad de Oriente) and Mario Cavani (Universidad de Oriente) Zero Convergent Solutions for a Class of p-Laplacian Systems . . . . . . . . . . . . . . 279 Mauro Marini, Serena Matucci (University of Florence), Pavel Řehák (Academy of Sciences of Czech Republic) Nonoscillatory Solutions for Nonlinear Discrete Systems . . . . . . . . . . . . . . . . . . . 289 Mauro Marini, Serena Matucci (University of Florence), Pavel Řehák (Academy of Sciences of Czech Republic) Mixed-hybrid Model of the Fracture Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 Jiří Maryška (Technical University of Liberec), Otto Severýn (Technical University of Liberec), and Martin Vohralík (Czech Technical University in Prague) Lp -estimates for solutions of Dirichlet and Neumann problems . . . . . . . . . . . . . 323 Alexander I. Nazarov (St. Petersburg State University) Multiple solutions for the Neumann problem for N. H. Inequalities . . . . . . . . . . 327 Nikolaos S. Papageorgiou (National Technical University of Athens) and George Smyrlis (National Technical University of Athens) Logistic Equation on Measure Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339 Zdeněk Pospíšil (Masaryk University Brno) Boundary stabilization of the Schrödinger equation . . . . . . . . . . . . . . . . . . . . . . . 349 Salah-Eddine Rebiai (Batna University) 7

Convergence, via summability, of formal solutions to Pfaﬃan systems . . . . . . . 357 Javier Sanz Computation and Continuation of Quasiperiodic Solutions . . . . . . . . . . . . . . . . 363 Frank Schilder and Werner Vogt (Technical University Ilmenau) Non-Uniqueness of Solution to Quasi-1D Compressible Euler Equations . . . . . 379 Pavel Šolín, Johannes Kepler University, Linz, and Karel Segeth, Mathematical Institute of the Academy of Sciences, Prague Some Remarks On the Terminal Value Problem in Hereditary Setting . . . . . . 391 Marko Švec Regularity of Minimizers in Optimal Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397 Delﬁm Torres (University of Aveiro, Portugal) Two-scale convergence with respect to measures in continuum mechanics . . . . 413 Jiří Vala (Technical University Brno) Irregular boundary value problems for ODE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437 Yakov Yakubov (Tel-Aviv University) Singular Solutions of the Briot-Bouquet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443 Hiroshi Yamazawa (Caritas College Yokohama) Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461

8

Equadiﬀ 10, August 27–31, 2001 Prague, Czech Republic

Equadiﬀ 10 CD ROM Papers, pp. 9–11

A Partial Generalization of Diliberto’s Theorem for Certain DEs of Higher Dimension. Ladislav Adamec Department of Mathematics, Faculty of Science, Masaryk University, Janáčkovo nám 2a, 662 95 Brno, Czech Republic, Email: [email protected]

Abstract. For analysis of bifurcation of planar systems is sometimes used a result ﬁrst obtained probably by Diliberto. This result is here partially extended to certain class of autonomous ordinary diﬀerential equations in R3 . MSC 2000. 37E99, 34C05, 34C30, 34D10 Keywords. Diﬀerential equations, bifurcations

1

Main results

Andronov and his coworkers were able, in their famous book [1], to derive an eﬀective theorem about continuation of periodic solutions of a two dimensional system. Chicone [2] achieved similar result but with substantially greater elegance using a result ﬁrst published by Diliberto [3]: Theorem 1. Let F = (p, q) be a C 1 2-vector function deﬁned on an open subset of R2 . Let (ϕ(t, x0 , y0 ), ψ(t, x0 , y0 )) be the solution of plane initial problem x˙ = p(x, y) y˙ = q(x, y)

x(0) = x0 , y(0) = y0 .

(1)

If |p(x0 , y0 )| + |q(x0 , y0 )| > 0, then the principal fundamental matrix Y (t) of (1) at t = 0 of the homogeneous variational equation u˙ =

∂p(ϕ(t, x0 , y0 ), ψ(t, x0 , y0 )) ∂p(ϕ(t, x0 , y0 ), ψ(t, x0 , y0 )) u+ v ∂x ∂y

This is the shortened preliminary version of the paper.

10

L. Adamec

v˙ =

∂q(ϕ(t, x0 , y0 ), ψ(t, x0 , y0 )) ∂q(ϕ(t, x0 , y0 ), ψ(t, x0 , y0 )) u+ v ∂x ∂y

is such that Y (t)F (x, y) = a(t, x, y)F ((ϕ(t, x, y), ψ(t, x, y)) + b(t, x, y)F (ϕ(t, x, y), ψ(t, x, y)) and Y (t)F (x, y) = F ((ϕ(t, x, y), ψ(t, x, y)), where b(t, x, y) =

p2 (x, y) + q 2 (x, y) × + q 2 (ϕ(t, x, y), ψ(t, x, y))

p2 (ϕ(t, x, y), ψ(t, x, y))

× exp 0

t

∂p(ϕ(s, x, y), ψ(s, x, y)) ∂q(ϕ(s, x, y), ψ(s, x, y)) + ds . (2) ∂x ∂y

In his book Chicone [2] was able to obtain an interesting geometrical identiﬁcation for the function b(t, x, y). Nowadays we are able to extend Diliberto’s result on many diﬀerential systems in Rn and the results will be published elsewhere. In this short announcement we shall limit ourselves to a three-dimensional system x˙ = f (x),

(3)

fulﬁlling the following hypotheses: H1 the function f : Rn → Rn is a C 1 function with open domain, H2 all solutions of (3) are deﬁned on [0, ∞), H3 the system (3) has a nondegenerate ﬁrst integral ([4, p.114]) g : Rn → Rn and g ∈ C2. If ϕ(t, x) is the solution of the initial problem (3), ϕ(t, 0) = x, then the ﬁrst integral g yields a two-dimensional submanifold M generated as the level set of g containing the point x. Clearly the unit surface normal n(x) := gradg(x)−1 gradg(x) is well deﬁned and is C 1 on M . Moreover we may suppose that H4 there are two C1 functions a1 (x), a2 (x) on M such that a1 (x) =a2 (x) = 1 and Tx M := span{a1 (x), a2 (x)}. Finally denoting the usual inner product as .|. and the usual cross-product as [.|.], we may state Diliberto’s theorem for three-dimensional systems (3) with a ﬁrst integral. Theorem 2. Let hypotheses H1, H2, H3, H4 be fulﬁlled and ϕ(t, x) denote the solution of the diﬀerential equation (3), ϕ(0, x) = x. If f (x) = 0, then the principal fundamental matrix Y (t) at t = 0 of the variational equation y˙ = Df (ϕ(t, x))y is such that Y (t)f (x) = f (ϕ(t, x)), Y (t)[n(x)|f (x)] = a(t, x)f (ϕ(t, x)) + b(t, x)[n(ϕ(t, x))|f (ϕ(t, x))],

11

Diliberto’s Theorem

and b(t, x) =

f (x)2 exp f (ϕ(t, x))2

t

a1 |(Df )a1

0

+ a2 |(Df )a2 − a1 |(Da1 )f + a2 |(Da2 )f (ϕ(s, x)) ds.

(4)

As an application let us present the following theorem concerning the Poincaré mapping of the system (3): Theorem 3. Let the hypotheses H1, H2, H3 and H4 be fulﬁlled. Let x1 ∈ M , f (x1 ) = 0 and x1 = ϕ(p, x1 ), where 0 < p < ∞ is the ﬁrst time with this property. Let Σ be a plane containing x1 and orthogonal to f (x1 ). Let Ψ : U ⊂ Σ → Σ be the Poincaré mapping. If v ∈ Tx1 Σ ∩ Tx1 M , then DΨ (x1 )v =

v1 |[n(x1 )|f (x1 )] b(p, x1 )[n(x1 )|f (x1 )]. f (x1 )2

References 1. A. A. Andronov, E. A. Leontovich, I. I. Gordon and A. G. Mayer: Theory of Bifurcation of Dynamical System on the Plane. John Wiley & Sons Inc., New York– London–Sydney, 1973. 2. C. Chicone: Ordinary Diﬀerential Equations with applications. Springer-Verlag, New York, 1999. 3. S. P. Diliberto: On Systems of Ordinary Diﬀerential equations. In Contributions to the Theory of Nonlinear Oscillations. Annals of Math. Studies 20, (1950), pp.1–38. Princeton: Princeton University Press. 4. Ph. Hartman: Ordinary Diﬀerential Equations., John Wiley & Sons Inc., New York– London–Sydney, 1964.

Equadiﬀ 10, August 27–31, 2001 Prague, Czech Republic

Equadiﬀ 10 CD ROM Papers, pp. 13–15

Quasilinear Elliptic Dirichlet Problem in Nonregular Domains Darya E. Apushkinskaya1 and Alexander I. Nazarov2 1

2

Department of Mathematics, Saarland University, Postfach 151150, D-66041 Saarbrücken, Germany Email: [email protected] Faculty of Mathematics and Mechanics, St. Petersburg State University, Bibliotechnaya pl. 2, Stary Peterhof, 198904 St. Petersburg, Russia Email: [email protected]

Abstract. We present the solvability result for the Dirichlet problem to nondivergent quasilinear elliptic equations of the second order in weighted Kondrat’ev spaces in the case when the boundary of a domain may include singularities — conical points or arbitrary codimensional edges.

MSC 2000. 35J25, 35B45, 35D05

Keywords. Dirichlet problem, quasilinear elliptic equations, domains with edges

We consider the boundary value problem −aij (x, u, Du)Di Dj u + a(x, u, Du) = 0

in Ω,

u|∂Ω = 0,

(1)

where Ω is a domain in Rn , (n 2), with compact closure Ω and with nonregular boundary ∂Ω. The term “nonregular” means that ∂Ω contains a (n − m)-dimensional submanifold M (an “edge” for m < n or a conical point for m = n), satisfying the following condition: for all x0 ∈ M there exist a neighborhood U (x0 ) ⊂ Rn and a diﬀeomorphism Ψ(x0 ) : U (x0 ) → Rn , such that

This work was partially supported by Russian Fund for Fundamental Research, grant no. 99-01-00684.

A note on results published in [1].

14

D. E. Apushkinskaya and A. I. Nazarov

(i) Ψ(x0 ) (U (x0 ) ∩ Ω) = {x ∈ Km (G) : |x | < ρ0 , |x | < ρ0 }. Here Km (G) = Km (G) × Rn−m , Km (G) stands for an open m-dimensional cone cutting on the unit sphere S m a domain G with smooth boundary, x = (x , x ), x ∈ Rm , x ∈ Rn−m and |x |, |x | denote corresponding Euclidean norms. Note also that G depends on x0 while ρ0 1 does not depend. (ii) Ψ(x0 ) (U (x0 ) ∩ ∂Ω) = {x ∈ ∂Km (G) : |x | < ρ0 , |x | < ρ0 }, (iii) Ψ(x0 ) (x0 ) = 0,

0 Ψ(x 0 ) (x ) = In ,

−1 (iv) the norms of Jacobians Ψ(x 0 ) (x) and (Ψ(x0 ) ) (Ψ(x0 ) (x)) are bounded uniformly with respect to x0 ∈ M and x ∈ U (x0 ), , x < θ < π } for all x0 ∈ M, and θ does not depend (v) Km (G) ⊂ {x ∈ Rm : x 1 2 0 on x . Setting d(x) = dist{x, M} we introduce the scale of weighted spaces Lr,(α) (Ω) with the norm |||u|||r,(α),Ω = u · (d(x))α Lr (Ω) ,

and the scale of Kondrat’ev spaces V2r,(α) (Ω) with the norm |||u|||V2r,(α) (Ω) = |||D(Du)|||r,(α),Ω + |||Du · (d(x))−1 |||r,(α),Ω + |||u · (d(x))−2 |||r,(α),Ω . Finally, the notation ∂Ω ∈ V2r,(α) with α < 1 − n/r is understood as follows: 2 ; 1) ∂Ω \ M ∈ Wr,loc 2) for all points x0 ∈ M the matrix D2 Ψ(x0 ) belongs to Lr,(α) (U (x0 )). Moreover the norms D2 Ψ(x0 ) r,(α) are bounded uniformly with respect to x0 ∈ M.

Assume that (aij ) in (1) is a symmetric matrix and the following natural structure conditions hold for all x ∈ Ω, z ∈ R1 , p ∈ Rn : ν|ξ|2 aij (x, z, p)ξi ξj ν −1 |ξ|2 ,

∀ξ ∈ Rn ,

|a(x, z, p)| µ|p| + b(x)|p| + Φ1 (x), 2

ν = const > 0,

µ = const > 0,

(A0) (A1)

b, Φ1 ∈ Lr,(α) (Ω), α < 1 − n/r, n < r < ∞; (2) ij ij ij ∂a (x, z, p) µ , ∂a (x, z, p) pk + ∂a (x, z, p) µ|p| + Φ2 (x), (A2) 1 + |p| ∂pk ∂z ∂xk Φ2 ∈ Lq,(α1 ) (Ω),

α1 < 1 − n/q,

n < q < ∞.

(3)

ν) Before stating the main result we need to introduce some notations. Let θ(θ, be the solution of the equation π = ν · ctg(θ), ctg(θ) θ ∈ 0, . 2 Let also Λ(m, θ) be the ﬁrst eigenvalue of the Dirichlet problem for the Laplace , x < θ} ∩ S m , while ω Bel’trami operator on the spherical “cap” {x ∈ Rm : x be 1 2 a positive solution of the equation ω + (m − 2)ω − Λ = 0.

Quasilinear Elliptic Dirichlet Problem in Nonregular Domains

15

Theorem 1 (Solvability in weighted spaces). Let the following conditions be fulﬁlled:

(a) r > max n, n−m α∈ 2− m , 1 − nr , ∂Ω ∈ V2r,(α) , ω −1 , r −ω (b) for all solutions u[τ ](·) ∈ V2r,(α) (Ω), τ ∈ [0, 1], of the family of problems: τ (−aij (x, u, Du)Di Dj u + a(x, u, Du)) − (1 − τ )∆u = 0 in Ω, u|∂Ω = 0

(4)

the estimate u[τ ] (·)Ω M0 holds true, (c) the conditions (A0)—(A2), (2)—(3) are fulﬁlled for |z| M0 , (d) the function a(·, z, p) regarded as an element of the space Lr,(α) (Ω) is continuous with respect to (z, p). Then for all τ ∈ [0, 1] the problem (4) has a solution u [τ ](·) ∈ V2r,(α) (Ω). In particular, u [1] (·) is a solution of the problem (1). For details and proof we refer the reader to [1].

References 1. D. E. Apushkinskaya and A. I. Nazarov, The Dirichlet problem for quasilinear elliptic equations in domains with smooth closed edges, (in Russian) Probl. Mat. Anal., No 21, (2000), 3–29; English transl. in J. Math. Sciences 105, No 5 (2001), 2299–2318.

Equadiﬀ 10, August 27–31, 2001 Prague, Czech Republic

Equadiﬀ 10 CD ROM Papers, pp. 17–20

Parallel realization of the ﬁnite diﬀerence method solution of the Poisson-Boltzmann equation Edik A. Ayrjan1 , Shura Hayryan2, Chin-Kun Hu3 , Imrich Pokorný4 and Igor V. Puzynin5

4

1 Laboratory of Information Technology, JINR, Dubna, 141980 Moscow region, Russia Email: [email protected] 2 Institute of Physics, Academia Sinica, Nankang, Taipei 11529, Taiwan Email: [email protected] 3 Institute of Physics, Academia Sinica, Nankang, Taipei 11529, Taiwan Email: [email protected] Department of Metal Forming FM, TU in Košiciach, Letná 9, 042 00, Košice, Slovakia Email: [email protected] 5 Laboratory of Information Technology, JINR, Dubna, 141980 Moscow region, Russia Email: [email protected]

Abstract. We demonstrate the parallel realization of ﬁnite diﬀerence solution of one problem by successive overrelaxation method (SOR) for the small number of processors. The solution was realized on supercomputer SPP2000 (Dubna, Russia). MSC 2000. 65Y05, 65M06, 65M55 Keywords. Parallel programming, ﬁnite diﬀerence, successive overrelaxation method, Poisson-Boltzmann equation

The paper is related to with the elaboration of eﬀective numerical algorithms for the multigrid solution of the following nonlinear Poisson-Boltzmann equation (PB) in the complex area: ∇[ε(r)∇φ(r)] − ε(r)κ(r)2 sinh[φ(r)] + This is the preliminary version of the paper.

4πρ(r) = 0. kT

18

E. A. Ayrjan, S. Hayryan, Ch. K. Hu, I. Pokorný and I. V. Puzynin

This equation describes a model of the energy of the electrostatic ﬁeld produced by the electrical charge placed at the center of some atoms of the protein molecule, where φ(r) is the dimensionless electrostatic potential in units of kT /q (k is the Boltzmann constant, T is the absolute temperature, and q is the charge on a proton), ε(r) is the dielectric constant, and ρ(r) is the ﬁxed charge density (in proton charge units). The term κ = 1/λ, where λ is the Debye length. The variables φ, ε, κ, and ρ are all functions of the position vector r. We solve this problem by the multigrid ﬁnite diﬀerence successive overrelaxation method (SOR). Iterations are stopped when the relative error in solution is less than some δ. Computations are organized using the multigrid approach by using the sequence of grids. This method provides a quick convergence as well as a better control and analysis of obtained approximations. The ﬁrst problem is solved on the mesh with step size h=1A by SOR iteration process. The obtained solution is interpolated on the mesh with a half grid size and a new iteration is performed. Similarly the solution on the ﬁne grid with a mesh size of 0.25A is obtained. The program PBSOLVE is constructed for numerical solution of the linearized Poisson - Boltzmann equation. Finite diﬀerence discretization and SOR iterations [1] on the sequence of grids (multigrid) are used to obtain the approximate electrostatic potential on the grid. The error control is provided by comparison of solutions on nested grids. A parallel version of the PBSOLVE program was written in the programming language FORTRAN77 by using MPI (Message Passing Interface) [6], and we used the multiprocessor computer system SP2000 with 8 processors. The improvement of the program is related to the treatment of the solutesolvent interface boundary. Two ways are possible to improve the mapping of the interface boundary onto the grid. The former is a multilevel mesh reﬁnement technique [2] . The method is intended for accurate electrostatic calculations over domains with local mesh reﬁnement patches. The latter is the exploitation of well-developed ﬁnite [3] and boundary element [4] techniques to complex molecular surface. To account exactly the behavior of the potential on the inﬁnity, the coupled boundary integral ﬁnite elements formulations of the problem [5] will be elaborated. A natural way of parallelization is to divide the area into the p peaces by planes (we used horizontal ones because we used FORTRAN), where p is the number of used processors. First, the processors work step by step. The other way of parallelization might be rearrange of cycles [7]. We obtain our results for the method inaccuracy 1, 0.25, and 0.0625; calculation inaccuracy - 10−4 ; and the best time ratio is for the 3 processors (2.82). This ratio decreases for more than 3 processors because the number of data transmits increases. This result we obtained when our program had worked in the non-solo regime. For the solo regime we obtained the following interesting result (see Tab. 1. for the molecule with 295 atoms): For the purpose of this paper we take the cubic solution area with the three equidistant grids (number of knots was 41, 81, and 161). Our parameters and calculate times are shown in the following table:

19

Parallel realization of the Poisson-Boltzmann equation

N. of proc. 1 2 3 4 5 6 7 Ast. time∗ 850.98 461.94 331.93 263.50 210.89 191.80 169.29 User time∗ 848.88 459.43 328.82 261.10 208.47 188.94 165.81 ∗∗ st 1 grid 97 97 97 96 96 95 95 ∗∗ nd 2 grid 165 165 163 166 165 166 167 ∗∗ rd 3 grid 309 308 311 312 312 316 316 Time rel. 1.00 1.84 2.56 3.23 4.05 4.44 5.03 ∗ ∗∗

8 172.86 155.74 95 167 319 4.92

The time is in sec. Number of iterations. Tab. 1.

The astronomic time of calculation is shown in the second line of Tab. 1. This time was obtained by MPI function MPI Wtime() which show the time between the start and the end of calculation by clock. Also this time is not a real calculation time of processors. A bit close to such time is a user time. We can obtain the user time by the function itimes(i) which was written by Sapozhnikov A. P. and Sapozhnikova T. F. in the programming language C. The diﬀerence between these times is based on the fact that our processor works with other system processes and sometimes it must wait. Numbers of iterations for our grides are shown in the 4th , 5th , and 6th lines respectively. Let the number of iterations for one processor be n . By our method the numbers of iterations for p processors (p = 1) can be at most n + p − 1 for the good convergence. In the real process this number sometimes is decreased. It is natural because when the last processor makes ith iteration the previous processors make (i + j)th iteration, where j is a diﬀerence of order numbers of the last processor and the actual one. The result of solving process for one processor can be diﬀerent from the result for several processors: – by the reason of previous paragraph (some part of data is from the highest iteration - for each processor except the last), – if the process converges slower for some data which are not in the last processor working area - the number of iterations can decrease. By the last line of Tab. 1. we can see that the ratio of times is increasing. This ratio is shown graphically on the following ﬁgure 1. The correctness of the algorithm was tested by solving the PB equation for a single spherical charge for which the exact solution is known. The performance of the algorithm was tested on the small peptide Metenkephalin. Acknowledgment. This research was supported by the grant VEGA 2/7192/20 and the grant RFFI 01-01-00726.

20

E. A. Ayrjan, S. Hayryan, Ch. K. Hu, I. Pokorný and I. V. Puzynin

y

✻

q

5 q

4

q

q

3

q

2 1

q

q q

0 1

2

3

4

5

6

7

8

✲ x

Fig. 1.

References 1. L. A. Hageman and D. M. Young, Applied iterative methods, Academic Press, New York, 1981. 2. D. Bai and A. Brandt, SIAM J. Sci. Stat. Comput., 8, 109, 1987. 3. O. C. Zienkiewicz, The ﬁnite element method, London, McGraw-Hill, 1977. 4. C. A. Brebbia, J. C. F. Telles, and L. C. Wrobel, Boundary element techniques: theory and applications in engineering, Springer-Verlag, 1984. 5. E. A. Ayrjan, E. P. Zhidkov a.o., Numerical algorithms for accelerators magnetic systems calculation, Particle Phys. Nucl, 1991. 6. M. Snir, S. W. Otto, S. Huss-Lederman, D. W. Walker, and J. Dongarra, MPI: The complete reference, The MIT Press, Massachusetts Institute of Technology, Cambridge (USA), 1997. 7. L. Gregushova, Method of paralleling of cycles for multiconveier numerical systems, (Russian), JINR, p11-87-900, Dubna, 1987.

Equadiﬀ 10, August 27–31, 2001 Prague, Czech Republic

Equadiﬀ 10 CD ROM Papers, pp. 21–23

Asymptotic behaviour of solutions of linear discrete equations Jaromír Baštinec1 and Josef Diblík2 1 Department of Mathematics, Faculty of Electrical Engineering and Computer Science, Brno University of Technology (VUT), Technická 8, 616 00 Brno, Czech Republic Email: [email protected] 2 Department of Mathematics, Faculty of Electrical Engineering and Computer Science, Brno University of Technology (VUT), Technická 8, 616 00 Brno, Czech Republic Email: [email protected]

Abstract. Asymptotic behaviour of a particular solutions of the linear discrete nonhomogeneous equation ∆u(k) = A(k)u(k) + g(k), k ∈ N (a) is considered, where ∆u(k) = u(k + 1) − u(k), N (a) = {a, a + 1, . . . }, a ∈ N is ﬁxed, N = {0, 1, . . . } and A, g : N (a) → R. MSC 2000. 39A10, 39A11 Keywords. Linear discrete equation, asymptotic formulae, oscillating solution

Let us consider the linear discrete nonhomogeneous equation ∆u(k) = A(k)u(k) + g(k), k ∈ N (a)

(1)

where ∆u(k) = u(k+1)−u(k), N (a) = {a, a+1, . . . }, a ∈ N is ﬁxed, N = {0, 1, . . . } and A, g : N (a) → R. Suppose A(k) = 0 for every k ∈ N (a). This is the preliminary version of the paper.

22

J. Baštinec and J. Diblík

Let us construct a formal series which satisﬁes equation (1). Deﬁne a sequence of functions f0 (k), f1 (k), . . . , fn (k), . . . , k ∈ N (a), as follows: f0 (k) = −

g(k) , A(k)

fp (k) =

∆fp−1 (k) , k ∈ N (a) A(k)

where p = 1, 2, . . . . Obviously, this sequence is well deﬁned for every k ∈ N (a). Deﬁne a formal series F S(k) := f0 (k) + f1 (k) + · · · + fn (k) + · · · .

(2)

Lemma 1. Suppose A(k) = 0 for every k ∈ N (a). Then the formal series F S(k) deﬁned by relation (2) is a formal solution of equation (1). Theorem 2. [1] Let us suppose that for every k ∈ N (a) and a ﬁxed p ∈ {0} ∪ N : 1) A(k) = 0. 2) fp+1 (k) < 0, ∆fp (k) < 0 and ∆fp+1 (k) > 0. Then there exists a particular solution upart = upart (k), k ∈ N (a) of the discrete linear nonhomogeneous equation (1) such that the inequalities p+1

fs (k) 0, ∆fp (k) > 0 and ∆fp+1 (k) < 0. Then there exists a particular solution upart = upart (k), k ∈ N (a) of the discrete linear nonhomogeneous equation (1) such that the inequalities p s=0

fs (k) < upart (k) <

p+1

fs (k)

s=0

hold for every k ∈ N (a). Example 4. Let us consider a linear discrete equation ∆u(k) = k 5 u(k) − k 6 .

(3)

In accordance with Theorem 3 (p = 0) there exists a particular solution upart = upart (k), k ∈ N (1) of the equation (3) such that the inequalities k < upart (k) < k + hold for every k ∈ N (1).

1 k5

Asymptotic behaviour of solutions of linear discrete equations

23

Acknowledgment. This investigation was supported by the grant 201/01/0079 of Czech Grant Agency (Prague) and by the Council of Czech Government MSM 2622000 13 of the Czech Republic.

References 1. J. Baštinec and J. Diblík, Asymptotic Formulae for Particular Solution of Linear Nonhomogeneous Discrete Equations, Computers and Mathematics with Applications (in the print). 2. J. Diblík, Retract principle for diﬀerence equations, Communications in Diﬀerence Equations, Proceedings of the Fourth International Conference on Diﬀerence Equations, Poznan, Poland, August 27–31, 1998. Eds.: S. Elaydi, G. Ladas, J. Popenda and J. Rakowski, Gordon and Breach Science Publ., 107–114, 2000.

Equadiﬀ 10, August 27–31, 2001 Prague, Czech Republic

Equadiﬀ 10 CD ROM Papers, pp. 25–35

Geothermal Flow in Porous Media Michal Beneš and Jiří Mikyška Czech Technical University in Prague, Faculty of Nuclear Sciences and Physical Engineering, Trojanova 13, 120 00 Prague 2, Czech Republic Email: [email protected] Email: [email protected]

Abstract. This contribution deals with numerical modelling of the geothermal ﬂow of groundwater in the vicinity of sources of thermal energy. The problem is described by a set of two partial diﬀerential equations — the heat transport equation with convection and the equation for pressure. These equations are coupled together in terms of dependency of density of the ﬂuid on temperature. As the density is assumed to be dependent on temperature only, the equation for pressure is of the elliptic type, even in the non-stationary case. The resulting system of equations is thus of parabolic-elliptic type. A suitable numerical scheme for approximation of solution to this system is proposed and it is tested on several numerical experiments which are presented in the conclusion.

MSC 2000. 65M60 65M25 76S05 Keywords. density-driven ﬂow, convection-diﬀusion equations, method of characteristics, ﬁnite element discretization

1

Introduction

Let Ω is a bounded domain in R2 representing a vertical cut through the soil which is fully saturated by water. In this domain, we require fulﬁlment of mass balance condition : ∂ρqz ∂ρ ∂ρqx + + = Q. (1) n ∂t ∂x ∂z

This work has been partly supported by the projects MSM 98/210000010 of the Czech Ministry of Education and 201/01/0676 of the Grant Agency of the Czech Republic.

This is an overview article summarizing degree thesis [5].

26

M. Beneš and J. Mikyška

Here, n denotes porosity, ρ is the density of the ﬂuid, qx and qz are components of the Darcy velocity vector @ q , and ﬁnally, Q is a source/sinks term. Equation (1) will be reﬀered as mass balance equation in the sequel. This equation was derived in [5] under assumption that the porosity is given function of spatial variables only — i.e. the soil is incompressible. In the equation (1), the dependency of @q on pressure p is explicitly given by the Darcy law: k (2) q = − (∇p − ρ@g), @ µ where µ denotes the coeﬃcient of dynamical viscosity of the ﬂuid and @g is a vector of gravity acceleration. The quantity k is the permeability of the porous medium which is assumed to be isotropic. In the whole text, we will suppose that the coordinate system (Oxz) is oriented so that the vector of gravitational acceleration points in the direction of negative part of z-axis. So @g = (0, g), where g = −9.81 m.s−2 . Under these assumptions the Darcy law can be written in components as k ∂p k ∂p , qz = − − ρg . (3) qx = − µ ∂x µ ∂z Substituting the Darcy law to the mass balance equation, we get single equation ∂ ∂ρ k ∂p ∂ k ∂p k 2 − − ρ g =Q−n , (4) ρ − ρ ∂x µ ∂x ∂z µ ∂z µ ∂t which will be denoted as equation for pressure in the whole text. Moreover, we add the equation of heat transport which is taken from [2] ρc

∂T + ρc@ q · ∇T = ∇ · (λ∇T ). ∂t

(5)

In this equation, T is the unknown temperature, c is the heat capacity per unit mass of the ﬂuid and λ denotes the homogenized coeﬃcient of heat conductivity. We suppose that the heat conductivity is scalar (i.e. the medium is isotropic), however, the extension for anisotropic case is possible. We use notation ρc = nρc + (1 − n)ρs cs , where ρs and cs denote the density and the heat capacity per unit mass of the soil, respectively. The equations are coupled by the dependency of the density of the ﬂuid on temperature ρ(T ) =

ρ0 , 1 + β1 (T − T0 ) + β2 (T − T0 )2

(6)

in which the coeﬃcients ρ0 ≡ ρ(T0 ), β1 , β2 , are given constants. Additionally, the density could depend on pressure, but this case is not considered here. As we assumed that neither density nor porosity depend on pressure, the equation for pressure (4) is an elliptic partial diﬀerential equation with respect to pressure. Therefore, we prescribe boundary conditions of the Dirichlet, Neumann or

27

Geothermal Flow in Porous Media

Newton type, i.e. p(x, z, t) = f1 (x, z, t)

∀t and ∀(x, z) ∈ S1 .

(ρqx nx + ρqz nz )(x, z, t) = f2 (x, z, t) ∀t and ∀(x, z) ∈ S2 . ∀t and ∀(x, z) ∈ S3 , (ρqx nx + ρqz nz )(x, z, t) = β(p − pout )(x, z, t)

(7) (8) (9)

Here pout denotes pressure at the outer side of boundary, β is the coeﬃcient of proportionality and the @n = (nx , nz ) denotes the unit vector of outer normal. The heat transport equation (5) is parabolic partial diﬀerential equation and therefore we supply one initial condition and the boundary conditions of the Dirichlet, Neumann or Newton type, i.e. ∀(x, z) ∈ Ω, T (x, z, 0) = T 0 (x, z) ∀t and ∀(x, z) ∈ S4 . T (x, z, t) = f4 (x, z, t) ∂T ∀t and ∀(x, z) ∈ S5 . −λ (x, z, t) = f5 (x, z, t) ∂@n ∂T ∀t and ∀(x, z) ∈ S6 , −λ (x, z, t) = γ(T − Tout )(x, z, t) ∂@n

(10) (11) (12) (13)

where Tout is the outer temperature and γ > 0 is the heat transfer coeﬃcient. The problem will be correctly formulated if the S1 , S2 , S3 and S4 , S5 , S6 are two (generally diﬀerent) decompositions of boundary ∂Ω.

2

Weak formulation

In the sequel, we will suppose that Ω is a bounded domain with the Lipschitz boundary. Let us introduce the space VT = {f ∈ C ∞ Ω : f |S4 = 0} and the enthalpy by the following deﬁnition T ρc(τ ) dτ.

H(T ) =

(14)

0

Moreover, let us denote the time interval (0, Θ) as I. At this moment, we are ready for the following deﬁnition. Suppose the following input data qualiﬁcation : ρs , cs , n, λ ∈ L∞ (Ω), c > 0, n ∈ (0, 1), qx , qz ∈ L2 (I; L2 (Ω)), f4 ∈ L2 (I; L1/2 (S4 )), f5 ∈ L2 (I; L2 (S5 )), γ ∈ L∞ (S6 ), Tout ∈ L2 (I; L2 (S6 )), T 0 ∈ W21 (Ω). Then we say that the mapping T ∈ L2 (I; W21 (Ω)) is the weak solution of the equation of heat transport if the following conditions hold T (0) = T 0 T (t)|S4 = f4 and the integral identity

a.e. in Ω,

(15)

a.e. in I,

(16)

28

M. Beneš and J. Mikyška

d (H(T ), v) + (ρc@ q · ∇T, v) + (λ∇T, ∇v) + dt

γT v dS S6

=−

f5 v dS + S5

γTout v dS

(17)

S6

holds for all v ∈ VT in D (I). Further, let us deﬁne the space of the functions which fulﬁll the homogeneous stable boundary conditions in sense of traces Vp = {f ∈ W21 (Ω) : f |S1 = 0}. At this moment, we can formulate the following deﬁnition. Let µ is a positive constant. Suppose the following input data qualiﬁcation : k ∈ L∞ (Ω), f2 ∈ L2 (I; L2 (S2 )), β ∈ L∞ (S3 ), pout ∈ L2 (I; L2 (S3 )) and Q,

∂ρ ∈ L2 (I; L2 (Ω)). ∂t

Suppose that there exists a function pΩ ∈ W21 (Ω) such that pΩ |S1 = f1 . Then we say that the mapping p ∈ L2 (I; W21 (Ω)) is the weak solution of the equation for pressure if a.e. in I, p(t) − pΩ ∈ Vp and the integral identity k ∂w k ρ ∇p · ∇w − ρ2 g dxdz + βpw dS µ µ ∂z Ω

= (Q, w) −

S3

f2 w dS + S2

∂ρ βpout w dS − n , w ∂t

(18)

S3

holds for all w ∈ Vp in D (I).

3

Discretization and numerical algorithm

We suggest the following combination of method of characteristics and the standard Galerkin scheme for approximation of heat transport equation (5) (A + ∆tk B)T(k+1) = AT(k) ◦ ϕ(k) + ∆tk F, where

Aij =

ρcNi Nj dxdz, Ω

∂Ni ∂Nj ∂Ni ∂Nj + dxdz + γNi Nj dS, ∂x ∂x ∂z ∂z Ω S6 Fi = − f5 Ni dS + γTout Ni dS.

Bij =

λ

S5

S6

(19)

29

Geothermal Flow in Porous Media

and the function ϕ(k) (x) := x − ∆tk ωh ∗

ρc@ q (tk−1 , x) nρc + (1 − n)ρs cs

(20)

is the Euler explicit approximation of characteristics. According the [3], the velocity ﬁeld has to be smoothed by convolution with molliﬁer ωh which is deﬁned as   κ exp |x|2 , for|x| < 1 1 x |x|2 −1 ωh (x) = 2 ω1 ( ), ω1 (x) = (21) 0 h h for|x| ≥ 1, and the coeﬃcient κ is chosen so that the integral of ω1 over the whole support is unitary. This smoothing guarantees that the approximated characteristics do not intersect each other. This scheme is nothing but the implicit standard Galerkin approximation of the heat transport equation without convection which is included in the right hand side in terms of method of characteristics. As the Galerkin scheme solves just the diﬀusion problem there are no problems with the artiﬁcial oscilations in the case of dominant convection. The matrix of the system of linear algebraical equations is symmetric and if we use the mass lumping technique then it will be diagonally dominant and therefore also positively deﬁnite which is advantageous for the numerical solution of this linear algebraical system. The equation for pressure (4) is discretized by the standard Galerkin approach. The same triangulation and the same linear basis functions as in the case of heat transport equation are used. Assuming that the heat transport equation has been solved before we can approximate the time derivative of density on the right hand side of (18) by the forward diﬀerence. The standard Galerkin discretization results to the following algebraical system for the unknown pressures at the nodes of the mesh in time tk+1 Ap(k+1) = F, (22) where ∂Ni ∂Nj k ∂Ni ∂Nj Aij = ρ + (23) dxdz + βNi Nj dS, µ ∂x ∂x ∂z ∂z Ω S3 ∂ρ k 2 ∂Ni Fi = QNi − n Ni + ρ g dxdz − f2 Ni dS + βpout Ni dS ∂t µ ∂z

Ω

S2

S3

are the coeﬃcents of the matrix A and vector F. As the pressure is approximed linearly on each element the pressure gradients are element-wise constant and thus, the Darcy velocity can be easily determined in terms of (3). Assume, that we are situated in the k−th time level and all the quantities at time tk are known - either from the initial condition or from the previous time step. The values of all examined quantities in time tk+1 can be obtained using the following steps:

30

M. Beneš and J. Mikyška

– Step 1. The solution of the heat transport equation with the initial condition T(k) gives us new distribution of temperature - T(k+1) . – Step 2. The new layer of densities is obtained by substitution of the new temperatures to (6). – Step 3. Solve the equation for pressure in order to obtain the new layer of pressures - p(k+1) . The time derivative of pressure on the right hand side of this equation can be approximated by the backward diﬀerence because we know the values of densities in time tk and tk+1 . – Step 4. Compute new Darcy’s velocities using (3). At this moment, we computed values of all the required quantities in time tk+1 . We can repeat this procedure from the step one or to stop the process if the simulation time is up.

4

Results

We simulate groundwater ﬂow in a rectangular domain of size 100 × 30 m. At the begining, the temperature inside of the considered domain is 10 ◦ C and the water does not move. The boundary conditions and the material properties will be described individualy depending on the problem solved. Problem 1. 1DirHill : Suppose that the soil in Ω is homogeneous and isotropic. The top and bottom boundary of Ω are impermeable. On the top part of boundary and on the sides, the zero heat ﬂux is prescribed. On the left and right side of the boundary, we prescribe the hydrostatic pressure. The water in domain Ω is heated from bottom - in this case we prescribe the temperature on the bottom side of the ∂Ω. This temperature grows up linearly from 10 ◦ C on the sides to the 90 ◦ C in the middle of the bottom part of the boundary ∂Ω. In the ﬁgure 1, we can see the situation in time of 1000 days. The colours represent the temperatures — white colour belong to the 10 ◦ C and black represents the 90 ◦ C. The other colours are associated with the temperatures in between. The arrows show distribution of the Darcy velocities. Problem 2. 3DirHill : This is an example of the similar problem as described in the previous subsection. The only diﬀerence is that the temperature on the bottom part of the ∂Ω is given by a cosinus function of x-coordinate such that the temperature changes between 10 and 90 ◦ C and has three maxima along the boundary. In the regions of these maxima, we can observe the ﬂow to the top of the aquifer where the water is cooled and then it ﬂows toward the bottom boundary in the regions of the minima of the temperature. The situation in Ω in time of 1000 days is shown in the ﬁgure 2. This example is useful for comparison with the problems 1Well and HorLayer whose results are shown in ﬁgures 3 and 4. The ﬁrst of the ﬁgures shows the same situation with added pumping while in the second ﬁgure there is a situation in which we added a horizontal layer of soil in which the permeability and the coeﬃcient of heat conductivity are higher than in the rest of the aquifer.

31

Geothermal Flow in Porous Media 0 −5 −10 −15 −20 −25 −30 −50

−40

−30

−20

−10

0

10

20

30

40

50

20

30

40

50

20

30

40

50

Fig. 1. Problem 1DirHill.

0 −5 −10 −15 −20 −25 −30 −50

−40

−30

−20

−10

0

10

Fig. 2. Problem 3DirHill

0 −5 −10 −15 −20 −25 −30 −50

−40

−30

−20

−10

0

10

Fig. 3. Problem 1Well.

32

M. Beneš and J. Mikyška

0 −5 −10 −15 −20 −25 −30 −50

−40

−30

−20

−10

0

10

20

30

40

50

Fig. 4. Problem HorLayer.

5

Analysis of the convergence

In this section, the convergence of the proposed numerical scheme is examined. The problem 1DirHill was chosen as a suitable problem used for testing. As the exact solution is not available, we computed a numerical solution on mesh 250 × 75 with timestep 1 day and this solution was used for the comparison. Then, the additional solutions of the same problem were computed on coarser meshes with larger timesteps. Our task was to measure the distances of the individual solutions from the solution on the ﬁnest mesh in the norms of several function spaces. The main problem is that we have to interpolate on two diﬀerent meshes. This is solved by the point-wise projection of the solution from the coarse mesh to the ﬁner mesh. Then, the computation of the norms of the diﬀerence is done on the ﬁner mesh. The results of the convergence analysis are contained in the tables 1 and 2. In the table 1, the symbol || · ||X denotes the X −norm of the diﬀerence of the temperature on the mesh 250 × 75 and the temperature projected from the coarser mesh to the 250 × 75−mesh. The same symbol used in table 2 has the analogous meaning, only the pressure diﬀerence is measured instead of the temperature. # Mesh Timestep || · ||L∞ (I;L2 (Ω)) || · ||L∞ (I;W 1 (Ω)) || · ||L∞ (I;L∞ (Ω)) 2

1 50 × 15

25 days

2 100 × 30

5 days

3 150 × 45 2 12 days 4 200 × 60 1 13 days

28.5641

43.1245

3.18548

8.33552

13.4872

0.84721

5.41337

5.89123

0.49784

4.09549

4.20282

0.39215

Table 1. The results of the convergence analysis for temperature

33

Geothermal Flow in Porous Media # Mesh Timestep || · ||L∞ (I;L2 (Ω)) || · ||L∞ (I;W21 (Ω)) || · ||L∞ (I;L∞ (Ω)) 1 50 × 15

25 days

438.598

658.184

66.8801

2 100 × 30

5 days

170.488

207.186

17.9188

days

108.380

124.451

9.99112

days

81.5763

93.0700

7.20534

3 150 × 45 4 200 × 60

2 12 1 13

Table 2. The results of the convergence analysis for pressure The log-log plots of the norms contained in the previous tables as a function of the mesh size are presented in the ﬁgures 5 and 6. The slope of the curves allows to estimate the experimental orders of the convergence (EOC s) for temperature and pressure in the norms of the corresponding function spaces. The EOC between two triangulations with the mesh sizes h1 and h2 is deﬁned as in [1] by EOC =

log E(h1 ) − log E(h2 ) , log h1 − log h2

(24)

2 "T.L2" "T.H1" "T.Linf" 1.5

1

0.5

0

-0.5 -0.3

-0.2

-0.1

0

0.1

0.2

0.3

Fig. 5. Log-log plot of the convergence curves for temperature

0.4

34

M. Beneš and J. Mikyška 3 "P.L2" "P.H1" "P.Linf"

2.8 2.6 2.4 2.2 2 1.8 1.6 1.4 1.2 1 0.8 -0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

Fig. 6. Log-log plot of the convergence curves for pressure

where the symbol E(h) denotes some of those norms of the diﬀerence of the solution on h-mesh and the ﬁnest mesh. The experimental orders of the convergence between the individual triangulations are summarized in the table 3.

EOC of ↓ between → #1 → #2 #2 → #3 #3 → #4 T in || · ||L∞ (I;L2 (Ω))

1.75

1.06

0.96

p in || · ||L∞ (I;L2 (Ω))

1.34

1.11

0.98

T in || · ||L∞ (I;W 1 (Ω))

1.65

2.03

1.17

p in || · ||L∞ (I;W21 (Ω))

1.64

1.25

1.00

T in || · ||L∞ (I;L∞ (Ω))

1.88

1.30

0.82

p in || · ||L∞ (I;L∞ (Ω))

1.87

1.43

1.13

2

Table 3. Experimental orders of the convergence for temperature and pressure

Geothermal Flow in Porous Media

6

35

Conclusion

The presented results show that the heat transport processes in porous media are relatively slow. The heat transport can be substantially faster in the fractures, so we intend to add the model of fracture ﬂow to the current model.

References 1. K. Deckelnick, G. Dziuk : Convergence of Numerical Schemes for the Approximation of Level Set Solutions to Mean Curvature Flow, ISSN 1439-961X, Preprint Nr. 17/2000-29.05.2000, Mathematische Fakultät, Freiburg, Germany. 2. U. Hornung : Homogenization and Porous Media, (Interdisciplinary Applied Mathmatics, Vol. 6), Springer-Verlag, New York, 1997. 3. J. Kačur : Application of Relaxation Schemes and Method of Characteristics to Degenerate Convection-Diﬀusion Problems, In W. Jäger, J. Nečas, O. John, K. Najzar, and J. Stará, editors, Partial Diﬀerential Equations - Theory and Numerical Solutions, pages 199–213, New York, 2000. 4. M. Beneš, J. Mikyška : Numerical Analysis of Thermal Flow and Heat Transport in Porous Media, in proceedings of SIMONA 2000, pp. 93 – 99, Liberec, September 2000. 5. J. Mikyška : Numerical Analysis of the Non-stationary 2-D Porous Media Flow and Heat Transport in the Vicinity of Sources of the Geothermal Energy, diploma thesis, FNSPE CTU Prague, 2001. 6. K. Rektorys : Variational Methods in the Engineering Problems and in Problems of Mathematical Physics, (in Czech) Academia, Prague, 1999.

Equadiﬀ 10, August 27–31, 2001 Prague, Czech Republic

Equadiﬀ 10 CD ROM Papers, pp. 37–43

On a class of forced nonlinear oscillators at resonance Denis Bonheure1 , Christian Fabry2 and Didier Smets3 1

2

3

Université Catholique de Louvain, Institut de Mathématique Chemin du Cyclotron, 2 , B-1348 Louvain-la-Neuve, Email: [email protected] Université Catholique de Louvain, Institut de Mathématique Chemin du Cyclotron, 2 , B-1348 Louvain-la-Neuve, Email: [email protected] Université Catholique de Louvain, Institut de Mathématique Chemin du Cyclotron, 2 , B-1348 Louvain-la-Neuve, Email: [email protected]

Pure et Appliquée, Belgium Pure et Appliquée, Belgium Pure et Appliquée, Belgium

Abstract. We present existence, non-existence and multiplicity results for periodic solutions of forced nonlinear oscillators at resonance, the nonlinearity being a bounded perturbation of a force deriving from an isochronous potential, i.e. a potential leading to free oscillations that all have the same period. The class of nonlinearities considered includes jumping nonlinearities, as well as singular forces of repulsive type. As particular cases of the existence results, we obtain conditions of Landesman-Lazer type. We also investigate the problem of boundedness of the solutions. MSC 2000. Primary:70K30, 65L10, 34C11 Keywords. Isochronous potential, resonance, Landesman-Lazer conditions, unbounded solutions

1

Introduction

In this note, we summarize some results obtained recently in [2] and [3]. We refer the reader to these papers for a more detailed exposition. We consider the following class of nonlinear oscillators x + V (x) + g(x) = p(t),

(1)

in situations of resonance. The following hypotheses, where k ∈ N∗ and a ∈ [−∞, 0), are assumed to hold throughout: This is an overview article.

38

D. Bonheure, Ch. Fabry and D. Smets

V : (a, +∞) → R is a 2π/k-isochronous, strictly convex potential whose derivative is locally lipschitzian; g : (a, +∞) → R is (Hk ) bounded and locally lipschitzian; p belongs to L1loc (R) and is 2πperiodic. By convention, we suppose that the minimum of V is reached at 0, so that V (0) = 0. By a 2π/k-isochronous potential, we mean that all nontrivial solutions of x + V (x) = 0 are of (minimal) period 2π/k. We also assume that V satisﬁes either k2 V (x) = and x→+∞ x 4 where a ∈ [−∞, 0), or (S)

lim

lim

x→a+

V (x) = α > 0 and x→+∞ x (in which case a = −∞). (N S)

lim

V (x) = +∞, x

V (x) =β>0 x→−∞ x lim

The ﬁrst case is referred to as the singular case, because when a = −∞, it corresponds to a repulsive singularity which, by convention, has been placed here on the negative side. In the second case, which is referred to as the non-singular case, V is asymptotic to a so-called jumping or asymmetric nonlinearity αx+ − βx− , where x+ = max{x, 0}, x− = max{−x, 0}. The isochronism assumption implies that 2 1 1 √ +√ = . (2) k α β We refer to [1] and [3] for examples of isochronous potentials. It is shown in [3] (see also Corollary 4 below) that perturbations of functions deriving from an isochronous potential cover a large class of nonlinearities. Consider equation (1) where V, g, p satisfy hypothesis (Hk ) for some k ∈ N∗ , and V satisﬁes either (S) or (N S). Let us deﬁne the function Φ by 2π Φ(θ) = p(t)ψ(t + θ) dt , (3) 0

where ψ denotes either | cos(kt/2)| in case (S) holds, or the solution of x + αx+ − βx− = 0, x(0) = 1, x (0) = 0, when (N S) holds. Notice that Φ, as ψ, is of period 2π/k. Let us also deﬁne 2π G(ρ) = g(ρψ(t))ψ(t) dt , 0

and its limits G+ = lim inf G(ρ) and G+ = lim sup G(ρ) . ρ→+∞

ρ→+∞

We show in [2], [3] and [4] that the function Φ plays a key role in the problem of existence of 2π-periodic solutions of (1) as well as in the problem of boundedness of the solutions of equation (1).

39

Nonlinear oscillators at resonance

2

Periodic solutions

Concerning the existence of periodic solutions, we prove in [3] the following theorem. Theorem 1. Let Φ, G+ , G+ be deﬁned as above. We have the following : (i) If there exists G∗ ∈ [G+ , G+ ], which is a regular value of Φ, and if the number of zeros of Φ − G∗ in [0, 2π/k) is diﬀerent from 2, equation (1) has at least one 2π-periodic solution. (ii) If there exist two regular values G1 , G2 ∈ [G+ , G+ ] of Φ such that the number of zeros of Φ − G1 and Φ − G2 in [0, 2π/k) are diﬀerent, equation (1) admits an unbounded sequence of 2π-periodic solutions. (iii) If [G+ , G+ ] contains no critical value of Φ, the set of 2π-periodic solutions of (1) is bounded. The existence condition includes the case of a function Φ − G∗ of constant sign. In particular, the result applies if max Φ < G+ or min Φ > G+ .

(4)

That situation has been treated by Krasnosel’skii and Mawhin [9] for perturbations of a linear oscillator. Arguing as in [9], it can be shown that, when g has a sublinear primitive, G+ = G+ = 0. Hence, if one considers, for instance, the equation x + αx+ − βx− + sin(x) = p(t) , where x+ = max{x, 0}, x− = max{−x, 0}, and α, β satisfy condition (2) with k = 1, or the equation x −

1 (x + 1) + + sin(x) = p(t) , 4(x + 1)3 4

for both of which g(x) = sin(x), it results from Theorem 1 that these equations have at least one 2π-periodic solution if the number of zeros (supposed to be simple), in [0, 2π), of the function Φ deﬁned by (3), is diﬀerent from 2. Several earlier results can be obtained as particular cases of the above theorem, corresponding to situations where the image of Φ does not intersect [G+ , G+ ]. Noting that ψ + lim inf g(x) ψ, G+ ≤ lim sup g(x) x→+∞

ψ>0

x→−∞

ψ<0

and writing a similar inequality for G+ , the following corollary is obtained, after computation of the integrals for ψ.

40

D. Bonheure, Ch. Fabry and D. Smets

Corollary 2. Suppose that k k Φ(θ) √ > lim sup g(x) − lim inf g(x) , for all θ ∈ [0, 2π], x→−∞ 2 α α β x→+∞ or Φ(θ) √ < 2 α

k k − lim sup g(x) , for all θ ∈ [0, 2π]. lim inf g(x) x→+∞ α β x→−∞

Then equation (1) has at least one 2π-periodic solution. Notice that β has to be considered as +∞ in the singular case, α being then equal to k 2 /4. The conditions appearing in the above corollary are conditions of LandesmanLazer type; they are clearly more restrictive than (4). On the other hand, if g admits limits g(±∞) at ±∞, it is immediate that √ g(+∞) g(−∞) + G+ = G = 2k α − , (5) α β still with β = +∞ in the singular case, so that the following corollary can be stated. Corollary 3. Assume that g admits limits at ±∞. If G+ = G+ , given by (5), is a regular value of Φ and if the number of zeros of Φ − G+ in [0, 2π/k) is diﬀerent from 2, equation (1) has at least one 2π-periodic solution. The situation of the above corollary has been treated by Fabry-Fonda [7] for equations with jumping nonlinearities. When g admits limits at ±∞, Corollary 3 provides existence conditions that are more general than conditions of LandesmanLazer type. Whether the Fredholm type conditions of Theorem 1 are necessary is probably false; on the other hand, existence of solutions for whatever p is also false. We prove in [3] that, for ε suﬃciently small, ε = 0, the equation x + V (x) = ε sin t

(6)

has no 2π-periodic solutions, assuming that V satisﬁes hypothesis (Hk ) with k = 1, and that V (x) satisﬁes either (S) or (N S). We assume moreover that V admits a derivative at 0. It does not seem easy to give simple conditions ensuring that a nonlinearity falls into the scope of the preceding theorems. Given an equation x + q(x) = p(t), it is not clear if the existence conditions can be easily stated in terms of q and p. In the singular case, we are able to prove the following. We denote by Q the primitive of q which is zero at zero.

41

Nonlinear oscillators at resonance

Corollary 4. Assume that p is locally integrable and 2π-periodic, q is locally lipschitzian, limx→−a Q(x) = +∞ and limx→+∞ (q(x) − x/4) = g∞ /4. Moreover, assume that there exists δ > 0 such that for every x ∈ (−a, −a + δ) : (i) q (x) > 0,

√ (ii) |q (x)| < Q(x)−3/2 |q(x)|3 / 2. Then, if the number of zeros of Φ − (g∞ − a) in [0, 2π) is not equal to 2, those zeros being simple, equation x + q(x) = p(t) admits at least one 2π-periodic solution. Corollary 4 provides an answer to a question raised by Del Pino and al. (Remark 1.2 in [6]). For the model equation x −

1 + βx = p(t), xν

with ν ≥ 1, Del Pino, Manásevich and Montero proved the existence of at least one 2π-periodic solution for β = k 2 /4 and p continuous 2π-periodic. Using a shift in the x coordinate, Corollary 4 can be applied to this model equation. Moreover we can show that for this model, g∞ = a. It gives then an explicit existence condition for 2π-periodic solutions in the resonant cases. On the other hand, as mentioned above, in the resonant cases a non-existence result is proved in [3]. For example, the equation 1 1 x − 3 + x = ε sin t x 4 has no 2π-periodic solution, at least for ε small.

3

Unbounded solutions

For the simpler equation x + αx+ − βx− = p(t),

(7)

with p smooth, it has been shown by Liu in [10] that all the solutions are bounded if Φ is of constant sign. This is also true for nonlinearities deriving from regular isochronous potentials, as proved in [4]. By contrast, it follows from results of Fabry and Mawhin [8] that, if the function Φ has zeros, all being simple, then the large amplitude solutions of (7) are unbounded either in the past or in the future (see also [5]). On the other hand, we show in [3] that the equation (6) where V is a smooth potential satisfying hypothesis (Hk ) with k = 1, and either (S) or (N S), has no 2π-periodic solution, at least for ε small. By a result of Massera [11], all the solutions of equation (6) are then unbounded for ε small. It is easy to check that for the forcing term p(t) = sin t, the function Φ has exactly two (simple) zeros in [0, 2π).

42

D. Bonheure, Ch. Fabry and D. Smets

All the above results suggest that, for equation (1), the boundedness of the solutions depends on whether Φ vanishes at some point or not, at least when G+ = G+ = 0. We show in [2] how to adapt the condition to the general case of arbitrary values of G+ and G+ . We prove the following result, which improves Theorem 2 of [8]. Theorem 5. Let Φ, G+ , G+ be deﬁned as above. Suppose that max Φ > G+ , min Φ < G+ and that [G+ , G+ ] does not contain any critical value of Φ. Then, there exists R > 0 such that all the solutions x(t) of (1) satisfying (x(0))2 + (x (0))2 > R are unbounded, either in the future or in the past. Notice that Theorem 5 does apply to the particular case G+ = G+ , which holds for a large class of functions g. When G+ = G+ , the assumption of Theorem 5 amounts to ask that Φ − G+ vanishes at some point, the zeros being simple. Hence, if one considers, for instance, the equation x + αx+ − βx− + sin(x) = p(t) ,

(8)

where α, β satisfy (2) with k = 1, or the equation x −

1 (x + 1) + sin(x) = p(t) , + 3 4(x + 1) 4

for both of which g(x) = sin(x), it results from Theorem 5 that if the function Φ deﬁned by (3) vanishes at some point in [0, 2π) (the zeros are supposed to be simple), then the large amplitude solutions of these equations are unbounded either in the past or in the future. In the particular case where k = 1 and p(t) = a + b cos t, it is easily computed that unbounded solutions are present when 3|a| < |b|. Equation (8) is already covered by the results of [8] but, when G+ = G+ , the result of Theorem 5 is new, even in the case of a harmonic oscillator. We refer to Proposition 1 of [3] for examples where the limits G+ and G+ are diﬀerent and where these can be easily computed from the limits of g.

References 1. S. Bolotin, R.S. MacKay, Isochronous potentials, preprint 2000. 2. D.Bonheure, C. Fabry, Unbounded solutions of forced isochronous oscillators at resonance, submitted for publication. 3. D. Bonheure, C. Fabry, D. Smets, Existence and non-existence results for periodic solutions of forced isochronous oscillators at resonance, submitted for publication. 4. D. Bonheure, C. Fabry, D. Smets, Littlewood’s problem for isochronous oscillators, in preparation. 5. W. Dambrosio, A note on the existence of unbounded solutions to perturbated asymmetric oscillator, Quaderni del Dipartimento di Matematica, N. 49/1999, Univ. di Torino. 6. M. Del Pino, R. Manásevich and A. Montero, T -periodic solutions for some second order diﬀerential equations with singularities, Proc. Roy. Soc. Edinburgh Sect. A 120 (1992), 231-243.

Nonlinear oscillators at resonance

43

7. C. Fabry and A. Fonda, Nonlinear resonance in asymmetric oscillators, J. Diﬀerential Equations 147 (1998), 58-78. 8. C. Fabry and J. Mawhin, Oscillations of a forced asymmetric oscillator at resonance, Nonlinearity 13 (2000), 493-505. 9. A.M. Krasnosel’skii and J. Mawhin, Periodic solutions of equations with oscillating nonlinearities, Math. Comput. Modelling 32 (2000), 1445-1455. 10. B. Liu , Boundedness in asymmetric oscillations, J. Math. Anal. Appl. , 231 (1999), 355-373. 11. J. Massera, The existence of periodic solutions of systems of diﬀerential equations, Duke Math. J. 17 (1950), 457-475.

Equadiﬀ 10, August 27–31, 2001 Prague, Czech Republic

Equadiﬀ 10 CD ROM Papers, pp. 45–51

Landesman–Lazer Type Conditions and Quasilinear Elliptic Equations Jiří Bouchala Department of Applied Mathematics, Faculty of Electrical Engineering and Computer Science, VŠB-Technical University, Tř. 17. listopadu, 708 33 Ostrava, Czech Republic, Email: [email protected]

Abstract. We study the existence of the weak solutions of nonlinear boundary value problem −∆p u = λ|u|p−2 u + g(u) − h(x) in Ω, u = 0 on ∂Ω, where Ω ⊂ RN is a smooth bounded domain, N ≥ 1, p > 1, g : R → R p ), ∆p is the p-Laplacian, i.e. is continuous function, h ∈ Lp (Ω) (p = p−1 ∆p u = div(|∇u|p−2 ∇u) and λ ∈ R. Our suﬃcient conditions generalize all previously published results.

MSC 2000. 35J20, 35P30, 47H15

Keywords. The p-Laplacian, Ekeland variational principle, saddle point theorem, the strong unique continuation property

1

Introduction. The variational eigenvalues.

We study the existence of the weak solutions of nonlinear boundary value problem −∆p u = λ|u|p−2 u + g(u) − h(x) in Ω, (1) u = 0 on ∂Ω,

Research supported by the Grant Agency of Czech Republic 201/00/0376 and grant of the Ministry of Education, Youth and Sports J17/98: 272400019

This is the preliminary version of the paper.

46

J. Bouchala

where Ω ⊂ RN is a smooth bounded domain, N ≥ 1, p > 1, g : R → R is p continuous function, h ∈ Lp (Ω) (p = p−1 ), λ ∈ R and ∆p is the p-Laplacian, i.e. ∆p u := div(|∇u|p−2 ∇u). We recall that u ∈ W01,p (Ω) is a weak solution of (1) if and only if p−2 p−2 |∇u| ∇u∇v dx = λ |u| uv dx + g(u)v dx − hv dx Ω

for all v ∈

Ω

Ω

Ω

W01,p (Ω).

It is possible to achieve that the weak solutions of our BVP (1) corresponding with the critical points of the functional 1 λ J(u) := |∇u|p dx − |u|p dx − G(u) dx + hu dx : W01,p (Ω) → R, p Ω p Ω Ω Ω where

G(t) :=

t

g(s) ds. 0

Now we are going to investigate how the choice of λ, g and h (and their relation) inﬂuence the geometry of our functional J. The great part in that has the information, if λ is the eigenvalue of the operator −∆p or not; i.e. if there exists a weak nontrivial solution of BVP −∆p u = λ|u|p−2 u in Ω, u = 0 on ∂Ω. Now we deﬁne the even functional |∇u|p dx I(u) := Ω : W01,p (Ω) \ {0} → R, p Ω |u| dx and for any k ∈ N we consider set Fk := A ⊂ {u ∈ W01,p (Ω) : uLp(Ω) = 1} :

there exists a continuous odd surjection h : S k → A ,

where S k represents the unit sphere in Rk . Pavel Drábek and Stephen B. Robinson proved in 1999 that for any k ∈ N the number λk := inf sup I(u) A∈Fk u∈A

is an eigenvalue of −∆p . This situation is very interesting, because it is not known if this represents a complete list of eigenvalues 1 but it is known that: 1

Nobody knows how to obtain all eigenvalues of −∆p ; we only know that we have complete list of eigenvalues if N = 1 or p = 2.

Landesman–Lazer Type Conditions and Quasilinear Elliptic Equations

47

• λ1 is the ﬁrst eigenvalue, λ1 = min{

Ω

|∇u|p dx; u ∈ W01,p (Ω),

Ω

|u|p dx = 1},

there exists a unique positive corresponding eigenfunction ϕ1 whose norm in W01,p (Ω) is 1, • λ2 is the second eigenvalue, • ∀k ∈ N \ {1, 2} : 0 < λ1 < λ2 ≤ λk ≤ λk+1 , • λk → +∞. Pavel Drábek and Stephen B. Robinson assumed in their paper that function g is bounded and they found some suﬃcient conditions for solvability of our BVP (1). Now we are going to generalize these results for some not bounded function g.

2

The case λ < λ1 .

Theorem 1. If we suppose in addition that lim g(x) p−1 x→±∞ |x|

= 0,

then the BVP (1) has at least one weak solution. (It follows from Ekeland variational principle (see [6] and [1]) that the energy functional J has a global minimum in this case.)

3

The case λ = λ1 .

Theorem 2. Let us deﬁne F (x) :=

p x

x

g(s) ds − g(x). 0

We suppose lim g(x) p−1 x→±∞ |x|

=0

and F (−∞) ϕ1 (x) dx < (p − 1) h(x)ϕ1 (x) dx < F (+∞) ϕ1 (x) dx, Ω

Ω

Ω

(2)

48

J. Bouchala

or F (+∞) ϕ1 (x) dx < (p − 1) h(x)ϕ1 (x) dx < F (−∞) ϕ1 (x) dx, Ω

Ω

(3)

Ω

where F (−∞) = lim sup F (x), F (+∞) = lim inf F (x), x→+∞

x→−∞

F (+∞) = lim sup F (x), F (−∞) = lim inf F (x). x→−∞

x→+∞

Then the BVP (1) has at least one weak solution. (If (2) is satisﬁed that the energy functional J has a saddle point geometry while if (3) holds this functional attains its global minimum . . . see [3].)

4

The case λk < λ < λk+1 .

Theorem 3. We suppose lim g(x) p−1 x→±∞ |x|

=0

and ∀v ∈ Ker(−∆p − λ) \ {0} : (p − 1) h(x)v(x) dx < F (+∞) Ω

v(x) dx + F (−∞)

{x∈Ω: v(x)>0}

v(x) dx,

(4)

v(x) dx,

(5)

{x∈Ω: v(x)<0}

or ∀v ∈ Ker(−∆p − λ) \ {0} : (p − 1) h(x)v(x) dx > F (+∞) Ω

v(x) dx + F (−∞)

{x∈Ω: v(x)>0}

{x∈Ω: v(x)<0}

and that ∀v ∈ Ker(−∆p − λ) \ {0}, v = 1 : (∀δ ∈ R+ ) (∃η(δ) ∈ R+ ) : meas{x ∈ Ω : |v(x)| ≤ η(δ)} < δ

(6)

(“the strong unique continuation property”). Then the BVP (1) has at least one weak solution. 2

2

Notice that if λ ∈ R is not an eigenvalue of the −∆p , i.e. if does not exist function v ∈ Ker(−∆p − λ) \ {0}, then the conditions (4), (5) and (6) are vacuously true.

49

Landesman–Lazer Type Conditions and Quasilinear Elliptic Equations

(Proof of this theorem is based on application a saddle point theorem for linked sets . . . see [1].)

The case λ = λk .

5

Theorem 4 ([1]). We suppose lim g(x) p−1 x→±∞ |x|

=0

and ∀v ∈ Ker(−∆p − λ) \ {0} : (p − 1) h(x)v(x) dx < F (+∞) Ω

v(x) dx + F (−∞)

{x∈Ω: v(x)>0}

v(x) dx,

(7)

v(x) dx,

(8)

{x∈Ω: v(x)<0}

or ∀v ∈ Ker(−∆p − λ) \ {0} : (p − 1) h(x)v(x) dx > F (+∞) Ω

v(x) dx + F (−∞)

{x∈Ω: v(x)>0}

{x∈Ω: v(x)<0}

and that ∀v ∈ Ker(−∆p − λ) \ {0}, v = 1 : (∀δ ∈ R+ ) (∃η(δ) ∈ R+ ) : meas{x ∈ Ω : |v(x)| ≤ η(δ)} < δ. Further, we assume that there exists sequence µn * λk (if we assume (7)) or µn + λk (if we assume (8)) such that ∀n ∈ N ∀v ∈ Ker(−∆p − µn ) \ {0}, v = 1 : (∀δ ∈ R+ ) (∃η(δ) ∈ R+ ) : meas{x ∈ Ω : |v(x)| ≤ η(δ)} < δ. Then the BVP (1) has at least one weak solution.

50

J. Bouchala

6

The one dimensional case.

At the ﬁnish we note: if we consider one dimensional problem and – for example – Ω = (0, π), i.e. if we consider BVP −(|u |p−2 u ) = λ|u|p−2 u + g(u) − h(x) in (0, π), (9) u(0) = u(π) = 0, then situation is easier: we know all eigenvalues of the p-Laplacian 3 and we know that any eigenfunction satisﬁes ”the strong unique continuation property”. Therefore we can rewrite our results in this form: Theorem 5. We suppose lim g(x) p−1 x→±∞ |x|

=0

and ∀v ∈ Ker(−∆p − λ) \ {0} : (p − 1)

π 0

h(x)v(x) dx < F (+∞)

π

v + (x) dx + F (−∞)

0

π 0

v − (x) dx,

or ∀v ∈ Ker(−∆p − λ) \ {0} : (p − 1)

π 0

h(x)v(x) dx > F (+∞)

π

v + (x) dx + F (−∞)

0

π 0

v − (x) dx,

where v + := max{0, v}, v − := min{0, v}. Then the BVP (9) has at least one weak solution u ∈ W01,p (0, π).

3

All eigenvalues are described by the equalities p kπp = kp λ1 , k ∈ N, λk := π where 1

πp := 2(p − 1) p

0

1

ds 1

(1 − sp ) p

.

Landesman–Lazer Type Conditions and Quasilinear Elliptic Equations

51

References 1. Bouchala J.,Strong resonance for quasilinear boundary value problems, PhD Thesis, FAV ZČU Plzeň, 2000. 2. Bouchala J.,Strong resonance problems for the p – Laplacian, Proceedings of Seminar in Diﬀerential Equations, ZČU Plzeň, 2000, 115–122. 3. Bouchala J., Drábek P., Strong resonance for some quasilinear elliptic equations, Journal of Mathematical Analysis and Applications 245, 2000, 7–19. 4. Drábek P., Robinson S.B., Resonance problems for the p – Laplacian, Journal of Functional Analysis 169, 1999, 189–200. 5. Tang C.L., Solvability for two–point boundary value problems, Journal of Mathematical Analysis and Applications 216, 1997, 368–374. 6. Willem M., Minimax theorems, Birkhäuser, Boston, 1996.

Equadiﬀ 10, August 27–31, 2001 Prague, Czech Republic

Equadiﬀ 10 CD ROM Papers, pp. 53–55

Extremality results for singular functional diﬀusion equations Alberto Cabada, José Angel Cid and Rodrigo L. Pouso Department of Mathematical Analysis, Univ. of Santiago de Compostela, Campus Sur s/n, 15782, Santiago de Compostela, SPAIN Email: [email protected] Email: [email protected] Email: [email protected]

Abstract. We study the existence of extremal positive solutions for initial value problems of the type ((k ◦ u)u ) (x) = f (x, u(x))u (x) + g(x, u(x))u(x) u(0) = 0,

a. e. in I = [0, α],

lim ((k ◦ u) u )(x) = 0,

x→0+

where k and g need not be continuous. MSC 2000. 34A36, 34A09 Keywords. Diﬀusion equations, discontinuous equations, singular equations

1

Introduction

In [1] we consider the ordinary diﬀerential equation ((k ◦ u)u ) (x) = f (x, u(x))u (x) + g(x, u(x))u(x) with initial data u(0) = 0,

a.e. in I = [0, α],

lim ((k ◦ u) u )(x) = 0,

x→0+

(1) (2)

for a ﬁxed α > 0. The function k is allowed to vanish for some values of u and, therefore, the equation becomes singular at those points. This type of diﬀerential operators naturally arises in diﬀusion processes [4]. This is an overview article.

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A. Cabada, J. A. Cid and R. Pouso

From a purely mathematical point of view one of the main diﬃculties involved in these equations is a lack of a priori bounds on the derivatives, which puts a stop to a Bernstein–Nagumo approach. Even in our case, where the dependence on the derivatives is linear, we have to give up obtaining nontrivial C 1 solutions. We deﬁne a solution of (1) – (2) as any element of the set

S = u ∈ W 1,1 (I) : u(x) > 0 for x ∈ (0, α] and (k ◦ u)u ∈ W 1,1 (I) , that satisﬁes the relations (1) – (2). Following the standard notation, W 1,1 (I) stands for the Sobolev space of L1 functions whose generalized derivative belongs to L1 (I). We note that for every element of W 1,1 (I) there exists a unique absolutely continuous function on I that equals it almost everywhere on I. We mean that u ∈ S satisﬁes (2) when the continuous functions that equal u and (k ◦ u)u a.e. fulﬁll the respective relations. On the other hand, we say that a solution u∗ ∈ S of (1)–(2) is the minimal solution in S if u∗ ≤ u for any other solution u ∈ S. Analogously we can deﬁne the concept of maximal solution. When both the minimal and the maximal solutions of (1)–(2) in S exist, we call them extremal solutions. Existence of extremal nontrivial solutions for (1)–(2) is proven in [1] under the following assumptions on the functions k, f and g: ∞ 1 k(ξ) k(ξ) dξ < ∞ and dξ = ∞. (k0) k ∈ L1loc [0, ∞), k(ξ) > 0 a.e. ξ ∈ [0, ∞), ξ ξ 0 1 (f0) f : [0, α] × [0, ∞) → R is continuous on [0, α] × [0, ∞); for a.e. ξ ∈ [0, ∞), f (·, ξ) is absolutely continuous on [0, α] and |D1 f (x, ξ)| ≤ B(ξ)

for a.e. x ∈ [0, α],

where B ∈ L1loc [0, ∞); there exists a null set N ⊂ [0, α] such that for each ﬁxed x ∈ [0, α]\N the derivative D1 f (x, ξ) exists for a.e. ξ ∈ [0, ∞). (f1) f (x, ξ) > 0 on [0, α]×[0, ∞) and for all x ∈ [0, α]\N we have that D1 f (x, ξ) ≤ 0 for a.e. ξ ∈ [0, ∞). (f2) f (x, ·) is nondecreasing in [0, ∞) for a.e. x ∈ [0, α]. (h0) There exists h1 ∈ L1 [0, α] such that f (x, ξ) ≤ h1 (x) for a.e. x ∈ [0, α] and all ξ ∈ [0, ∞). (h1) There exists h2 ∈ L1 [0, α] such that for all x ∈ [0, α]\N we have D1 f (x, ξ) ≥ h2 (x) for a.e. ξ ∈ [0, ∞) (Notice that h2 has to be nonpositive a.e. so that this assumption and (f1) are consistent). (g0) g : [0, α] × [0, ∞) → R is such that g(·, h(·)) is a measurable function on [0, α] whenever h is continuous on [0, α]. (g1) For a.e. x ∈ [0, α], g(x, ξ)ξ is nondecreasing with respect to ξ ∈ [0, ∞). (g2) There exists ψ ∈ L1 [0, α] such that 0 ≤ g(x, ξ) ≤ ψ(x) for a.e. x ∈ [0, α] and all ξ ∈ [0, ∞).

55

Discontinuous Diﬀusion Equations

The main diﬃculty that we have to overcome in [1] concerns how to turn problem (1) – (2) into a problem of ﬁnding ﬁxed points of an equivalent integral operator. Such an operator is constructed by using the formula

b

f (s, h(s))h (s)ds =

a

h(b)

b

h(s)

f (a, ξ)dξ − h(a)

D1 f (s, ξ)dξds, a

h(b)

which extends both the formula of change of variables and that of integration by parts. From the assumed conditions, the so constructed operator is, in general, not continuous. Despite this, we deduce existence of extremal ﬁxed points by using the results proved in [3] for this type of operators. The previous existence result can be extended to cover a wider class of functional equations. In fact, we consider in [2] the solvability of functional equations of the form d (k(u(x), u)u (x)) = f (x, u(x), u)u (x) + g(x, u(x), u)u(x) a.e. on [0, α], dx

(3)

together with the initial conditions u(0) = 0, limx→0+ k(u(x), u) u (x) = 0. Roughly speaking, the main idea can be sketched as follows: we require the equation d (k(u(x), v)u (x)) = f (x, u(x), v)u (x) + g(x, u(x), v)u(x) a.e. on [0, α], dx

(4)

to satisfy the conditions (k0) − (g2) in an uniform way for v in a suitable set of functions. As a consequence we have, by virtue of the main result in [1], that the corresponding problem has extremal solutions. Subsequently we consider an iteration scheme based on the previous type of problems that leads to our existence result. The key assumption on the functionals k(u, ·), f (x, u, ·) and g(x, u, ·) is monotonicity (the reader is referred to [2] for the details).

References 1. A. Cabada, J. A. Cid, and R. L. Pouso, Positive solutions for a class of singular diﬀerential equations arising in diﬀusion processes, manuscript. 2. A. Cabada, J. A. Cid, and R. L. Pouso, Existence results and approximation methods for functional ordinary diﬀerential equations with singular diﬀusion–type diﬀerential operators, to appear in Comput. Math. Appl. 3. S. Heikkilä and V. Lakshmikantham, Monotone Iterative Techniques for Discontinuous Nonlinear Diﬀerential Equations, Marcel Dekker, New York, 1994. 4. S. Itˆ o, Diﬀusion equations, Translations of Mathematical Monographs, 144, A.M.S., Providence, Rhode Island, 1992.

Equadiﬀ 10, August 27–31, 2001 Prague, Czech Republic

Equadiﬀ 10 CD ROM Papers, pp. 57–66

Attractors of nonautonomous and stochastic diﬀerential inclusions Tomás Caraballo1, José Antonio Langa1 , Valery S. Melnik2 and José Valero3 1

Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla, Spain Email: [email protected] Email: [email protected] 2 Institute of Applied System Analysis, Pr. Pobedy 37, 252056-Kiev, Ukraine Email: [email protected] 3 Universidad Cardenal Herrera CEU, Campus de Elche Comissari 3, 03203 Elche, Alicante, Spain Email: [email protected]

Abstract. In this paper we study the existence of pullback global attractors for multivalued processes generated by diﬀerential inclusions. First, we deﬁne multivalued dynamical processes, prove abstract results on the existence of global attractors and study their topological properties (compactness, conectedness). Further, we apply the abstract results to nonautonomous diﬀerential inclusions of the reaction-diﬀusion type in which the forcing term can grow polynomially in time, and to stochastic diﬀerential inclusions as well.

MSC 2000. 35B40, 35B41, 35K55, 35K57, 37L55

Keywords. Attractor, asymptotic behaviour, diﬀerential inclusion, reaction-diﬀusion equations, nonautonomous dynamical systems

1

Introduction

In this paper we study the existence of pullback global attractors for multivalued processes generated by diﬀerential inclusions. The theory of pullback attractors has been developed for stochastic and nonautonomous systems in which the trajectories can be unbounded when times rises to inﬁnite. In such systems the classical This is the preliminary version of the paper.

58

T. Caraballo, J. A. Langa, V. S. Melnik and J. Valero

theory of global attractors is not applicable. Hence, a diﬀerent approach has been considered [4,5,6,9]. A new diﬃculty appears if the solution corresponding to each initial state can be non-unique. The classical results on attractors in the autonomous and nonautonomous cases are generalized to the multivalued case in [7] and [8], respectively, with applications to evolution inclusions. In [1,2,3] the study of multivalued dynamical systems is extended to the stochastic case, generalizing in this way the results of [4,5]. In this paper we are mainly concerned with nonautonomous multivalued dynamical systems in which the trajectories can be unbounded in time and also with nonautonomous stochastic multivalued dynamical systems. In the second section we deﬁne multivalued dynamical processes, prove abstract results on the existence global attractors and study their topological properties (compactness, conectedness). In the third section we apply the abstract results to nonautonomous diﬀerential inclusions of the reaction-diﬀusion type in which the forcing term can grow polynomially in time. In the fourth section we give applications to stochastic diﬀerential inclsuions with additive and multiplicative noises.

2

Attractors for multivalued processes

In this section we shall deﬁne multivalued dynamical processes in metric spaces. Maps of this kind appear in diﬀerential equations for which, although we are able to prove the existence of at least one global solution for each inital condition in some phase space, we do not know if it is unique or not. Hence, multivalued processes generalize the concept of processes, for which the uniqueness property holds. Let X be a complete metric space with the metric denoted by ρ and let P (X) non-empty (B (X) , Cv (X)) be the set of all non-empty (non-empty bounded,

bounded closed and convex) subsets of X. Let us denote Rd = (t, s) ∈ R2 : t ≥ s , dist (A, B) = sup inf ρ (x, y), distH (A, B) = max{dist(A, B), dist(B, A)}, for any A, B ⊂ X.

x∈Ay∈B

Definition 1. The map U : Rd × X → P (X) is called a multivalued dynamical process (MDP) on X if: 1. U (t, t, ·) = I is the identity map; 2. U (t, s, x) ⊂ U (t, τ, U (τ, s, x)), for all x ∈ X, s ≤ τ ≤ t. The MDP U is called strict if: U (t, s, x) = U (t, τ, U (τ, s, x)) , for all x ∈ X, s ≤ τ ≤ t. Consider a parameter set Σ = Σ1 × Σ2 . If {Uσ : σ ∈ Σ} is an arbitrary family of MDP, then for any σ2 ∈ Σ2 the map UΣ1 ,σ2 : Rd × X → P (X) deﬁned by Uσ1 ,σ2 (t, s, x) . UΣ1 ,σ2 (t, s, x) = σ1 ∈Σ1

Attractors of nonautonomous inclusions

59

is a MDP. Suppose that we are given a one-parameter group T (h) : Σ → Σ, where Σ = Σ1 × Σ2 , h ∈ R and T (h) = (T1 (h) , T2 (h)) , Ti (h) : Σi → Σi , i = 1, 2. This is called the shift operator. In the sequel we shall assume: (T 1) For any (t, s) ∈ Rd , x ∈ X, σ ∈ Σ, h ∈ R the following inclusion holds: Uσ1 ,σ2 (t, s, x) ⊂ UT1 (h)σ1 ,T2 (h)σ2 (t − h, s − h, x) . Lemma 2. Condition (T 1) implies Uσ1 ,σ2 (t, s, x) = UT1 (h)σ1 ,T2 (h)σ2 (t − h, s − h, x) . Lemma 3. T (h) Σ = Σ, for all h ∈ R. Definition 4. Let (T 1) hold. Then the family of sets {ΘΣ1 (σ2 )}σ2 ∈⊀2 is called a Σ1 -uniform global attractor of the MDP {Uσ } if: 1. ΘΣ1 (σ2 ) is Σ1 -uniformly attracting at time 0 for any σ2 ∈ Σ2 , that is, lim dist (UΣ1 ,σ2 (0, s, B) , ΘΣ1 (σ2 )) = 0, for any B ∈ B (X) .

s→−∞

(1)

2. It is semi-invariant, that is, ΘΣ1 (T2 (t) σ2 ) ⊂ UΣ1 ,σ2 (t, s, ΘΣ1 (T2 (s) σ2 )) , for any (t, s) ∈ Rd , σ2 ∈ Σ2 . 3. It is minimal, that is, for any σ2 ∈ Σ2 and any closed Σ1 -uniformly attracting set Y (σ2 ) at time 0, we have ΘΣ1 (σ2 ) ⊂ Y (σ2 ). Theorem 5. Let X be a complete metric space in which every compact set is nowhere dense, (T 1) hold and let for any B ∈ B (X), σ2 ∈ Σ2 there exist a compact set D (σ2 , B) ⊂ X such that lim dist (UΣ1 ,σ2 (0, s, B) , D (σ2 , B)) = 0.

s→−∞

(2)

Then the following statements hold: 1. If for all τ ≤ 0 and σ2 ∈ Σ2 the graph of the map x -→ UΣ1 ,σ2 (0, τ, x) ∈ P (X) is closed, then there exists the Σ1 -uniform global attractor {ΘΣ1 (σ2 )}. Moreover, ΘΣ1 (σ2 ) = ωΣ1 (0, σ2 , B) = X, B∈B(X)

where ωΣ1 (t, σ2 , B) = s≤t τ ≤s UΣ1 ,σ2 (t, τ, B), and further for each σ2 ∈ Σ2 , ΘΣ1 (σ2 ) is a Lindelöf, normal space. It is locally compact in some topology τ⊕ , which is stronger than the topology induced by X in ΘΣ1 (σ2 ).

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T. Caraballo, J. A. Langa, V. S. Melnik and J. Valero

2. If, in addition, Σ1 is a compact metric space, the map Σ1 × X . (σ1 , x) -−→ Uσ1 ,σ2 (0, τ, x) ∈ P (X) is upper semicontinuous for any τ ≤ 0, σ2 ∈ Σ2 , Uσ has connected values for any σ ∈ Σ, (0, τ ) ∈ Rd , x ∈ X, Σ1 is a connected space and ΘΣ1 (T2 (τ ) σ2 ) ⊂ B1 (σ2 ) , for all τ ≤ 0, where B1 (σ2 ) is a bounded connected set for any σ2 ∈ Σ2 , then the set ΘΣ1 (σ2 ) is connected for any σ2 ∈ Σ2 . Theorem 6. Let us suppose that for all (0, s) ∈ Rd and σ2 ∈ Σ2 the graph of the map x -→ UΣ1 ,σ2 (0, s, x) ∈ P (X) is closed. If, moreover, for any σ2 ∈ Σ2 there exists a compact set D (σ2 ) , which is Σ1 -uniformly attracting at time 0, then the set ΘΣ1 (σ2 ) = ωΣ1 (0, σ2 , B) B∈B(X)

is the Σ1 -uniform global attractor of Uσ . Moreover, the sets ΘΣ1 (σ2 ) are compact and, if the conditions of the second statement in Theorem 5 hold, then they are connected. Proposition 7. Let the MDP Uσ be strict, Σ1 be a compact metric space and let the map Σ1 × X . (σ1 , x) -−→ Uσ1 ,σ2 (0, τ, x) ∈ P (X) be lower semicontinuous. Then the global attractors obtained in Theorems 5 and 6 are invariant, that is, ΘΣ1 (T2 (t) σ2 ) = UΣ1 ,σ2 (t, τ, ΘΣ1 (T2 (τ ) σ2 )), for all τ ≤ t, σ2 ∈ Σ2 .

3

Applications to nonautonomous evolution inclusions

Let Ω ⊂ Rn be a bounded open subset with smooth boundary ∂Ω. Consider the parabolic inclusion  n  ∂u p−2 ∂u ∂u ∂  − ∈ f1 (t, u) + f2 (t, u) + g1 (t) + g2 (t) ,  i=1 ∂xi  ∂xi ∂xi  ∂t (3) in Ω × (τ, T ) ,   = 0, u |  ∂Ω   u |t=τ = uτ , where τ ∈ R, p ≥ 2, fi : R × R → Cv (R), i = 1, 2, g1 ∈ L∞ (R, L2 (Ω)) , g2 ∈ Lloc 2 (R, L2 (Ω)) and the following conditions hold: (F1) There exists C ≥ 0 such that distH (f1 (t, u) , f1 (t, v)) ≤ C |u − v| , for all t ∈ R, u, v ∈ R.

61

Attractors of nonautonomous inclusions

(F2) For any t, s ∈ R and u ∈ R, it holds distH (f1 (t, u) , f1 (s, u)) ≤ l (|u|) α (|t − s|) , where α is a continuous function such that α (t) → 0, as t → 0+ , and l is a continuous nondecreasing function. Moreover, there exist K1 , K2 ≥ 0 such that |l (u)| ≤ K1 + K2 |u| , for all u ∈ R. (F3) There exist D ∈ R+ , v0 ∈ R for which |f1 (t, v0 )|+ ≤ D, for all t ∈ R, where |f1 (t, v0 )|+ =

sup

|ζ|.

ζ∈f1 (t,v0 )

(F4) There exist α1 (t) , α2 (t) ≥ 0, α1 (·) , α2 (·) ∈ Lloc 2 (−∞, ∞) , such that sup

|y| ≤ α1 (t) + α2 (t) |u|, for all u, t ∈ R.

y∈f2 (t,u)

(F5) For each t ∈ R, the map f2 (t, ·) is upper semicontinuous. (F6) For each s ∈ R, the map f2 (·, s) is measurable. (F7) If p = 2, there exist U > 0 and M ≥ 0 such that yu ≤ (λ1 − U) u2 + M, for all u ∈ R, t ∈ R, y ∈ f1 (t, u) + f2 (t, u) , where λ1 is the ﬁrst eigenvalue of −∆ in H01 (Ω). (F8) There exist R1 , R2 , R3 > 0 such that R3

g2 (t)L2 (Ω) ≤ R1 + R2 |t|

, for a.a. t ∈ R.

(F9) If p > 2, there exist R4 , R5 , R6 > 0 such that R6

|αi (t)| ≤ R4 + R5 |t|

, for a.a. t ∈ R, i = 1, 2.

First let us construct the sets Σ1 , Σ2 . Denote by W the space Cv (R) endowed with the Hausdorﬀ metric ρ (x, y) = dist H (x, y). The space W is complete. For any ψ ∈ W let |ψ|+ = max |y| . Deﬁne y∈ψ

also the space

M = ψ ∈ C (R, W ) : |ψ (v)|+ ≤ D1 + D2 |v| , where the constants D1 , D2 are such that |y| ≤ D1 + D2 |u| , for all u ∈ R, t ∈ R, y ∈ f1 (t, u) The hull of (f1 , g1 ) will be denoted by Σ1 = H (f1 ) × H (g1 ), where H (f1 ) = clC(R,M) {f1 (· + h) : h ∈ R}, H (g1 ) = clLloc {g1 (· + h) : h ∈ R} . The set 2,w (R,L2 (Ω)) loc Σ1 is compact in the space C (R, M) × Lloc 2,w (R, L2 (Ω)), where L2,w (R, L2 (Ω))

62

T. Caraballo, J. A. Langa, V. S. Melnik and J. Valero

is the space Lloc 2 (R, L2 (Ω)) endowed with the weak topology. Then the set Σ1 is a compact metric space and T1 (h) Σ1 = Σ1 , for all h ∈ R, where T1 (h) σ1 (t) = σ1 (t + h) . For the set Σ2 we put Σ2 = (f2 (· + h) , g2 (· + h)) . h∈R

It is clear that T2 (h) Σ2 = Σ2 , for all h ∈ R, where T2 (h) σ2 (t) = σ2 (t + h). Now let X = L2 (Ω) with the norm ·X and the scalar product (·, ·). Consider the abstract evolution inclusion du dt (t) ∈ A (u (t)) + Fσ (t, u (t)) , t ∈ [τ, ∞) , (4) u (τ ) = uτ , where σ = (σ1 , σ2 ) ∈ Σ and A : D (A) ⊂ X → 2X , Fσ : R × X → 2X , are multivalued maps deﬁned as follows: n ∂ A (u) = ∂x i i=1

! ∂u p−2 ∂u 1,p (Ω) : A (u) ∈ L (Ω) , , D (A) = u ∈ W 2 0 ∂xi ∂xi

Fσ1 (t, u) = {y ∈ X : y (x) ∈ fσ1 (t, u (x)) + gσ1 (t) , a.e. on Ω} , Fσ2 (t, u) = {y ∈ X : y (x) ∈ fσ2 (t, u (x)) + gσ2 (t) , a.e. on Ω} Fσ (t, u) = Fσ1 (t, u) + Fσ2 (t, u) . The operators A and Fσ satisfy the following properties: (A1) The operator A is m-dissipative, i.e. (ξ1 − ξ2 , y1 − y2 ) ≤ 0 , for any y1 , y2 ∈ D(A), ξi ∈ A(yi ), i = 1, 2, and Im(A − λI) = X, for all λ > 0. (A2) D (A) = L2 (Ω) and A generates a compact semigroup S. (S1) Fσ : R × X → Cv (X), for all σ ∈ Σ. (S2) For any ﬁxed t ∈ R and σ ∈ Σ the map u -−→ Fσ (t, u) is w-upper semicontinuous, that is, for any ε > 0 there exists δ > 0 such that if u − vX < δ, then dist (Fσ (t, u) , Fσ (t, v)) < ε. (S3) For any σ ∈ Σ there exist β1, β2 ≥ 0, β1 , β2 ∈ Lloc 2 (−∞, ∞) (depending on σ2 but not on σ1 ), such that Fσ (t, u)+ ≤ β1 (t) + β2 (t) uX , for all u ∈ Xand a.a. t ∈ R. (S4) For any (T, τ ) ∈ Rd , x ∈ X, σ ∈ Σ, the map t -→ Fσ (t, x) has a measurable selection.

63

Attractors of nonautonomous inclusions

Definition 8. The continuous function uσ (·) ∈ C ([τ, T ] , X) is called an integral solution of (4) if uσ (τ ) = uτ and there exists l (·) ∈ L1 ([τ, T ] , X) such that l (t) ∈ Fσ (t, uσ (t)), a.e. on (τ, T ), and for any ξ ∈ D(A), v ∈ A(ξ) one has uσ (t) −

ξ2X

≤ uσ (s) −

ξ2X

t

(l(r) + v, uσ (r) − ξ) dr, t ≥ s.

+2

(5)

s

It follows from (A1) − (A2) , (S1) − (S4) that for any uτ ∈ L2 (Ω) there exists at least one integral solution uσ to (4) for any T > τ [10, Theorem 2.1]. For a ﬁxed σ ∈ Σ let Dσ,τ (x) be the set of all integral solutions corresponding to the initial condition u (τ ) = x. We shall deﬁne the map Uσ : Rd × X → P (X) by Uσ (t, τ, x) = {z : there exists u (·) ∈ Dσ,τ (x) such that u (t) = z} . Proposition 9. For each σ ∈ Σ, h ∈ R, τ ≤ s ≤ t, x ∈ X we have Uσ (t, s, Uσ (s, τ, x)) = Uσ (t, τ, x) , UT (h)σ (t, τ, x) = Uσ (t + h, τ + h, x) . Hence, Uσ is a multivalued process for each σ ∈ Σ and condition (T 1) holds. Theorem 10. If (F 1)−(F 9) hold and g1 ∈ L∞ (R, L2 (Ω)), g2 ∈ Lloc 2 (R, L2 (Ω)), then the family of MDP Uσ has the Σ1 − uniform global compact attractor ΘΣ1 (σ2 ). Let us consider now the connectivity of the global attractor. Theorem 11. In the conditions of Theorem 10, let f2 ≡ 0 and let there exist a non-decreasing map C (t) such that g2 (t)X ≤ C (t), for a.a. t ∈ R. Then the set ΘΣ1 (σ2 ) is connected in X for each σ2 ∈ Σ2 .

4 4.1

Stochastic non-autonomous evolution inclusions Additive white noise case

Consider the following non-autonomous diﬀerential inclusion perturbed by an additive white noise  m dwi (t)  ∂u i=1 φi dt , on D × (τ, T ) , ∂t − ∆u ∈ f (t, u) + g1 (t) + g2 (t) + u |∂D = 0,  u |t=τ = uτ ,

(6)

where τ ∈ R, D ⊂ Rn is an open bounded set with smooth boundary ∂D, wi (t) are independent two-sided, i.e. t ∈ R, real Wiener processes with wi (0) = 0, φi ∈ D(A) (where A (u) = ∆u, D (A) = H01 (Ω) ∩ H 2 (Ω)), i = 1, ..., m, f :

64

T. Caraballo, J. A. Langa, V. S. Melnik and J. Valero

loc R×R → mCv (R), i = 1, 2, g1 ∈ L∞ (R, L2 (D)) , g2 ∈ L2 (R, L2 (D)) . We write ζ (t) = i=1 φi wi (t). Consider the Wiener probability space (Ω, F , P) deﬁned by

Ω = {ω = (w1 (·) , ..., wm (·)) ∈ C (R, Rm ) | ω (0) = 0} , equipped with the Borelσ−algebra F and the Wiener measure P. Each ω ∈ Ω m generates a map ζ (·) = i=1 φi wi (·) ∈ C (R, L2 (D)) such that ζ (0) = 0. Suppose that f satisﬁes (F1)–(F3), (F7), whereas g2 satisﬁes (F8). Firstly, let us construct the sets Σ1 , Σ2 . The set Σ1 will be deﬁned in the same way as in the previous section. For the set Σ2 we write ˜2 × Ω, Σ ˜ 2 = ∪ g2 (· + h) . Σ2 = Σ h∈R

We deﬁne the map θs : Ω → Ω as follows θs ω = (w1 (s + ·) − w1 (s) , ..., wm (s + ·) − wm (s)) ∈ Ω. " " Then m the function ζ corresponding to θs ω is deﬁned by ζ (τ ) = ζ (s + τ ) − ζ (s) = i=1 φi (wi (s + τ ) − wi (s)) . The operator T1 is deﬁned as before. We deﬁne the shift operator T2 : Σ2 → Σ2 as ˜2 , ω ∈ Ω. T2 (h)σ2 = T2 (h)(˜ σ2 , ω) = (˜ σ2 (· + h), θh ω), for all σ ˜2 ∈ Σ Thus, T2 (h) Σ2 = Σ2 , for all h ∈ R. To study (6), we make the change of variable v (t) = u (t)−ζ (t). Then inclusion (6) turns, for each ω ∈ Ω ﬁxed, into dv m i=1 ∆φi wi (t) , dt ∈ ∆v (t) + f (t, v (t) + ζ (t)) + g1 (t) + g2 (t) + (7) v |∂D = 0, v (τ ) = vτ = uτ − ζ (τ ) . Now let X = L2 (Ω). Consider the abstract evolution inclusion dv(t) dt ∈ A (v (t)) + Fσ (t, v (t)) , t ∈ [τ, ∞) , v (τ ) = vτ = uτ − ζ (τ ) ,

(8)

where σ = (σ1 , σ2 ) ∈ Σ, A is deﬁned as before and Fσ : R × X → 2X is deﬁned as follows: Fσ (t, ω, u) = gσ2 (t) + Fσ1 (t, u + ζ (t)) + Aζ (t) , where Fσ1 is as in the previous section. As before, the operators A, Fσ satisfy (A1)–(A2), (S1)–(S4), so that for any vτ ∈ L2 (Ω) there exists at least one integral solution to (8) for any T > τ [10, Theorem 2.1]. For a ﬁxed σ ∈ Σ let Dσ,τ (x) be the set of all integral solutions corresponding to the initial condition v (τ ) = x. We deﬁne the map Uσ : Rd × X → P (X) by Uσ (t, τ, x) = {z + ζ(t) : there exists v (·) ∈ Dσ,τ (x − ζ (τ )) such that v (t) = z} . Theorem 12. In the preceedings conditions, the family of MDP Uσ has the Σ1 − uniform global compact attractor ΘΣ1 (σ2 ).

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4.2

Multiplicative white noise case

Finally, consider the following non-autonomous diﬀerential inclusion perturbed by a linear multiplicative white noise in the Stratonovich sense   ∂u ∂t − ∆u ∈ f (t, u) + g1 (t) + g2 (t) + u ◦ u |∂D = 0,  u |t=τ = uτ ,

dw(t) dt ,

on D × (τ, T ) , (9)

where τ ∈ R, D ⊂ Rn is and open bounded set with smooth boundar ∂D, f : R × R → Cv (R), i = 1, 2, g1 ∈ L∞ (R, L2 (D)) , g2 ∈ Lloc 2 (R, L2 (D)) . Consider the Wiener probability space (Ω, F , P) deﬁned by Ω = {ω = w (·) ∈ C (R, R) | ω (0) = 0} , equipped with the Borel σ−algebra F and the Wiener measure P. Suppose again that f satisﬁes (F1)–(F3), (F7), whereas g2 satisﬁes (F8). ˜ 2 × Ω and T1 , T2 exactly as in the previous We deﬁne Σ = Σ1 × Σ2 = Σ1 × Σ section, with θs : Ω → Ω θs ω = (w (s + ·) − w (s)) ∈ Ω. To study (9), we make the change of variable v (t) = γ(t)u (t) , with γ(t) = γ (ω, t) = e−w(t) (we shall omit ω). Then inclusion (9) turns into dv dt

∈ ∆v (t) + γ(t)f t, γ −1 (t)v(t) + γ(t)(g1 (t) + g2 (t)), v |∂D = 0, v (τ ) = vτ = γ (τ ) uτ .

(10)

Now let X = L2 (Ω). Consider dv(t) dt

∈ A (v (t)) + Fσ (t, v (t)) , t ∈ [τ, ∞) , v (τ ) = vτ ,

(11)

where σ = (σ1 , σ2 ) ∈ Σ, A is deﬁned as before, and Fσ : R × X → 2X is deﬁned as Fσ (t, ω, u) = γ (t) gσ2 (t) + γ (t) Fσ1 t, γ −1 (t) u , where Fσ1 is as in the previous section. As before, the operators A, Fσ satisfy (A1)–(A2), (S1)–(S4). We deﬁne the map Uσ : Rd × X → P (X) by

Uσ (t, τ, x) = γ −1 (t) z : there exists v (·) ∈ Dσ,τ (γ (τ ) x) such that v (t) = z . Theorem 13. In the preceedings conditions, the family of MDP Uσ has the Σ1 − uniform global compact attractor ΘΣ1 (σ2 ).

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References 1. Caraballo, T., Langa, J.A. and Valero, J.: Global attractors for multivalued random semiﬂows generated by random diﬀerential inclusions with additive noise, C.R. Acad. Sci., Paris, Série I, 331 (2001), 131-136. 2. Caraballo, T., Langa, J.A. and Valero, J.: Global attractors for multivalued random dynamical systems, Nonlinear Anal. (to appear). 3. Caraballo, T., Langa, J.A. and Valero, J.: Global attractors for multivalued random dynamical systems generated by random diﬀerential inclusions with multiplicative noise, J. Math. Anal. Appl., 260 (2001), 602-622. 4. Crauel, H. and Flandoli, F., Attractors for random dynamical systems, Prob. Theory Related Fields, 100 (1994), 365-393. 5. Crauel, H., Debussche, A. and Flandoli, F,: Random attractors, J. Dynamics Diﬀerential Equations, 9 (1997), 307-341. 6. Kloeden, P. E. and Schmalfuss, B., Asymptotic behaviour of nonautonomous diﬀerence inclusions, Systems & Control Letters, 33 (1998), 275-280. 7. Melnik, V.S. and Valero, J.: On attractors of multivalued semi-ﬂows and diﬀerential inclusions, Set-Valued Anal., 6 (1998), 83-111. 8. Melnik, V.S. and Valero, J.: On global attractors of multivalued semiprocesses and nonautonomous evolution inclusions, Set-Valued Anal., 8 (2000), 375-403. 9. Schmalfuss, B., Attractors for the nonautonomous dynamical systems, Proceedings of Equadiﬀ 99, Berlin (Fiedler B., Gröger K. and Sprekels J. editors), World Scientiﬁc, Singapore, 2000, 684-689. 10. Tolstonogov, A.A. and Umansky Ya.I.: On solutions of evolution inclusions II, Siberian Math. J., (4) 33 (1992) 693-702 (English translation in Siberian Math. Journal, (4) 33 (1992), 693-702).

Equadiﬀ 10, August 27–31, 2001 Prague, Czech Republic

Equadiﬀ 10 CD ROM Papers, pp. 67–76

Approximation of attractors for multivalued random dynamical systems Tomás Caraballo1, José Antonio Langa2 and José Valero3 1 Dpto. Ecuaciones Diferenciales y Análisis Numérico Facultad de Matemáticas. C/ Tarﬁa s/n. 41012. Sevilla. Spain. Email: [email protected] 2 Dpto. Ecuaciones Diferenciales y Análisis Numérico Facultad de Matemáticas. C/ Tarﬁa s/n. 41012. Sevilla. Spain. Email: [email protected] 3 Comissari 3, 03203 Elche, Alicante. Spain. Email: [email protected]

Abstract. The concept of global attractor for stochastic partial diﬀerential inclusions has been recently introduced as a joint generalization of the theory of random attractors for random dynamical systems and global attractors for multivalued semiﬂows. We present a general result on the upper semicontinuity of attractors for multivalued random dynamical systems. In particular, our theory shows how the random attractor associated to a small random perturbation of a (deterministic) partial diﬀerential inclusion approximates the global attractor of the limiting problem. Some applications ilustrate the results. MSC 2000. 58F39, 60H15, 35K55 Keywords. multivalued maps, random dynamical system, global attractor

1

Introduction

When a phenomenon from Physics, Chemistry, Biology, Economics can be modelled by a system of diﬀerential equations where the existence of global solutions can be assured, one of the most interesting problems is to know what is the asymptotic behaviour of the system when time grows to inﬁnite. In this context, the concept of global attractor has become a very useful tool to describe the long-time This is an overview article.

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behaviour of many important diﬀerential equations (Ladyzhenkaya [13], Babin and Vishik [4], Hale [12], Temam [16]). Most of this theory has been successfully and deeply developed for autonomous deterministic partial diﬀerential equations. Some diﬃculties appear when we have to work without uniqueness of solutions in the system or when the model is better described by, for instance, a diﬀerential inclusion. In these cases, it has been shown that the theory of multivalued ﬂows makes suitable the treatment of the asymptotic behaviour of these diﬀerential equations and inclusions, and the concept of global attractor can be generalized to this situation (Ball [5], Melnik and Valero [14]). When a random term is added to the deterministic equation, the corresponding stochastic partial diﬀerential equation must be treated in a diﬀerent way. The new and rapidly growing theory of random dynamical systems (Arnold [1]) has become the appropriate tool for the study of many important random and stochastic diﬀerential equations. In this framework, Crauel and Flandoli [11] (see also Schmalfuss [15]) introduced the concept of random attractor as a proper generalization (see Caraballo et al. [7] for a justiﬁcation of this fact) of the corresponding (deterministic) global attractor. Recently, Caraballo et al. [8,9] have generalized the theory of attractors to the case of stochastic partial diﬀerential inclusions. A global attractor in this case is a family of compact sets, invariant for the corresponding multivalued random dynamical system and attracting all bounded sets “from −∞” at any ﬁxed ﬁnal time (see Section 2). The theory has been successfully applied to some stochastic diﬀerential inclusions with additive and multiplicative noise. We study in this note the relationship between the random attractor for multivalued random dynamical systems and the global attractor for (deterministic) multivalued semiﬂows. Indeed, given a multivalued semiﬂow with an associated global attractor, we can add a small random perturbation so that the new stochastic multivalued dynamical system has a random attractor. Although the two attracting sets are too diﬀerent (a compact set for the deterministic situation and an unbounded familiy of sets in the random case), we are able to prove a result on the upper semicontinuity of the attractors when we make the random perturbation tend to zero. In other words, when the perturbations are small enough, each compact set of the random attractor is within a small neighbourhood of the global attractor associated to the multivalued semiﬂow. This result, together with that in Caraballo et al. [7], in our opinion, comes to reinforce the concept of random attractor as a proper generalization of global attractors for deterministic diﬀerential equations and inclusions. We apply this result to a stochastic partial diﬀerential inclusion generated by additive white noise.

2

Multivalued dynamical systems (MDS)

In this section, we summarize the main concepts and results on dynamical systems related to problems containing multivalued functions and stochastic terms. These situations have led to diﬀerent branches in the theory of dynamical systems

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and they have become the proper frameworks for the study of their qualitative properties. 2.1

Multivalued semiflows and global attractors

Let (X, dX ) be a complete and separable metric space with the Borel σ-algebra B (X), R+ = [0, +∞) and P (X), B(X), C(X), K(X) be the set of all nonempty, nonempty bounded, nonempty closed and nonempty compact subsets of X, respectively. Definition 1. The map G : R+ × X → P (X) is called a multivalued semiﬂow if it satisﬁes: i) G(0, ·) = I (the identity map in X). ii) G(t + s)x ⊆ G(t)G(s)x, for all t, s ∈ R+ , x ∈ X, where G(t)D = d∈D G(t)d, D ⊂ X. Definition 2. Given a multivalued semiﬂow G and A, B ⊂ X, it is said that A attracts B with respect to G if dist(G(t)B, A) → 0 as t → +∞, where ‘dist’ denotes the Hausdorﬀ semidistance deﬁned by dist(C, D) = sup inf d(c, d). c∈C d∈D

The set A is said to be attracting for G if it attracts every B ∈ B(X). Definition 3. A set A ⊂ X is called a global attractor associated to G if it is attracting and negatively semi-invariant, that is, A ⊂ G(t)A, for all t ∈ R+ . Remark 4. In some applications, the global attractor will be a compact subset of the phase space X and invariant for the semiﬂow (i.e. A = G(t)A for all t ∈ R+ ) (Melnik and Valero [14]). Note that this concept is similar to that of global attractor for single-valued semiﬂows (or dynamical systems) associated, for example, to partial diﬀerential equations (Hale [12], Temam [16]). For the sake of completeness, we shall recall the deﬁnitions of upper and lower semicontinuity. Definition 5. G(t) is said to be upper semicontinuous if given x ∈ X and a neighbourhood of G(t)x, O(G(t)x), there exists δ > 0 such that if dX (x, y) < δ then G(t)y ⊂ O(G(t)x). On the other hand, G(t) is called lower semicontinuous if given xn → x (n → +∞) and y ∈ G(t)x, there exists yn ∈ G(t)xn such that yn → y. It is said to be continuous if it is upper and lower semicontinuous.

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Remark 6. Note that these two deﬁnitions are not equivalent in general, as can be seen in easy examples (Aubin and Frankowska [3], Section 1.4). Definition 7. G(t) is said to be w-upper semicontinuous if for all x0 ∈ X we have that for any ε > 0 there exists δ (ε) > 0 such that dist (G (t) y, G (t) x0 ) ≤ ε, for any y satisfying dX (y, x0 ) ≤ δ (ε). Remark 8. Obviously, any upper semicontinuous map is w-upper semicontinuous. The converse is true when G (t) has compact values (see Aubin & Cellina [2]). The following is a general result for the existence of global attractors for multivalued semiﬂows (see Melnik and Valero [14] and also Ball [5] for a similar result): Theorem 9. Let G(t) : X → C(X) be an upper semicontinuous map, for all t ∈ R+ . Suppose there exists a compact attracting set K ⊂ X. Then, there exists a global compact attractor A. Moreover, it is the minimal closed attracting set. 2.2

Multivalued random dynamical systems (MRDS) and random attractors

Let (Ω, F , P) be a probability space and θt : Ω → Ω a measure preserving group of transformations in Ω such that the map (t, ω) -→ θt ω is measurable and satisfying θt+s = θt ◦ θs = θs ◦ θt ;

θ0 = Id.

The parameter t takes now values in R endowed with the Borel σ-algebra B (R) . Definition 10. A set valued map G : R+ ×Ω ×X → C(X) is called a multivalued random dynamical system (MRDS) if it is measurable (that is, the inverse image of an open set is measurable; see Aubin and Frankowska [3, Deﬁnition 8.1.1]) and satisﬁes: i) G(0, ω) = Id on X; ii) G(t + s, ω)x = G(t, θs ω)G(s, ω)x, ∀t, s ∈ R+ , x ∈ X and ω ∈ Ω (perfect cocycle property) . Remark 11. We note that throughout this contribution all the results are obtained for ω in a θ-invariant subset of Ω of probability one, which does not depend on the time variable t. In order to avoid any confusion we shall write “for all ω ∈ Ω” instead of “for P-a.a.” when the time variable appears. Definition 12. A closed random set D is a map D : Ω → C(X), which is measurable. The measurability must be understood in the sense of Castaing and Valadier [10] for measurable multifunctions, that is, {D(ω)}ω∈Ω is measurable if given x ∈ X the map ω ∈ Ω -→ dist(x, D(ω))

Approximation of attractors for MRDS

71

is measurable, where dist(x, D(ω)) = inf d∈D(ω) dX (x, d). A closed random set D(ω) is said to be negatively (resp. strictly) invariant for the MRDS if D(θt ω) ⊂ G(t, ω)D(ω) (resp. D(θt ω) = G(t, ω)D(ω)), ∀t ∈ R+ , ω ∈ Ω. Remark 13. This concept of measurability and the previous one are equivalent (see Aubin and Frankowska [3, Theorem 8.3.1]). Suppose the following conditions for the MRDS G: (G1) There exists an absorbing random compact set B(ω), that is, for every bounded set D ⊂ X, there exists tD (ω) such that for all t ≥ tD (ω) one has G(t, θ−t ω)D ⊂ B(ω), P − a.s. (1) (G2) G(t, ω) : X → C(X) is upper semicontinuous, for all t ∈ R+ and ω ∈ Ω. Definition 14. The closed random set ω -→ A(ω) is said to be a global random attractor for the MRDS G if: i) G(t, ω)A(ω) ⊇ A(θt ω), for all t ≥ 0, ω ∈ Ω (that is, it is negatively invariant); ii) For all bounded D ⊂ X, lim dist(G(t, θ−t ω)D, A(ω)) = 0, P − a.s.;

t→+∞

iii) A (ω) is compact P − a.s. We have the following theorem on the existence of random attractors for MRDS (Caraballo et. al. [8]): Theorem 15. Let (G1)–(G2) hold, the map (t, ω) -−→ G(t, ω)D be measurable, for all D ∈ B(X), and the map (t, ω, x) -→ G(t, ω)x have compact values. Then there exists the global random attractor A(ω) for G, strictly invariant and measurable with respect to F . It is unique and the minimal closed attracting set.

3

Upper semicontinuity of attractors

Suppose that we have a deterministic multivalued dynamical system G0 : R+ × X → P (X) such that it satisﬁes the conditions of Theorem 9, so that there exists a global attractor A associated to it. Assume also that G0 has compact values. Now we perturb the multivalued system by adding a random term depending on a small parameter σ ∈ (0, 1], so that we obtain a MRDS Gσ : R+ × Ω × X → C(X). We shall assume the following condition:

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(H1) For all ω ∈ Ω, t ∈ R+ it holds dist(Gσ (t, ω)x, G0 (t)x ) → 0, as σ * 0, uniformly on compact sets of X. We also assume that for all σ ∈ (0, 1] Gσ satisﬁes (G1)–(G2), so that there exists a random attractor Aσ (ω) ⊂ Kσ (ω), where Kσ (ω) is a compact absorbing set for Gσ , and that the following condition holds: (H2) There exists a compact set K ⊂ X such that lim dist(Kσ (ω), K) = 0, for all ω ∈ Ω.

σ0

Then we can prove the following result: Theorem 16. Assume that (G1)–(G2), (H1)–(H2) hold and G0 satisifes the conditions of Theorem 9 and has compact values. Then, lim dist(Aσ (ω), A) = 0, for all ω ∈ Ω.

σ0

4

Applications: a partial diﬀerential inclusion perturbed by additive noise

Let X be a real separable Hilbert space with the scalar product ·, · and the norm ·. Consider the following stochastic diﬀerential inclusion which can be regarded as a random perturbation of a deterministic diﬀerential inclusion with the small parameter σ ≥ 0, # m du ∈ Au (t) + F (u (t)) + σ i=1 φi dwdti (t) , t ∈ (0, T ) , (2) dt u (0) = u0 , where A : D (A) → X is a linear operator, φi ∈ D (A) and wi (t) are independent two-sided (i.e. t ∈ R) real Wiener processes with wi (0) = 0, i = 1, ..., m. Let us introduce the following conditions: (A) The operator A is m-dissipative, i.e.

Ay, y ≤ 0 , ∀y ∈ D(A), and Im(A − λI) = X, ∀λ > 0. (F1) F : X → Cv (X), where Cv (X) is the set of all non-empty, bounded, closed, convex subsets of X. (F2) The map F is Lipschitz on D(A), i.e. ∃C ≥ 0 such that distH (F (y1 ), F (y2 )) ≤ C y1 − y2 , ∀y1 , y2 ∈ D(A), where distH (·, ·) denotes the Hausdorﬀ metric of bounded sets, i.e. it holds distH (A, B) = max{dist(A, B), dist(B, A)}.

Approximation of attractors for MRDS

73

It is well known that for σ = 0, under some additional assumptions, diﬀerential inclusion (2) generates a multivalued semiﬂow G0 which has the global compact attractor A (see Melnik & Valero [14]). Now we consider the random perturbation for σ > 0 and prove that we can construct a multivalued random dynamical system having a random attractor Aσ (ω) which approximates the deterministic one in the upper semicontinuity sense. m Let us denote ζ (t) = i=1 φi wi (t) and consider the Wiener probability space (Ω, F , P) deﬁned by Ω = {ω = (w1 (·) , ..., wm (·)) ∈ C (R, Rm ) | ω (0) = 0} , equipped with the Borelσ−algebra F and the Wiener measure P. Each ω ∈ Ω generates a map ζ (·) = m i=1 φi wi (·) ∈ C (R, X) such that ζ (0) = 0. We make the change of variable v (t) = u (t) − σζ (t). Inclusion (2) turns into #

m dv ∈ Av (t) + F (v (t) + σζ (t)) + σ i=1 Aφi wi (t) , dt v (0) = v0 = u0 .

(3)

We shall deﬁne the multivalued map F"σ : [0, T ] × Ω × X → Cv (X) , $σ (t, ω, x) = F (x + σζ (t)) + σAζ (t) . F It is easy to obtain from (F2) the existence of constants D1 , D2 such that F (x)+ ≤ D1 + D2 x ,

(4)

+

where F (x) = sup y. Hence, y∈F (x)

% %+ %$ % %Fσ (t, ω, x)% ≤ D1 + D2 x + D2 σ ζ (t) + σ Aζ (t) = nσ (t, ω, x) .

(5)

$σ satisﬁes the next property: It follows that F (F3) For any x ∈ X there exists n (·) ∈ L1 (0, T ) depending on x, ω and σ such that %+ % % %" %Fσ (t, ω, x)% ≤ nσ (t) , a.e. in (0, T ) . $σ satisﬁes conditions (F1)–(F2) for any On the other hand, it is clear that F ﬁxed t ∈ [0, T ] and ω ∈ Ω, where the constant C does not depend on t or ω or σ. Consider also the equation   dv(t) = Av(t) + f (t), dt (6)  v(0) = v0 , where f (·) ∈ L1 ([0, T ], X).

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Definition 17. The continuous function v : [0, T ] → X is called an integral solution of problem (6) if: 1. v(0) = v0 ; 2. ∀ξ ∈ D(A), one has 2

2

v(t) − ξ ≤ v(s) − ξ + 2

t

f (τ ) + Aξ, v(τ ) − ξ dτ, t ≥ s.

(7)

s

Definition 18. The function u : [0, T ] → X is called a strong solution of problem (6) if: 1. u(·) is continuous on [0, T ] and u(0) = u0 ; 2. u(·) is absolutely continuous on any compact subset of (0, T ) and almost everywhere (a.e.) diﬀerentiable on (0, T ); 3. u(·) satisﬁes (6) a.e. on (0, T ). It is well known (see Barbu [6, p.124]) that any strong solution of problem (6) is an integral solution. Definition 19. The function vσ : [0, T ]× Ω → X is said to be an integral solution of problem (3) if for any ω ∈ Ω one has: 1. vσ (·) = vσ (·, ω) : [0, T ] → X is continuous. 2. vσ (0) = v0 ; $σ (t, ω, vσ (t)) a.e. on (0, T ), 3. For some selection f ∈ L1 ([0, T ], X), f (t) ∈ F inequality (7) holds. If condition (A) holds and f ∈ L1 ([0, T ], X), then for all v0 ∈ D(A) there exists a unique integral solution v(·) of (6) for each T > 0 (see Barbu [6, p.124]). We shall denote this solution by v(·) = I(v0 )f (·). Since T > 0 is arbitrary, each solution can be extended on [0, ∞). Let us denote by Dσ (v0 , ω) the set of all integral solutions of (3) such that v(0) = v0 (D0 (v0 ) for σ = 0). We deﬁne the maps Gσ : R+ × Ω × D(A) → P (D(A)), ϑs : Ω → Ω as follows Gσ (t, ω)v0 = {vσ (t) + σζ (t) | vσ (·) ∈ Dσ (v0 , ω)}, θs ω = (w1 (s + ·) − w1 (s) , ..., wm (s + ·) − wm (s)) ∈ Ω. Then the function w " corresponding to θs ω is deﬁned by ζ" (τ ) = ζ (s + τ ) − ζ (s) = m φ (w (s + τ ) − wi (s)) . i i=1 i By similar computations to the ones in Caraballo et al. [8] but taking care of the new parameter σ, we have the following results: Theorem 20. Let (A), (F1), (F2) hold and the semigroup S (t, ·) generated by the operator A be compact. Then, Gσ generates a MRDS.

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Proposition 21. Let (A), (F1), (F2) hold. Suppose that each integral solution of (3), vσ (·) = I (u0 ) f (·) is a strong solution of (6). Let there exist constants δ > 0, M ≥ 0 such that

y, u ≤ (−δ + α) u2 + M, ∀u ∈ D (A) , y ∈ F (u) ,

(8)

where α ≥ 0 is the biggest constant such that 2

Au, u ≤ −α u .

(9)

Then there exists a random radius rσ (ω) > 0 such that for any bounded set B ⊂ D (A) we can ﬁnd T (B) ≤ −1 for which +

Gσ (−1 − t0 , θt0 ω) u0 ≤ rσ (θ−1 ω) , P − a.s., ∀t0 ≤ T (B) , ∀u0 ∈ B. Theorem 22. Let the conditions of Proposition 21 hold, the semigroup S (t, ·) generated by the operator A be compact and the multivalued map Gσ (1, ω) be P-a.s. compact (that is, it maps bounded sets into precompact ones). Then Gσ satisﬁes (G1)–(G2) and has the minimal global random attractor Aσ (ω). Moreover, it is invariant and measurable with respect to F . Remark 23. Note that in the case where σ = 0 the semiﬂow G0 satisﬁes the conditions of Theorem 9, the global attractor obtained in Theorem 22 is deterministic and coincides with the global attractor A of Melnik & Valero [14]. Proposition 24. In the conditions of Theorem 22 we have that lim dist(Gσ (t, ω)x, G0 (t)x) = 0, for all ω ∈ Ω, t ∈ R+ ,

σ0

uniformly on compact subsets of D (A). For (H2) note that a compact absorbing set for Gσ is given by Kσ (ω) = Gσ (1, θ−1 ω)B(rσ (θ−1 ω)), where B(rσ (θ−1 ω)) is a ball in H with radius rσ (θ−1 ω) (see Caraballo et. al. [8]). Lemma 25. In the conditions of Theorem 22 we have for the compact set K = G0 (1, B (R)) dist (Kσ (ω) , K) → 0, as σ * 0, P-a.s. We have proved that (H2) holds. As a consequence of Theorem 22, Proposition 24, Lemma 25 and Theorem 16 we obtain: Theorem 26. In the conditions of Theorem 22 we have dist (Aσ (ω) , A) → 0, as σ * 0, P-a.s. Remark 27. The theory can also be applied to diﬀerential inclusions perturbed by multiplicative noise (Caraballo et al. [9]).

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References 1. Arnold L., Random Dynamical Systems, Springer Monographs in Mathematics (1998). 2. Aubin J.P., Cellina A., Diﬀerential Inclusions, Springer-Verlag, Berlin (1984). 3. Aubin J.P., Frankowska H., Set-Valued Analysis, Birkhäuser, Boston (1990). 4. Babin A.V., Vishik M.I., Attractors of partial diﬀerential equations and estimate of their dimension, Russian Math. Surveys 38 (1983), 151-213. 5. Ball J.M., Continuity properties and global attractors of generalized semiﬂows and the Navier-Stokes equations, J. Nonlinear Sci. 7 (1997), 475-502. 6. Barbu V., Nonlinear Semigroups and Diﬀerential Equations in Banach Spaces, Editura Academiei, Bucuresti (1976). 7. Caraballo T., Langa J.A., Robinson J., Upper semicontinuity of attractors for small random perturbations of dynamical systems, Com. Partial Diﬀerential Equations 23 (1998), 1557-1581. 8. Caraballo, T. Langa, J.A., Valero, J., Global attractors for multivalued random dynamical systems, Non. Anal. TMA, to appear. 9. Caraballo, T. Langa, J.A., Valero„ J., Global attractors for multivalued random dynamical systems generated by random diﬀerential inclusions with multiplicative noise, J. Math. Anal. and Appl. 260 (2001), 602-622. 10. Castaing C., Valadier M., Convex Analysis and Measurable Multifunctions, LNM Vol. 580, Springer (1977). 11. Crauel H., Flandoli F., Attractors for random dynamical systems, Prob. Theory Related Fields 100 (1994), 365-393. 12. Hale J., Asymptotic Behavior of Dissipative Systems, Math. Surveys and Monographs, AMS, Providence (1988). 13. Ladyzhenskaya O., Attractors for Semigroups and Evolution Equations, Accademia Nazionale dei Lincei, Cambridge University Press, Cambridge (1991). 14. Melnik V., Valero J., On Attractors of multivalued semi-ﬂows and diﬀerential inclusions, Set-Valued Anal. 6 (1998), 83-111. 15. Schmalfuss B., Backward cocycle and attractors of stochastic diﬀerential equations, in V. Reitmann, T. Redrich and N. JKosch (eds.), International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractor Approximation and Global Behaviour (1992), 185-192. 16. Temam R., Inﬁnite Dimnesional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York (1988). 17. Tolstonogov A.A., On solutions of evolution inclusions.I, Sibirsk. Mat. Zh. 33, 3 (1992), 161-174 (English translation in Siberian Math. J., 33, 3 (1992)).

Equadiﬀ 10, August 27–31, 2001 Prague, Czech Republic

Equadiﬀ 10 CD ROM Papers, pp. 77–82

Distributed Delayed Competing Predators Mario Cavani Departmento de Matemáticas, Escuela de Ciencias, Universidad de Oriente, Cerro Colorado, Cumaná 6101, Venezuela, Email: [email protected]

Abstract. We propose a model of competing species where the dynamic of the predators depend on the past history of the prey by mean of distributed delay that takes an average of the Michaelis -Menten functional response of the prey population. We show that the system is pointwise dissipative and the existence of a global attractor of the solutions of this model.

MSC 2000. 39B82, 34K60

Keywords. predator-prey models, distributed delays

1

Statement of the model

In this paper we improve the model of two competing predators species proposed by S. B. Hsu, S. P. Hubbell and P. Waltman in [6] and [5]. There, the authors have considered a renewable resourse with reproductive properties, a more classic prey, and the predators compete purely exploitatively for the prey without interference between rivals. Our purpose here, in a more realistic fashion, is to introduce a distributed delay in the equations in the same way that in [2] and [9]. Thus we start with the model

Research supported by Consejo de Investigación, Univeridad de Oriente, Proyecto No. C.I. 5-1003-1036/01

This is the preliminary version of the paper.

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S(t) m1 x1 (t)S(t) m2 x2 (t)S(t) − S (t) = S(t) 1 − − K a1 + S(t) a2 + S(t) x1 (t) =

m1 x1 (t)S(t) − D1 x1 (t) a1 + S(t)

x2 (t) =

m2 x2 (t)S(t) − D2 x2 (t) . a2 + S(t)

that have been analyzed in [6]. In the equations x1 (t), x2 (t) denote the predator populations that compete by the prey specie, S(t) at the time t. Both species have access to the prey and compete only by lowering the population of shared prey. For death rates it is assumed that the number dying is proportional to the number currently alive. In the absence of predators the prey grows logistically and the predator’s functional response obey the Michaelis-Menten kinetic too so called the Holling type curve. The parameter mi is the maximum growth (birth) rate of the ith predator, Di is the death rate for the ith predator, ai is the half-saturation constant for the ith predator, which is the prey density at which the functional response of the predator is half maximal. The parameter K is the carrying capacity for the prey population. We shall consider the following modiﬁcation of the predator-prey model considering that there are signiﬁcative lags in the system by including a distributed delay to describe the time lag involved in the process of conversion of prey consumed into predators. More precisely,we are assuming in a more realistic fashion that the present level of the predators aﬀects instantaneously the growth of the prey, but the growth of the predator is inﬂuenced by the amount of prey in the past. We are supposing that the predator grow up depending on the weight average time of the function of Michaelis-Menten of S over the past per predator. Thus, the model takes the form S(t) m1 x1 (t)S(t) m2 x2 (t)S(t) − S (t) = S(t) 1 − − K a1 + S(t) a2 + S(t) t m1 S(τ ) α1 x1 (τ ) exp(−α1 (t − τ )) x1 (t) = −D1 x1 (t) + dτ a1 + S(τ ) −∞ t m2 S(τ ) α2 x1 (τ ) exp(−α2 (t − τ )) dτ. x2 (t) = −D2 x2 (t) + a2 + S(τ ) −∞

(1)

In the literature the kernel, K(u) = α exp(−αu),

(2)

is called the weak kernel and is frequently used in biological modeling and clearly implies that the inﬂuence of the past is fading away exponentially and the number

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1 might be interpreted as the measure of the inﬂuence of the past. So, to smaller αi αi , longer is the interval in the past in which the values of S are taken into account (see [3], [7]). This kernel has the property that t ∞ αi exp(−αi (t − τ ))dτ = αi exp(−αi s)ds = 1. (3) −∞

0

We show that this system is pointwise dissipative and therefore that there exists an global attractor for the solutions of the model.

2

An equivalent system

The adequate space for the initial data of our problem and some related notations is as follow: 3 denote the Banach space of the bounded and continuous function Let BC+ mapping from the interval (−∞, 0] to R3+ , the 3-dimensional vectors with positive coordinates . From the general theory of integral diﬀerential equations(see [1] and 3 , there exists a unique solution [8]), for any initial data φ = (φ0 , φ1 , φ2 ) ∈ BC+ π(φ; t) := (S(φ; t), x1 (φ; t), x2 (φ; t)) for all t > 0 and π(φ; 0) |(−∞,0] = φ. Through3 out, we will use (S(t), x1 (t), x2 (t)) to denote the solution π(φ; t) with φ ∈ BC+ , when no confusion arises. By a positive solution π(φ; t) or (S(t), x1 (t), x2 (t)) of 3 system (1), we shall mean the solution has initial condition φ ∈ BC+ and each component of the solution is positive for all t > 0. However, the integrodiﬀerential system (1) can be associated to an system ordinary diﬀerential equations in the following way: We introduce the two new unknown functions, yi , deﬁned by t mi S(τ ) yi (t) = αi xi (τ ) exp(−αi (t − τ )) dτ, i = 1, 2. (4) ai + S(τ ) −∞ and using the the form of the weak kernel (2) and the linear trick chain technique (see [7]) we get the new system of ordinary diﬀerential equations, S(t) m1 x1 (t)S(t) m2 x2 (t)S(t) − S (t) = S(t) 1 − − K a1 + S(t) a2 + S(t) x1 (t) = −D1 x1 (t) + y1 (t) x2 (t) = −D2 x2 (t) + y2 (t) m1 x1 (t)S(t) − y1 (t) a1 + S(t) m2 x2 (t)S(t) − y2 (t) . y2 (t) = α2 a2 + S(t) y1 (t) = α1

(5)

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Thus, system (1) is “equivalent” to this system of ﬁve-dimensional ordinary differential equations. We understand the relationship between the systems (1) and (5) as follows : If (S, x1 , x2 ) : [0, ∞) → R3+ is the solution of (1) correspond˜ x˜1 , x˜2 ) : (−∞, 0] → R3 , ing to the continuous and bounded initial functions (S, + 5 then (S, x1 , x2 , y1 , y2 ) : [0, ∞) → R+ is solution of (5) with the initial conditions ˜ S(0) = S(0), x1 (0) = x˜1 0 , x2 (0) = x˜2 0 and 0 mi S(τ ) αi xi (τ ) exp(−αi (t − τ )) yi (0) = dτ, i = 1, 2. ai + S(τ ) −∞ Conversely, if (S, x1 , x2 , y1 , y2 ) is any solution of (5), deﬁned on the entire real line and bounded on (−∞, 0], then yi , i = 1, 2, is given by (2) so (S, x1 , x2 ) satisﬁes (4). In most of the results of the paper we will consider the “equivalent system” (5).

3

Pointwise dissipativity and existence of an global attractor

The ﬁrst lemma is basic for the well-posedness of the model (1). The result indicates that the model (1) possesses the property that positive initial data yield positive solutions. 3 with φ0 > 0, φi > 0, i = 1, 2, the Lemma 1 (Positivity). For any φ ∈ BC+ solution π(φ; t) remains positive for all t > 0.

Proof. Clearly, t m2 x2 (τ ) S(τ ) m1 x1 (τ ) − 1− − dτ , S(t) = S(0) exp K a1 + S(τ ) a2 + S(τ ) 0 and so S(t) > 0 for all t > 0 since S(0) > 0. To show that xi (φ; t) > 0 for all t > 0, we suppose the contrary, that it is not true. Let t¯ = inf {t > 0 : xi (φ; t) = 0 and xi (θ) > 0, 0 ≤ θ < t} < ∞. Then xi (t¯) = 0 and xi (t¯) ≤ 0 and.But from (1), we have xi (t¯) = −Di xi (t¯) +

t¯

−∞

αi xi (τ ) exp(−αi (t − τ ))

t¯

= −∞

αi xi (τ ) exp(−αi (t − τ ))

mi S(τ ) ai + S(τ )

mi S(τ ) ai + S(τ )

dτ > 0.

This is a contradiction. Therefore, xi (φ; t) > 0 for any t > 0.

dτ

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This lemma implies that the set E = {(S, x1 , x2 , y1 , y2 ) ∈ R5 | S > 0, x1 > 0, x2 > 0, y1 > 0, y2 > 0} is invariant under the ﬂow induced by the system (5). The following theorem has to do with the property of pointwise dissipativity. Theorem 2 (Pointwise Dissipativity). All positive solutions of model (1) are bounded for t > 0. Moreover, system (5) is pointwise dissipative and the absorbing set (into which every solution eventually enters and remains) is given by B = [0, K] × [0, L(D1 )] × [0, L(D2 )] × [0, where L(D) =

K K + 1] × [0, + 1], p p

(6)

1 K + + 1 and p = min {1, α1 , α2 }. pD D

Proof. Of the equation for S in (5) it is easy to see that we can use an similar argument as in the proof of Lemma 3.1 in [6] to obtain the boundedness of S(t). Precisely, for suﬃciently small ε > 0 there exists T depending only on S(0), such that S(t) < K + ε, for t > T . Now let y2 y1 + W =S+ α1 α2 then, for t > T we have, y1 y + 2 α1 α2 S = S(1 − ) − y1 − y2 , K y1 y2 < K + ε − S − α1 ( ) − α2 ( ), α1 α2 y1 y2 < K + ε − p{S − − }, α1 α2

W = S +

where p = min{1, α1 , α2 }. And so, W < K − pW. This clearly implies the uniform boundedness of y1 and y2 . Note too that taking into account the equations for xi in the equations (5), the boundedness of yi (t) implies the boundedness of xi (t). Taking into account the previous diﬀerential inequality and the equations for xi in the system (5), we can get a number T1 = T1 (ε, P0 ), P0 = (S0, x10 , x20 , y10 , y20 ), such that 0 < S(t) < K + ε, 0 < xi (t) < Mi +

ε ε K +1+ , , 0 < yi (t) < pDi p p

(7)

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K 1 + + 1, for all t > T1 .Thus we have the pointwise dissipativity pDi Di for the systems (5).

where Mi =

The following corollary give us the existence of an global attractor. Corollary 3. The solution of the system 1 have a global attractor. Proof. It is an inmediate consequence of the theorem 3.4.6, page 39, in [4].

References 1. T. A. Burton, Volterra Integral and Diﬀerential Equations, Academic Press, New York, 1983. 2. M. Cavani, M. Lizana, H. L. Smith, Stable Periodic Orbits for a Predator-Prey Model with Delay, J. of Math. Anal. and Appl., 249 (2000), 324-339. 3. J. M. Cushing, Integrodiﬀerential Equations and Delay Models in Population Dynamics 20, Springer-Verlag, Heidelberg, 1977. 4. J. K. Hale, Asymptotic behavior of dissipative systems, Mathematical Surveys and Monographs No. 25, American Mathematical Society, Providence, Rhode Island:1988. 5. S. B. Hsu, S. P. Hubbell, P. Waltman, A contribution to the theory of Competing Predators, Ecological Monographs, 48 (1978), 337-349. 6. S. B. Hsu, S. P. Hubbell, P. Waltman, Competing Predators, SIAM J. Appl. Math. 35 (1978), 617-625. 7. N. MacDonald, Time Lags in Biological Models, Lecture Notes in Biomathematics 27, Springer-Verlag, Heidelberg,1978. 8. R. K. Miller, Nonlinear Volterra Integral Equations, Benjamin, New York, 1971. 9. G. Wolkowicz, H. Xia, S. Ruan, Competition in the Chemostat: A distributed delay model and its global asymptotic behavior, SIAM J. Appl. Math. 57(1997), 1281-1310.

Equadiﬀ 10, August 27–31, 2001 Prague, Czech Republic

Equadiﬀ 10 CD ROM Papers, pp. 83–101

Some Applications of Parabolic Diﬀerential Equations of Allen-Cahn type in Image Processing Vladimír Chalupecký1 and Michal Beneš2 1

2

Department of Mathematics, Czech Technical University in Prague, Trojanova 13, 120 00 Praha, Czech Republic Email: [email protected] Department of Mathematics, Czech Technical University in Prague, Trojanova 13, 120 00 Praha, Czech Republic Email: [email protected]

Abstract. In this contribution, we present several algorithms for image processing based on some modiﬁcations of the Allen-Cahn equation. These algorithms include noise removal, pattern recovery and shape recovery and are motivated by similar models based on the level set formulation of motion by mean curvature. The equations are solved by semi-implicit ﬁnite difference method. Also, results of some numerical experiments are included.

MSC 2000. 35K55, 35K65, 65M06, 68U10

Keywords. Image processing, ﬁnite diﬀerence method, mean curvature, Allen-Cahn equation

1

Introduction

In this contribution, we will present some algorithms for image processing by means of modiﬁcations of the Allen-Cahn equation, which is closely connected with notion of mean curvature motion.

This work has been partly supported by grants of the Grant Agency of Czech Republic number 201/01/0676 and of Ministry of Education of Czech Republic number MSM 98/210000010.

This is an overview article summarizing the degree thesis of the ﬁrst author.

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The family of curves Γ t : S → R2 , t ∈ (0, T ), S ⊂ R, undergoes motion in the normal direction and the velocity is a function of curvature κ. The isotropic motion law then is υΓ = −κ + F. (1) Equations of this type have recieved a lot of attention during recent years, both from the theoretical and practical point of view, as there is a wide range of possible applications. They can be solved by directly discretizing the parameterized curve and these methods are then called direct. Another approach are indirect methods, where the evolving curve is a level set of a higher dimensional function, that is the solution of a partial diﬀerential equation. Level set methods are based on a degenerate parabolic equation, that can be directly derived from the equation (1). Level sets of the solution of this equation move in the normal direction with speed a function of curvature, as it was introduced in the work of Osher and Sethian [20]. In order to solve this equation numerically, some kind of regularization needs to be introduced. The Allen-Cahn equation [3] can be considered as another possible regularization of level-set methods. It originates from the theory of phase transitions [3,4,5,6,7,8], where the evolving curve represents an interface separating solid and liquid phases. The dependence of normal velocity of this interface is given by surface tension eﬀects. With level set methods, singularities such as corners, splitting and merging of curves seem to be handled automatically, which is a great advantage. However, from the computational point of view, the level set methods are more computationally expensive than numerical methods based on the direct approach due to the discretization of a 2D domain. The range of applications of the Allen-Cahn in image processing includes among other noise removal, image segmentation, shape recovery and morphing. The goal of this work is to investigate properties of this equation within the context of image processing. Several models for image processing based on the Allen-Cahn equation will be shown followed by the description of numerical approximations of these models. Results of numerical experiments will be demonstrated on some artiﬁcial and real images. This work is organised as follows. In the Section 2, we will shortly describe the Allen-Cahn equation and its diﬀerent forms. In the Section 3, a number of possible applications of the equation in some of the tasks in image processing will be described. The Section 4 is devoted to numerical schemes for solving the AllenCahn equation based on semi-implicit ﬁnite diﬀerence methods. Some remarks on used iterative solvers together with results of several numerical experiments will be presented in the Section 5.

2

Allen-Cahn equation

The Allen-Cahn equation is a well-known reaction-diﬀusion equation, that originates from the theory of phase transitions (see [3,4,5,6,7,8] and references therein). It gives rise to a sharp interface between two domains Ω0t and Ω1t (both depending

Parabolic Equations of Allen-Cahn Type in Image Processing

85

on time), that moves by its mean curvature according to the law (1). The general form of this equation with zero Neumann boundary condition is following ∂p ξ2 = ξ 2 ∆p + f (p; ξ), ∂t ∂p = 0, ∂n ∂Ω p|t=0 = p0 ,

x ∈ Ω,

t > 0,

(2) (3) (4)

where ξ is a given constant and n denotes the outer normal to the domain Ω. The term ξ plays a special role in the model. It determines the width of the interface domain between the domains Ω0t and Ω1t , which is of order ξ. For ξ → 0+ , the motion law for this interface converges to the law (1). Each form of this equation then diﬀers in the choice of the function f (p; ξ) and three possible choices will be described now. Model 1 This is the original form of the function f (p; ξ) and it has the following form [8] f (p; ξ) = ap(1 − p)(p − 0.5) + ξF, (5) where a and F are given constants. However, this model has certain limitations. The behaviour of the solution is given by three zeros of the polynom f (p; ξ). For certain values of ξ and F this polynom looses these three zeros and thus the correct behaviour (see the ﬁgure 2). Another limitation is that the range of the solution is not 0, 1. Model 2 Another form of the function f (p; ξ), that was proposed in [18], is f (p; ξ) = ap(1 − p)(p − 0.5 + ξF ).

(6)

The solution of the Allen-Cahn equation with this form of f (p; ξ) stays between 0 and 1 for all times. However, the constants ξ and F must be chosen so that the term p − 0.5 + ξF has a zero between 0 and 1. Model 3 Better properties has the Allen-Cahn equation with the following form of the function f (p; ξ), which was proposed in [4,5] f (p; ξ) = ap(1 − p)(p − 0.5) + ξ 2 F · |∇p|.

(7)

This form is motivated by the level set formulation of the motion law (1). The solution of this equation should stay within 0, 1 for all t and it approximates the motion law (1) more accurately. Comparison of this model with the corresponding sharp interface law was performed in [7]. In the subsequent sections, the AllenCahn equation solely in this form will be used.

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0.8

0.6

0.4

0.2

0

−0.2

−0.4

−0.6

−0.8

−1 −0.5

0

0.5

1

1.5

Fig. 1. Function f0 (p) = 2p(1 − p)(p − 0.5)

3

Allen-Cahn equation in image processing

In this section, we describe applications of the Allen-Cahn equation in a stage of image processing that is usually called image pre-processing. The aim of image pre-processing is to improve an image and enhance features of objects in it, so that any further analysis, such as object detection and recognition, is more reliable or even possible. A typical ﬁeld of human activity, that uses tools of image processing extensively, is medicine. Visual data from methods like echocardiography or MRI are often disturbed by noise and some image enhacement is thus necessary. Noise removal therefore makes very important part of image pre-processing, but not a sole one. Another important task encountered in image processing is segmentation. The goal of image segmentation is to divide the rectangular domain of the image into ﬁnitely many regions in which a certain property has constant value. This property can be value of intensity function or a speciﬁc high-frequency pattern, often called a texture. Boundaries of these regions are edges, along which the intensity gradient is by deﬁnition “large”. We will address this problem in the Sections 3.1 and 3.2. In the Section 3.3, we will describe how the Allen-Cahn equation can be applied in image morphing. By image morphing we mean a task with two given images, the initial image and the ﬁnal image, and we seek a continous transformation between these two images. Apparently, there are inﬁnitely many such transformations and it is up to the user to choose the appropriate one. Our model performs transformation, in which the level sets of the solution move by mean curvature.

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87

In the following we suppose, that the input image is a real function p0 (x, y) p0 : Ω → 0, 1, representing intensity of a gray-scale image. Ω ⊂ R2 represents a spatial domain (rectangular in practice). The multiscale image analysis (as introduced in [1]), associates with p0 (x, y) a family p(x, y, t) of transformed images depending on an abstract parameter t, that is called a scale. As it has been proved in [1], under basic assumptions the family can be represented as a solution of a second order parabolic partial diﬀerential equation ∂p = F (p, ∇p, ∇2 p, t) ∂t

(8)

p(x, y, 0) = p0 (x, y).

(9)

with the initial condition In our case, the equation (8) will be represented by the Allen-Cahn equation. For other possible forms of equation (8) representing multiscale analysis see [1,2]. 3.1

Noise removal

In this Section, when referring to the Allen-Cahn equation, we consider the following form 1 ∂p = ξ∆p + ap(1 − p)(p − 0.5) + ξF · |∇p|, ∂t ξ ∂p = 0, ∂n ∂Ω p|t=0 = p0 . ξ

(10)

We already know, that the Alle-Cahn equation has some very special properties. With increasing scale t, the Allen-Cahn equation forms regions, in which the solution is close to 0 or 1, and an interface domain of a small width (determined by the parameter ξ), where the solution changes rapidly. This interface moves in the normal direction at a speed that is proportional to its mean curvature. We can use these properties in image processing as follows. Noise in an image is a disturbation, whose level sets are curves with high mean curvature. By a level set of a function p(x, t) at level c we mean the following set L = {x ∈ R2 | p(x) = c}. Thus, if we were able to move these level sets by their mean curvature, the features in an image with high curvature (such as noise) would rapidly shrink to a point, while at the same time important features with lower mean curvature would change only a little. From the properties of the Allen-Cahn equation follows, that this can be succesfully accomplished by applying this equation on the noisy image as an

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initial condition. Moreover, we are able to control, which features in the image will be kept by means of the forcing term F. If we look at the motion law (1) υΓ = −κ + F, we can see, that if the term F is equal to mean curvature, the normal velocity is zero. This means that such an object in the image will be left unchanged, while objects with higher curvature will shrink and objects with lower curvature grow. More precisely, let us consider a simple example with a circle. A circle has curvature equal to 1/r, where r is its radius. Thus, if we put F = 1/r, this circle will remain unchanged during the whole evolution of the equation. For F > 1/r, the circle will shrink and vice versa. With F = 0, the radius of the circle with initial radius r0 at time t is given by & r(t) = r0 2 − 2t The process of shrinking of a circle can be seen at Figure (3.1). The previous

0.8 t=0.3 t=0.4 t=0.45 t=0.499

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8 0.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

Fig. 2. Shrinking of a circle with r0 = 1 by its mean curvature. relation implies a certain disadvantage. As the system evolves, not only the noise is removed, but also other objects in the image, that might be important. In time, every object in the image becomes convex due to the inﬂuence of mean curvature and then shrinks to a point. In the end, the system gets to the steady state, which in the case of images would be a black or white image, depending on the intial image and value of the parameter F . This implies, that the evolution has to be stopped after a certain time. However, no stopping criterion is known, which means, that the stopping time has to be determined experimentally. Another disadvantage is

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89

implied by the properties of the Allen-Cahn equation itself. Its solution tends to values 0 and 1 for any initial condition and any choice of parameters, thus making the equation unsuitable for processing of gray-scale images. For computational experiments, see Section 5. 3.2

Shape recovery

After removing the noise, the subsequent aim is often detection of objects in the image or recovery of their boundaries. Geometric active contour models turn out to be a very useful tool for edge detection in gray-scale images (e.g. [10,11,12,16,17]). The idea behind these methods is quite simple. Given an initial curve, that encloses all the objects to be recovered, we want to move this curve in the direction of its normal vector ﬁeld so that it adheres to the edges in the image. Besides edge detection, active contour models have been used for segmentation, shape modelling and visual tracking. A number of approaches have been proposed for active contour models in the past few years. Caselles et al. [10] and Malladi et al. [19] proposed the following model based on the level set formulation ∇u ∂u = g(x, y)|∇u| ∇ +ν . (11) ∂t |∇u| The function g(x, y) depends on the image I, in which we want to ﬁnd boundaries, and is used as a “stopping term”. In [10,19], the term g=

1 1 + λ|∇Gσ ∗ I|n

was chosen (in [10] n = 1, in [19] n = 2), where Gσ is the Gaussian function. The level sets of u move in their normal direction with speed a function of curvature as it was introduced in [20]. The initial curve given as a level set of the initial condition u0 is put inside or outside the object to be recovered. The stopping term g should ensure, that the speed of shrinking of the curve is small near a boundary, thus stopping the curve at the boundary. However, due to the term ν, this model does not guarantee to stop the motion at the boundary and the edge can be crossed. An improved model has been proposed by Kichenassamy et al. [17]. The model they propose is ∇u ∂u = g|∇u| ∇ + ν + ∇g · ∇u. (12) ∂t |∇u| Again, the shrinking (or growing) of the edge-seeking curve is an inhomogeneous process. The curve in those parts of the image I, where there is no edge, shrinks due to the term ν. When the curve approaches an edge in the image I, the function g is “small”, thus suppressing the inﬂuence of curvature. The convection term ∇g · ∇u attracts the curve to the edge and its form causes, that the curve will not go past an edge, to which it has adhered (see Figure 3.2).

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1 p

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Fig. 3. Explanation of the forcing term in the shape recovery model in onedimensional case. Motivated by the just described models, let us now describe our model based on the Allen-Cahn equation ∂p 1 ξ = g ξ∆p + f0 (p) + ξK, ∂t ξ ∂p = 0, ∂n ∂Ω p|t=0 = p0 , where

# K= g=

F · |∇p| ∇p · ∇g

if if

∇g · ∇p < β ∇g · ∇p ≥ β

(13)

(14)

1 . 1 + λ|∇Gσ ∗ I|2

We modify the Allen-Cahn equation by introducing the function g and by modifying the forcing term F in the same way as in the level set model so that, as the edge-seeking curve evolves, it is attracted to the boundaries in the image, in which we want to recover boundaries (we have denoted this image as I). We supply the initial edge-seeking curve as a boundary of C ⊂ Ω , that contains all objects to be recovered. The initial condition for the modiﬁed Allen-Cahn equation then is p0 = 1 − χC . Let us explain the terms appearing in this equation in more detail. With the choice for the forcing term given by (14), two cases may arise. The parameter β plays a role of a “switch”; it determines, which form of the forcing term should be used:

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1. ∇g · ∇p ≥ β – the edge-seeking curve is given by the level set at 0.5 of the solution p. Then, this case arises, when this curve gets “close” to an edge in the image I. To explain the form of the forcing term in more detail, let us demonstrate the idea in the one-dimensional case as shown at Figure (3.2). As it can be seen from the last ﬁgure, on each side of the edge the forcing term has a diﬀerent sign. As the curve approaches a region with higher gradient in image I, it is attracted by the edge. Moreover, the form of ∇g · ∇p in this case does not allow the curve to go past this edge. If it crossed the edge, the opposite sign of ∇g · ∇p would guarantee, that the curve would be “dragged” back. From these facts, the role of the function g is apparent – at a boundary with high curvature, we need to suppress its inﬂuence. In the other case, the force keeping the curve at the boundary would be smaller than the inﬂuence of curvature and the curve would cross the edge. Then, the boundary could not be recovered. 2. ∇g · ∇p < β – in this case, the edge-seeking curve does not approaches an edge in I and the forcing term has the same form as in the Allen-Cahn equation. This has two reasons. Firstly, if the curve moved only by its mean curvature, the evolution might be too slow. Secondly, during the evolution of the solution, some parts of the curve may become straight. If the term K had the same form as in the previous case, it would be equal or close to zero. Mean curvature in such a part would be zero as well and thus there would not be any force, that would move the curve to the desired boundary. This would result in a situation, when the exact shape of the object could not be recovered. On the other hand, this feature might be sometimes desirable, for instance in the case we wanted to recover a missing part of the boundary. 3.3

Pattern recovery

Pattern recovery in our context essentially diﬀers from shape recovery. In shape recovery, we have one image in which we want to ﬁnd boundaries of objects in it. In pattern recovery, by means of the Allen-Cahn equation with a modiﬁed forcing term we want to obtain a continuous transformation of one image into another. Obviously, there are inﬁnitely many such transformations, but this model chooses the one, in which the level sets move by mean curvature. The model we propose is 1 ∂p = ξ∆p + f0 (p) + ξF (p; p1 ) · |∇p|, ξ ∂t ξ ∂p = 0, ∂n ∂Ω p|t=0 = p0 , F (p; p1 ) = b(p1 − p), b > 0,

(15)

where p0 denotes the initial image (initial condition), p1 is the ﬁnal image and b > 0 is a constant. The equation for pattern recovery consists in modifying the

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forcing term in the type 3 of the Allen-Cahn equation (2). The evolution is then an inhomogeneous process - the forcing term is modiﬁed so that is depends on the solution p and on the ﬁnal image p1 . With this setting, level sets of the solution move to level sets in the ﬁnal image in the apropriate direction, depending on whether the value of the solution is greater or less than the value of the ﬁnal image in a given point. By means of the constant b, we can control speed of the transformation. This model can be easily extended to processing of color images, the equation can be applied to each color channel separately.

4

Numerical schemes

In this section, numerical schemes for the models presented in the Section 3 will be discussed. These schemes have been used to compute the numerical results presented in the Section 5. To approximate the parabolic equtions from the Section 3, the ﬁnite diﬀerence method with the semi-implicit time discretization has been used. The nonlinearity introduced into the models by the polynom f0 (p) was treated from the previous discrete time step, while the linear terms are handled implicitly. In this way, we obtain a linear algebraic system, that can be eﬃciently solved by some iterative method. By using an explicit scheme, we would have to face the stability problems and by using a fully implicit scheme we would have to solve a system of nonlinear equations. Notations Let us ﬁrst introduce notations used throughout this section. Let h and τ be discrete space and time (in our context this variable is called a scale) steps. By Ω we denote the rectangular domain of the image, by nx and ny we denote the number of pixels in the direction of x and y axis respectively and by nT the number of discrete scale steps. Let us denote pj = p(·, jτ ),

pj − pj−1 , pk,s = p(kh, sh), τ ωh = {[ih, jh]|i = 1, . . . , nx − 2; j = 1, . . . , ny − 2},

δpj =

ω ¯ h = {[ih, jh]|i = 0, . . . , nx − 1; j = 0, . . . , ny − 1}, pks − pk−1,s pk+1,s − pks , px,ks = , px¯,ks = h h pks − pk,s−1 pk,s+1 − pks , py,ks = , py¯,ks = h h 1 px¯x,ks = 2 (pk+1,s − 2pks + pk−1,s ), h and ∇h p = [1/2(px¯ + px ), 1/2(py¯ + py )], ∆h p = px¯x + py¯y , as mappings from ωh to R2 or R, respectively.

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4.1

93

Allen-Cahn equation

The semi-implicit scheme for the Allen-Cahn equation, used for the numerical experiments, is 1 j−1 ξδpjh = ξ∆h pjh + f0 (pj−1 h ) + ξF · |∇h ph | on ωh , ξ

(16)

with initial condition p0h = p0 . The solution is a map ph : {1, . . . , nT } × ω ¯ h → R. The zero Neumann boundary condition is handled by reﬂecting the image at the boundaries, for instance at the left boundary of the image holds pj1,s − pj−1,s =0 2h

⇒

pj1,s = pj−1,s

s = 0, . . . , ny − 1.

If we number the pixels by rows from left to right, we can write the scheme using matrix notation Apj = g j−1 ,

A ∈ Rn,n

j = 1, . . . , nT ,

(17)

where n = nx · ny . In this case, pj denotes the vector of unknows and g j−1 is the right-hand side of the linear algebraic system, made of terms from the previous scale step. The matrix A is a sparse positive deﬁnite matrix with a special penta-diagonal structure. The boundary condition causes, that the matrix is not symmetric. However, if we multiply the rows of (17) corresponding to the pixels at boundaries by 1/2 and the rows corresponding to the pixels at corners by 1/4, the matrix A becomes symmetric, which allows us to use eﬃcient iterative solvers like conjugate gradient method. 4.2

Shape recovery

The semi-implicit scheme for the shape recovery equation is 1 on ωh , ξδpjh = ξ∆h pjh + f0 (pj−1 h ) + ξK ξ # F · |∇h pjh | if ∇h pjh · ∇h gh < β K= if ∇h pjh · ∇h gh ≥ β ∇h pjh · ∇h gh 1 gh = , 1 + λ(Gσ ∗ I)2h with initial condition p0h = p0 . The image, in which we want to recover boundaries, is denoted by I.

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Pattern recovery

The semi-implicit FDM scheme of this model is 1 j−1 j−1 ξδpjh = ξ∆h pjh + f0 (pj−1 h ) + ξF (ph )|∇h ph | on ξ F (p) = b(p1 − p), b > 0,

ωh ,

with initial condition p0h = p0 .

5

Computational results

As it has been mentioned in the previous section, the semi-implicit FDM discretization leads to solving a large sparse algebraic system. In most cases, the matrix of this system is symmetric, which allows one to use eﬃcient iterative solvers like conjugate gradient method, possibly with some kind of preconditioning. In the case of the shape recovery model, the matrix is not symmetric, which inﬂuences the choice of the solver. For instance, possible choices then could be bi-conjugate gradient method or classical relaxation methods like Gauss-Seidel or SOR. In fact, in most cases, the use of the Gauss-Seidel method is suﬃcient due to the special properties of the Allen-Cahn equation. With lowering the parameter ξ (and thus increasing the accuracy of the model), one needs to lower the discrete scale step as well. The solution between two scale levels does not diﬀer much and the solution from the previous scale level can be used as a good initial guess for the solver. Number of required iterations then is not large, thus the convergence rate even of the Gauss-Seidel method is very good. Compared to the CG method, the advantage of the Gauss-Seidel method is in lower memory requirements and in simplicity of the algorithm. # of iter 1st step 5th step 20th step 50th step Total time Gauss-Seidel 23 23 23 22 153.22 CG No precond 28 28 26 24 250.45 CG Jacobi 21 20 17 17 252.85 CG IC(0) 17 17 16 15 373.38

Table 1. Comparison of iterative solvers for the scheme (16); gray-scale image size: 512x512; implicit scheme

Despite the fact, that for the presented numerical computations the GaussSeidel method has been used, it is interesting to compare this method with CG

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without preconditioning and with Jacobi preconditioner and the IC(0) preconditioner. See Table (1) for comparison of these two types of preconditioners. The results in this table were computed by solving the Allen-Cahn equation of the type 3 on AMD 650MHz with 128MB memory. In each column there is a number of solver iterations at a given scale. The total time, shown in the last column, is amount of time spent in the solver during the evolution.

5.1

Computational experiments

Example 1 Allen-Cahn equation with gradient term. Gray-scale image (size 512x 512 pixels), contamined with salt and pepper noise. Parameters: h = 0.05, τ = 0.0005, ξ = 0.05, a = 2, F = 0. Image in the upper left corner is the original image, followed by images after 10, 20 and 30 iterations. Two bottom images show level set at 0.5 in the original and ﬁnal image. Example 2 Allen-Cahn equation with gradient term. Binary image (size 180x180 pixels), contamined with salt and pepper noise. Parameters: h = 0.05, τ = 0.0005, ξ = 0.05, a = 2, F = 10. Image in the upper left corner is the original image contamined with additive noise, followed by images after 4, 8 and 12 iterations. Two bottom images show level set at 0.5 in the original and ﬁnal image. It is apparent, that the Allen-Cahn equation gives excellent results on binary images, even if their structure is quite complicated. The noise vanishes very quickly. Example 3 Pattern recovery model. Initial condition: binary image (size 180x180 pixels), ﬁnal condition: binary image (size 180x180 pixels). Parameters: h = 0.05, τ = 0.0005, ξ = 0.05, a = 2, b = 100. The sequence consists of images after 10, 20, 30, 40 and 50 iterations. Example 4 Shape recovery of a binary image (size 180x180 pixels) contamined by salt and pepper noise. Parameters: h = 0.05, τ = 0.0005, ξ = 0.07, a = 2, b = 100, λ = 1000, β = 10−10 , F = 15. In the upper left corner, the image I is shown and in the upper right corner, level set at 0.5 of the ﬁnal image is shown. Then, initial condition together with images after 300, 600, 800 iterations is depicted. We can see, that a part of the letter Q with high curvature has not been recovered exactly. Example 5 Shape recovery of multiple objects in a binary image (size 256x256 pixels) contamined by salt and pepper noise. Parameters: h = 0.05, τ = 0.0005, ξ = 0.07, a = 2, b = 100, λ = 5000, β = 10−9 , F = 15. In the upper left corner, the image I is shown and in the upper right corner, level set at 0.5 of the ﬁnal image is shown. Then, initial condition together with images after 400, 800, 1200 iterations is depicted. We can conclude, that the presented model gives satisfactory results for binary images even in the presence of noise.

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Fig. 4. Example 1

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Fig. 5. Example 2

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Fig. 6. Example 3

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Fig. 7. Example 4

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Fig. 8. Example 5

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References 1. L. Alvarez, F. Guichard, P.-L. Lions and J.-M. Morel, Axioms and fundamental equations of image processing, Arch. Rational Mech. Anal. 123 (1993), 199–257. 2. L. Alvarez,P. L. Lions and J. M. Morel, Image selective smoothing and edge detection by nonlinear diﬀusion. II, SIAM J. Numer. Anal. 29 (1992), 845–866. 3. S. M. Allen and J. W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metall. 27 (1979), 1085–1095. 4. M. Beneš, On a computational comparison of phase-ﬁeld and sharp-interface model of microstructure growth in solidiﬁcation, Acta Techn. CSAV 41 (1996), 597–608. 5. M. Beneš, Quantitative analysis of phase-ﬁeld model of phase transitions, Proceedings on Prague Mathematical Conference 1996, 1996, pp. 23–28. 6. M. Beneš, Analysis of equations in the phase-ﬁeld model, Proc. of Equadiﬀ 9, Brno, (1997), 17–35. 7. M. Beneš and K. Mikula, Simulation of anisotropic motion by mean curvature - comparison of phase ﬁeld and sharp interface approaches, Acta Math. Univ. Comenianae LXVII (1998), 17–42. 8. G.Caginalp, An analysis of a phase ﬁeld model of a free boundary, Arch. Rational Mech. Anal. 92 (1986), 205–245. 9. G. Caginalp, The dynamics of a conserved phase ﬁeld system: Stefan-like, Hele-Shaw, and Cahn-Hilliard models as asymptotic limits, IMA J. Appl. Math. 44 (1990), 77–94. 10. V. Caselles, F. Catté, T. Coll and F. Dibos, A geometric model for active countours, Numer. Math. 66 (1993), 1–31. 11. V. Caselles and B. Coll, Snakes in movement, SIAM J. Numer. Anal. 33 (1996), 2445–2456. 12. V. Caselles, R. Kimmel and G. Sapiro, Geodesic active contours, Int. J. Comp. Vision. (1997). 13. L. C. Evans and J. Spruck, Motion of level sets by mean curvature I., J. Diﬀ. Geom. 33 (1991), 635–681, 14. A. Handlovičová, K. Mikula and F. Sgallari, Variational numerical methods for solving nonlinear diﬀusion equations arising in image processing, Journal of Visual Communication and Image Representation (2001) (to appear). 15. K. Mikula and J. Kačur Solution of nonlinear diﬀusion appearing in image smoothing and edge detection, Appl. Numer. Math. 17 (1995), 47–59. 16. M. Kass, A. Witkin and D. Terzopoulos, Snakes: active contour models, Int. J. Comp. Vision 1 (1987) 321–331. 17. S. Kichenassamy, A. Kumar, P. Olver, A. Tannenbaum, and A. Yezzi, Jr., Conformal curvature ﬂows: from phase transitions to active vision, Arch. Rational Mech. Anal. 134 (1996), 275–301. 18. R. Kobayashi, Modeling and numerical simulations of dendritic crystal growth, Physica D 63 (1993), 410–423. 19. R. Malladi, J. A. Sethian and B. Vemuri, Shape modelling with front propagation: a level set approach, IEEE Trans. on Pattern Analysis and Machine Intelligence 17 (1995), 158–175. 20. S. J. Osher and J. A. Sethian, Front propagating with curvature dependent speed: algorithms based on the Hamilton-Jacobi formulation, J. Comp. Phys. 79 (1988), 12–49.

Equadiﬀ 10, August 27–31, 2001 Prague, Czech Republic

Equadiﬀ 10 CD ROM Papers, pp. 103–108

A general controllability theorem Veronika Chrastinová

Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, Technical University, Žižkova 17, 662 37 Brno, Czech Republic, Email: [email protected]

Abstract. The report concerns a fundamental result from the theory of quite general systems Ω of smooth partial diﬀerential equations. The existence of a unique “composition series” of the kind Ω0 ⊂ Ω1 ⊂ . . . ⊂ Ω consisting of “factorsystems” of Ω is explained. Here Ωk is the maximal system of diﬀerential equations“induced” by Ω such that the formal solution of Ωk depends on the choice of arbitrary functions of k variables (on constants if k = 0). This is a well-known result only in the particular case of underdetermined systems of ordinary diﬀerential equations. Then Ω1 = Ω and Ω0 involves all ﬁrst integrals, hence Ω0 is trivial if and only if Ω is a controllable system. In full generality, we may speak of a “multidimensional controllability” composition series of Ω. MSC 2000. 35A30, 58A17, 58E99 Keywords. Underdetermined, determined and overdetermined diﬀerential equations, controllability, composition series.

1

Introduction

The article concerns certain aspects of the formal theory of general compatible systems Ω of smooth partial diﬀerential equations ∂ i1 +...+in uk , . . . = 0 (i = 1, . . . , I) Ω : fi x1 , . . . , xn , u1 , . . . , um , . . . , i1 ∂x1 . . . ∂xinn

This research has been conducted at the Department of Mathematics and Descriptive Geometry as part of the research project CEZ : MSM J22 : 261100009.

This is the preliminary version of the paper.

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which can be closely related to the classical calculus of variations. Let us therefore brieﬂy recall the general Lagrange problem of the calculus of variations: a solution u1 , . . . , um of Ω in a domain D and satisfying certain boundary conditions is to be determined such that a given integral ∂ i1 +...+in uk 1 m f x1 , . . . , xn , u , . . . , u , . . . , i1 , . . . dx1 . . . dxn ∂x1 . . . ∂xinn D attains the extremal value. It is well-known that the sought extremal solution u1 , . . . , um obey the familiar necessary conditions, the Euler-Lagrange system d2 d ∂F ∂F ∂F + − ... = 0, − · · k k ∂u dxi ∂(∂u /∂xi ) dxi dxj ∂(∂ 2 uk /∂xi ∂xj ) F =f+ λi fi , λi = λi (x1 , . . . , xn ) , k = 1, . . . , m , where λi are certain unknown coeﬃcients, the Lagrange multipliers. In the favourable case when the system Ω is empty, the functions fi and hence the multipliers are absent and then the theory very simpliﬁes – this case can be found in current textbooks. In general, the presence uf uncertain multipliers λi causes many diﬃculties, especially for the so-called degenerate variational problems. The question may be asked whether the multipliers λi can be explicitly expressed in terms of the functions uk and their derivatives. In the one-dimensional case for n = 1, x = x1 of ordinary diﬀerential equations Ω the answer to the question is known: this is possible if and only if the system Ω does not admit any nontrivial ﬁrst integrals, i. e. , functions that are constant on every solution of Ω. In the terminology of the optimal regulation theory, Ω is called a controllable system in this case. Our idea is as follows: in order to investigate the nature of Lagrange multipliers in the multidimensional case with the number of independent variables n > 1, it is desirable to study the generalized controllability concept for a general system Ω.

2

Classiﬁcation of equations

Eventually passing to the proper topic, let us recall some concepts concerning the formal theory of general systems Ω. Such a system need not have any actual solutions at all but it always has the formal solutions represented by formal power series, i. e. , the values of derivatives of the unknown functions calculated at a ﬁxed point. We shall denote Ω = Ων if this formal solution depends on the choice of certain number µ = µ(Ω) ≥ 1 of arbitrary functions of ν = ν(Ω) independent variables (on constants in ν = 0). Recall that the formal solutions may be identiﬁed with the actual solutions in favourable cases, e. g. , for analytical systems Ω. For better clarity, let us mention some examples. First assume the ordinary diﬀerential equations, hence n = 1, x = x1 . They can be always represented be a ﬁrst order system and then two subcases are distinguished:

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1. Determined system Ω = Ω0 : dui = fi (x, u1 , . . . , um ) ; dx

i = 1, . . . , m ;

• number of equations equals the number of unknown functions, • solution depends on constants u1 (x0 ) = c1 , . . . , um (x0 ) = cm , • investigated in common theory of ordinary diﬀerential equations. 2. Underdetermined system Ω = Ω1 with 0 ≤ a ≤ m: dum dui dua+1 = fi x, u1 , . . . , ua , ua+1 , . . . , um , ,... ; dx dx dx

i = 1, . . . , a ;

• number of equations is less than the number of unknown functions, • ua+1 (x), . . . , um (x) can be arbitrarily chosen, • needful in the calculus of variations. Second, let us state only very particular examples for the case of two independent variables n = 2, denoting x = x1 , y = x2 . Then three subcases should be distinguished: overdetermined, determined or underdetermined system Ω. It is deﬁned by the property that the solution depends either on mere constants, or on the choice of certain functions of one independent variable, or on certain arbitrary functions of two variables. 1. Overdetermined system Ω = Ω0 : ∂uj = fj (x, y, u1 , . . . , um ) ; ∂x

∂uj = gj (x, y, u1 , . . . , um ) ; ∂y

j = 1, . . . , m ;

• number of equations is greater than the number of unknown functions, • values u1 (x0 , y0 ) = c1 , . . . , um (x0 , y0 ) = cm uniquely determine the solution, • occuring e. g. in the theory of Lie groups and diﬀerential geometry. 2. Determined system Ω = Ω1 (Cauchy - Kowalewska system): ∂um ∂uj ∂u1 = fj (x, y, u1 , . . . , um , ,..., ), ∂x ∂y ∂y

j = 1, . . . , m ;

• number of equations equals the number of unknown functions, • initial values u1 (x0 , y), . . . , um (x0 , y) determine the solution, • investigated in the common theory of partial diﬀerential equations (especially in matematical physics). 3. Underdetermined system Ω = Ω2 : ∂um ∂u1 ∂um ∂uj ∂ua+1 = fj (x, y, u1 , . . . , um , ,..., , ,..., ); ∂x ∂x ∂x ∂y ∂y j = 1, . . . , a (0 ≤ a < m)

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• number of equations is less than the number of unknown functions, • ua+1 (x, y), . . . , um (x, y) can be arbitrarily chosen, • needful in the calculus of variations. We shall not state other examples for general n > 2. As for the terminology, let us only recall that always ν ≤ n and it is useful to classify three subcases: Ω = Ωn underdetermined systems (calculus of variations), Ω = Ωn−1 determined systems (mathematical physics), Ω = Ωa , a < n−1 overdetermined systems (Lie groups, diﬀerential geometry). Unfortunately, this terminology is not commonly accepted.

3

Controllability theorem

After these somewhat lengthy preparations, let us eventually pass to the main result: Controllability theorem. Let Ω = Ων be a system such that the formal solution depends on a certain number of arbitrary functions of ν variables, on constants if ν = 0. Then for every a, 0 ≤ a < ν, there exists a unique maximal subsystem Ωa ⊂ Ω which is induced in a certain sense by Ω such that its solutions depend on the choice of arbitrary functions of a independent variables. So we have a unique composition series Ω0 ⊂ Ω1 ⊂ . . . Ωa ⊂ . . . ⊂ Ων

(a < ν) .

The initial term Ω0 involves all ﬁrst integrals F i of Ων , i. e. , the system Ω0 can be represented by certain Pfaﬃan equations dF i = 0 where F i are appropriate functions. This exactly corresponds to the controllability concept for the particular case n = 1, ν = 1 of ordinary diﬀerential equations, but in general we have a far going generalization of this well-known achievement. If all Ω0 , . . . , Ων−1 are trivial, we may speak of the generalized controllable system Ω. We cannot discuss here the proof in more details because rather unusual tools must be employed and instead refer to the literature below. Let us note only that the terms Ωa can be determined by a pure algebraic computation. No deep existence theorems concerning the solutions of partial diﬀerential equations are necessary in the proofs and the same algorithm can be applied if one tries to calculate the Lagrange multipliers of a variational problem.

4

Two examples

Let us conclude with two quite simple examples for the case ot two independent variables (n = 2). We use slightly simpliﬁed notation here.

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A general controllability theorem

1. We start with the determined system of equations ∂u ∂u ∂v ∂v 1 Ω = Ω : F x, y, u, v, , , , ∂x ∂y ∂x ∂y ∂u ∂u ∂v ∂v , , , = G x, y, u, v, = 0, ∂x ∂y ∂x ∂y ∂F /∂ux ∂F /∂uy ∂F /∂vx ∂F /∂vy =2 rank ∂G/∂ux ∂G/∂uy ∂G/∂vx ∂G/∂vy with two unknown functions u = u1 (x, y), v = u2 (x, y). This is really a determined system, the solution depends on the choice of one independent variable. Assuming, e. g. , ∂F /∂ux = 0 for certainty, then the composition series Ω0 ⊂ Ω1 = Ω is nontrivial if and only if the original system can be adapted as Ω = Ω1 : ux − A − Cvx = uy − B − Cvy = 0 ; here A, B, C are functions of x, y, u, v such that Ay − Bx + Au B − Bu A = Cy − Bv + Cu B − Bu C = Av − Cx + Au C − Cu A = 0 . In this case, the diﬀerential form du − C dv − A dx − B dy = µ df

(f = f (x, y, u, v))

is a multiple of total diﬀerential of a function f such that f ≡ const on every solution of Ω. The subsystem Ω0 ⊂ Ω is represented by the Pfaﬃan equation df = 0 and it can be expressed by the overdetermined system Ω0 : ∂u/∂x = A(x, y, u, v) ,

∂u/∂y = B(x, y, u, v) ,

∂u/∂v = C(x, y, u, v)

for the unknown function u = u(x, y, v). In classical terms, for every solution u, v of the original system Ω there exists (in general not unique) solution u of Ω0 such that u(x, y) = u(x, y, v(x, y)). 2. The second example concerns the underdetermined system of a single diﬀerential equation Ω = Ω2 : ∂v/∂y = f (∂u/∂x, ∂u/∂y, ∂v/∂x) with two unknown functions u, v. We have the underdetermined case since the function u = u(x, y) can be arbitrarily chosen in advance. The underdetermined system Ω0 ⊂ Ω is always trivial, and, as a rule, the maximal determined system Ω1 ⊂ Ω2 is trivial, too. However in the exceptional case when f = G (∂u/∂x, ∂v/∂x) ∂u/∂y + H (∂u/∂x, ∂v/∂x) ,

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H = h(G) ,

∂v/∂x − G∂u/∂x = g(G)

for appropriate functions g, h, G, H, the system Ω1 is nontrivial and consists of the equations Ω1 : ∂v/∂x = g (∂v/∂u) ,

∂v/∂y = h (∂v/∂u)

for the unknown function v = v(x, y, u). In classical terms, this result reads: every solution u = u(x, y) , v = v(x, y) of the original system Ω2 satisﬁes the identity v(x, y) = v(x, y, u(x, y)) for appropriate (not unique) solution v of Ω1 .

References 1. Chrastina J., Tryhuk V., On the Structure of Diﬀerential Equations, to appear. 2. Tryhuk V., Chrastinová V., On the Mayer Problem, to appear. 3. Chrastina J., Examples from the Calculus of Variations I, II, III, IV, Mathematica Bohemica 2000, 2001. 4. Chrastina J., The Formal Theory of Diﬀerential Equations, Folia Fac. Scient. Nat. Univ. Masarykianae Brunensis, Mathematica 6, 1998. 5. Lenin V. I., Krok vpřed – dva kroky vzad, VUML Heritage Publishers, 2000.

Equadiﬀ 10, August 27–31, 2001 Prague, Czech Republic

Equadiﬀ 10 CD ROM Papers, pp. 109–118

Bifurcation of periodic points and normal form theory in reversible diﬀeomorphisms. Maria-Cristina Ciocci Universiteit Gent, Vakgroep Zuivere Wiskunde en Computeralgebra, Krijgslaan 281, 9000 Gent, Belgium. Email: [email protected]

Abstract. We survey a number of results on the bifurcation of periodic points from a ﬁxed point in parametrized families of reversible diﬀeomorphisms; such problems arise for example when stufying subharmonic branching in reversible systems. We provide a structure preserving reduction result which can be used to study ’branching phenomena’ near a ﬁxed point. We also brieﬂy discuss how one can determine the stability of bifurcating periodic orbits using normal form theory. Here an improvement of a previous normal form result is given. As an application we give the analysis of the branching of subharmonic solutions from a primary branch of periodic solutions of a reversible system.

MSC 2000. 34C, 58F

Keywords. Periodic Points, reversible mappings, subharmonic branching, normal forms.

1

Set up and basic reduction result

Consider a m-parameter family Φλ of reversible (local) diﬀeomorphisms on Rn having a ﬁxed point at the origin, i.e. Φ : Rn × Rm → Rn , (x, λ) -→ Φλ (x) is such that (H) - Φ(0, λ) = 0 for all λ ∈ Rm ; - Aλ := Dx Φλ (0) ∈ L (Rn ) is invertible for all λ ∈ Rm ; (R) there exists a linear involution R ∈ L (Rn ) (i.e. R2 = Id) such that R · Φλ · R = Φ−1 λ , This is an overview article.

∀λ ∈ Rm .

(1)

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Such reversible diﬀeomorphisms arise for example as stroboscopic maps for periodic non-autonomous time-reversible systems, or as Poincaré-maps for symmetric periodic solutions of autonomous reversible systems. We study then the following bifurcation problem. (P) Given an integer q ≥ 1 and some λ0 ∈ Rm , ﬁnd all the solutions (x, λ) ∈ Rn × Rm near (0, λ0 ) of the equation Φqλ (x) = x;

Φqλ = Φλ ◦ · · · ◦ Φλ (q-times).

(2)

Without loss of generality we can set λ0 = 0. Let A0 = S0 + N0 be the unique semisimple-nilpotent decomposition S0 semisimple, N0 nilpotent and of A0 (i.e. S0 N0 = N0 S0 ). Setting N0 := log I + S0−1 N0 we also see that A0 can be written in a unique way as A0 = S0 eN0 , (3) with N0 nilpotent and S0 N0 = N0 S0 ; we call (3) the semisimple-unipotent decomposition of A0 . One can easily verify that RS0 R−1 = S0−1

and RN0 = −N0 R.

(4)

Introduce then the so-called reduced phase space for our problem; this is a subspace of Rn deﬁned by U := ker (S0q − I). (5) Note that U is the generalized eigenspace corresponding to those eigenvalues of A0 which are q-th roots of unity. Since S0 is semisimple we have that Rn = U ⊕ Im (S0q − I); also, U is invariant under R. Moreover, S0 generates on U a Zq -action. It is shown in [2] that problem (P) reduces via an adapted LiapunovSchmidt method to solving an appropriate determining equation (see equation (6) below). A particular feature of this reduction is that it does not require any restriction on the eigenvalues of A0 , i.e. higher multiplicities and nilpotencies are allowed. Also, the symmetry of the reduced diﬀeomorphism results in a very easy form of the bifurcation equations. In the following theorem we summarize this reduction result, referring to [2] for more details. Theorem 1. Assume (H) and ﬁx some q ≥ 1. Then there exist smooth mappings x∗ : U × Rm −→ Rn and Φr : U × Rm −→ U , such that the following hold (we set Φr,λ := Φr (·, λ)): 1. Φr (0, λ) = 0, Du Φr (0, 0) = A0 |U , x∗ (0, λ) = 0, and Du x∗ (0, 0)· u = u, (for all u ∈ U ); 2. Φr,λ (S0 u) = S0 Φr,λ (u), ∀ (u, λ) ∈ U × Rm i.e. Φr,λ is Zq -equivariant; 3. for all suﬃciently small (x, λ) ∈ R2n × Rm we have that x is a q-periodic point of Φλ if and only if x = x∗ (u, λ) for some suﬃciently small u ∈ U which itself is a q-periodic point of Φr,λ ; 4. for all suﬃciently small (u, λ) ∈ U × Rm we have that u is a q-periodic point of Φr,λ if and only if Φr,λ (u) = S0 u, (6) i.e. all small q-periodic orbits of Φr (., λ) are necessarily Zq -orbits.

Periodic points in reversible diﬀeomorphisms

111

Moreover, if (R) is satisﬁed we have 5. x∗ (Ru, λ) = Rx∗ (u, λ) , 6. R ◦ Φr,λ ◦ R = Φ−1 r,λ , i.e. Φr,λ is R-reversible. This theorem establishes a one-to-one relation between the small q-periodic orbits of Φλ and those of the reduced diﬀeomorphism Φr,λ which lives on a reduced space U . This reduced diﬀeomorphism also retains the additonal structures of the original one, and, as we will explain in section 2, it can be approximated up to any order by using normal forms. Moreover, one can also prove the following. Proposition 2. For suﬃciently small (u, λ) ∈ U ×Rm the equation (6) is equivalent to the equation (7) B (u, λ) := Φr,λ S0−1 u − S0 Φ−1 r,λ (u) = 0. Observe that this equation is Dq -equivariant: indeed, we have that B (S0 u, λ) = S0 B (u, λ) and B (Ru, λ) = −RB (u, λ) ,

(8)

with S0 and R generating a Dq -action on U , see properties (4).

2

Approximating the reduced mapping Φr,λ

One of the possibilities to calculate the reduced mapping Φr,λ is to use normal form theory. The normal form techniques as well as the Lyapunov-Schmidt-like reductions are very popular tools for studying bifurcations. Here we provide a theorem on a reversible normal form for diﬀeomorphisms. More details and proofs can be found in [2]. These proofs are mainly inductive and are based on a combined use of the adjoint action of the group of diﬀeomorphisms satisfying (H) on itself and the implicit function theorem. Some technical results are of course needed to take care of the reversible structure1 ; to this purpose the following is a crucial Lemma. Lemma 3. Let S0 be reversible and semisimple. Then there exists a scalar product

·, · on Rn such that when we denote the transpose of a linear operator A ∈ GL− (n, R) with respect to this scalar product by AT the following holds: (i) the involution R is orthogonal, i.e. RT R = IRn ; (ii) a linear operator A ∈ L(Rn ) commutes with S0 if and only if it commutes with S0T . Here we use x, Ay = AT x, y for all x, y ∈ Rn . For a detailed proof see [2] or [6] where a similar statement is proved when S0 satisﬁes S0 R = −RS0 . Denoting by the exponential the time-one map, one proves the following. 1

The set of reversible diﬀeomorphisms does not form a Lie group.

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Theorem 4. For each k ≥ 1 there exists a neighbourhood ωk of the origin in Rm and a parameter dependent near identity transformation which brings Φλ in the form (9) Φλ = S0 eN0 +Xλ + Rk+1 , with Rk+1 (x, λ) = O xk+1 uniformily for λ ∈ ωk , and with Xλ a smooth parameter-dependent vectorﬁeld on Rn such that Xλ (0) = 0, DX0 (0) = 0 and Xλ (S0 x) = S0 Xλ (x),

T

T

etN0 Xλ = Xλ etN0 .

(10)

Moreover, the vectorﬁeld Xλ is reversible: Xλ (Rx) = −RXλ (x).

(11)

F Then we call ΦN = S0 eN0 +Xλ the normal form of Φλ up to order k. λ This normal form can be used to approximate the reduced mapping Φr,λ : if Φλ is in normal form up to order k then k+1 k+1 F and x∗ (u, λ) = u + O u . (u) + O u Φr,λ (u) = ΦN λ

Just to give an idea on how to deal with the reversibility in normal forms, we prove Theorem 4 in the case of linear reversible operators. We use the following notations GL± (n, R) := {A ∈ GL(n, R)| RAR = A±1 } gl±R (n, R) := {A ∈ gl(n, R)| RA = ±AR}. Notice that gl+R (n, R) is a Lie algebra and the corresponding Lie group is then GL+ (n, R). Also, for Ψ ∈ GL(n, R) deﬁne the operator Ad(Ψ ) : GL(n, R) → GL(n, R) by (12) Ad(Ψ )Φ := Ψ · Φ · Ψ −1 , ∀Φ ∈ GL(n, R), and for ψ ∈ gl(n, R) deﬁne ad(ψ) ∈ L gl(n, R) by ad(ψ)A = ψA − Aψ,

∀A ∈ gl(n, R).

(13)

Observe that for each Ψ ∈ GL(n, R) the automorphism Ad(Ψ ) on GL(n, R) induces a linear mapping on gl(n, R) obviously given by Ad(Ψ ) · A := Ψ · A · Ψ −1 ,

∀A ∈ gl(n, R).

Notice also that if Ψ ∈ GL+R (n, R) then Ad(Ψ ) : gl± (n, R) → gl±R (n, R), while if ψ ∈ gl−R (n, R) then ad(ψ) : gl±R (n, R) → gl∓R (n, R). We start with the following consequence of Lemma 3, which is crucial in the proof Theorem 4. Corollary 5. Let A0 = S0 eN0 be the SU-decomposition of A0 ∈ GL− (n, R) and let ·, · be a scalar product as in as in Lemma 3. Then also AT0 , S0T belong to

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GL− (n, R), and N0 T belongs to gl−R (n, R). Moreover, the following direct sum splitting holds: ker Ad(S0 ) − I ∩ gl−R (n, R) = adN0 gl+R (n, R) ∩ ker(Ad(S0 ) − I) (14) ⊕ ker Ad(S0 ) − I ∩ gl−R (n, R) ∩ kerad N0T . For the proof we again refer to [2]. We have now all the ingredients to prove the main result on normal forms of linear reversible operators. Proposition 6. Let A0 = S0 eN0 be the SU-decomposition of a given R-reversible operator on Rn . Then there exist a neighbourhood U of A0 in GL− (n, R) and a mapping Ψ : U → GL+ (n, R) such that Ψ (A0 ) = I and Ad Ψ (A) · A = S0 eN0 +B(A) , ∀A ∈ U, (15) with B(A0 ) = 0 and B(A) ∈ ker Ad(S0 ) − I ∩ ker ad(N0T ) ∩ gl− (n, R), i.e. B(A) commutes with S0 and N0T . Proof. Consider the direct sum splitting (14) and denote by π the linear projection of ker Ad(S0 )−I ∩gl−R (n, R) onto the ﬁrst component and parallel to the second component. Referring to a previous normal form result in [3], (see also [2]), we may assume the existence of a neighbourhood V of A0 in GL− (n, R) such that all A ∈ V can, via an appropriate near identity transfomation, be written in the form A = S0 eC(A) , for some smooth C : V → ker Ad(S0 ) − I ∩ gl−R (n, R) such that C(A0 ) = N0 . Knowing that if Ψ ∈ ker Ad(S0 ) − I ∩ GL+ (n, R) then Ad(Ψ ) · (A) = S0 eAd(Ψ )·C(A), with Ad(Ψ ) · C(A) ∈ ker Ad(S0 ) − I ∩ gl−R (n, R), then we only have to determine Ψ , dependent on A, such that Ad(Ψ ) · C(A) = N0 + B, with B ∈ ker Ad(S0 ) − I ∩ker ad(N0T ) ∩gl− (n, R). To do so we deﬁne a mapping F : ker Ad(S0 ) − I ∩ gl+R (n, R) × V → adN0 ker Ad(S0 ) − I ∩ gl+R (n, R) by

F (ϕ, A) := π Ad(Ψ ) · C(A) − N0 .

It follows that F (I, A0 ) = 0 and DΨ F (I, A0 ) = π · ad(N0 )|

ker Ad(S0 )−I ∩gl+R (n,R)

.

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The linear operator DΨ F (I, A0 ) is surjective from ker Ad(S0 )−I ∩gl+R (n, R) onto adN0 gl+R (n, R) ∩ ker Ad(S0 ) − I . Hence we can invoke the Implicit Function Theorem to conclude that there exista neighbourhood U ⊂ V ⊂ GL− (n, R) of A0 and a smooth mapping Ψ : U → ker Ad(S0 ) − I ∩ GL+R (n, R) with Ψ (A0 ) = I and such that F (Ψ (A), A) = 0, for all A ∈ U. By deﬁnition of F and π it follows that B(A) := Ad Ψ (A) · C(A) − N0 ∈ ker Ad(S0 ) − I ∩ ker ad(N0T ) ∩ gl−R (n, R), which proves the proposition.

3

Generic bifurcation of periodic points

In this section we show how the foregoing reduction can be used to describe a simple type of bifurcation which occurs generically when studying branching of solutions at a symmetric periodic solution of an autonomous time-reversible system. To this purpose, consider a 2k-dimensional autonomous time-reversible vector ﬁeld such that dim F ix(R) = k. Assume that the system has a non-constant Rsymmetric periodic solution γ0 with period T0 . We are interested in other periodic orbits of the system nearby the given γ0 . In order to study such orbits we consider the Poincaré map Φ associated to γ0 . It turns out that Φ is a local diﬀeomorphism satisfying (H) and (R) with n = 2k − 1 and m = 0. Cfr. [2,5]. The ﬁxed point 0 of Φ corresponds to γ0 , other ﬁxed points correspond to periodic orbits of the system close to γ0 with minimal period close to T0 . Finally, q-periodic orbits of Φ correspond to so-called subharmonic solutions of the system, that is, periodic orbits near γ0 with minimal period near qT0 . We ﬁrst study the periodic orbits near γ0 with minimal period near T0 by looking for ﬁxed points of Φ near 0 (i.e. q = 1 in (P)). Assume the simplest possible situation: (H1) the operator A0 has +1 as simple eigenvalue. It follows that ker(S0 − I) is one-dimensional, moreover Ru = u, for all u ∈ ker(S0 − I), [2,3]. Then via the reduction we obtain the following result. Theorem 7. Assume (H), (R), (H1). Then there exists a smooth mapping x∗ : ker(S0 − I) → Rn such that a) x∗ (0) = 0; b) Rx∗ (u) = x∗ (u), ∀u ∈ ker(S0 − I); c) Φ(x∗ (u)) = x∗ (u), for all suﬃciently small u ∈ ker(S0 − I). Moreover, Φ has in some neighbourhood of the origin no other ﬁxed points than those on the curve {x∗ (u)| |u| < u0 }.

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Periodic points in reversible diﬀeomorphisms

We shall call this branch of ﬁxed point the primary branch. The matrix DΦ (x∗ (u)) is reversible, and hence if µ ∈ C is an eigenvalue, so is µ−1 . It follows that if DΦ (x∗ (u)) has for u = u0 a pair of simple eigenvalues on the unit circle, these eigenvalues will stay on the unit circle for all u near u0 . Assuming that they move with non-zero speed it follows that along the primary branch one may ﬁnd symmetric ﬁxed points at which the linearization has eigenvalues which are q-th roots of unity, for some q ≥ 3. As shown in the next paragraph, this situation leads to branching of periodic points for Φ, which means subharmonic branching for the original system. Take q ≥ 3 in (P) and assume the following (H2)- A0 has a pair of simple eigenvalues χq , χq , with χq = exp 2iπ pq and q ≥ 3, 0 < p < q, gcd (p, q) = 1; - A0 has, besides 1, χq and χq , no other eigenvalues µ ∈ C such that µq = 1. One can easily see that the continuation of the eigenvalue χq can be written as eiαq (λ) χq , with αq (λ) ∈ R and αq (0) = 0. More precisely, the eigenvalues will move along the unit circle as we move along the primary branch. We assume the transversality condition (T) αq (λ) = 0. Notice that dim ker (S0q − I) = dim U = 3 and that ker (S0 − I) ⊂ ker (S0q − I). Also, denoting by V the S0 -invariant complement of ker (S0 − I) in U , we can identify U with the direct product R × C, where ker (S0 − I) ∼ = R and V ∼ = C. Moreover, S0 |C acts as multiplication by χq and R|C acts as z -→ z. It follows that the reduced bifurcation mapping (7) on U = R × C takes the form: B(α, z) = b0 (α, z), b1 (α, z) , (16) with b0 : R × C → R and b1 : R × C → C such that b0 (0, 0) = 0,

b1 (0, 0) = 0

b0 (α, χq z) = b0 (α, z),

b1 (α, χq z) = χq b1 (α, z)

b0 (α, z) = −b0 (α, z),

b1 (α, z) = −b1 (α, z).

Compare with [2,3]. Using a result on the normal form of complex Dq -equivariant functions, see [4,1], it follows that the non-trivial solutions of the bifurcation equation not lying on the primary branch are the solutions of b1 (α, z) = iθ1 (α, z) z + iθ2 (α, z) z q−1 = 0

(17)

with θi : R × C → R (i = 1, 2) smooth, real-valued and Dq -invariant functions. Using polar coordinates and some generically satisﬁed conditions one obtains the existence of exactly two R-symmetric branches of q-periodic orbits bifurcating from the ﬁxed point-branch. These branches have the form

αi (ρ)) , χjq z"i (ρ) 0 < ρ < ρ0 , 0 ≤ j ≤ q − 1 , (i = 1, 2) (18) γi = ("

116

M. C. Ciocci π

with z"1 (ρ) := ρ, z"2 (ρ) := ρei q , while the functions α = α "i (ρ) are the solutions of the equation i θ1 α, z"i (ρ) + (−1) ρq−2 θ2 α, z"i (ρ) = 0 (i = 1, 2), (19) "i (ρ) , z"i (ρ) gives then two branches "i (ρ) := x∗ α such that α "i (0) = 0. Setting x "i (0) of q-periodic points of Φ bifurcating from the ﬁxed point, since α "i (0) , x = (0, 0). We conclude with some remarks: (a) the greater is q the closer are the two branches q−2 in parameter space. Indeed, "2 (ρ)| = O ρ 2 ; one can show that |" α1 (ρ) − α (b) if θ2 (0, 0) = 0 there are no other q-periodic orbits (close to (0, 0)) than the two branches we have found. For similar bifurcation results in the symplectic case we refer to [4].

4

Stability

In this section we show how one can obtain some information on the stability of bifurcating periodic orbits. When x ∈ Rn generates a q-periodic orbit of Φλ then the stability of this orbit is determined by the eigenvalues of DΦqλ (x): the orbit is stable if all eigenvalues are inside the unit circle, and unstable if some eigenvalues are outside the unit circle. When the periodic orbit is symmetric (i.e. invariant under R) then together with µ ∈ C also µ−1 will be an eigenvalue of DΦqλ (x); in such case the orbit will be unstable if some eigenvalue is oﬀ the unit circle, and there will be a weak form of stability if all eigenvalues are on the unit circle. When applying this to bifurcating periodic orbits we have to determine the eigenvalues of DΦqλ (x∗ (u, λ)) for all small (u, λ) ∈ U × Rm which satisfy (6). For (u, λ) = (0, 0) this operator reduces to Aq0 , which has eigenvalues 1 on U and eigenvalues away from 1 on V := Im (S0q − I); it follows that for small (u, λ) the operator DΦqλ (x∗ (u, λ)) will have some eigenvalues near 1, with total multiplicity equal to dim U , and all other eigenvalues uniformly bounded away from 1. We call the eigenvalues near 1 the critical eigenvalues. If the non-critical eigenvalues of Aq0 are all simple and on the unit circle then the critical eigenvalues of DΦqλ (x∗ (u, λ)) will determine the stability of the corresponding periodic orbit. To approximate these critical eigenvalues, deﬁne a smooth mapping D : U × Rm → L (Rn ) by D (u, λ) := DΦλ x∗ S0q−1 u, λ · · · DΦλ (x∗ (S0 u, λ)) · DΦλ (x∗ (u, λ)) . (20) When (u, λ) satisﬁes (6) then Φλ x∗ (u, λ) = x∗ (S0 u, λ) and DΦqλ (x∗ (u, λ)) = D (u, λ); therefore it is suﬃcient to determine the critical eigenvalues of D (u, λ) for such (u, λ) . As an example, it turns out that the stability of the symmetric periodic orbits along the branch γi (see (18)), is determined by the number "i (ρ) , " z"i (ρ) , λ τi (ρ) := tr D (21)

Periodic points in reversible diﬀeomorphisms

where

" (ρ) = exp DX (" " u D " (ρ) , λ u (ρ)) + O ρk ; λ(ρ)

117

(22)

see [2,3] for the details. The calculations show that one of the two branches is stable, the other is unstable, (cfr. [2,3]).

5

1:1 resonance: some remarks

Returning to the situation described in section 3 the following scenario can happen. The mapping Φ has a ﬁxed point at which the linearization of Φ has two pairs of simple complex conjugate eigenvalues on the unit circle close to each other. Then, when moving along the corresponding primary branch, phenomena such as ’collision’ and ’splitting’ may happen, that is: the eigenvalues collide into a pair of non-semisimple double eigenvalues on the unit circle and then plitt oﬀ the circle. Introducing an external parameter λ ∈ R, one may arrange things such that for some value of the parameter, λ = λ0 , the collision happens at a q-th root of unity. It is then natural to ask what kind of bifurcation scenario emerges when solving problem (P) for a one-parameter family of reversible mappings satisfying (H), (R) and the following: (H3)- A0 has simple eigenvalue 1 and eigenvalues χq , χq with algebraic multiplicity 2 (and geometric multiplicity 1), - A0 has no other eigenvalues on the unit circle. Application of the reduction result of section 1 shows that we are left with a 5dimensional problem on U , dim U = 5. Also in this case ker (S0 − I) ⊂ U can be identiﬁed with R and its 4-dimensional S0 -invariant complement in U with C × C. So the reduced bifurcation equation consists of one real equation and two complex ones. A combined use of the Normal Form Theorem 4 and Implicit Function Theorem allows us to solve one of the two complex equation; we are then left with a problem similar to that of subharmonic branching analized in section 3. In a forthcoming paper we will describe the full bifurcation picture.

References 1. Maria-Cristina Ciocci. Subharmonic branching in Hamiltonian systems. Master thesis University ”La Sapienza” of Rome, 1997. 2. Maria-Cristina Ciocci. PhD Thesis, in preparation. University of Gent. 3. Maria-Cristina Ciocci. On the stability of bifurcating periodic orbits in reversible and symplectic mappings, Proceedings Equadiﬀ ’99 Berlin, World Scientiﬁc (2000) vol.1, pag. 166-168. 4. M.C. Ciocci, A. Vanderbauwhede, University of Gent . Bifurcation of periodic orbits for symplectic mappings. Journal of Diﬀerence Equations and Applications, 1998.

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5. J. Knobloch and A. Vanderbauwhede. A general reduction method for periodic solutions in consrvative and reversible systems. Journal of Dynamics and Diﬀ. Eqns., 8 (1996) 71-102. 6. J. Knobloch and A. Vanderbauwhede. Hopf bifurcation at k-fold resonances in reversible systems. Preprint TechnischeUniversit at Ilmenau, No. M 16/95. 7. A.Vanderbauwhede. Normal forms and versal unfoldings of symplectic linear maps. World Scientiﬁc Ser.in Appl. Anal. 4 (1995) 685-700. 8. A.Vanderbauwhede. Subharmonic bifurcation at multiple resonances. Proceedings of the Mathematics Conference, Birzeit University, Palestine 1998. World Scientiﬁc (2000) 254-276 9. A.Vanderbauwhede. Hopf bifurcation for equivariant conservative and time-reversible systems. Proceedings of the Royal Society of Edimburgh, 116A, 103-128, 1990. 10. A.Vanderbauwhede. Bifurcation of subharmonic solutions in time-reversible systems. Journal of Applied Mathematics and Physics, vol 37, July 1986. 11. A.Vanderbauwhede. Stability of bifurcating solutions. Publications de l’UER Mathematiques Pures et Appliques, Université de Lille, vol.5 (2)(1983).

Equadiﬀ 10, August 27–31, 2001 Prague, Czech Republic

Equadiﬀ 10 CD ROM Papers, pp. 119–127

Inﬁnitely Many Solitary Waves in Three Space Dimensions Pietro d’Avenia and Lorenzo Pisani Dipartimento Interuniversitario di Matematica Università degli Studi di Bari Via Orabona, 4 70125 BARI (Italy) Email: [email protected] Email: [email protected]

Abstract. In this paper we give some existence results obtained by V. Benci, P. d’Avenia, D. Fortunato, A. Masiello and L. Pisani about a model of Lorentz-invariant nonlinear ﬁeld equation in three space dimensions which gives rise to topological solitary waves. MSC 2000. 58E15, 35J70, 35Q51, 35Q60 Keywords. Topological solitary waves, Maxwell equations, concentrationcompactness

1

Introduction

A solitary wave is a solution of a wave equation whose energy is ﬁnite and travels as a localized packet; a soliton is a solitary wave which preserves its shape after interaction, having so a particle-like behavior. The soliton solutions occur in many questions of mathematical physics (nonlinear optics, classical and quantum ﬁeld theory, plasma physics), chemistry and biology (see [7,8,9,10]). For some equations, the existence of soliton solutions is guaranteed by topological constraints. In one space dimension, the simplest example of topological solitons is given by the sine-Gordon equation ψtt − ψxx + sin ψ = 0.

Research supported by M.I.U.R.

This is an overview article.

(1)

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If we look for ﬁnite-energy solutions, we put the asymptotic conditions ψ (−∞, t) = 2hπ h, k ∈ Z ψ (+∞, t) = 2kπ and just the diﬀerence h − k represents the topological constraint. It is well known that the only static solutions of the sine-Gordon equation uK (x) = 4 arctan (ex+a ) + 2hπ

uAK (x) = 4 arctan e−x+a

+ 2h π

give rise to one-soliton solutions. Indeed, since the sine-Gordon equation is Lorentz-invariant, we consider the “travelling” solutions x − vt ψK (x, t) = uK √ 1 − v2 x − vt ψAK (x, t) = uAK √ 1 − v2 with v ∈ R, |v| < 1. Moreover, there exist other solutions of (1) which represent the superposition of these basic solutions. This kind of results lead Derrick to look for stable, time-independent, localized solutions of the nonlinear wave equation ψtt − ∆ψ + V (ψ) = 0 in three space dimensions. In [6] he proves that the corresponding static equation −∆u + V (u) = 0 has not ﬁnite-energy stable solutions. In the same paper he proposes several ways to avoid this diﬃculty. One of them is the following. The equation (2) is the Euler-Lagrange equation related to the action S1 = L1 dxdt where, 1 L1 = − σ − V (ψ) , 2 being 2

2

σ = |∇ψ| − |ψt | .

(2)

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121

Derrick suggests to take a Lagrangian density L1 = −α (σ) − V (ψ) which gives rise to Lorentz-invariant equation, but he concludes that such kind of Lagrangian density lead to a very complicated diﬀerential equation. Indeed, in the 60ties, the methods of Nonlinear Analysis were not suﬃciently developed to face quasilinear equations. In this review, we recall some existence results concerning a nonlinear wave equation which is similar to the one proposed by Derrick.

2

The model

Let us consider the internal parameter space

M = R4 \ ξ where ξ = (1, 0, 0, 0) . The topological solitary waves introduced in [3] are ﬁelds ψ : R3 × R → M. The authors of [3] take the Lagrangian density ε 1 σ + σ 3 − V (ψ) L1 = − 2 3 where ε > 0 and V : M → R. We notice that this Lagrangian density is the one related to (2) plus the correction term ε − σ3 . 6 The Euler-Lagrange equation related to the action S1 = L1 dxdt is

∂ (3) 1 + εσ 2 ψt − ∇ 1 + εσ 2 ∇ψ + V (ψ) = 0. ∂t Since M has non-trivial topology (π3 (M) = Z), if we suppose that the ﬁelds ψ are smooth and lim ψ (x, t) = 0, (4) |x|→∞

we can classify them in the following way.

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Definition 1. For every t ∈ R, the topological charge of ψ (·, t) is ch (ψ (·, t)) = deg ((P ◦ ψ) (·, t) , K (ψ (·, t)) , N ) where

ξ−ξ +ξ P : ξ -→ ξ ξ − ξ

is the projection into the unitary sphere centered in ξ, N = 2ξ is the north pole of this sphere and

K (ψ (·, t)) = x ∈ R3 | |ψ (·, t) (x)| > 1 . If (4) is uniform with respect to t, the charge does not depend on t. The existence results of topological solitary waves can be generalized to an arbitrary number of space dimensions with a more general choice of Lagrangian density as in [1]. The three space dimensional case is necessary to interpret the ﬁleds ψ as charged relativistic particles.

3

The static solutions

The static solutions u = u (x) of (3) solve −∇ 1 + ε |∇u|4 ∇u + V (u) = 0 or, brieﬂy,

−∆u − ε∆6 u + V (u) = 0.

Assume that (V1) V ∈ C 2 (M, R) ; (V2) V (ξ) ≥ V (0) = 0 and the Hessian matrix V (0) is non-degenerate; (V3) there exist c, r > 0 such that −6 |ξ| < r ⇒ V ξ + ξ ≥ c |ξ| . We can obtain solutions of (5) looking for critical points of the functional 1 ε 2 6 |∇u| + |∇u| + V (u) dx. E (u) = 6 R3 2 In order to get

R3

1 ε 2 6 |∇u| + |∇u| dx < +∞, 2 6

(5)

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Inﬁnitely Many Solitary Waves in Three Space Dimensions

we consider the Banach space H = H 1 R3 , R4 ∩ W 1,6 R3 , R4 . By Sobolev embedding theorems we can say that the elements of H are continuous functions which go to zero at inﬁnity. For every u in the open subset

Λ = u ∈ H | ∀x ∈ R3 : u (x) = ξ the topological charge is well deﬁned and the condition (V3) implies that if u ∈ / Λ, then E (u) = +∞. Although the functional E is weakly lower semicontinuous, we cannot minimize it in the connected components ΛK = {u ∈ Λ ch (u) = K} since the domain R3 is not compact and ΛK are not weakly closed. The ﬁrst existence result can be stated as follows (Theorem 2.2 of [3]). Theorem 2. If V satisﬁes (V1), (V2), (V3), then there exists a weak solution of (5) obtained as minimum of the functional E in Λ∗ = {u ∈ Λ | ch (u) = 0} . This result is proved by using a Splitting Lemma in the spirit of concentrationcompactness principle. Moreover we have the following result. 3. If V satisﬁes (V1), (V2), (V3) and for every g ∈ O (3) , and ξ = Theorem ξ0, ξ1 , ξ2, ξ3 ∈ M V ξ0, g · ξ1, ξ2 , ξ3 = V ξ0 , ξ1, ξ2, ξ3 , then for every N ∈ Z there exists uN non-trivial solution of (5) such that ch (uN ) = N. For N = 0, the existence of a non-trivial solution is proved in [1]. The authors use a suitable invariance of the functional E in order to avoid the lack of compactness. For N = 0 the existence of a nontrivial solution is obtained in [5] using the Hopf invariant. A function which satisﬁes all these conditions is 4

V (ξ) =

ω02

2

|ξ|

|ξ| + ξ − ξ 6

! .

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4 4.1

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Interaction with the electromagnetic ﬁeld Charge density and current density

If we want to interpret the ﬁelds ψ as charged relativistic particles, it is natural to attribute to the topological charge the meaning of electric charge. In [2], the authors introduce the ﬁelds J (ψ) : R4 → R3 ρ (ψ) : R4 → R with the meaning of charge density and current density generated by ψ. Let 3 'k ∧ . . . ∧ dξ 3 ηk (ξ) dξ 0 ∧ . . . ∧ dξ η= k=0

be the unique 3-form closed but not exact on M, where the hatted symbols are omitted, k 1 (−1) ξ k − ξ¯k ηk (ξ) = |Σ| ξ − ξ¯4 and |Σ| is the measure of the unitary sphere in R4 . Let ψ ∗ η denote the pullback of η by ψ. The Hodge operator applied to ψ ∗ η gives a 1-form on R4 ∗ (ψ ∗ η) .

(6)

The ﬁelds (J, ρ) introduced in [2] are the components of (6). Indeed we can verify that ρ (ψ) dx ch (ψ (·, t)) = R3

(see Appendix of [4]). On the other hand, since η is closed, the pullback ψ ∗ η is closed and then d (ψ ∗ η) = 0. This can be written as the continuity equation ∇J +

∂ρ = 0. ∂t

Remark 4. When we consider static ﬁelds u = u (x), since in the expression of j Ji (ψ) appears the factor ∂ψ ∂t , we have J = 0, namely, as it is natural, in this case there is not electric current.

Inﬁnitely Many Solitary Waves in Three Space Dimensions

4.2

125

The system solitary wave-e.m. field

Let (A, φ) denote the gauge potential associated to the electromagnetic ﬁeld. Using (A, φ), we can deﬁne the electric ﬁeld E and the magnetic induction ﬁeld B as follows E = − (At + ∇φ)

(7)

B = ∇ × A.

(8)

By (7) and (8) we get immediately the ﬁrst two Maxwell equations ∇ × E + Bt = 0 ∇ · B = 0. In the vacuum, with a suitable choice of the physical constants, the second pair of the Maxwell equations ∇E = 4πρ ∇ × B − Et = 4πJ can be obtained as the Euler-Lagrange equations related to the action Semf = (L2 + L3 ) dxdt where

1 2 2 |E| − |B| 8π is the Lagrangian density of the electromagnetic ﬁeld and L2 =

L3 = (JA) − ρφ. Hence, if we consider the system solitary wave-electromagnetic ﬁeld, the total action is S = S (ψ, A, φ) = S1 (ψ) + Semf (ψ, A, φ) . The Euler-Lagrange equations in the static case are − ∆u − ε∆6 u + V (u) = G ∇ × (∇ × A) = 0 −∆φ = 4πρ (u)

(9) (10) (11)

where G derives from the interaction term and depends on u, φ (and their derivatives). We have the following results: Theorem 5. If V satisﬁes (V1), (V2), (V3), then there exist u ∈ H and φ ∈ D1,2 R3 , R such that

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– ch (u) = 0; – (u, 0, φ) is solution of (9, 10, 11). (Theorem 1.1 of [2]) 6. If V satisﬁes (V1), (V2), (V3), and for every g ∈ O (3) , and ξ = Theorem ξ0, ξ1 , ξ2, ξ3 ∈ M V ξ0, g · ξ1, ξ2 , ξ3 = V ξ0 , ξ1, ξ2, ξ3 , then for every N ∈ Z there exist uN ∈ H and φN ∈ D1,2 R3 , R such that – ch (uN ) = N ; – (uN , 0, φN ) is a non-trivial solution of (9, 10, 11). (Theorem 7 of [4]) The static solutions of (9, 10, 11) are obtained as critical points of the functional 1 ε 1 2 6 2 |∇u| + |∇u| + V (u) dx − |∇φ| dx + φρ (u) dx f (u, φ) = 2 6 2 R3 R3 R3 which is strongly indeﬁnite. Then the solutions are obtained using the reduced functional J (u) = f (u, Φ [u]) where Φ [u] is implicitly deﬁned by ∂f = 0. ∂φ Remark 7. If u ¯ is the non-trivial static solution of (5) having ch(¯ u) = 0 found in Theorem 2 of [5], an immediate calculation shows that (¯ u, 0, 0) is a solution of (9, 10, 11) (Theorem 4 of [5]). This is a further conﬁrmation of the model’s coherence: indeed an uncharged particle does not create any electromagnetic ﬁeld.

References 1. V. Benci, P. d’Avenia, D. Fortunato, L. Pisani, Solitons in Several Space Dimension: Derrick’s Problem and Inﬁnitely Many Solutions, Arch. Rat. Mech. Anal. 154 (2000), 297-324. 2. V. Benci, D. Fortunato, A. Masiello, L. Pisani, Solitons and the electromagnetic ﬁeld, Math. Z. 232 (1999), 73-102.

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3. V. Benci, D. Fortunato, L. Pisani, Soliton like solution of a Lorentz invariant equation in dimension 3, Reviews in Mathematical Physics 3 (1998), 315-344. 4. P. d’Avenia, D. Fortunato, L. Pisani, Topological Solitary Waves with Arbitrary Charge and Electromagnetic Field, preprint. 5. P. d’Avenia, L. Pisani, Remarks on the Topological Invariants of a class of Solitary Waves, to appear on Nonlinear Analysis TMA. 6. G. H. Derrick, Comments on nonlinear wave equations as models for elementary particles, J. Math. Phys. 5 (1964), 1252-1254. 7. R. K. Dodd, J. C. Eilbeck, J. D. Gibbon, H. C. Morris, Solitons and Nonlinear Wave Equations, Academic Press, London, New York, 1982. 8. S. Kichenassamy, Non linear wave equations, Marcel Dekker Inc., New York, Basel, Hong Kong, 1996. 9. R. Rajaraman, Solitons and instantons, North Holland, Amsterdam, Oxford, New York, Tokyo, 1988. 10. G. B. Witham, Linear and nonlinear waves, John Wiley and Sons, New York, 1974.

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Equadiﬀ 10 CD ROM Papers, pp. 129–131

Positive and oscillating solutions of equation x(t) ˙ = −c(t)x(t − τ ) Josef Diblík

Department of Mathematics, Faculty of Electrical Engineering and Computer Science, Brno University of Technology (VUT), Technická 8, 616 00 Brno, Czech Republic Email: [email protected]

Abstract. Positive and oscillating solutions of delayed equation x(t) ˙ = −c(t)x(t − τ ) +

with c ∈ C(I, R ), I = [t0 , ∞), R+ = (0, ∞) and 0 < τ = const are studied. MSC 2000. 34K15, 34K25 Keywords. Linear diﬀerential equations with delay, positive solution, oscillating solution

Let us consider the equation x(t) ˙ = −c(t)x(t − τ )

(1)

where c ∈ C(I, R ), I = [t0 , ∞), R = (0, ∞) and 0 < τ = const. Deﬁne lnk t = ln(lnk−1 t), k ≥ 1 where ln0 t ≡ t for t > expk−2 1 where expk t ≡ (exp(expk−1 t)), k ≥ 1, exp0 t ≡ t and exp−1 t ≡ 0. (Instead of expressions ln0 t, ln1 t is only t and ln t written in the sequel.) Moreover, deﬁne so called critical functions for (1) +

ck (t) ≡

+

τ 1 τ τ τ + + + + ···+ 2 2 2 eτ 8et 8e(t ln t) 8e(t ln t ln2 t) 8e(t ln t ln2 t . . . lnk t)2

with k ≥ 0.

Research supported by grant 201/99/0295 of Czech Grant Agency (Prague) and by the Council of Czech Government MSM 2622000 13.

This is the preliminary version of the paper.

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Theorem 1. [1] A) Let us assume that c(t) ≤ ck (t) for t → ∞ and an integer k ≥ 0. Then there is a positive solution x = x(t) of Eq. (1). Moreover, & x(t) < νk (t) ≡ e−t/τ t ln t ln2 t . . . lnk t as t → ∞. B) Let us assume that c(t) > ck−1 (t) +

θτ 8e(t ln t ln2 t . . . lnk t)2

(2)

for t → ∞, an integer k ≥ 1 and a constant θ > 1. Then all solutions of Eq. (1) oscillate. Theorem 2. [1] Assume that the inequality (2) holds for t → ∞, an integer k ≥ 1 and a constant θ > 1. Then each solution of Eq. (1) has at least one zero on each interval (p − τ, q) for q = expk−2 (lnk−2 p)exp(π/ζ) , ζ 2 < (θ − 1)/4, (ζ is a positive constant) and p suﬃciently large. Theorem 3. [2] Let there exists a positive solution x ˜ of (1) on I. Then there are positive solutions x1 and x2 of (1) on I satisfying the relation x2 (t) = 0. t→∞ x1 (t) lim

(3)

Moreover, every solution x of (1) on I is represented by the formula x(t) = Kx1 (t) + O(x2 (t)) where K ∈ R depends on x. Definition 4. [3] Let x1 and x2 be ﬁxed positive solutions of the delayed equation (1) on I, with the property (3). Then (x1 , x2 ) is called a pair of dominant and subdominant solutions on I. Let us consider the equation (1) in the case when the coeﬃcient c is equal to a critical function, i.e., in the case of equation x(t) ˙ = −ck (t)x(t − τ ),

k ≥ 0; t ≥ t0 > expp−1 1.

(4)

Theorem 5. [3] Let k ≥ 0 be ﬁxed. Then for any ﬁxed constants δ1 > 2 and δ2 < 0 there are a t0 , and a pair (x1 , x2 ) of dominant and subdominant solutions of (4) on I satisfying the two-sided estimates ( ( 1 t e−t/τ t ln t ln2 t · · · lnp t ln2p+1 t < x1 (t) < e−t/τ t ln t ln2 t · · · lnp t lnδp+1 and e−t/τ on I.

( & 2 t ln t ln2 t · · · lnp t lnδp+1 t < x2 (t) < e−t/τ t ln t ln2 t · · · lnp t

Positive and oscillating solutions

131

References 1. Diblík J., Positive and oscillating solutions of diﬀerential equations with delay in critical case, J. Comput. Appl. Math, 88 (1998), 185–202. 2. Diblík J., Behaviour of solutions of linear diﬀerential equations with delay, Arch. Math. (Brno), 34 (1998), 31–47. 3. Diblík J. and Koksch N., Positive solutions of the equation x(t) ˙ = −c(t)x(t − τ ) in the critical case, J. Math. Anal. Appl., 250 (2000), 635–659. 4. Domshlak Y., Stavroulakis I.P., Oscillation of ﬁrst-order delay diﬀerential equations in a critical case, Appl. Anal., 61 (1994), 359–371.

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Inequalities for solutions of systems with “pure” delay Josef Diblík1 , Denis Ja. Khusainov2 and Violeta G. Mamedova3 1

Department of Mathematics, Faculty of Electrical Engineering and Computer Science, Brno University of Technology (VUT), Technická 8, 616 00 Brno, Czech Republic Email: [email protected] 2 Department of Complex Systems Modelling, Faculty of Cybernetics, Kiev University, Vladimirskaja 64, 252 033 Kiev, Ukraine, Email: [email protected] 3 Department of Complex Systems Modelling, Faculty of Cybernetics, Kiev University, Vladimirskaja 64, 252 033 Kiev, Ukraine, Email: [email protected]

Abstract. A result concerning the estimation of a solution of a linear system with “pure” delay is formulated.

MSC 2000. 34K20, 34K25

Keywords. Inequalities for solutions, “pure” delay

Let us consider the following system of diﬀerential equations with delay x(t) ˙ = B(t)x(t − τ ),

x ∈ Rn , t ≥ t0 , τ > 0, τ = const.

(1)

We suppose that the corresponding system without delay (i.e. the system (1) with τ = 0) has the fundamental matrix of solutions Φ(t, t0 ), normed in t = This is the preliminary version of the paper.

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t0 . Investigation of the system (1) is performed with the aid of nonautonomous Liapunov function of quadratic form v(x, t) = xT H ∗ (t, t0 )x with

H ∗ (t, t0 ) = [Φ(t, t0 )−1 ]T Φ−1 (t, t0 ).

¯ Suppose, moreover, Theorem 1. Let matrix B(t) be bounded, i.e. B(t) ≤ B. that there exists a constant k > 0 such that γ(t, t0 , k) ≥ 0

for t ≥ t0 + τ

with γ(t, t0 , k) = & = λmax [H ∗ (t, t0 )] ×

#

kλmin [H ∗ (t, t0 )] ¯2 − 2B λmax [H ∗ (t, t0 )]3/2

t−τ t−2τ

e−k(s−t)/2 ds & λmin [H ∗ (s, t0 )]

) .

Then for solution x(t) of the system (1), determined by continuous initial function ϕ(t) on initial interval [t0 − τ, t0 ], the following inequalities hold: ¯ − t0 )]x(t0 ) |x(t)| < [1 + B(t if t0 ≤ t ≤ t0 + τ and * ¯ ]x(t0 ) × |x(t)| < [1 + Bτ

1 t λmax [H0∗ ] exp kτ − [γ(s, t , k) − k] ds 0 λmin [H ∗ (t, t0 )] 2 t0 +τ

with x(t0 ) =

max

t∈[t0 −τ,t0 ]

ϕ(t), λmax [H0∗ ] =

max

t0 −τ ≤s≤t0 +τ

{λmax [H ∗ (s, t0 )]}

if t ≥ t0 + τ . Acknowledgment. The ﬁrst author has been supported by the plan of investigations MSM 2622000 13 of the Czech Republic.

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Solutions of a singular Cauchy problem for a nonlinear system of diﬀerential equations Josef Diblík1 and Miroslava R˚ užičková2 1 Department of Mathematics, Faculty of Electrical Engineering and Computer Science, Brno University of Technology (VUT), Technická 8, 616 00 Brno, Czech Republic Email: [email protected] 2 Department of Appl. Mathematics, Faculty of Science, University of Žilina, J. M. Hurbana 15, 010 26 Žilina, Slovak Republic Email: [email protected]

Abstract. The solvability of the singular Cauchy problem for the system of nonlinear diﬀerential equations g(x)y = A(x)α(y) − ω(x), y(0+ ) = 0 is investigated. MSC 2000. 34C10, 34C15, 34B15 Keywords. Positive solution, nonlinear system, singular Cauchy problem

Let us consider the system of nonlinear diﬀerential equations g(x)y = A(x)α(y) − ω(x)

(1)

y(0+ ) = 0.

(2)

and initial Cauchy problem T

Here y = (y1 , . . . , yn ) is the vector of unknown functions; α(y) = (α1 (y1 ), . . . , αn (yn ))T is a nonlinearity vector with entries αi , i = 1, . . . , n; A(x) is n × n matrix with elements aij (x), i, j = 1 . . . , n; ω(x) = (ω1 (x), . . . , ωn (x))T and g(x) = This is the preliminary version of the paper.

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diag(g1 (x), . . . , gn (x)) is a diagonal matrix with diagonal entries indicated. The symbol Is indicate an interval of the form (0, s] with a ﬁxed s > 0. The system (1) is considered under the following main assumptions: (C1 ) (C2 ) (C3 ) (C4 ) (C5 ) (C6 )

gi ∈ C(Ix0 , R+ ), i = 1, . . . , n with R+ = (0, ∞); α ∈ C 1 (Iy0 , Rn ), α(y) 6 0 on Iy0 , α (y) 6 0 on Iy0 and α(0+ ) = 0; ω ∈ C 1 (Ix0 , Rn ); aij ∈ C 1 (Ix0 , R), aii (x) = 0, i, j = 1, . . . , n and detA(x) = 0 on Ix0 ; αi (y) ≤ M αi (y), i = 1, . . . , n on Iy0 with a constant M ∈ R+ ; Ω(x) ≡ A−1 (x)ω(x) 6 0, Ω (x) 6 0 on Ix0 and Ω(0+ ) = 0.

The problem (1), (2) is a singular problem if assumptions (C1 )–(C6 ) hold and if, in additional, gi (0+ ) = 0 for at least one i ∈ {1, . . . , n}. The latter condition is implicitly contained in the assumptions of Theorem 2. Definition 1. A function y = y(x) ∈ C 1 (Ix∗ , Rn ) with 0 < x∗ ≤ x0 is said to be a solution of the problem (1), (2) on interval Ix∗ if y satisﬁes (1) on Ix∗ and y(0+ ) = 0. Theorem 2. Suppose that conditions (C1 ) − (C6 ) are satisﬁed. Let A) for i = 1, . . . , p ≤ n: ωi (x) < 0, ωi (x) < 0, x ∈ Ix0 ,

(3)

aij (x) ≥ 0, j = i, j = 1, . . . , n and aij (x) ≥ 0, j = 1, . . . , n, x ∈ Ix0 ,

(4)

and ωi (δx) > ωi (x) + δM gi (x)

Ωi (δx) , x ∈ Ix0 Ωi (δx)

for a constant δ ∈ (0, 1); B) for i = p + 1, . . . , n: ωi (x) > 0, ωi (x) > 0, x ∈ Ix0 ,

(5)

aij (x) ≤ 0, j = i, j = 1, . . . , n and aij (x) ≤ 0, j = 1, . . . , n, x ∈ Ix0 ,

(6)

and ωi (Kx) > ωi (x) + KM gi (x)

Ωi (Kx) , x ∈ Ix0 Ωi (Kx)

for a constant K > 1. Then there exists (n − p)-parametric family of solutions of the problem (1), (2), having positive coordinates, on an interval Ix∗ ⊆ Ix∗∗ with x∗∗ ≤ min{x0 K −1 , y0 }.

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137

Consequence. If Theorem 2 holds then there exist (n − p)-parametric family of solutions y = y ∗ (x) of the problem (1), (2) each of which satisﬁes on interval Ix∗ the inequalities ϕ(δx) 7 y ∗ (x) 7 ϕ(Kx). Consider the linear system g(x)y = A(x)y − ω(x).

(7)

Theorem 3 (Linear case). Suppose that conditions (C1 ), (C3 ), (C4 ), (C6 ), (3) – (6) are satisﬁed. Let, moreover, ωi (δx) > ωi (x) + δgi (x)Ωi (δx), x ∈ Ix0 , i = 1, . . . , p ≤ n for a constant δ ∈ (0, 1) and ωi (Kx) > ωi (x) + Kgi (x)Ωi (Kx) , x ∈ Ix0 , i = p + 1, . . . , n for a constant K > 1. Then there exists (n − p)-parametric family of solutions y = y ∗ (x) of the problem (7), (2), having positive coordinates on an interval Ix∗ ⊂ Ix0 , each of which satisﬁes here the inequalities Ω(δx) 7 y ∗ (x) 7 Ω(Kx). Example 4. Let us consider a linear singular problem of the type (7), (2): x2 y1 = −5y1 + y2 + y3 + x + x2 , y1 − 5y2 + y3 + x + x2 , x2 y2 = 2 x y3 = −2y1 − 3y2 + 2y3 − x + 3x2 , y1 (0+ ) = y2 (0+ ) = y3 (0+ ) = 0. This problem has (by Theorem 3) one-parametric family of positive solutions. Really, the general solution of system considered is expressed by means of relations y1 = x + 11C1 exp(6/x) + C2 exp(3/x) + C3 exp(−1/x), y2 = x − 10C1 exp(6/x) + C2 exp(3/x) + C3 exp(−1/x), y3 = 3x − C1 exp(6/x) + C2 exp(3/x) + 5C3 exp(−1/x) with arbitrary constants C1 , C2 and C3 . By Theorem 2 there exist one-parametric family of positive solutions of nonlinear problem x3 y1 = −5y12 + y25 + y33 + x + x2 , x4 y2 = y12 − 5y25 + y33 + x + x2 , 5 x y3 = −2y12 − 3y25 + 2y33 − x + 3x2 , y1 (0+ ) = y2 (0+ ) = y3 (0+ ) = 0. Acknowledgment This work was supported by the grants 201/99/0295 of Czech Grant Agency, 1/5254/01 of Slovak Grant Agency; Slovak-Czech project 022(25)/2000 and by the plan of investigations MSM 262200013 of the Czech Republic.

Equadiﬀ 10, August 27–31, 2001 Prague, Czech Republic

Equadiﬀ 10 CD ROM Papers, pp. 139–141

An existence criterion of positive solutions of p-type retarded functional diﬀerential equations Josef Diblík1 and Zdeněk Svoboda2 1

Department of Mathematics, Faculty of Electrical Engineering and Computer Science, Brno University of Technology (VUT), Technická 8, 616 00 Brno, Czech Republic Email: [email protected] 2 Department of Mathematics, Military Academy Brno(VA Brno), Kounicova 65, 612 00 Brno PS 13, Czech Republic, Email: [email protected]

Abstract. A criterion and conditions for existence of positive solutions of p-type retarded functional diﬀerential equations are presented.

MSC 2000. 34K20, 34K25

Keywords. Positive solution, delayed equation, p - function

The function p ∈ C[R × [−1, 0], R] is called a p -function if it has the following properties: p(t, 0) = t, p(t, −1) is a nondecreasing function of t and there exists a σ ≥ −∞ such that p(t, ϑ) is an increasing function in ϑ for each t ∈ (σ, ∞) (see [2]). For t ∈ [t0 , t0 + A) with A > 0 we deﬁne yt (ϑ) = y(p(t, ϑ)), −1 ≤ ϑ ≤ 0. Consider the system y(t) ˙ = f (t, yt ) (1) where f ∈ C[[t0 , t0 + A) × C, Rn ] with C = [[−1, 0], Rn]. This system is called the system of p -type retarded functional diﬀerential equations ([2]). We say that the functional g ∈ C(Ω, R) is strongly decreasing (increasing) with respect to the second argument on Ω ⊂ R × C if for each (t, ϕ), (t, ψ) ∈ Ω with ϕ(p(t, ϑ)) 7 ψ(p(t, ϑ)), ϑ ∈ [−1, 0): g(t, ϕ) − g(t, ψ) > 0 (< 0). Let k 6 0 and µ This is the preliminary version of the paper.

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be constant vectors, µi = −1, i = 1, . . . , p and µi = 1, i = p + 1, . . . , n. Let λ(t) denote a real vector with continuous entries on [ p∗ , ∞), p∗ = p(t∗ , −1). Put t t t T (k, λ)(t) ≡ keµ p∗ λ(s)ds = k1 eµ1 p∗ λ1 (s)ds , . . . , kn eµn p∗ λn (s)ds . Theorem 1. Suppose Ω = [t∗ , ∞) × C, f ∈ C(Ω, Rn ) is locally Lipschitzian with respect to the second argument and, moreover: (i) f (t, 0) ≡ 0 if t ≥ t∗ . (ii) The functional fi is strongly decreasing if i = 1, . . . , p and strongly increasing if i = p + 1, . . . , n with respect to the second argument on Ω. Then for the existence of a positive solution y = y(t) on [ p∗ , ∞) of the system (1) a necessary and suﬃcient condition is that there exists a vector λ ∈ C([ p∗ , ∞), Rn ), such that λ 6 0 on [t∗ , ∞), satisfying the system of integral inequalities µi −µi pt∗ λi (s)ds λi (t) ≥ e · fi (t, T (k, λ)t ) , i = 1, . . . , n ki for t ≥ t∗ , with a positive constant vector k. Consider the equation

t

y(t) ˙ =−

K(t, s)y(s)ds,

(2)

τ (t)

where K : [t∗ , ∞) × [ p∗ , ∞) → R+ is a continuous function, and τ : [t∗ , ∞) → [ p∗ , ∞) is a nondecreasing function with τ (t) < t. Theorem 2. The equation (2) has a positive solution y = y(t) on [ p∗ , ∞) if and only if there exists a function λ ∈ C([ p∗ , ∞), R), such that λ(t) > 0 for t ≥ t∗ and t t K(t, s)e s λ(u)du ds λ(t) ≥ τ (t) ∗

on the interval [t , ∞). Let us consider a partial case of Eq. (2) when τ (t) ≡ t−l, l ∈ R+ and K(t, s) ≡ c(t) for t ∈ [t∗ , ∞). Then Eq. (2) takes the form t y(s) ds. (3) y(t) ˙ = −c(t) t−l

Theorem 3. For the existence of a solution of Eq. (3), positive on [t∗ − l, ∞), the inequality c(t) ≤ M, t ∈ [t∗ , ∞) is suﬃcient for M = α(2 − α)/l2 = const with a constant α being the positive . root of the equation 2 − α = 2e−α . (The approximate values are α = 1, 5936 and . 2 M = 0, 6476/l .) This work has been supported by the plan of investigations MSM 2622000 13 of the Czech Republic and by the grant 201/99/0295 of Czech Grant Agency (Prague).

An existence criterion of positive solutions . . .

141

References 1. J. Diblík, A criterion for existence of positive solutions of systems of retarded functional diﬀerential equations. Nonl. Anal., TMA 38 (1999), 327–339. 2. L. H. Erbe, Q. Kong, B. G. Zhang, Oscillation Theory for Functional Diﬀerential Equations. Marcel Dekker, New York, 1995. 3. I. Györi, G. Ladas, Oscillation Theory of Delay Diﬀerential Equations. Clarendon Press, Oxford, 1991.

Equadiﬀ 10, August 27–31, 2001 Prague, Czech Republic

Equadiﬀ 10 CD ROM Papers, pp. 143–157

Fieldless Methods for the Simulation of Stationary and Nonstationary Induction Heating Ivo Doležel1 , Pavel Šolín1 and Bohuš Ulrych2 1

3

Institute of Electrical Engineering of the ASCR, Dolejškova 5, 18200 Prague, Czech Republic Email: [email protected] 2 Institute of Electrical Engineering of the ASCR, Dolejškova 5, 18200 Prague, Czech Republic Email: [email protected] Department of Electrical Engineering, Westbohemian University, Sady pětatřicátník˚ u 35, 30614 Pilsen, Czech Republic Email: [email protected]

Abstract. The paper gives a survey of a class of novel stationary and nonstationary methods for the simulation of the induction heating of nonferromagnetic metal bodies in harmonic electromagnetic ﬁelds. One of the main advantages of the presented method is the elimination of the surrounding air from the electromagnetic model, which strongly reduces the necessity of meshing and simpliﬁes the computation. The task is formulated either as a stationary or as a non-stationary quasi-coupled problem, with respecting the temperature dependencies of all important material parameters. Distribution of the eddy currents and Joule losses in the metal body is solved by a system of second-kind Fredholm integral equations. Existence and uniqueness of solution for the continuos as well as discrete problem is shown. Convergence results for the numerical scheme are presented. The theoretical analysis is supplemented with examples motivated in the engineering practise. .

MSC 2000. 31A10, 45F15, 35K05, 35K55

Keywords. induction heating, second-kind Fredholm integral equation, heat transfer equation, collocation schemes

This research was supported by the Grant No. 102/00/0933 of the Grant Agency of the Czech Republic.

This is the preliminary version of the paper.

144

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I. Doležel, P. Šolín and B. Ulrych

Introduction

Mathematical modelling of the induction heating belongs to relatively well explored disciplines. The model consists of two second-order (generally non-linear) partial diﬀerential equations of the elliptic and/or parabolic types, whose solution yields distribution of the electromagnetic ﬁeld, eddy currents, corresponding Joule losses and consequent temperature rise of the heated body. Sometimes, however, certain diﬃculties have to be overcome for obtaining correct results. We can mention, for example, the temperature dependent parameters of the materials involved, speciﬁc arrangements of the heaters etc. Nevertheless, in most geometries the ﬁeld equations supplemented with correct boundary conditions may be solved by existing FEM-based professional programs (FLUX, ANSYS, MARC), and the results well correspond to the physical reality. In case of thin metal slabs of negligible thickness or general three-dimensional metal bodies surrounded by (possibly moving) inductors of often very complicated shapes, the basic complication consists in the geometrical incommensurability of particular subregions in the investigated area (geometry of the metal body versus 3D inductor and/or practically unbounded air). Using methods such as FEM can lead to serious problems associated with generation of the discretisation grid that may result in unacceptable errors occurring particularly at the electromagnetic ﬁeld calculation. The paper oﬀers an alternative method for direct determining the local Joule losses (that represent the input data for the consequent thermal calculation) based on solution of a system of second-kind Fredholm integral equations for the eddy current density in the metal body. Detailed knowledge of the 3D electromagnetic ﬁeld is, therefore, unnecessary (here the atribute ﬁeldless of the proposed method is originated) and the mentioned problems are avoided. The temperature ﬁeld distribution is then solved by means of the non-stationary heat transfer equation with a special procedure for correcting values of the temperature dependent material properties.

2

Heating of thin non-ferromagnetic metal slabs (2D case)

A thin non-ferromagnetic slab Ω1 of sizes a, b and h is heated by an inductor formed by two equal coils Ω2 and Ω2 . positioned symmetrically with respect to the plate (Fig. 1). Both coils carrying identical harmonic currents Iext and Iext of angular frequency ω produce a ﬁeld perfectly perpendicular to the slab. Thickness h of the slab is very small with respect to the other measures, so that the electromagnetic quantities may be considered independent of co-ordinate y. The inductor contains no ferromagnetic parts. Due to absence of non-linearities within the investigated domain all quantities of the electromagnetic ﬁeld may be expressed in terms of their phasors.

145

Fieldless Methods for the Simulation of Induction Heating Ie

W

x t

2

u p p e r c o il y z b

P

A

rP

r

r x

P Q'

W

p la te

A (Q ) A

Q

Q

z

h

d l

J

R Q

J e (R )

e z

J R

e x

x

a I e 'x

P ' t

W '2

d l'

lo w e r c o il

1

Fig. 1. The investigated arrangement

3

The coupled electromagnetic-thermal model (2D case)

Let Q be a point within the slab lying in plane y = 0. Phasor A of the vector potential at this point is given by superposition of three components excited by the ﬁeld currents Iext and Iext and the eddy currents in the slab

A(Q) = A(P Q) + A(P Q) + A(RQ) = µ0 = 4π µ0 = 4π

I ext Ω2

I ext Ω2

dl(P ) + I ext rP Q

dl(P ) + I ext rP Q

Ω2

Ω2

dl (P ) + rP Q

dl (P ) + rP Q

Jeddy (R) dV rRQ

Ω1

S

!

Jeddy (R) hdS rRQ

= ! .

(1)

Here, µ0 denotes the permeability of vacuum, dl and dl are vectors denoting the elementary lengths of conductors of the ﬁeld coils, dV the elementary volume of the slab and S the cross-section of the slab in plane y = 0. All remaining symbols follow from Fig. 1. Phasors A(P Q) and A(P Q) have generally three components, according to the shape of the coil. In our case these components in direction x are equal, in direction z as well, and components in direction y eliminate one another. Equation (1) may be now rewritten as follows Jeddy (R) dl(P ) µ0 +h dS . (2) A(Q) = 2I ext 4π rRQ Ω2 rP Q Ω1 For the next considerations it is useful to express vector potential A(Q) in the slab in terms of the eddy current density Jeddy (Q). Starting from the second

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Maxwell equation ∂rotA ∂B =− ∂t ∂t and interchanging the order of the operators we get rotE = −

E=−

(3)

∂A − gradϕ, ∂t

(4)

where ϕ denotes the scalar potential. Applying this equation to the slab that is not connected to any external source of voltage (ϕ = 0) and rewriting it in terms of the corresponding phasor quantities we ﬁnally obtain E = −jωA ⇒ Jeddy = −jωγA.

(5)

Hence, using the symbolics from (1), A(Q) =

j J (Q), ωγ eddy

(6)

where γ denotes the electrical conductivity of the slab and ω the angular frequency of the ﬁeld currents Iext (Iext ). Substitution of (6) into (2) provides the basic integral equation for Jeddy jJeddy (Q) − κ1 S

where κ1 =

Jeddy (R) dS = κ2 I ext rRQ

ωγµ0 h , 4π

κ2 =

Ω2

dl(P ) , rP Q

(7)

ωγµ0 . 2π

This phasor equation may easily be subdivided into two equations (for the components in directions x and z) of the complex character. The speciﬁc average Joule losses wJa in the metal body are then given by formula ∗ Jeddy · Jeddy , (8) wJa = γ ∗ where Jeddy is the complex conjugate to Jeddy . The non-stationary distribution of the temperature in the metal body is generally described (for example [2]) by equation ∂T − wJa , (9) div(λ grad T ) = ρc ∂t where λ denotes the thermal conductivity, ρ the speciﬁc mass of the heated material, c its speciﬁc heat and wJa the speciﬁc Joule losses given by (8). The boundary condition along the whole surface of the body reads (radiation is not considered)

−λ

∂T = α(T − Text ), ∂n

(10)

Fieldless Methods for the Simulation of Induction Heating

147

where α denotes the coeﬃcient of the convective heat transfer, Text the temperature of the surrounding medium (moving or quiet air) and n direction of the outward normal. As this simpler 2D case is analogous to the next (slightly more complex) 3D one concerning the discretization, analysis of solvability and uniqueness both of the continuous and discrete problem and also concerning the convergence of the numerical scheme, we shall perform these considerations only for the three-dimensional model.

4

Heating of non-ferromagnetic metal bodies (3D case)

A bounded metal body Ω1 with a Lipschitz-continuous boundary is heated by an inductor formed by a system of conductors and/or coils Ω2 (Fig. 2). For simplicity, let the conductors and coils carry identical harmonic current Iext of angular frequency ω. The inductor contains no ferromagnetic parts.

Fig. 2. The investigated arrangement

Due to absence of non-linearities within the investigated domain all quantities of the electromagnetic ﬁeld may be expressed in terms of their phasors.

5

The coupled electromagnetic-thermal model (3D case)

Let Q ∈ Ω1 . Phasor A of the vector potential at this point is given by superposition of two components excited by the ﬁeld currents Iext and the eddy currents Jeddy in Ω1 : Jeddy (R) dl(P ) µ0 + dV . (11) A(Q) = A(P Q) + A(RQ) = I ext 4π rRQ Ω2 rP Q Ω1

148

I. Doležel, P. Šolín and B. Ulrych

Similarly as in the previous case, the second Maxwell equation yields A(Q) =

j J (Q) ωγ eddy

(12)

where γ denotes the electrical conductivity of the metal and ω the angular frequency of the ﬁeld currents Iext . Substitution of (12) into (11) provides the basic integral equation for Jeddy Jeddy (R) dl(P ) j Jeddy (Q) − κ dV = κ I ext (13) r RQ Ω1 Ω2 rP Q

where κ = ωγµ0 h/(4π). Here, µ0 denotes the permeability of vacuum, dl and dl are vectors denoting the elementary lengths of conductors of the ﬁeld coils and dV the elementary volume of Ω1 . All remaining symbols follow from Fig. 2. Coupling with the heat transfer equation occurs in the same way as in the previous case, using Joule losses as source terms. The system (13) is discretized (see also [3]) using a ﬁrst-order collocation scheme based on a discretitation of Ω1 with piecewise linear approximation of all components of Jeddy in each cell. The only diﬃculty is with the diagonal coeﬃcients of the corresponding matrix that are given by improper integrals. Their values are, however, ﬁnite and may be, with some eﬀort, determined analytically. The arising dense system of linear equations can be solved e.g. by the Gauss elimination. The heat transfer equation (9) (formally the same in the 2D and 3D case) is semi-discretized in space using the method of lines and integrated in time using higher-order explicit Runge-Kutta schemes. Temperature-dependent material parameters are adjusted automatically during the time-evolution.

6

Analysis of solvability and uniqueness (3D case)

The phasor equation (13) may easily be subdivided into three identical equations (for the components in spatial directions x, y, z) of a complex form. For the xcomponent, we obtain J eddy,x (R) dx(P ) dV = κ I ext . (14) j J eddy,x(Q) − κ rRQ Ω1 Ω2 rP Q Using a substitution

T −Im{J eddy,x} Re{J eddy,x } , (v), κ κ dx (Re{I ext }, Im{I ext })T , Fx (v) = Ω2 |u − v|

Lx (v) =

(15)

(16)

we can rewrite (14) into an operator form (I + K)Lx = Fx

(17)

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Fieldless Methods for the Simulation of Induction Heating

with

k(v, w)Lx (w)dw,

(KLx )(v) = Ω1

κ M, k(v, w) = |v − w|

M=

0 −1 1 0

(18) .

(19)

It is easy to see that the operator K: C(Ω1 ) → C(Ω1 ) is compact and self adjoint in a weighted L2 (Ω1 )-norm with a weighting factor 1/κ. Therefore, under a technical assumption – let us denote it by (A−1 ) – that the homogeneous equation (I + K)Lx = 0 has only trivial solution (i.e. −1 is not an eigenvalue of K), the Fredholm alternative immediately yields the solvability, uniqueness and continuous dependence on the right-hand-side for (17). An analogous conclusion holds, of course, for the remaining spatial components. Let us remark that the assumption (A−1 ) is obviously satisﬁed from the physical point of view. However, its correct mathematical proof exhibits some diﬃculties and is still in progress. There are no problems with the exiatence and uniqueness of solution for the parabolic heat transfer equation (9) in a weak sense as all the temperaturedependent material parameters are Lipschitz-continuous functions. Analysis of the solvability and uniqueness of the discrete problem is performed in an analogous way.

7

Convergence of the numerical scheme (3D case)

Let us consider the continuous problem (17). For simplicity, let us further consider that the domain Ω1 is covered by the discretization mesh exactly (Ω1,h ≡ Ω1 ). With a function κh obtained by elementwise averaging the function κ from (14), we can write the discrete problem for the eddy currents Jeddy in Ω1 as (I + Kh )Lx,h = Fx,h .

(20)

with (Kh Lx,h )(v) =

kh (v, w)Lx,h (w)dw, Ω1

κh M, kh (v, w) = |v − w|

M=

0 −1 1 0

(21)

.

(22)

Subtracting (20) from (17) we obtain that Lerr = Lx − Lx,h is governed by Lerr = (I + K)−1 [Ferr − (K − Kh )Lx,h ]

(23)

where obviously Ferr = Fx − Fx,h → 0 as the grid diameter h → 0, K − Kh → 0 as h → 0 from the deﬁnition of κh and Lx,h is bounded from the compactness of (I + Kh )−1 . Note that again we used the technical assumption (A−1 ) from the previous section together with a similar assumption (A−1,h ) for the discrete problem.

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I. Doležel, P. Šolín and B. Ulrych

Finally, convergence in the source terms of the heat transfer equation (9) yields also the convergence of the whole discrete coupled model for all ﬁnite times. Obviously, when integrating explicitly in time one has to fulﬁl the classical stability condition for the time step 8t ∈ O(h2 ) for parabolic equation.

8

Example: Temperature-Dependent Material Parameters for Copper

Fig. 3. Temperature-dependent γ, λ and ρc for copper

151

Fieldless Methods for the Simulation of Induction Heating

9

Example 1: Heating of a thin copper plate

The suggested methodology has been applied to an arrangement depicted in Fig. 4. A copper slab of sizes 0.3×0.6 m was heated by two helicoidal inductors (number of turns N = 10) of the indicated geometry.

y I

e x t

= 3 0 0 + j×0 A

in d u c to r

N = 1 0

0 ,0 0 0 5

0 ,0 2

p la te 0

0 ,0 1

u p p e r c o il

x

s y m m e tric a lly p la c e d lo w e r c o il z

S /m

0 ,3

6

0, 05

g = 1 ,2 ×1 0

c o il x 0 1

Fig. 4. The investigated arrangement

The dependencies of parameters γ and λ on temperature T have been shown in Fig. 3. The temperature dependence of product ρc is considered linear: ρc = 3.6312· 106 + 934.5 · (T − 20) J/deg m3 . Parameter α = 25 W/m2 deg and Text = 20 ◦ C. All numerical computations have been performed by a special user program package developed by the authors and written in C++. Several results for parameters f = 1000 Hz and h = 0.0005 m are shown in Figs. 5 to 7. Fig. 5 depicts the distribution of the temperature (at various time

152

I. Doležel, P. Šolín and B. Ulrych

levels) in the slab along line z = 0.05 m and Fig. 6 the same distribution along line z = 0.15 m. Fig. 7 shows the time evolution of temperature T at the “hottest” point of the slab (co-ordinates 0.25 m and z = 0.15 m).

T (°C )

3 0 0

t = 2 0 0 s

2 5 0

t = 1 6 0 s

2 0 0

t = 1 2 0 s

1 5 0

t = 8 0 s t = 4 0 s

1 0 0

t = 0 s

5 0 0 0

0 .1

0 .2

0 .3

0 .4

0 .5

x ( m ) 0 .6

Fig. 5. Distribution of temperature along line z = 0.05 m

5 0 0 4 5 0 4 0 0 3 5 0 3 0 0 2 5 0 2 0 0 1 5 0 1 0 0 5 0 0

T (°C ) t = 2 0 0 s t = 1 6 0 s t = 1 2 0 s t = 8 0 s t = 4 0 s

0

t = 0 s

0 .1

0 .2

0 .3

0 .4

0 .5

x ( m ) 0 .6

Fig. 6. Distribution of temperature along line z = 0.15 m

153

Fieldless Methods for the Simulation of Induction Heating 5 0 4 5 4 0 3 5 3 0 2 5 2 0 1 5 1 0

T (°C ) 0

0

0 0

0 0

0 0

0 5 0 0

0

5 0

1 0 0

1 5 0

2 0 0

2 5 0

t (s)

3 0 0

Fig. 7. Evolution of the temperature at point [0.25, 0.15] m

10

Example 2: Heating of a brass prism

A prismatic brass bar (Fig. 8) is heated by a coil-shaped inductor formed by a hollow tubular water-cooled conductor. The basic arrangement of the system is obvious from parts A and B of Fig. 8. The ﬁeld current in the inductor I = 550 A, its frequency f = 150 kHz. As the inductor is formed by a massive conductor, it was substituted by 8 thinner conductors located at points indicated in part C of Fig. 8. Each of these conductors carries current Ik = 68.75 A, k = 1, 2, . . ., 8. The starting temperature Tstart of the body and temperature Text of the surrounding air is 20 ◦ C. Coeﬃcient of the convective heat transfer is 25 W/m2 . The discretisation grid covering the body consists of 750 cubic elements sized 5 × 5 × 5 mm. In the following Fig. 9 we present a series of color plots for the temperature evolution in the investigated metal body. The Fig. 10 shows steady-state temperature cutlines at the time t = 90 min corresponding to its axis and to one of its longest edges.

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I. Doležel, P. Šolín and B. Ulrych

Fig. 8. The investigated arrangement

155

Fieldless Methods for the Simulation of Induction Heating

Fig. 9. Temperature distribution after 1, 2, 5, 10, 30 and 60 minutes.

533.5 533 532.5 532 531.5 531 530.5 530 529.5 0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

534 533.5 533 532.5 532 531.5 531 530.5 530 529.5 529

Fig. 10. Temperature cutlines after 90 minutes of heating (axis and edge).

156

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I. Doležel, P. Šolín and B. Ulrych

Example 3: Heating of a brass frame

In this section we present another example related to the induction heating of a brass frame. The outer measures of the frame are 25 × 25 × 150 mm. Its geometry and the discretization grid consisting of cubic elements sized 1 × 1 × 1 mm are shown in Fig. 11. The inductor is formed by a single coil with N = 6 turns, radius r = 19 mm, which carries a harmonic current I = 500 A of frequency f = 2000 Hz. The length-increment corresponding to one turn of the coil is 8l = 13 mm. Remaining parameters are the same as in the previous example.

Fig. 11. Brass frame, geometry and mesh.

Fieldless Methods for the Simulation of Induction Heating

157

Fig. 12. Temperature distribution after 1, 3, 5, 9, 13, 15 seconds.

The Fig. 12 shows the temperature evolution during the ﬁrst 15 seconds of the heating. After approximately 15 seconds, the steady state is reached The reader may notice that in this case the temperature distribution is not uniform (as e.g. in the last example), which may indicate that, from the engineering point of view, the inductor may undergo some additional re-arrangements.

References 1. Engl, H. W., Integralgleichungen (Springer, Wien, 1997, in German). 2. Lykov, A. V., Theory of Heat Conduction (Moscow, 1967, in Russian). 3. Šolín, P., Doležel, I., Škopek, M., Ulrych, B., Stationary Temperature Field in a Nonmagnetic Thin Plate Heated by Transversal Electromagnetic Field, Acta Technica CSAV 45 (2000) 105–128.

Equadiﬀ 10, August 27–31, 2001 Prague, Czech Republic

Equadiﬀ 10 CD ROM Papers, pp. 159–168

Topological Properties of Nonlinear Evolution Equations Vladimír Ďurikovič1 and Monika Ďurikovičová2 1

Department of Applied Mathematics SS, Cyril and Methodius University nám. J. Herdu 2, 917 00 Trnava, Slovak Republic Email: [email protected] 2 Department of Mathematics of Slovak Technical University, nám. Slobody 17, 812 31 Bratislava, Slovak Republic Email: [email protected]

Abstract. The generic properties of solutions of the second order ordinary diﬀerential equations were studied by L. Brüll and J. Mawhin in [2], J. Mawhin in [5] and by V. Šeda in [9]. Such questions were solved for nonlinear diﬀusional type problems with the Dirichlet, Neumann and Newton type conditions by V. Ďurikovič, Ma. Ďurikovičová in [4]. In the present paper we study the set structure of classic solutions, bifurcation points and the surjectivity of an associated operator to a general second order nonlinear evolution problem by the Fredholm operator theory. MSC 2000. 35K20, 35K60, 47F05, 47A53, 47H30 Keywords. Initial-boundary value problem, Fredholm operator, Hölder space, bifurcation point, surjectivity

1

The formulation of problem and basic notations

Throughout this paper we assume that the set Ω ⊂ Rn for n ∈ N is a bounded domain with the suﬃciently smooth boundary ∂Ω. The real number T is positive and Q := (0, T ] × Ω, Γ := (0, T ] × ∂Ω. We use the notation Dt for ∂/∂t and Di for ∂/∂xi and Dij for ∂ 2 /∂xi ∂xj where i, j = 1, . . . , n and D0 u for u. The symbol cl M means the closure of a set M in Rn . We consider the nonlinear diﬀerential equation possibly a non-parabolic type Dt u − A(t, x, Dx )u + f (t, x, u, D1 u, . . . , Dn u) = g(t, x) This is an overview article.

(1.1)

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V. Ďurikovič and M. Ďurikovičová

for (t, x) ∈ Q, where the coeﬃcients aij , ai , a0 for i, j = 1, . . . , n of the second order linear operator A(t, x, Dx )u =

n

aij (t, x)Dij u +

i,j=1

n

ai (t, x)Di u + a0 (t, x)u

i=1

are continuous functions from the space C(cl Q, R). The function f is from the space C(cl Q × Rn+1 , R) and g ∈ C(cl Q, R). Together with the equation (1.1) we consider the following general homogeneous boundary condition B3 (t, x, Dx )u|Γ :=

n

bi (t, x)Di u + b0 (t, x)u|Γ = 0,

(1.2)

i=1

where the coeﬃcients bi for i = 1, . . . , n and b0 are continuos functions from C(cl Γ, R). Furthermore we require for the solution of (1.1) to satisfy the homogeneous initial condition (1.3) u|t=0 = 0 on cl Ω. In the following deﬁnitions we shall use the notations

ust,µ,Q :=

|u(t, x) − u(s, x)| , |t − s|µ (t,x),(s,x)∈cl Q sup

(1.4)

t=s

uyx,ν,Q :=

|u(t, x) − u(t, y)| , ν |x − y| (t,x),(t,y)∈cl Q sup

(1.5)

x=y

f s,y,v t,x,u s,y,v(s,y)

f t,x,u(t,x)

:= |f (t, x, u0 , u1 , . . . , un ) − f (s, y, v0 , v1 , . . . , vn )| , := |f [t, x, u(t, x), D1 u(t, x), . . . , Dn u(t, x)]− −f [s, y, v(s, y), D1 v(s, y), . . . , Dn v(s, y)]| ,

where x = (x1 , . . . , xn ) and y = (y1 , . . . , yn ) are from Rn , µ, ν ∈ R and |x − y| = n

1 = (xi − yi )2 2 . i=1

The concept of a domain with a locally smooth boundary is given in the following deﬁnition. Definition 1.1. Let r ∈ (1, ∞) and Ω ⊂ Rn be a bounded domain. We say that the boundary ∂Ω belongs to the class C r , r ≥ 1 if: (i) There exists a tangential space to ∂Ω at any point from the boundary ∂Ω. (ii) Assume y ∈ ∂Ω and let (y; z1 , . . . , zn ) be a local orthonormal coordinate system with the center y and with the axis zn oriented like the inner normal to ∂Ω at the point y. Then there exists a number b > 0 such that for every

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Topological Properties of Nonlinear Evolution Equations

y ∈ ∂Ω there exists a neighbourhood O(y) ⊂ Rn of the point y and a function F ∈ C r (cl B, R) such that the part of boundary ∂Ω ∩ O(y) = {(z , F (z )) ∈ Rn , z = (z1 , . . . , zn−1 ) ∈ B}, where B = {z ∈ Rn−1 ; |z | < b}. Here C r (cl B, R) is a vector space of the functions u ∈ C l (cl B, R) for l = [r] with the ﬁnite norm + k ,y Dx u x,α,B , sup Dxk u(x) + ||u||l+α = 0≤k≤l

x∈cl B

k=l

whereby α = r − [r] ∈ [0, 1) and r = l + α. Further, we shall need the following Hölder spaces — see [3, p. 147]. Definition 1.2. Let α ∈ (0, 1). (1+α)/2,1+α

1. By the symbol Ct,x (cl Q, R) we denote the vector space of continuous functions u : cl Q → R which have continuous derivatives Di u for i = 1, . . . , n on cl Q and the norm ||u||(1+α)/2,1+α,Q :=

n

sup

i=0 (t,x)∈cl Q

|Di u(t, x)| + st,(1+α)/2,Q + +

n

s

Di ut,α/2,Q +

i=1

n

y

Di ux,α/2,Q

(1.6)

i=1

is ﬁnite. (2+α)/2,2+α (cl Q, R) means the vector space of continuous func2. The symbol C(t,x) tions u : cl Q → R for which there exist continuous derivatives Dt u, Di u, Dij u on cl Q, i, j = 1, . . . n and the norm ||u||(2+α)/2,2+α,Q := +

n

n

sup

|Di u(t, x)| +

i=0 (t,x)∈cl Q n

|Dij u(t, x)| +

sup

i,j=1 (t,x)∈cl Q n

sup

< Di u >st,(1+α)/2,Q + < Dt u >st,α/2,Q +

i=1

< Dij u >st,α/2,Q + < Dt u >yx,α,Q +

+

i,j=1

|Dt u(t, x)| +

(t,x)∈cl Q

n

< Dij u >yx,α,Q

(1.7)

i,j=1

is ﬁnite. (3+α)/2,3+α

3. The symbol Ct,x (cl Q, R) means the vector space of continuous functions u : cl Q → R for which the derivatives Dt , Di u, Dt Di u, Dij u, Dijk u, i, j, k = 1, . . . , n are continuous on cl Q and the norm

162

||u||(3+α)/2,3+α,Q :=

V. Ďurikovič and M. Ďurikovičová

n

sup

i=0 (t,x)∈cl Q n

sup

+

n

|Di u(t, x)| +

|Dij u(t, x)| +

sup

i,j=1 (t,x)∈cl Q n

|Dt Di u(t, x)| +

i=0 (t,x)∈cl Q

sup

|Dijk u(t, x)| +

i,j,k=1 (t,x)∈cl Q

s

+ Dt ut,(1+α)/2,Q +

n

s

Dij ut,(1+α)/2,Q +

i,j=1

+

+

n

s

Dt Di ut,α/2,Q +

n

s

Dijk ut,α/2,Q +

i=1

i,j,k=1

n

n

y

Dt Di ux,α,Q +

i=1

y

Dijk ux,α,Q

(1.8)

i,j,k=1

is ﬁnite. The above deﬁned norm spaces are Banach ones and we call them Hölder spaces. Definition 1.3. (The smoothness condition (S31+α ).) Let α ∈ (0, 1). We say that the diﬀerential operator A(t, x, Dx ) from (1.1) and B3 (t, x, Dx ) from (1.2), respectively satisﬁes the smoothness condition (S31+α ) if (i) the coeﬃcients aij , ai , a0 from (1.1) for i, j = 1, . . . , n belong to the space (1+α)/2,1+α (cl Q, R) and ∂Ω ∈ C 3+α and Ct,x (ii) the coeﬃcients bi from (1.2) for i = 1, . . . , n belong to the space (2+α)/2,2+α Ct,x (cl Γ, R). Definition 1.4. (The complementary condition (C).) If at least one of the coefﬁcients bi for i = 1, . . . , n of the diﬀerential operator B3 (t, x, Dx ) in (1.2) is not zero we say that B3 (t, x, Dx ) satisﬁes the complementary condition (C). In the following part we shall reformulate the problem (1.1), (1.2), (1.3) to the operator equation F3 u = A3 u + N3 u = g using several assumptions from Definition 1.5. 1. Fredholm conditions (A3 .1) Consider the operator A3 : X3 → Y3 , where A3 u = Dt u − A(t, x, Dx )u, u ∈ X3

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and the operators A(t, x, Dx ) and B3 (t, x, Dx ) satisfy the smoothness condition (S31+α ) for α ∈ (0, 1) and the complementary condition (C). Here we consider the vector spaces (3+α)/2,3+α

D(A3 ) := {u ∈ Ct,x

B3 (t, x, Dx )u|Γ = 0, u|t=0 (x) = 0

(cl Q, R); for x ∈ cl Ω}

and (1+α)/2,1+α

H(A3 ) := {v ∈ Ct,x

(cl Q, R); B3 (t, x, Dx )v(t, x)|t=0,x∈∂Ω = 0}

and Banach subspaces of the given Hölder spaces X3 = D(A3 ), ||.||(3+α)/2,3+α,Q and

Y3 = H(A3 ), ||.||(1+α)/2,1+α,Q .

(A3 .2) There is a second order linear homeomorphism C3 : X3 → Y3 with C3 u = Dt u − C(t, x, Dx )u, u ∈ X3 , where C(t, x, Dx )u =

n i,j=1

cij (t, x)Dij u +

n

ci (t, x)Di u + c0 (t, x)u

i=1

satisfying the smoothness condition (S31+α ). The operator C3 is not necessarily parabolic one. 2. Local Hölder and compatibility conditions. Let f := f (t, x, u0 , u1 , . . . , un ) : cl Q × Rn+1 → R, α ∈ (0, 1) and let p, q, pr for r = 0, 1, . . . , n be nonnegative constants. Here, D represents any compact subset of (cl Q) × Rn+1 . For f we need the following assumptions: (N3 .1) Let f ∈ C 1 (cl Q × Rn+1 , R) and let the ﬁrst derivatives ∂f /∂xi , ∂f /∂uj be locally Hölder continuous on cl Q × Rn+1 such that ) s,y,v n

∂f /∂xi t,x,u α/2 α + q|x − y| + pr |ur − vr | ≤ p|t − s|

∂f /∂uj s,y,v t,x,u r=0 for i = 1, . . . , n and j = 0, 1, . . . , n and any D. (N3 .2) Let f ∈ C 3 (cl Q × Rn+1 , R) and let the local growth conditions for the third derivatives of f hold on any D:  ,t,x,v + 3  ∂ f /∂τ ∂xi ∂uj t,x,u     + 3 ,t,x,v    ∂ f /∂τ ∂uj ∂uk t,x,u   n + 3 ,t,x,v  ≤ ∂ f /∂xi ∂xl ∂uj t,x,u ps |us − vs |βs   s=0 + 3 ,t,x,v   ∂ f /∂xi ∂uj ∂uk t,x,u     + 3 ,t,x,v   ∂ f /∂uj ∂uk ∂ur t,x,u 

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where βs > 0 for s = 0, 1, . . . , n and i, l = 1, . . . , n; j, k, r = 0, 1, . . . , n. (N3 .3) The equality of compatibility n

bi (t, x)Di f (t, x, 0, . . . , 0) + b0 (t, x)f (t, x, 0, . . . , 0)|t=0,

x∈S

=0

i=1

holds. 3. Almost coercive condition. Let for any bounded set M3 ⊂ Y3 there exist a number K > 0 such that for all solutions u ∈ X3 of the problem (1.1), (1.2), (1.3) with the right hand side g ∈ M3 , the following alternative holds: (F3 .1) Either (α3 ) u(1+α)/2,1+α,Q ≤ K, f := f (t, x, u0 ) : cl Q×R → R and the coeﬃcients of the operators A3 and C3 (see (1.1) and (A3 .2) satisfy the equations aij = cij , ai = ci

for i, j = 1, . . . , n, a0 = c0 on cl Q

or (β3 ) u(2+α)/2,2+α,Q ≤ K, f := f (t, x, u0 , u1 , . . . , un ) : cl Q × Rn+1 → R and the coeﬃcients of the operators A3 and C3 satisfy the relations aij = cij

for i, j = 1, . . . , n

and ai = ci for at least one i = 1, . . . , n

on cl Q. Remark 1.6. 1. Especially, the condition (A3 .2) is satisﬁed for the diﬀusion operator C3 u = Dt u− u, u ∈ X3 or for any uniformly parabolic operator C3 with suﬃciently smooth coeﬃcients. However the operator C3 is not necessarily uniform parabolic. 2. The local Hölder conditions in (N3 .1) and (N3 .2) admit suﬃciently strong growths of f in the last variables u0 , u1 , . . . , un . For example, they include exponential and power type growths. Definition 1.7. 1. A couple (u, g) ∈ X3 × Y3 will be called the bifurcation point of the mixed problem (1.1), (1.2), (1.3) if u is a solution of that mixed problem and there exists a sequence {gk } ⊂ Y3 such that gk → g in Y3 as k → ∞ and the problem (1.1), (1.2), (1.3) for g = gk has at least two diﬀerent solutions uk , vk for each k ∈ N and uk → u, vk → u in X3 as k → ∞. 2. The set of all solutions u ∈ X3 of (1.1), (1.2), (1.3) (or the set of all functions g ∈ Y3 ) such that (u, g) is a bifurcation point of the problem (1.1), (1.2), (1.3) will be called the domain of bifurcation (the bifurcation range) of that problem.

Topological Properties of Nonlinear Evolution Equations

165

Under the previous hypotheses we have proved the fundamental lemas: Lemma 1.8. The following implications are true: (1) (A3 .1), (A3 .2) imply that the operator A3 : X3 → Y3 is a linear bounded Fredholm operator of the zero index. (2) (N3 .1), (N3 .2) imply that the Nemitskij operator N3 : X3 → Y3 deﬁned by (N3 u)(t, x) = f [t, x, u(t, x), D1 u(t, x), . . . , Dn u(t, x)] for u ∈ X3 and (t, x) ∈ cl Q is completely continuous. (3) (A3 .1), (A3 .2), (N3 .1), (N3 .3), (F3 .1) imply that the operator F3 = A3 + N3 : X3 → Y3 is coercive. (4) (N3 .2), (N3 .3) imply that N3 ∈ C 1 (X3 , Y3 ) and is completely continuous. Lemma 1.9. Let A3 : X3 → Y3 be the linear operator satysfying (A3 .1), (A3 .2) and let N3 : X3 → Y3 be the Nemitskij operator satysfying (N3 .1), (N3 .3) and F3 = A3 + N3 : X3 → Y3 . Then: (i) The function u ∈ X3 is a solution of the initial-boundary value problem (1.1), (1.2), (1.3) for g ∈ Y3 if and only if F3 u = g. (ii) The couple (u, g) ∈ X3 × Y3 is the bifurcation point of the initial-boundary value problem (1.1), (1.2), (1.3) if and only if F3 (u) = g and u ∈ Σ, where Σ means the set of all points of X3 at which F3 is not locally invertible.

2

Generic properties for continuous operators

Aplying Theorem (Ambrosetti). Let F ∈ C(X, Y ) be a proper mapping. Then the cardinal number card F −1 ({q}) of the set F −1 ({q}) is constant and ﬁnite (it may be zero) for each q taken from the same (connected) component of the set Y − F (Σ). Here Σ means the set of all points u ∈ X for which F is not locally invertible. and Theorem (S. Smale and F. Quinn). If F : X → Y is a Fredholm mapping of class C q , q > max(ind F, 0) and either X has a countable basis (Smale) or F is σ-proper (Quinn), then the set RF of all regular values of F is residual in Y . If F is proper, then RF is open and dense in Y . we can prove the main results for the nonlinear problem (1.1), (1.2), (1.3). Here X and Y are Banach spaces either both real or complex.

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Theorem 2.1. Under the assumptions (A3 .1), (A3 .2) and (N3 .1), (N3 .3) the following statements hold for the problem (1.1), (1.2), (1.3): (a) The operator F3 = A3 + N3 : X3 → Y3 is continuous. (b) For any compact set of the right hand sides g ∈ Y3 from (1.1), the corresponding set of all solutions is a countable union of compact sets. (c) For u0 ∈ X3 there exists a neighbourhood U (u0 ) of u0 and U (F3 (u0 )) of F3 (u0 ) ∈ Y3 such that for each g ∈ U (F3 (u0 )) there is a unique solution of (1.1), (1.2), (1.3) if and only if the operator F3 is locally injective at u0 . Moreover, if (F3 .1) is assumed, then: (d) For each compact set of Y3 the corresponding set of all solutions is compact (possibly empty). Theorem 2.2. If the hypotheses (A3 .1), (A3 .2), (N3 .1), (N3 .3) and (F3 .1) are satisﬁed, then for the initial-boundary value problem (1.1), (1.2), (1.3) the following statements hold: (e) For each g ∈ Y3 the set S3g of all solutions is compact (possibly empty). (f) The set R(F3 ) = {g ∈ Y3 ; there exists at least one solution of the given problem } is closed and connected in Y3 . (g) The domain of bifurcation D3b is closed in X3 and the bifurcation range R3b is closed in Y3 . F3 (X3 − D3b ) is open in Y3 . (h) If Y3 − R3b = ∅, then each component of Y3 − R3b is a nonempty open set (i.e. a domain). The number n3g of solutions is ﬁnite, constant (it may be zero) on each component of the set Y3 − R3b , i.e. for every g belonging to the same component of Y3 − R3b . (i) If R3b = ∅, then the given problem has a unique solution u ∈ X3 for each g ∈ Y3 and this solution continuously depends on g as a mapping from Y3 onto X3 . (j) If R3b = ∅, then the boundary of the F3 - image of the set of all points from X3 in which the operator F3 is locally invertible, is a subset of the F3 - image of all points from X3 in which F3 is not locally invertible, i.e. ∂F3 (X3 − D3b ) ⊂ F3 (D3b ) = R3b

3

Generic properties for C 1 -diﬀerentiable operator

In case the Nemitskij operator N3 ∈ C 1 (X, Y ), we get stronger results. Using the theorem on a local C 1 -diﬀeomorphism Theorem (E. Zeidler). Let F : (U (u0 ) ⊂ X) → Y be a C 1 -mapping. Then F is a local C 1 -diﬀeomorphism at u0 if and only if u0 is a regular point of F . and

Topological Properties of Nonlinear Evolution Equations

167

Theorem (R. S. Sadyrchanov). Let dim Y ≥ 3 and let F : X → Y be a Fredholm mapping of the zero index. If u0 is an isolated singular point of F , then the mapping F is localy invertibly at u0 . we obtain main results for C 1 -diﬀerentiable operators. Theorem 3.1. Assume that the hypotheses (A3 .1), (A3 .2), (N3 .2), (N3 .3) hold. Then the open set Y3 − R3b is dense in Y3 and thus the range of bifurcation R3b of initial-boundary value problem (1.1), (1.2), (1.3) is nowhere dense in Y3 . Also we shall investigate the linear problem in h ∈ X3 for some u ∈ X3 : A3 h(t, x)+

n ∂f [t, x, u(t, x), D1 u(t, x), . . . , Dn u(t, x)] Dj h(t, x) = g(t, x) (3.1) ∂uj j=0

with the conditions (1.2), (1.3). Theorem 3.2. Assume that the hypotheses (A3 .1), (A3 .2), (N3 .2), (N3 .3) and (F3 .1) hold. Then (a) The number of solutions of (1.1), (1.2), (1.3) is constant and ﬁnite (it may be zero) on each connected component of the open set Y3 − F (S3 ), i.e. for any g belonging to the same connected component of Y3 − F3 (S3 ). Here S3 means the set of all critical points of problem (1.1), (1.2), (1.3). (b) Let u0 ∈ X3 be a regular solution of (1.1), (1.2), (1.3) with the right hand side g0 ∈ Y3 . Then there exists a neighbourhood U (g0 ) ⊂ Y3 of g0 such that for any g ∈ U (g0 ) the initial-boundary value problem (1.1), (1.2), (1.3) has one and only one solution u ∈ X3 . This solution continuously depends on g. The associated linear problem (3.1), (1.2), (1.3) for u = u0 has a unique solution h ∈ X3 for any g from a neighbourhood U (g0 ) of g0 = F3 (u0 ). This solution continuously depends on g. (c) Denote by G3 the set of all right hand sides g ∈ Y3 of equation (1.1) for which the corresponding solutions u ∈ X3 of the problem (1.1), (1.2), (1.3) are its critical solutions. Then G3 is closed and nowhere dense in Y3 . (d) If the singular points set of the initial-boundary value problem (1.1), (1.2), (1.3) is empty, then this problem has unique solution u ∈ X3 for each g ∈ Y3 . It continuously depends of the right hand side g. Corollary 3.3. Let the hypotheses of Theorem 3.2 hold and (i) the linear homogeneous problem (3.1), (1.2), (1.3) (for g = 0) has only zero solution h = 0 ∈ X3 for any u ∈ X3 . Then the initial-boundary value nonlinear problem (1.1), (1.2), (1.3) has a unique solution u ∈ X3 for any g ∈ Y3 . This solution u continuously depends on g. Moreover linear problem (3.1), (1.2), (1.3) has a unique solution h ∈ X3 for any u ∈ X3 and for each right hand side g ∈ Y3 of (3.1) and this solution continuously depends on g.

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Theorem 3.4. Suppose that the hypotheses (A3 .1), (A3 .2), (N3 .2), (N3 .3) and (F3 .1) hold together with the condition (i) Each point u ∈ X3 is either a regular point or an isolated critical point of problem (1.1), (1.2), (1.3). Then to each g ∈ Y3 there exists one and only one solution u ∈ X3 of the problem (1.1), (1.2), (1.3) and it continuously depends on g.

References 1. A. Ambrosetti, Global inversion theorems and applications to nonlinear problems. Conferenze del Seminario di Matematica dell’ Università di Bari, Atti del 3◦ Seminario di Analisi Funzionale ed Applicazioni, in A Survey on the Theoretical and Numerical Trends in Nonlinear Analysis, Gius. Laterza et Figli, Bari 1976 pp. 211 to 232. 2. L. Brüll and J. Mawhin, Finiteness of the set of solutions of some boundary value problems for ordinary diﬀerential equations. Archivum Mathematicum (Brno) 24, 1988, pp. 163–172. 3. V. Ďurikovič, An initial-boundary value problem for quasi–linear parabolic systems of higher order. Ann. Polon. Math. XXX, 1974, pp. 145–164. 4. V. Ďurikovič and Ma. Ďurikovičová, Some generic properties of nonlinear second order diﬀusional type problem. Archivum Mathematicum (Brno) 35, 1999, pp. 229 to 244. 5. J. Mawhin, Generic properties of nonlinear boundary value problems. Diﬀerential Equations and Mathematical Physics, Academic Press Inc., New York 1992, pp. 217 to 234. 6. F. Quinn, Transversal approximation on Banach manifolds. In Proc. Sympos. Pure Math. (Global Analysis) 15 (1970), pp. 213–223. 7. R. S. Sadyrchanov, Selected Questions of Nonlinear Functional Analysis. Publishers ELM, Baku 1989 (in Russian). 8. S. Smale, An inﬁnite dimensional version of Sard’s theorem. Amer. J. Math. 87, 1965, pp. 861–866. 9. V. Šeda, Fredholm mappings and the generalized boundary value problem. Diﬀerential and Integral Equation 8, Nr.1, 1995, pp. 19–40. 10. E. Zeidler, Nonlinear Functional Analysis and its Application I, Fixed-Point Theorems. Springer-Verlag, Berlin, Heidelberg, Tokyo 1986.

Equadiﬀ 10, August 27–31, 2001 Prague, Czech Republic

Equadiﬀ 10 CD ROM Papers, pp. 169–176

A doubly degenerate elliptic system González Montesinos María Teresa1 and Ortegón Gallego Francisco2 1

2

Departamento de Matemáticas, Universidad de Cádiz, Facultad de CC.EE. y Empresariales, Avda. Duque de Nájera, 6, 11002 Cádiz, Spain Email: [email protected] Departamento de Matemáticas, Universidad de Cádiz, Facultad de Ciencias 11510 Puerto Real (Cádiz) Spain, Email: [email protected]

Abstract. We consider the steady state of the thermistor problem, a coupled set of nonlinear elliptic equations governing the temperature and the electric potential. We study the existence of weak solutions under the assumption that the two diﬀusion coeﬃcients are not bounded below far from zero, arising to a degenerate system.

MSC 2000. 35J70, 35J20, 35J60

Keywords. Degenerate elliptic systems, nonlinear elliptic equations, thermistor problem

1

Introduction

The heat produced by an electrical current passing through a conductor device is governed by the so-called thermistor problem. This problem consists of a system of nonlinear parabolic-elliptic system describing the temperature, u, and the electric potential ϕ ([1,7]). Here, we consider the steady state case, resulting in a coupled nonlinear elliptic system. Let J be the current density, Q the heat ﬂux and E = −∇ϕ the electric ﬁeld; then by Ohm’s and Fourier’s law we have J = σ(u)E,

Q = −a(u)∇u,

This is the preliminary version of the paper.

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where a(u) and σ(u) are, respectively, the thermal and electric conductivities. Also, from the usual conservation laws ∇ · J = 0, ∇ · Q = E · J we obtain  −∇ · (a(u)∇u) = ∇ · (σ(u)ϕ∇ϕ) en Ω,    ∇ · (σ(u)∇ϕ) = 0 en Ω, u = 0 sobre ∂Ω,    ϕ = ϕ0 sobre ∂Ω,

(1)

where Ω is an open, bounded and smooth enough set in RN , N ≥ 1. Usually, the right hand side of the equation for the temperature is written as σ(u)|∇ϕ|2 , which is equal to ∇ · (σ(u)ϕ∇ϕ) thanks to the equation veriﬁed by ϕ; this is true, for instance, if ϕ ∈ H 1 (Ω). The steady state themistor problem has been studied by several authors along the last two decades. Among them, we refer to Cimatti ([3,4,5,6]) and Cimatti– Prodi [8]. In these papers, the authors have obtained some existence results of weak solutions in both, two and three dimensions, using the so-called Diesselhorst transformation, and under the conditions u = u0 on ∂Ω, and u0 being a constant value, or u = u0 ≥ um > 0 on ∂Ω, together with the hypothesis 0 < am ≤ a(u), or a(u) = a0 constant, or even under the Wiedemann–Franz law (that is, a(s) = Lsσ(s), L > 0 a constant value) with metallic conduction, and certain assumptions on σ(u). We notice that in all these papers is assumed that a(s) ≥ a0 > 0, for all s. In the present work we show an existence result of a weak solution to the steady state thermistor problem in divergence form (1) under the general assumption that both a(s) and σ(s) are not bounded below far from zero. In this way, system (1) becomes doubly degenerate; in particular, we cannot expect the regularity ϕ ∈ H 1 (Ω) ∩ L∞ (Ω), or that u belongs to some Sobolev space. We point out that the technique we use here is not based en the derivation of L∞ -estimates for the temperature.

2

Setting of the problem

We consider the steady state thermistor problem in divergence form (1) under the following hypotheses on data: (H.1) σ ∈ C(R) and 0 < σ(s) ≤ σ ¯ , for all s ∈ R. +∞ r ∞ (H.2) a ∈ C(R) ∩ L (R), 0 a(s) ds = +∞, and A(r) = 0 a(s) ds is a strictly increasing function. (H.3) ϕ0 ∈ H 1 (Ω). (H.4) There exist an integer M > 1 and a function α : [M, +∞) → R such that α(s) > 0, for all s ≥ M , α is non-increasing and σ(s) ≥ α(s) > 0.

171

A doubly degenerate elliptic system

(H.5) Let p ∈

2N N +2 , 2

+∞

M

if N ≥ 2, p ∈ (1, 2) if N = 1 and p = 2 − p, then ds

α(s)p/p A(s

− 1)q¯/2

 ∗  if N ≥ 3, q¯ = 2 < +∞, with q¯ ∈ [2, +∞) if N = 2,   q¯ ∈ [1, +∞) if N = 1.

(2)

The main result of this work now follows Theorem 1. Under assumptions (H.1)–(H.5), problem  −∆A(u) = ∇ · (σ(u)ϕ∇ϕ) in D (Ω),    ∇ · (σ(u)∇ϕ) = 0 in Ω, u=0 on ∂Ω,    on ∂Ω, ϕ = ϕ0

(3)

has a weak solution (u, ϕ) in the following sense N if N ≥ 2, q = 2 if N = 1, A(u) ∈ W01,q (Ω), N −1 ϕ − ϕ0 ∈ W01,p (Ω), σ(u)1/2 ∇ϕ ∈ L2 (Ω), ∇A(u)∇ξ = − σ(u)ϕ∇ϕ∇ξ, for all ξ ∈ D(Ω), Ω Ω σ(u)∇ϕ∇φ = 0, for all φ ∈ H01 (Ω).

∀q <

(4) (5) (6) (7)

Ω

Furthermore, the term ∇ · (σ(u)ϕ∇ϕ) is a Radon measure and u ≥ 0 almost everywhere in Ω. 2.1

Approximate problems

1 1 Let n ∈ N and introduce the functions an (s) = a(s) + , σn (s) = σ(s) + , then n n we set the approximate problem given as follows  −∇ · (an (un )∇un ) = σn (un )|∇ϕn |2 in Ω,    in Ω, ∇ · (σn (un )∇ϕn ) = 0 (8) u = 0 on ∂Ω,  n   on ∂Ω. ϕn = Tn (ϕ0 ) where Tn (s) = min(|s|, n) sign s. By virtue of the classical existence results ([1]), problem (8) has a solution such that un ∈ H01 (Ω), ϕn − ϕ0 ∈ H01 (Ω) ∩ L∞ (Ω). 2.2 Since

Estimates and passing to the limit σn (un )∇ϕn ∇φ = 0, for all φ ∈ H01 (Ω), Ω

(9)

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taking φ = ϕn − ϕ0 yield 2 σn (un )|∇ϕn | = σn (un )∇ϕn ∇ϕ0 Ω

Ω

≤

1/2

σn (un )|∇ϕn |2 Ω

hence

1/2 σn (un )|∇ϕ0 |2

,

Ω

σn (un )|∇ϕn |2 ≤ σ ˜ Ω

|∇ϕ0 |2 ≤ σ ˜ ϕ0 H 1 (Ω) = C(˜ σ , ϕ0 ) = C1 ,

(10)

Ω

therefore, (fn ) = σn (un )|∇ϕn |2 is bounded in L1 (Ω). r Let vn = An (un ), An (r) = 0 an (s) ds and consider the elliptic problem −∆vn = fn in Ω, vn = 0 on ∂Ω. From Boccardo–Gallou¨et estimates ([2,9]), we deduce that (vn ) is bounded in W01,q (Ω), for all q <

N if N ≥ 2, q = 2 if N = 1. N −1

(11)

In this way, there exist a subsequence (vm ) ⊂ (vn ) and v ∈ W01,q (Ω) such that vm \ v in W01,q (Ω)-weakly.

(12)

Since the embeddings W01,q (Ω) ]→ Lr (Ω), for all r < NN−2 if N ≥ 2, or W01,q (Ω) = ¯ if N = 1, are compacts, we may also assume that H01 (Ω) ]→ C(Ω) vm → v in Lr (Ω)-strongly, if N ≥ 2, ¯ vm → v in C(Ω)-strongly, if N = 1, vm → v a.e. in Ω.

(13) (14) (15)

Moreover, since fn ≥ 0 in Ω, then vn ≥ 0 in Ω. Since An is strictly increasing, we also have un ≥ 0 in Ω. Now, we show that (A(un )) ⊂ H01 (Ω) is bounded in W01,q (Ω). Indeed, |∇A(un )| = |a(un )∇un | ≤ |an (un )∇un | = |∇An (un )| and by virtue of (11), (A(un )) is also bounded in W01,q (Ω); then there exist a subsequence (A(um )) ⊂ (A(un )) and z ∈ W01,q (Ω) such that A(um ) \ z in W01,q (Ω)-weakly, N if N ≥ 2, A(um ) → z in Lr (Ω)-strongly, for all r < N −2 ¯ A(um ) → z in C(Ω)-strongly if N = 1, A(um ) → z a.e. in Ω.

(16) (17) (18) (19)

173

A doubly degenerate elliptic system

But, since A is bijective, from (19) we deduce um → A−1 (z) = u a.e. in Ω,

(20)

with u ≥ 0 a.e. in Ω. Thanks to the deﬁnition of σn together with (20) we obtain σm (um ) → σ(u) a.e. in Ω.

(21)

Also, by virtue of (H.1), (σn (un )) is bounded in L∞ (Ω), and taking into account (21), we have σm (um ) → σ(u) in L∞ (Ω)-weakly-∗.

(22)

Now, we seek for estimates to the sequence (ϕn ) in some Sobolev space W 1,p (Ω), with 1 < p < 2. By virtue of (H.5), p2 is the conjugate exponent of p2 . Applying Young’s inequality and taking into account (10), we obtain p /2 p/2 p −p/p |∇ϕn | ≤ σn (un ) σn (un )|∇ϕn |2 Ω

Ω

p/2

≤ C1

Ω

σn (un )−p/p

p /2 .

Ω

Let’s show the following estimate σn (un )−p/p ≤ C2 .

(23)

Ω

˜ , for all s ∈ R, it yields From 0 < σ(s) ≤ σn (s) ≤ σ

σ ˜ −p/p ≤ σn (s)−p/p ≤ σ(s)−p/p , for all s ∈ R, hence σn (un )−p/p ≤ σ(un )−p/p ≤ Ω

{|un |≤M}

Ω

σ(un )−p/p +

{un >M}

σ(un )−p/p .

Thanks to (H.1), σ −1 is bounded on compact sets of R, in particular, there exists a constant value CM > 0 such that min|s|≤M σ(s) = CM , and this implies that −p/p

σ(un )−p/p χ{|un |≤M} ≤ CM , and −p/p σ(un )−p/p ≤ CM |Ω| = C(M, p, p , Ω) = C3 . {|un |≤M}

On the other hand, by virtue of (H.4), we deduce σ(un )−p/p ≤ α(un )−p/p ≤ {un >M}

≤

i≥M

{i≤un
{un >M}

α(i + 1)−p/p ≤

i≥M

i≥M

{i≤un
α(un )−p/p

α(i + 1)−p/p |{un ≥ i}|

(24)

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In order to derive some estimate to |{un ≥ i}|, we ﬁrst study |{vn = An (un ) ≥ i}|. To do so, we take Ti (vn ) as a test function in the equation of un ; then ∇vn ∇Ti (vn ) = σn (un )|∇ϕn |2 Ti (vn ) ≤ C1 i, Ω

Ω

the left hand side can be written as Sobolev’s inequality we have

Ω

∇vn ∇Ti (vn ) =

2/¯q q¯

Ii,n ≥ C

|Ti (vn )| Ω

=C

i

2

Ω

≥C

!2/¯q

{vn ≥i}

|Ti (vn )| 2/¯ q

= Ci2 |{vn ≥ i}|

q¯

{vn ≥i}

|∇Ti (vn )| = Ii,n . By

!2/¯q

q¯

,

where q¯ = 2∗ = 2N/(N − 2) and C = C(Ω, N ), if N ≥ 3, q¯ ∈ [2, +∞) and C = C(Ω, q¯), if N ≤ 2. Consequently, 2/¯ q

|{vn ≥ i}|

≤

C1 C1 i = , i2 C iC

q¯/2 C1 C4 = q¯/2 . Since un ≥ 0 in Ω, An (un ) ≥ A(un ) iC i in Ω, {A(un ) ≥ i} ⊂ {vn = An (un ) ≥ i} and

which yields, |{vn ≥ i}| ≤

|{A(un ) ≥ i}| ≤ |{vn ≥ i}| ≤ hence

C4 , iq¯/2

{un ≥ A−1 (i)} ≤ C4 , iq¯/2

this can be expressed as |{un ≥ l}| ≤

C4 . A(l)q¯/2

Therefore, thanks to (2) in (H.5) and (24), we have σ(un )−p/p ≤ α(i + 1)−p/p {un >M}

i≥M

+∞

≤ C4 M−1

C4 A(i)q¯/2

ds = C5 . α(s + 1)p/p A(s)q¯/2

This shows (23) and we deduce that p/2 p /2 |∇ϕn |p ≤ C1 C2 = C6 , Ω

(25)

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A doubly degenerate elliptic system

which means that, ϕn − ϕ0 is bounded in W01,p (Ω). We then take a subsequence (ϕm ) ⊂ (ϕn ) and ϕ ∈ W 1,p (Ω) such that ϕm

ϕm \ ϕ in W 1,p (Ω)-weakly, → ϕ in Lr¯(Ω)-strongly, for all r¯ < p∗ if N ≥ 2, ¯ ϕm → ϕ in C(Ω)-strongly, if N = 1, ϕm → ϕ a.e. in Ω.

(26) (27) (28) (29)

From (H.5), p > 2N/(N +2) which implies that p∗ = N p/(N −p) > 2. In particular ϕm → ϕ in L2 (Ω)-strongly. (30) 1/2 2 N Thanks to (10) σn (un ) ∇ϕn is bounded in L (Ω) ; and there exist a subse quence σm (um )1/2 ∇ϕm ⊂ σn (un )1/2 ∇ϕn and Φ ∈ L2 (Ω)N such that σm (um )1/2 ∇ϕm \ Φ in L2 (Ω)N -weakly.

(31)

From (22) and (26) it is deduced that Φ = σ(u)1/2 ∇ϕ ∈ L2 (Ω)N . Moreover, taking into account (H.1), (22) and (31), we also have σm (um )∇ϕm \ σ(u)∇ϕ in L2 (Ω)N -weakly.

(32)

Consequently, ∇ · (σ(u)∇ϕ) ∈ H −1 (Ω) and σ(u)∇ϕ∇φ = 0, for all φ ∈ H01 (Ω),

∇ · (σ(u)∇ϕ) , φ = − Ω

Going back to (9) and taking φ = ϕn ξ, with ξ ∈ D(Ω). Then σn (un )∇ϕn ∇(ϕn ξ) = σn (un )|∇ϕn |2 ξ + σn (un )∇ϕn ϕn ∇ξ 0= Ω Ω Ω = σn (un )|∇ϕn |2 ξ − ∇ · (σn (un )ϕn ∇ϕn ) ξ, Ω

and so,

Ω

σn (un )|∇ϕn |2 = ∇ · (σn (un )ϕn ∇ϕn ) en D (Ω);

(33)

from the equality σm (um )ϕm ∇ϕm ∇ξ = σm (um )1/2 ϕm σm (um )1/2 ∇ϕm ∇ξ. Ω

Ω

and by virtue of (21), (30) and (31), passing to the limit in m → ∞, it yields σ(u)1/2 ϕσ(u)1/2 ∇ϕ∇ξ = σ(u)ϕ∇ϕ∇ξ, for all ξ ∈ D(Ω), Ω

Ω

so, σm (um )|∇ϕm |2 = ∇ · (σm (um )ϕm ∇ϕm ) → ∇ · (σ(u)ϕ∇ϕ) en D (Ω). Since σm (um )|∇ϕm |2 ≥ 0 is bounded in L1 (Ω), we conclude that ∇ · (σ(u)ϕ∇ϕ) is a positive Radon measure. This ends up the proof of theorem 1.

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Remark 2. It is interesting to know if the equality ∇ · (σ(u)ϕ∇ϕ) = σ(u)|∇ϕ|2 holds in our setting. There are cases where this holds true (for instance in N = 1). In the general case and with the regularity deduced here for u and ϕ, we do not know if this equality still holds ([10]).

Acknowledgements This research has been partially supported by Ministerio de Educación y Cultura under DGICYT grant, project PB98–0583, and by Consejería de Educación y Ciencia de la Junta de Andalucía.

References 1. Antontsev S. N., Chipot M. The thermistor problem: existence, smoothness, uniqueness, blowup, SIAM J. Math. Anal., 25(4), (1994) 1128–1156. 2. Boccardo L., Gallou¨et T. Non–linear elliptic and parabolic equations involving measure data. Journal of Functional Analysis, 87:149–169, 1989. 3. Cimatti G. A bound for the temperature in the thermistor problem, IMA Journal of Applied Mathematics, 40, (1988) 15–22. 4. Cimatti G. Remark on existence and uniqueness for the thermistor problem under mixed boundary conditions, Quaterly of Applied Mathematics, XLVII(1), (1989) 117–121. 5. Cimatti G. The stationary thermistor problem with a current limiting device, Proceedings of the Royal Society of Edinburgh, 116A, (1990) 79–84. 6. Cimatti G. On two problems of electrical heating of conductors, Quaterly of Applied Mathematics, XLIX(4), (1991) 729–740. 7. Cimatti G. Existence of weak solutions for the nonstationary problem of the Joule heating of a conductor, Ann. Mat. Pura Appl. (IV), CLXII, (1992) 33–42. 8. Cimatti G., Prodi G. Existence results for a nonlinear elliptic system modelling a temperature dependent electrical resistor, Ann. Mat. Pura Appl., 152(4), (1988) 227–236. 9. Gómez Mármol M. Estudio matemático de algunos problemas no lineales de la mecánica de ﬂuidos incompresibles, PhD. Thesis, Universidad de Sevilla, (1998). 10. González Montesinos M. T. PhD. Thesis, Universidad de Cádiz (to appear).

Equadiﬀ 10, August 27–31, 2001 Prague, Czech Republic

Equadiﬀ 10 CD ROM Papers, pp. 177–181

Thermal Eﬀects in an Elastic Plate-beam Structure Marié Grobbelaar-Van Dalsen Department of Mathematics and Applied Mathematics, University of Pretoria, 0002 South Africa, Email: [email protected]

Abstract. We consider a linear model for a 2 − D hybrid elastic structure consisting of a thermo-elastic plate which has a beam attached to its free end. We show that the interplay of parabolic dynamics and hyperbolic dynamics in the model yields analyticity for the entire system. This result provides an easy route to uniform stability.

MSC 2000. 35Q72,73D35

Keywords. thermo-elastic, plate, plate-beam, coupled, dynamic boundary conditions.

1

Introduction and Statement of the Problem

We consider well-posedness of the following model, P r (P ), for the transversal vibrations of a hybrid structure consisting of a thin rectangular thermo-elastic plate which is clamped along three edges, while to its free edge a thin beam with ends clamped to the adjoining clamped edges of the plate, is attached: wtt + ∆2 w + α∆θ = 0 in ΩT ∂w on ∂ΩT − ΓT w=0= ∂n βθt − η∆θ − α∆wt = 0 in ΩT θ = 0 on ∂ΩT − ΓT This is an overview article.

178

M. Grobbelaar

wtt − [wxxx + (2 − ν)wxyy ] + wyyyy − α

βθt + η

∂θ + bθyy = 0 on ΓT ∂n ∂w = 0 on ΓT ∂n w = 0 = wy at ∂ΓT

∂θ − κθyy − bwyyt = 0 on ΓT ∂n θ = 0 at ∂ΓT ,

w(x, y, 0) = w0 (x, y), wt (x, y, 0) = w1 (x, y), θ(x, y, 0) = θ0 (x, y) in Ω w(a, y, 0) = µ0 (y), wt (a, y, 0) = µ1 (y), θ(a, y, 0) = θ1 (y) on Γ. Here Ω denotes the interior of the plate with corner points (0, 0), (a, 0), (a, ^) and (0, ^), while Γ is the line joining (a, 0) and (a, ^) and ∂Γ its end-points. The constitutive equations in P r (P ) are “contact” equations in the sense that the deﬂections as well as the temperatures of the plate and the beam match at the interface for t > 0, but not necessarily initially. Thus the 1−D biharmonic equation and heat equation along Γ form a system of dynamic boundary conditions for the thermo-elastic plate equations. By allowing for interaction between the plate and the beam, the partial diﬀerential equations along Γ contain additional terms: the third order space derivatives of the displacement variable w in the beam equation represent the combined shear force and twisting moment exerted by the plate on the beam, while the conormal derivative of the thermal variable θ in the heat equation along Γ reﬂects the ﬂux of heat from the plate to the beam across the interface Γ.

2

Implicit Evolution Equation for P r (P )

We formulate P r (P ) as an implicit evolution problem, P r (AEP ), of the form Find U such that d (BU (t)) + AU (t) = 0, U ∈ D ⊂ X, t > 0 dt lim BU (t) = y ∈ Y t→0+

with A and B operators from a Banach space X to a second Banach space Y. The construction of a unique solution of P r (AEP ) with representation U (t) = S(t)y entails the construction of a double family of evolution operators [4], viz.

{S(t), E(t)} = {S(t) : Y → X|t > 0}, {E(t) : Y → Y |t > 0}, with E(t) =: BS(t) a semigroup in Y. The evolution from an initial state in Y to a solution in the space X, is generated by the jointly closed operator pair −A, B : D → Y × Y in which R(B) is dense in Y.

3

Mathematical Setting for P r (P )

We deﬁne the following spaces and operators:

179

Thermo-elastic Hybrid Structure

X0 =: L2 (Ω) with inner product (, )0 and norm .0 . H m (Ω) = H m,2 (Ω) denotes the usual Sobolev spaces with inner products (, )m and norms .m when m > 0 and the Hilbert space L2 (Ω) when m = 0. (, )m,Γ and .m,Γ denote the inner products and norms in H m (Γ ). For u ∈ H m (Ω) we denote the trace of u on Γ by γu. We deﬁne the following subspaces of X0 : X1 =: {w ∈ H 1 (Ω)w = 0 on ∂Ω − Γ, γw ∈ H01 (Γ )}. ∂w 2 X2 =: {w ∈ H 2 (Ω)w = 0 = ∂w ∂n on ∂Ω − Γ, ∂n = 0 on Γ, γw ∈ H0 (Γ )}. The spaces Xi , i = 0, 1, 2 are endowed with the inner products (, )i and the norms .i . For X2 we also use the equivalent inner product ((, ))2 given by a(w, z) = (wxx , zxx )0 +2(1−ν)(wxy , zxy )0 +(wyy , zyy )0 +ν(wxx , zyy )0 +ν(wyy , zxx )0 . The associated norm will be denoted by |||.|||2 . Y0 =: X0 × L2 (Γ ). The (usual) inner product and norm are denoted by (, )Y0 and .Y0 . The domains D1 and D2 are deﬁned by ∂w 4 2 D1 =: {w ∈ H 4 (Ω)w = 0 = ∂w ∂n on ∂Ω − Γ, ∂n = 0 on Γ, γw ∈ H (Γ ) ∩ H0 (Γ )}. D2 =: {θ ∈ H 2 (Ω)θ = 0 on ∂Ω − Γ, γθ ∈ H 2 (Γ ) ∩ H01 (Γ )}. The operators A, B and Cj , j = 1, 2, 3 from X0 into Y0 are deﬁned by , + Aw =: ∆2 w, −[γ(wxxx + (2 − ν)wxyy )] + (γw)yyyy , Bw =: w, γw , w ∈ D1 = D(A). , + ∂θ C1 θ =: α∆θ, −αγ ∂n + b(γθ)yy , θ ∈ D2 , C2 w˙ =: C3 θ =:

1 β 1 β

−α∆w, ˙ −b(γ w) ˙ yy , w˙ ∈ X2 , + , ∂θ −η∆θ, ηγ ∂n − κ(γθ)yy , θ ∈ D2 .

Observing that R(B) = { w, γw , w ∈ D1 } is a proper subset of Y0 , we deﬁne the following subsets of X1 × H01 (Γ ) and X2 × H02 (Γ ): Y1 = C^(B[X1 ]), Y2 = C^(B[X2 ]), with closures taken in Y0 . 1

Y1 will be endowed with the norm BwY1 = (∇w20 + (γw)y 20,Γ ) 2 and Y2 with 1 the norm |||Bw|||Y2 ≡ ((Bw, Bw))Y2 = (a(w, w) + (γw)yy 20,Γ ) 2 . To cast P r (P ) in the abstract form P r (AEP ), we deﬁne product spaces equipped with product space inner products and norms, viz. the “ﬁnite energy” space XE and its accompanying space YE and a weaker space X , with accompanying space Y:

180

M. Grobbelaar

XE =: X2 × (X0 )2 , YE =: Y2 × (Y0 )2 . In XE we deﬁne the domain ˙ θ), w ∈ D1 , w˙ ∈ X2 , θ ∈ D2 } DE =: {UE = (w, w, The linear domain DE are now deﬁned by  A and B onthe common  operators  0 −B 0 B 0 0 AUE =:  A 0 C1  UE , BUE =:  0 B 0  UE , U ∈ DE . 0 C2 C3 0 0 B To deﬁne the weaker spaces X , Y we ﬁrst deﬁne 12 H =: {w ∈ X0 A w = 0}, WX0 = H ⊥ . [1][2] 1 Z =: {y = y1 , y2 ∈ Y0 A 2 y1 = 0}, WY0 = Z ⊥ . X =: (WX0 ∩ X0 ) × (X0 )2 , Y =: (WY0 ∩ Y0 ) × (Y0 )2 . 1

To deﬁne a domain D in X , we introduce the variable U =: (u, w, ˙ θ), Bu =: −A 2 w. D =: {U = (u, w, ˙ θ), u ∈ (WX0 ∩ H 2 (Ω)), w˙ ∈ X2 , θ ∈ D2 .} The linear operators L and M from X    1 0 A2 0 B 1 LU =:  −A 2 0 C1  U, MU =:  0 0 0 C2 C3

to Y are now deﬁned by  0 0 B 0  U, U ∈ D. 0 B

P r (P ) may now be cast in the form of implicit evolution problems, viz. P r (AEP )I : d (BUE (t)) + AUE (t) = 0, UE ∈ DE , t > 0, dt lim BUE (t) = G ∈ YE t→0+

or P r (AEP )II : d (MU ) + LU = 0, U ∈ D, t > 0, dt lim+ MU (t) = F ∈ Y. t→0

4

Main Results

Th e reader is referred to [3] for the detailed proofs. Lemma 1. Re {(AUE , BUE )YE } = Re {−((B w, ˙ Bw))Y2 + (Aw, B w) ˙ Y0 + (C1 θ, B w) ˙ Y0 + (C2 w˙ + C3 θ, Bθ)Y0 } 2 2 = η∇θ0 + κ(γθ)y 0,Γ , Re{(LU, MU )Y } = η∇θ20 + κ(γθ)y 20,Γ .

Thermo-elastic Hybrid Structure

181

We prove Theorem 2. The operator pair −A, B generates a unique uniformly bounded double family S, E = {S(t) : YE → XE |t > 0}, {E(t) : Y → Y|t > 0} of evolution operators. Thus P r (AEP )I has unique solution UE ∈ C((0, ∞); DE ) with representation UE (t) = S(t)G for any G ∈ R(B) and each t ∈ (0, ∞). Corollary 3. P r (P ) in (w, wt , θ) can be associated with a uniformly bounded ˙ θ) evolution operator S(t) : YE → DE ⊂ XE in the sense that S(t)G = UE (t) = (w, w, solves P r (AEP )I for any G = (G1 , G2 , G3 ) = ( g1 , γg1 , g2 , γg2 , g3 , γg3 ) such that   g1 ∈ D1 , γg1 ∈ H 4 (Γ ) ∩ H02 (Γ ) g2 ∈ X2 , γg2 ∈ H02 (Γ )  g3 ∈ D2 , γg3 ∈ H 2 (Γ ) ∩ H01 (Γ ). The restriction that each Gi , i = 1, 2, 3, of G is of the form gi , γgi , may be interpreted as meaning that the initial displacement, velocity and temperature in the plate and the beam should match along Γ. Theorem 4. The operator pair −L, M generates a unique analytic uniformly bounded double family S, E = {S(t) : Y → X |t > 0}, {E(t) : Y → Y|t > 0} of evolution operators. Thus P r (AEP )II has unique solution U ∈ C((0, ∞); D) with representation U (t) = S(t)F for any F ∈ Y and each t ∈ (0, ∞). Corollary 5. P r (P ) in (w, wt , θ) can be associated with an analytic evolution operator S(t) : Y → D ⊂ X in the sense that S(t)F = U (t) = (u, w, ˙ θ) solves P r (AEP )II for any F = ( f1 , f2 , g1 , g2 , h1 , h2 ) such that   f1 ∈ X0 , 0 = E( f1 , γf1 ), f2 ∈ L2 (Γ ) g1 ∈ X0 , g2 ∈ L2 (Γ )  h1 ∈ X0 , h2 ∈ L2 (Γ ) with 2E( f1 , γf1 ) = a(f1 , f1 ) + (γf1 )yy 20,Γ the elastic potential energy. With the aid of Lemma 1 we obtain uniform stability for P r (AEP )II : Theorem 6. There exist constants M, σ > 0 such that for t > 0, the unique solution U ∈ C((0, ∞); D) of P r (AEP )II , represented as U (t) = S(t)F for any F ∈ Y, satisﬁes S(t)F X ≤ M exp(−σt)F Y .

References 1. Grobbelaar-Van Dalsen, M., Fractional Powers of a closed Pair of Operators, Proc. Roy. Soc. Edinburgh 102A 1986, 149–158. 2. Grobbelaar-Van Dalsen, M., On Fractional Powers of a Pair of Matrices and a Platebeam Problem, Appl. Anal. 72(3–4) 1999, 369–390. 3. Grobbelaar-Van Dalsen, M., Thermal Eﬀects in a two-dimensional Hybrid Elastic Structure, J. Math. Anal. Appl., to appear. 4. Sauer, N., Empathy Theory and the Laplace Transform, Linear Operators Banach Center Publications 38 1997, 325–338.

Equadiﬀ 10, August 27–31, 2001 Prague, Czech Republic

Equadiﬀ 10 CD ROM Papers, pp. 183–189

Solvability of Some Higher Order Two-Point Boundary Value Problems Maria do Rosário Grossinho1 and Feliz Manuel Minhós2 1

Dep. Matemática, ISEG, Universidade Técnica de Lisboa, Rua do Quelhas, 6, 1200-781 LISBOA, Portugal, CMAF, Universidade de Lisboa, Av. Prof. Gama Pinto, 2 1649-003 LISBOA Codex, Portugal Email: [email protected] 2 Dep. Matemática. Universidade de Évora Rua Rom˜ ao Ramalho, 7000 ÉVORA, Portugal, and CIMA-UE, Universidade de Évora, Rua Rom˜ ao Ramalho, 7000 ÉVORA, Portugal, Email: [email protected]

Abstract. This paper is concerned with the study of some nonlinear n-th order diﬀerential equation u(n) (t) = f (t, u(t), u (t), ..., u(n−1) (t)), with two-point boundary conditions, via upper and lower solutions.

MSC 2000. 34B15

Keywords. Higher order two-point BVP, upper and lower solutions, Leray-Schauder degree, Nagumo-type conditions

1

Motivation

Consider the n-th order nonlinear diﬀerential equation 2k+1

u(n) (t) = arctan u(n−2) (t) − [u (t)]

Supported by FCT.

This is the preliminary version of the paper.

2 u(n−1) (t) ,

(1)

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t ∈ [0, 1], k ∈ N, and the two-point boundary conditions u(i) (0) = 0, i = 0, ..., n − 3, a u(n−2) (0) − b u(n−1) (0) = A, c u(n−2) (1) + d u(n−1) (1) = B.

(2)

We observe that the results contained in the work [7] for higher order nonlinear diﬀerential problems cannot be applied to study the above problem. In fact, there, the equations involve nonlinearities that do not depend on the (n − 1)-th order derivative of the solution. More precisely, [7] concerns equations of the following type u(n) (t) + f (t, u(t), u (t), ..., u(n−2) (t)) = 0. Motivated by the above facts, we study the equation u(n) (t) = f (t, u(t), u (t), ..., u(n−1) (t)),

(3)

with the boundary conditions (2), where f : [0, 1] × Rn → R is a continuous function, a, b, c, d, A, B ∈ R and a, b, c and d satisfy b, d ≥ 0, a2 + b > 0 and c2 + d > 0. Then we apply it to solve problem (1)–(2). The arguments used follow some ideas contained in [1] and [4], for second order problems, and [2] for third order. In Section 2, we establish an existence result for problem (3)–(2) relying on the existence of upper and lower solutions. The function f is supposed to satisfy some Nagumo-type conditions. We sketch brieﬂy the proof and refer [3] for details. In Section 3, we consider the problem (1)–(2), with a, b, c and d non-negative constants. We exhibit an upper and a lower solution for this problem and show 2k+1 2 (xn−1 ) satisﬁes Nagumo-type that f (t, x0 , ..., xn−1 ) = arctan (xn−2 ) − (x0 ) conditions. Then an existence result is derived by applying the theorem of Section 2. We end Section 3 with more one applied problem.

2

Existence Result

We begin by deﬁning lower and upper solutions for problem (3)–(2) and Nagumotype conditions. Definition 1. (i) A function α(t) ∈ C n (]0, 1[) ∩ C n−1 ([0, 1]) is a lower solution of problem (3)–(2) if

and

α(n) (t) ≥ f (t, α(t), α (t), ..., α(n−1) (t))

(4)

α(i) (0) = 0, i = 0, ..., n − 3, a α(n−2) (0) − b α(n−1) (0) ≤ A, c α(n−2) (1) + d α(n−1) (1) ≤ B.

(5)

Solvability of Some Higher Order Two-Point Boundary Value Problems

185

(ii)A function β(t) ∈ C n (]0, 1[) ∩ C n−1 ([0, 1]) is an upper solution of problem (3)–(2) if β (n) (t) ≤ f (t, β(t), β (t), ..., β (n−1) (t)) (6) and

β (i) (0) = 0, i = 0, ..., n − 3, a β (n−2) (0) − b β (n−1) (0) ≥ A, c β (n−2) (1) + d β (n−1) (1) ≥ B.

(7)

Definition 2. Let E ⊂ [0, 1] × Rn . A continuous function g : E → R satisﬁes the Nagumo-type conditions in E if there exists a real continuous function hE : R+ 0 → ]0, +∞[, such that |g(t, x0 , ..., xn−1 )| ≤ hE (|xn−1 |), ∀(t, x0 , ..., xn−1 ) ∈ E, with

+∞ 0

s ds = +∞ . hE (s)

(8)

(9)

The following lemma will play a crucial role in establishing a priori estimates for the solutions of (3)–(2). Lemma 3. Let f : [0, 1] × Rn → R be a continuous function verifying Nagumotype conditions (8) and (9) in E = {(t, x0 , ..., xn−1 ) ∈ [0, 1] × Rn : γi (t) ≤ xi ≤ Γi (t), i = 0, ..., n − 2} , where γi (t) and Γi (t) are continuous functions such that, for each i and every t ∈ [0, 1], γi (t) ≤ Γi (t). Then there is r > 0 (depending only on hE , γn−2 and Γn−2 ) such that every solution u(t) of (3)–(2) and verifying γi (t) ≤ u(i) (t) ≤ Γi (t), for i = 0, ..., n − 2 and every t ∈ [0, 1], satisﬁes % % % (n−1) % %u % < r. ∞

The following theorem contains an existence result. Some information about the location of the solution and its i-derivatives, with i = 1, ..., n − 2, is also given. Theorem 4. Let f : [0, 1] × Rn → R be a continuous function. Suppose that there are lower and upper solutions of (3)–(2), α(t) and β(t), respectively, such that, for t ∈ [0, 1], (10) α(n−2) (t) ≤ β (n−2) (t)

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and that f satisﬁes Nagumo-type conditions (8) and (9) in E∗ = (t, x0 , ..., xn−1 ) ∈ [0, 1] × Rn : α(i) (t) ≤ xi ≤ β (i) (t), i = 0, ..., n − 2 , where by α(0) and β (0) we mean α and β. If f veriﬁes f (t, α(t), ..., α(n−3) (t), xn−2 , xn−1 ) ≥ f (t, x0 , ..., xn−1 ) ≥ ≥ f (t, β(t), ..., β (n−3) (t), xn−2 , xn−1 ),

(11)

for every (t, x0 , ..., xn−1 ) ∈ [0, 1] × Rn such that α(i) (t) ≤ xi ≤ β (i) (t) with i = 0, ..., n − 3, then the problem (3)–(2) has at least a solution u(t) ∈ C n ([0, 1]) satisfying α(i) (t) ≤ u(i) (t) ≤ β (i) (t), for i = 0, ..., n − 2 and t ∈ [0, 1]. Remark: If the function f (t, x0 , ..., xn−1 ) is decreasing on (x0 , ..., xn−3 ) then (11) is satisﬁed. Proof. We sketch brieﬂy the proof. For i = 0, ..., n − 2 deﬁne the auxiliary continuous functions  (i)  β (t) if xi > β (i) (t) δi (t, xi ) = xi if α(i) (t) ≤ xi ≤ β (i) (t)  (i) α (t) if xi < α(i) (t). For λ ∈ [0, 1], consider the homotopic equation u(n) (t) = λ f (t, δ0 (t, u(t)), ..., δn−2 (t, u(n−2) (t)), u(n−1) (t))+ +u(n−2) (t) − λ δn−2 (t, u(n−2) (t)),

(12)

with the boundary conditions u(i) (0) = 0, i = 0, ..., n − 3, u(n−2) (0) = λ [A − a δn−2 (0, u(n−2) (0)) + b u(n−1) (0)+ +δn−2 (0, u(n−2) (0))], (n−2) (1) = λ [B − c δn−2 (1, u(n−2) (1)) − d u(n−1) (1)+ u +δn−2 (1, u(n−2) (1))]. Take r1 > 0 such that for every t ∈ [0, 1], −r1 < α(n−2) (t) ≤ β (n−2) (t) < r1 , f (t, α(t), ..., α(n−2) (t), 0) − r1 − α(n−2) (t) < 0, f (t, β(t), ..., β (n−2) (t), 0) + r1 − β (n−2) (t) > 0

(13)

Solvability of Some Higher Order Two-Point Boundary Value Problems

187

A − a β (n−2) (0) + β (n−2) (0) < r1 ,

and

A − a α(n−2) (0) + α(n−2) (0) < r1 , B − c β (n−2) (1) + β (n−2) (1) < r1 , B − c α(n−2) (1) + α(n−2) (1) < r1 . The proof is based on the following steps (see [3] for details) Step 1.Every solution u(t) of problem (12)–(13) satisﬁes (i) u (t) < r1 , ∀t ∈ [0, 1], for i = 0, ..., n − 2 and independently of λ ∈ [0, 1]. This statement follows easily by using the deﬁnitions of upper and lower solutions combined with the condition (11). Step 2. There is r2 > 0 such that, for every solution u(t) of problem (12)–(13), (n−1) (t) < r2 , ∀t ∈ [0, 1], u independently of λ ∈ [0, 1]. This assertion can be derived by using Step 1 and the auxiliar Lemma 3. Step 3. For λ = 1, problem (12)–(13) has at least a solution u1 (t). This statement follows by applying Leray-Schauder degree theory. Step 4. The function u1 (t) is a solution of (3)–(2). By using the deﬁnitions of upper and lower solutions and condition (11), it can be shown that every solution of the problem (12)–(13) lies between α and β, and therefore is a solution of (3)–(2). By integration one can easily deduce the location result that concerns the derivatives of u1 (t).

3

Applications

Application 1. Consider the diﬀerential equation (1) and the boundary conditions u(i) (0) = 0, i = 0, ..., n − 3, a u(n−2) (0) − b u(n−1) (0) = A, c u(n−2) (1) + d u(n−1) (1) = B, for A, B ∈ R, a, b, c, d ≥ 0 such that a + b > 0 and c + d > 0.

(14)

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M. R. Grossinho and F. M. Minhós

The function f (t, x0 , ..., xn−1 ) = arctan (xn−2 ) − (x0 )2k+1 (xn−1 )2 is continuous and decreasing on x0 . If A and B are such that |A| ≤ a and |B| ≤ c then functions α, β : [0, 1] → R deﬁned by α(t) = −

tn−2 tn−2 and β(t) = (n − 2)! (n − 2)!

are, respectively, lower and upper solutions of the problem (1)–(14). Moreover, the function f satisﬁes the Nagumo-type conditions (8) and (9) in tn−2 n E = (t, x0 , ..., xn−1 ) ∈ [0, 1] × R : |x0 | ≤ , (n − 2)! + given by hE (x) = π2 + x2 . As conditions (10) and (11) are for hE : R+ 0 → R satisﬁed then, by Theorem 4, there is at least a solution u(t) for (1)–(14) such that tn−2−i tn−2−i ≤ u(i) (t) ≤ , − (n − 2 − i)! (n − 2 − i)!

for i = 0, ..., n − 2. Observe that in this case the estimation for u(n−2) does not depend on n since by the above inequality −1 ≤ u(n−2) (t) ≤ 1. Next application shows a non-uniform estimation for u(n−2) . Application 2. For n ≥ 2, consider the equation (n−2) ( 2 u (t) k u(n−1) (t) + 1 − arctan(u(t)), u(n) (t) = arctan (n − 2)!

(15)

with k ∈ N, and the boundary conditions (14). If A, B ∈ R are such that |A| ≤ a(n − 2)! and |B| ≤ c(n − 2)!, then functions α, β : [0, 1] → R given by α(t) = −tn−2 and β(t) = tn−2 are, respectively, lower and upper solutions for (15)-(14), verifying (10). The function ( xn−2 k 2 f (t, x0 , ..., xn−1 ) = arctan (xn−1 ) + 1 − arctan(x0 ) (n − 2)! is continuous. Moreover, it satisﬁes (11) and the Nagumo-type conditions (8) and (9) with ( π k π 2 h(x) = + (x) + 1, 2 2

Solvability of Some Higher Order Two-Point Boundary Value Problems

189

in every subset E ⊂ [0, 1] × Rn . So, by Theorem 4, there is at least a solution u(t) for (15)–(14) such that, for every t ∈ [0, 1], −(n − 2)...(n − i − 1) tn−2−i ≤ u(i) (t) ≤ (n − 2)...(n − i − 1) tn−2−i , with i = 0, ..., n − 3, and −(n − 2)! ≤ u(n−2) (t) ≤ (n − 2)!.

References 1. C. De Coster, La méthode des sur et sous solutions dans l’étude de probl` emes aux limites. Université Catholique de Louvain. Faculté des Sciences. Départment de Mathématique. Février 1994. 2. M.R. Grossinho and F. Minhós, Existence result for some third order separated boundary value problems. Journal of Nonlinear Analysis: Series A. Theory and Methods, 47, (2001), 2407-2418. 3. M.R. Grossinho and F. Minhós, Upper and lower solutions for higher order boundary value problems, preprint CMAF (2001). 4. C. De Coster and P. Habets. Upper and lower solutions in the theory of ODE boundary value problems: classical and recent results. Institut de Mathématique Pure et Appliquée. Université Catholique de Louvain. Recherches de Mathématique no 52. April 1996. 5. J. Mawhin, Topological degree methods in nonlinear boundary value problems, Regional Conference Series in Mathematics, no 40, American Mathematical Society, Providence, Rhode Island. (1979). 6. M. Nagumo, Über die diﬀerentialgleichung y = f (t, y, y ), Proc. Phys-Math. Soc. Japan, 19, (1937), 861-866. 7. F. H. Wong, An application of Schauder’s ﬁxed point theorem with respect to higher order BVPs, Proceedings of the American Mathematical Society, 126, (1998), 23892397.

Equadiﬀ 10, August 27–31, 2001 Prague, Czech Republic

Equadiﬀ 10 CD ROM Papers, pp. 191–202

Non existence of standing waves for hyperbolic Davey-Stewartson Systems Guzmán-Gómez Marisela Departament of Basic Sciences, Autonomus Metropolitan University-Azc. Av. San Pablo # 180, Col. Reynosa Tamaulipas 02000 México, D.F. México Email: [email protected]

Abstract. We consider the problem of existence of standing wave solutions of the Davey-Stewartson (DS) system in the hyperbolic-hyperbolic case. We extend the result of non existence of standing wave solutions for the elliptic-hyperbolic case of the (DS) system ([8]). We show that there are no solutions of the form eiwt v(x, y) with v ∈ H 1 (R2 ) and homogeneous boundary conditions on ϕ if b = 0. We ﬁnish with a result about non-existence of standing wave solutions which are smooth but with non-homogeneous boundary conditions on the velocity potential for both elliptic-hyperbolic and hyperbolic-hyperbolic cases. MSC 2000. 35Q55, 76B15 Keywords. Davey-Stewartson System, Standing Waves

1

Introduction

The Davey–Stewartson (DS) system models the evolution of water waves in a three dimensional ﬂow that travels predominantly in one direction. The system can be written in the form: iut + δuxx + uyy = λ|u|2 u + buϕx , ϕxx + mϕyy = (|u|2 )x ,

(x, y) ∈ R, t ∈ R,

(1) (2)

for the (complex) wave amplitude u(x, y, t) and the (real) mean velocity potential ϕ. The coeﬃcients (δ, λ, m, b) depend on the ﬂuid depth, surface tension and gravity and can take both signs [1,4,5]. The parameters λ and δ are normalized such that, |λ| = |δ| = 1. This is the ﬁnal form of the paper.

192

M. Guzmán-Gómez

The character of the solution depends strongly on the signs of the above coeﬃcients. It is useful to classify the system as elliptic-elliptic, elliptic-hyperbolic, hyperbolic-elliptic and hyperbolic-hyperbolic according to the respective sign of (δ, m): (+, +), (+, −), (−, +) and (−, −) ([6]). It has been known since the work of Ghidaglia and Saut ([6]) that the initial value problem of (DS) systems in the elliptic-elliptic and hyperbolic-elliptic cases has a unique solution in the spaces L2 (R2 ), H 1 (R2 ) and H 2 (R2 ). The Cauchy problem for the DS system in the elliptic-hyperbolic and the hyperbolic-hyperbolic cases has been studied by Hayashi and Saut [9]. The boundary conditions that have been imposed are, for the wave amplitude u: u(x, y, t), Dα u → 0 as x2 + y 2 → ∞,

(3)

and for the mean velocity ϕ are of radiation type: lim ϕ(ξ, η, t) = 0,

ξ→−∞

lim ϕ(ξ, η, t) = 0

η→−∞

(4)

where (ξ, η) are the characteristic coordinates: ξ=

√ 1 (x + −my), 2

η(x, y) =

√ 1 (x − −my). 2

(5)

More general boundary conditions for ϕ may be the following: lim ϕ(ξ, η, t) = f (η),

ξ→−∞

lim ϕ(ξ, η, t) = g(ξ)

η→−∞

(6)

with lim f (ξ) = lim g(ξ) = 0,

ξ→−∞

ξ→−∞

(7)

and f, g ∈ L∞ (R). Standing wave solutions for the DS system have been studied in the elliptic-elliptic and hyperbolic-elliptic cases. By extending the analysis developed for standing wave solutions of the Nonlinear Schrödinger equation iut + uxx + uyy = λ|u|2 u.

(8)

Cipolatti [2] proved existence, regularity and behavior at inﬁnity of standing wave solutions in the elliptic-elliptic case, (δ = 1, m > 0). Moreover, he showed the existence and uniqueness of ground states (positive solutions). In [3], Cipolatti proved that the ground states are unstable. Ghidaglia and Saut ([7]) gave necessarily conditions for existence of standing waves in the hyperbolic–elliptic case (δ = −1, m > 0). They showed that solutions of the form eiωt v(x, y) exists only if λ = −1 and b > 1. Recently, Guzmán-Gómez ([8]) showed that for elliptic-hyperbolic Davey–Stewartson system (δ = 1, m < 0), and boundary conditions as in (4) there are not standing wave solutions. This study was rather diﬀerent from Cipollati [2]

193

Non existence of standing waves

due to the lack of regularizing eﬀect for the velocity potential ϕ which satisﬁes a hyperbolic equation if m < 0. In [8], the author proved that if u(x, y, t) = eiωt v(x, y)

(9)

ϕ(x, y, t) = φ(x, y), ω ∈ R, v ∈ H 1 and ϕ(x, y) ∈ L∞ (R2 )

(10) (11)

is a solution of the system (1)-(2), m < 0, v satisﬁes weakly the elliptic equation (1 −

1 1 )(vxx + vyy ) + 2(1 + )vxy − ωv = F. m m

(12)

Due to the ellipticity of (12), if F ∈ L2 (R2 ), then v ∈ H 2 (R2 ). Once v is regular enough and decays at inﬁnity it can be concluded that v must be zero. The aim of this paper is to show ﬁrst that the hyperbolic-hyperbolic case of the (DS) system has no solutions of the form (9)-(11). Also, we approach the problem for non-homogeneous boundary conditions on ϕ (6) and obtain conditions on f and g for which standing wave classical solutions does not exist. This latter result is valid for both: elliptic-hyperbolic and hyperbolic-hyperbolic cases. In this work, we notice that if there is a solution of the system (1)-(2), in the form (9) − (10), with v ∈ C0∞ (R2 ) then v is necessarily zero. We then extend the result to H 1 (R2 ) by density. This technique is more general that the one used in [8]; we do not need the regularity eﬀect of the correspondent equation (12) and the density argument is valid for both: δ = 1 and δ = −1. The non existence of standing wave classical solutions follow from the proof of Theorem 6 where homogeneous boundary conditions on ϕ are considered; we approach the problem of existence of standing wave solutions in the classical sense but with non-homogeneous boundary conditions and provide conditions on f and g that no standing wave solutions may exist. Here H k (R2 ) denotes the Sobolev space of square integrable functions with square integrable derivatives up to order k and 2 2 α 2 u2 = uL2 (R2 ) , uH k = u2 + D u2 , and f, g = fg R2

|α|≤k

The paper is organized as follows: In section 2 we solve the wave equation (2) for the velocity potential ϕ in terms of u and substitute it in equation (1) to obtain a single equation of Schrödinger type with a nonlocal term (eq. 16). We also provide the main estimates for the nonlocal term that will be used in section 3. In section 3 we show that if u(x, y, t) = eiωt v(x, y), v ∈ H 1 (R2 ) is a weak solution of the (DS) system then v satisﬁes weakly (22); we obtain some estimates for the linear and nonlinear part of equation (22) to conclude that if {vn } ⊂ C0∞ (R2 ), then

+∞ +∞

lim

n→+∞

{vn } → v in H 1 (R2 )

−∞

−∞

(vn2 )x (x, y)dy

2 dx = 0.

(13)

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M. Guzmán-Gómez

We then prove the main Theorem (6), that is, we show that v(x, y) = 0 a.e. In section 4 we prove Theorem 7. We show that under certain conditions on the boundary conditions for ϕ if there is a classical solution of the form eiωt v(x, y) then v(x, y) = 0∀(x, y) ∈ R2 .

2

Velocity Potential

We begin transforming the coupled system (1)-(2) into a single equation with a nonlocal term by solving equation (2) and substituting it in equation (1). In terms of the characteristic variables (ξ, η) , (5), we can rewrite the equation for the mean velocity as 1 (14) ϕξη = (|u|2 )ξ + (|u|2 )η . 4 We will consider boundary conditions of radiation type (4) for ϕ. A similar problem can be stated with the boundary conditions deﬁned at +∞ instead of at −∞, leading to the same results. Integrating equation (14), we obtain ϕ(ξ, η) = =

1 4 1 4

ξ

−∞

η

(|u|2 )ξ + (|u|2 )η (ξ , η )dξ dη −∞ !

−∞

η

|u(ξ, η )|2 dη +

ξ

−∞

|u(ξ , η)|2 dξ

.

Rewriting equation (1) in terms of the ξ-η variables and using the above expression for ϕ we obtain 1 1 iut + δ − (uξξ + uηη ) + 2 δ + uξη m m ! ξ η b b 2 2 2 = λ+ √ |u|ξ dη + |u|η dξ . |u| u + √ u 2 −m 4 −m −∞ −∞

(15)

By the ξ, η by x, y, and deﬁning the new parameters α = renaming variables 1 1 δ−m , β =2 δ+ m , γ = λ + 2√b−m , U = 4√b−m we rewrite equation (15) as y iut + α(uxx + uyy ) + βuxy = γ|u|2 u + Uu (|u|2 )x (x, y )dy −∞ x + (|u|2 )y (x , y)dx .

(16)

−∞

In the next lemma we state the main estimate we will use for the second term of the right hand side of (16).

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Non existence of standing waves

Lemma 1. Let f, g ∈ H 1 (R2 ) and h ∈ L2 (R2 ). Then % % y % % % % gh(x, y )dy % ≤ f H 1 gH 1 h2 . a) %f −∞ % x %2 % % b) % gh(x , y)dx % %f % ≤ f H 1 gH 1 h2 . −∞

(17) (18)

2

Proof. We only prove a). We notice that % y %2 % % % % % 2 %f % (gh)(x, y )dy % ≤ f L2y L∞ % x % −∞

f 2L2y L∞ x

∞

= −∞

%2 % %

−∞

2

where

y

(gh)(x, y )dy %

(19)

L2x L∞ y

2

(essupx |f (x, y)|) dy.

Thanks to the Sobolev inequality uL∞(R) ≤ uH 1 (R) , ∞ 2 = f (x, y)2L∞ dy f L2y L∞ x x −∞

≤

∞

2 ! ∂f (x, y) dx dy |f (x, y)|2 dx + ∂x −∞ −∞

−∞

∞

∞

≤ f 2H 1 (R2 ) . Also, % % % %

y

−∞

%2 % (gh)(x, y )dy % %

L2x L∞ y

(20)

+∞

= −∞

essupy

y

−∞

2 (gh)(x, y )dy dx

2 |gh(x, y)|dy dx −∞ −∞ ∞ ∞ ∞ ≤ |g(x, y)|2 dy |h(x, y)|2 dy dx

+∞

+∞

≤

−∞

−∞

−∞

h22 . ≤ g2L2y L∞ x

(21) A @

Using (19), (20), and (21), (17) is obtained.

3

Standing Wave Solutions

We look for time-periodic solutions of equation (16) in the form u(x, y, t) = eiωt v(x, y) where v is real valued and belongs to H 1 (R2 ). Therefore, the function v must solve the following equality y x 3 2 2 (v )x dy + vy dx , α(vxx + vyy ) + βvxy = ωv + γv + Uv (22) −∞

−∞

196

M. Guzmán-Gómez

1 1 , β = 2(−1 + m where α = − 1 + m ), m < 0. In this paper we only consider weak solutions of equation (22), that is, v ∈ H 1 (R2 ) that satisﬁes equation −α vx fx − α vy fx − β vy fx = ω vf + γ v3 f R2 R2 R2 R2 y x 2 2 +U v (v )x (x, y )dy + (v )y (x , y)dx f, ∀f ∈ H 1 (R2 ). (23) R2

−∞

−∞

In [8], thanks to the regularity eﬀect of the elliptic equation (12),(m < 0), the authors proved that any weak solution of (16), belongs to H 2 (R2 ) and they can conclude that v = 0. In the hyperbolic-hyperbolic (DS) system we cannot use that v ∈ H 2 (R2 ). Instead, we use that C0∞ (R2 ) is dense in H 1 (R2 ), {vn } ⊂ C0∞ (R2 ), {vn } → v in H 1 (R2 ) and with the help of standard Sobolev estimates and lemma 1 we obtain that +∞

n→+∞

−∞

2

+∞

lim

−∞

(vn )2x

= 0.

We then prove the main theorem. We deﬁne by L and N to be the corresponding linear and nonlinear part of equation (23), that is, L(u) = α(uxx + uyy ) + βuxy − ωu, m < 0, y x (u2 )x dy + (u2 )y dx . N (u) = λu3 + U −∞

−∞

We may conclude that v ∈ H 1 (R2 ) is a weak solution of (22) if and only if

L(v), f = N (v), f

∀f ∈ L2 (R2 ).

(24)

We will use that whenever {vn } ⊂ C0∞ (R2 ), vn → v in H 1 (R2 ) lim (L + N )(vn ), vn x = (L + N )(v), vx

n→+∞

(25)

If v is a weak solution of equation (22), the right hand side of equality (25) is zero. Also, L(vn ), (vn )x = 0 ∀n > 0; on the other hand, after several integration by parts, we can prove that limn→+∞ N (vn ), vn x = 0 only if vn → 0, that is v = 0 a.e. . Equality (25) is a consequence of the following limits: lim L(vn ) − L(v)2 = 0,

n→+∞

lim N (vn ) − N (v)2 = 0.

n→+∞

To prove the two limits above is the purpose of the following two propositions. Proposition 2. Let v ∈ H 1 (R2 ) be a weak solution of equation (22),and {vn } ⊂ C0∞ (R2 ) such that vn − vH 1 → 0, then a) L(v) ∈ L2 (R2 ), b) lim L(vn ) − L(v)2 = 0. n→+∞

197

Non existence of standing waves

Proof. Let v ∈ H 1 (R2 ) be a weak solution of equation 22. Thank’s to the Sobolev embedding H 1 (R2 ) ⊂ L6 (R2 ), v 3 ∈ L2 (R2 ) and from Lemma 1 y x 2 2 U v vx (x, y)dy + v vy (x, y)dx ∈ L2 (R2 ), −∞

−∞

therefore N (v) ∈ L2 (R2 ). From (24) | L(v), f | ≤ N (v)2 f 2 ,

∀f ∈ L2 (R2 ),

hence L(v) ∈ L2 (R2 )

and L(v)2 ≤ N (v)2 ,

a) follows. Now we prove b): Let {vn } ⊂ C0∞ (R2 ) with limn→+∞ vn − vH 1 = 0. For any f ∈ H 1 (R2 )

L(vn ), f = −α vnx fx − α vny fy − β vny fx − ω vn f R2

R2

R2

R2

therefore, lim L(vn ), f = −α

n→+∞

R2

vx fx − α

= L(v), f .

R2

vy fy − β

R2

vy fx − ω

vf R2

Because H 1 (R2 ) is dense en L2 (R2 ), lim L(vn ), f = L(v), f

n→+∞

for any f ∈ L2 (R2 ).

(26)

Equation (26) together with a) implies that lim L(vn ) − L(v)2 = 0.@ A

n→+∞

Proposition 3. Let v ∈ H 1 (R2 ) be a weak solution of equation (22), and {vn } ⊂ H 1 (R2 ) such that limn→+∞ vn − vH 1 = 0 then lim N (vn ) − N (v)2 = 0.

(27)

n→+∞

Proof. To prove (27) is enough to show the following three limits: a) lim (vn )3 − v 3 2 = 0, n→+∞ % x % x % % 2 2 % (v ) dx − v (v ) dx b) lim % v n y y n % % = 0, n→+∞ −∞ −∞ % y %2 y % % 2 2 c) lim % (vn )y dy − v (v )x dy % %vn % = 0. n→+∞

−∞

−∞

2

198

M. Guzmán-Gómez

The ﬁrst limit follows from Cauchy-Schwartz inequality and the Sobolev embedding H 1 (R2 ) ⊂ Lp (R2 ), ∀p > 2. (vn )3 − (v)3 2 = (vn2 − vn v + v 2 )(vn − v)2 ≤ (vn )2 − vn v + v 2 4 vn − v4 ≤ 2 vn 28 + v28 vn − vH 1 . To prove limit b) we use Lemma 1: % % % y y % % % 2 2 % %vn %vn (v ) − v (v ) ≤ x% n x % % −∞

−∞

y

% % % % %(vn − v) (vn2 − v 2 )x % + % %

−∞

2

y

% % (v 2 )x % %

−∞

2

2

≤ vn 1H vn + v1H vn − vH 1 + vn − v1H v2H 1 ≤ C vn 2H 1 + v2H 1 vn − vH 1 . A @

Limit c) follows similarly. Lemma 4. There exists a positive constant C such that

2

+∞ +∞

−∞

−∞

dx ≤ Cf 4H 1

(f 2 )x (x, y)dy

(28)

for any f ∈ H 1 (R2 ). Proof. We observe that by Cauchy-Schwartz inequality

+∞ +∞

2 (f )x dxdy ≤ 4

+∞ +∞

2

−∞

−∞

2

2

f dy

−∞

≤ 4 sup x∈R

−∞ +∞ 2

(fx ) dy dx

+∞

−∞ +∞ +∞

f (x, y)dy

−∞

−∞

−∞

(fx )2 (x, y)dxdy.(29)

We use the inequality gL∞ (R) ≤ CgH 1 (R) , A @

(see (20)) to estimate the right hand side of (29) and obtain (28). Lemma 5. Let f be in H 1 (R2 ) and {fn } ⊂ C0∞ (R2 ) such that lim fn − f H 1 = 0,

n→+∞

therefore

+∞ +∞

lim

n→+∞

−∞

−∞

(fn )2x (x, y)dy

2

+∞ +∞

dx = −∞

−∞

(f )2x (x, y)dy

2 dx.

(30)

199

Non existence of standing waves

Proof. Let Fn (x) =

+∞

−∞

(fn2 )x (x, y)dy

+∞

−∞

Fn2 (x) − F 2 (x) dx =

+∞

(f )2 (x, y)dy

and F (x) = −∞ +∞

(Fn (x) + F (x))(Fn (x) − F (x))dx ≤ Fn L2 (R) + F L2 (R) Fn − F L2 (R) . −∞

From Lemma 4, Fn 2L2 (R) ≤ Cfn 4H 1 and F 2L2 (R) ≤ Cf 4H 1 , therefore

+∞

−∞

Fn2 (x) − F 2 (x) dx ≤ C fn 2H 1 + f 2H 1 Fn − F L2 (R) .

Now we estimate Fn − F L2 (R) : +∞ 2 Fn − F L2 (R) ≤ −∞

+∞

+∞

−∞

(fn2 )x (x, y)dy

−

(31)

+∞ 2

−∞

(f )x (x, y)dy dx

2 +∞ = (fn fnx − f fx )(x, y)dy dx 2 −∞ −∞ +∞ +∞ fn (fnx − fx )(x, y)dy ≤4

−∞

−∞

+∞

2

+ fx (fn − f )(x, y)dy dx −∞ ≤ 8 fn 22 + fx 22 fn − f 2H 1 . Combining (31) and (32) and using that fn − f H 1 → 0, limit (30) follows.

(32) A @

Now we prove the main theorem. Theorem 6. Let v ∈ H 1 (R2 ) be a weak solution of equation (22) with b = 0, then v(x, y) = 0 almost every where. Proof. Let {vn } ⊂ C0∞ (R2 ) such that {vn } → v in H 1 (R2 ), then v satisﬁes Eq.(24) and

L(v) − L(vn ), f + L(vn ), f = N (v) − N (vn ), f + N (vn ), f ∀f ∈ L2 (R2 ). Therefore,

L(v) − L(vn ), vnx + L(vn ), vnx = N (v) − N (vn ), vnx + N (vn ), vnx . Thanks to {vn } ⊂ C0∞ (R2 ), L(vn ), vnx = 0 and | N (vn ), vnx | ≤ L(v) − L(vn )2 vnx 2 + N (v) − N (vn )2 vnx 2 .

(33)

200

M. Guzmán-Gómez

Because v ∈ H 1 (R2 ) and {vn } → v in H 1 (R2 ), there exists a positive constant M , independent of n, such that vn H 1 ≤ M. (34) Combining (33), (34) with Propositions 2 and 3, we obtain lim N (vn ), vnx = 0.

(35)

n→+∞

On the other hand,

N (vn ), vnx = γ

R2

vnx (vn )3 +U

R2

vnx vn

x

(vn2 )y +U

−∞

R2

vnx vn

y

(vn2 )x . (36)

−∞

We observe that {vn } ⊂ C0∞ (R2 ) implies that the ﬁrst integral in the right hand side of (36) is zero. Integrating by parts the second term in the right hand side of (36) we obtain x x 2 U 2 U vnx vn (x, y (vn )y (x , y)dx = (vn2 )y (x , y)dx dy v (x, y) x 2 R2 n −∞ −∞ R2 x U ∞ 2 2 = lim v (x, y) (vn )y (x , y)dx dy 2 −∞x→+∞ n −∞ U (vn )2 (vn2 )y . (37) − 2 R2 Because vn ∈ C0∞ (R2 ) and for any y ∈ R,

x

−∞

(vn2 )y (x , y)dx

≤2

∞

−∞

12 |vn (x, y)| dx 2

∞

−∞

12 |vn y (x, y)| dx , 2

the right hand side of equation (37) is zero and equality (36) becomes y

N (vn ), vnx = U vnx vn (x, y) (vn2 )x (x, y )dy . −∞ R2 y y U (vn2 )xy (x, y )dy (vn2 )x (x, y )dy = 2 R2 −∞ −∞ 2 y ∂ U 2 = (vn )x (x, y )dy dy 4 R2 ∂y −∞ 2 +∞ U +∞ = (vn2 )x (x, y)dy dx. 4 −∞ −∞

(38)

Equation (38) together with equation (35) implies that

+∞ +∞

lim

n→+∞

−∞

−∞

(vn2 )x dy

2 dx = 0.

(39)

201

Non existence of standing waves

and together with Lemma 5,

+∞ +∞

−∞

−∞

2

(v 2 )x dy

dx = 0.

(40)

Hence,

+∞ 2

−∞

(v )x (x, y)dy = 0

+∞

v 2 (x, y)dy = constant.

almost everywhere and −∞

Because v ∈ L2 (R2 ) the constant is necessarily zero and Theorem 6 follows.

4

A @

Standing Wave Solutions. Non-homogeneous boundary conditions.

In this section we prove the non-existence of standing wave solutions of elliptichyperbolic and hyperbolic-hyperbolic cases of the Davey-Stewartson system for classical solutions with some non-homogeneous boundary conditions of the mean velocity potential. The Davey-Stewartson system with non-homogeneous boundary conditions (6) can be written in the form: y (|u|2 )x (x, y )dy iut + α(uxx + uyy ) + βuxy = γ|u|2 u + Uu −∞ x 2 + (|u| )y (x , y)dx + buf (y) + bug (x).(41) −∞

A standing wave solution for the equation (41) is a function v ∈ C 2 (R2 ) that satisﬁes y x (v 2 )x dy + vy2 dx α(vxx + vyy ) + βvxy = ωv + γv 3 + Uv

−∞

+ bv(f (y) + g (x)), where α = δ −

1 m , β = 2(δ +

1 m ),

−∞

(42)

m < 0.

Theorem 7. Let f and g be bounded functions in C 2 (R) such that f (x) and g (x) are also bounded . If f (x) ≤ 0 ∀x ∈ R or g (x) ≤ 0∀x ∈ R. lim f (x) = lim g(x) = 0 x→−∞

x→−∞

(43) (44)

If v ∈ H 2 (R2 ) is a classical solution of equation (42) with b = 0, then v(x, y) = 0 ∀(x, y) ∈ R2 .

202

M. Guzmán-Gómez

Proof. We consider that g (x) ≤ 0 ∀x ∈ R. The proof follows the ideas of the proof of Theorem 6. We use that v is a solution in the classical sense with v ∈ H 2 (R2 ). We take the L2 inner product of vx with each term of equation (42), integrate by parts and, similarly as in the proof of Theorem 6 we obtain that

2

+∞ +∞

0= −∞

−∞

(v 2 )x dy

dx −

b 2

v 2 (g (x))

(45)

Therefore, using the assumption on f , we conclude that

2

+∞ +∞ 2

−∞

−∞

(v )x dy

dx = 0

(46)

therefore we conclude similarly as in the proof of Thereom 6 that v(x, y) = 0 a.e., because v is continuous, v(x, y) = 0 ∀(x, y) ∈ R2 . A @

References 1. Ablowitz M., Segur H. On the evolution of packets of water waves, J. Fluid Mech. 92 1976, 691–715. 2. Cipollati R. On the existence of standing waves for the Davey-Stewartson system, Communications on Partial Diﬀerential Equations 17 (1992), 967–988. 3. Cipollati R. On the instability of ground states for a Davey-Stewartson system, Ann. Inst. H. Poincare, Phys. Theor. 58 (1994), 85–104. 4. Davey A., Stewartson K. On three-dimensional packets of surface waves, Proc. Royal Soc. London A 338 1974, 807–836. 5. Djordjevic V.D., Redekopp L.G. On two-dimensional packets of capillary-gravity waves J. Fluid Mech. 79 1977, 703–714. 6. Ghidaglia J-M., Saut J-C. On the initial value problem for the Davey-Stewartson systems Nonlinearity, 3 1990, 475–506. 7. Ghidaglia J-M., Saut J-C. Non existence of travelling wave solutions to non elliptic nonlinear Schrödinger equations J. Nonlinear Sci. 6, 1993, 139–145. 8. Guzmán-Gómez M. Non existence of standing waves for a Davey-Stewartson system Aportaciones Matemáticas, 25 1999, 71–78. 9. Hayashi N., Saut J-C. Global existence of small solutions to the Davey-Stewartson and the Ishimori systems. Diﬀ. and Integral Eqs., 8 1995, 1657-1675.

Equadiﬀ 10, August 27–31, 2001 Prague, Czech Republic

Equadiﬀ 10 CD ROM Papers, pp. 203–216

Heteroclinics for a class of fourth order conservative diﬀerential equations P. Habets1 , L. Sanchez2 , M. Tarallo3 and S. Terracini4

2

1 Institut de Mathématique Pure et Appliquée, Chemin du Cyclotron, 2, 1348 Louvain la Neuve, Belgium, Email: [email protected] Universidade de Lisboa, Centro de Matemática e Aplica¸co ˜es Fundamentais, Avenida Professor Gama Pinto, 2, 1649-003 Lisboa, Portugal, Email: [email protected] 3 Università degli studi di Milano, Via Saldini 50, Milano, Italy, Email: [email protected] 4 Politecnico di Milano, Via Bonardi 9, Milano, Italy, Email: [email protected]

Abstract. We prove the existence of heteroclinics for a 4th order O.D.E. related to the extended Fisher-Kolmogorov equation. Those solutions are obtained by minimization of a functional over a convenient set of functions. In particular, we obtain heteroclinic connections between the extreme equilibria for a (double well) potential with three minima at the same level. MSC 2000. 34B15, 34C37 Keywords. heteroclinic, double well potential

1

Introduction

In the study of ternary mixtures containing oil, water and amphiphile, a modiﬁcation of a Ginzburg-Landau model yields for the free energy a functional of the form (see[2]) F (u) = [c(∇2 u)2 + g(u)|∇u|2 + f (u)] dx dy dz

Research supported by Funda¸ca ˜o para a Ciˆencia e a Tecnologia.

This is the preliminary version of the paper.

204

P. Habets, L. Sanchez, M. Tarallo and S. Terracini

where the scalar order parameter u is related to the local diﬀerence of concentrations of water and oil. The function g(u) quantiﬁes the amphiphilic properties and the “potential” f (u) is the bulk free energy of the ternary mixture. In some relevant situations g may take negative values to an extent that is balanced by the positivity of c and f . The admissible density proﬁles may therefore be identiﬁed with critical points of F in a suitable function space. In the simplest case where the order parameter depends only on one spatial direction, u = u(x) is deﬁned on the real line and (after scaling) our functional becomes +∞ 1 [ [(u2 ) + g(u)u2 ] + f (u)] dx. (1) F (u) = −∞ 2 The corresponding Euler-Lagrange equation is 1 uiv − g(u)u − g (u)u2 + f (u) = 0. 2

(2)

When g ≡ const = β, we recognize here the well known extended Fisher-Kolmogorov equation. If the potential f (u) has two nondegenerate minima, say, at ±1, with f (±1) = 0, one question of great interest is the existence of a heteroclinic that connects these two equilibria. Our purpose is to obtain such a solution to (2) in cases where g may change sign. However, we shall be interested also in the case where f possesses three minima at the same level, since this is the framework where description of the three distinct phases becomes possible (see [2,4,5]). Let us recall that the case g ≡ const = β = 0, has attracted the attention of several authors. Peletier and Troy (see [7] and related papers in its references) have extensively dealt with the case β > 0 and the model potential f (u) = 14 (u2 − 1)2 ; they have shown that a heteroclinic connecting ±1 exists for all values of β > 0. The cases β 2 < 8 (“saddle-focus” case) and β 2 ≥ 8 (“saddle-node” case) need a diﬀerent treatment. Kalies and VanderVorst [6] have considered an even potential in the saddle-foci case. In [3] Kalies, Kwaspisz and VanderVorst have classiﬁed heteroclinic connections (with β > 0) between the two consecutive equilibria according to their homotopy type. Jan Bouwe van den Berg [1] has proved that if β 2 ≥ 8 the heteroclinic is asymptotically stable. More recently Smets and van den Berg have considered the case β < 0 for which they prove (in a saddle-foci case), by a version of the mountain pass theorem, that at each equilibria there arise homoclinic solutions. Here we start by considering the simple case g ≡ 0 and address two problems that, however, seem not to be covered by the existing literature. First (section 2) we prove the existence of the heteroclinic without the assumption of symmetry for f . Second (section 3), we look at the case where the potential f is symmetric, having not two, but three equilibria (0 and ±1) at the same minimum level: we shall see that a heteroclinic connecting ±1 still exists in this case. For simplicity, we deal with a potential that reduces to a quadratic function near ±1, see assumption (F1). However, by using the Hartmann-Grobman theorem, we could instead consider any nondegenerate minima at these points.

Heteroclinics for a class of fourth order conservative diﬀerential equations

205

Of course, in the case g ≡ β = 0 our functional becomes J (u) =

+∞ −∞

1 [ (u2 ) + f (u)] dx. 2

(3)

and its corresponding Euler-lagrange equation is uiv + f (u) = 0.

(4)

Finally, in section 4 we shall introduce a “compatibility condition” relating g and f in order to consider the original problem (1), (2). That condition allows g to take negative values somewhere between ±1 as required in the the theory of ternary mixtures; it is consistent with physical arguments (see [2]) and from the mathematical point of view, it enables us to reduce the problem to the simpler case g ≡ 0, since we are then able to construct a functional similar to J that bounds F from below. A variety of numerical results leading to solutions of equations of type (2), obtained by minimization of the free energy F , can be found in the thesis of H. Leit˜ao [4]. The authors are indebted to H. Leit˜ ao for having brought this problem to their attention. We aknowledge also the interest and useful discussions with D. Bonheure.

2

Potentials with a single well

We consider a potential f ∈ C 2 (R) such that for some 0 < a < 1/2 and α > 0, f (u) = 2α4 (u − 1)2 ,

(F 1)

4

2

f (u) = 2α (u + 1) ,

∀u ∈ (1 − a, 1 + a), ∀u ∈ (−1 − a, −1 + a),

f (u) = 0 if and only if u = ±1

(F 2) and (F 3)

lim inf f (u) > 0. |u|→∞

Lemma 1. Given an interval [a, b] ⊂ R and a function u ∈ H 2 (a, b) such that u(a) = A, u(b) = B, u (a) = A1 , u (b) = B1 the following inequality holds: a

b

u2 dx ≥

4 B−A B−A [(B1 − A1 )2 + 3( − A1 )( − B1 )] b−a b−a b−a

and equality holds if and only if u is a 3rd degree polynomial.

206

P. Habets, L. Sanchez, M. Tarallo and S. Terracini

We introduce the space E consisting of all functions u deﬁned in R such that u ∈ C 1 (R), u ∈ L2 (R) and lim u(x) = ±1.

x→±∞

In order to minimize J in E we shall now obtain estimates on functions u ∈ E in terms of an upper bound of the values J (u). Lemma 2. There are constants K and N , depending only on C, such that, for any function u ∈ E, J (u) ≤ C implies u∞ ≤ K, u ∞ ≤ N. Proof. According to the assumptions on f there exist K > 0 and A > 0 such that K2 K > C and A f (u) > C ∀|u| ≥ . A3 2 We claim that u∞ ≤ K. Otherwise, either the set {x : |u(x)| ≥ K/2} has measure greater than A and +∞ C f (u) dx > A = C, J (u) ≥ A −∞ a contradiction, or we can pick up an interval (c, d) such that d − c < A, u(c) = K |u|∞ , u(d) = K 2 , u(x) ≥ 2 , ∀x ∈ (c, d) and therefore by lemma 1

d

J (u) ≥ c

2 u2 u(d) − u(c) u(d) − u(c) dx ≥ [u (d)2 + 3( − u (d))( )] 2 d−c d−c d−c

4 u(d) − u(c) 2 K2 [ ] ≥ 3 > C, d−c d−c A again a contradiction. Hence the ﬁrst part of the statement is proved. Next choose N such that ≥

N > 4K, , N 2 > 8C. We show that u cannot attain the value N and one shows analogously that it cannot attain the value −N . For suppose that u (t0 ) = N ; then there exists t1 ∈ 0) it turns out that (t0 , t0 + 1) such that u (t1 ) = N2 . Hence, letting m = u(t1t1)−u(t −t0 J (u) ≥

t1

t0

2 N2 N N u2 dx ≥ > C, [( )2 + 3(m − N )(m − )] ≥ 2 t1 − t0 2 2 8

which is impossible.

Heteroclinics for a class of fourth order conservative diﬀerential equations

207

Lemma 3. If u ∈ E and J (u) < ∞ then lim u (x) = 0.

|x|→∞

Proof. This is a variation on the above argument. If, for instance, there exists U > 0 and a sequence xn → +∞ with u (xn ) ≥ U for all n then, since u(+∞) = 1, ∀δ > 0 we can pick up an interval (t0 , t1 ) such that u (t0 ) ≥ U, u (t1 ) = 2U , u(t1 ) ≤ u(t0 )+δ and t1 − t0 < 2Uδ . As before, we derive (with the same meaning for m) t1 U3 4U U U 2J (u) ≥ u2 dx ≥ [( )2 + 3(m − U)(m − )] ≥ . 2δ 2 2 8δ t0 The main idea in the next lemma is that there is an upper bound, depending only on the value of J , for the time it takes for a function u ∈ E to travel in the (u, u )-plane from a neighborhood of (−1, 0) to a neighborhood of (1, 0). Lemma 4. Let C > 0 and U > 0 be given. Then there exists R > 0 such that for any function u ∈ E with J (u) ≤ C, there exist x1 and x2 > x1 that satisfy |u (xi )| ≤ U

|u(xi ) − (−1)i | ≤ U,

and

x2 − x1 ≤ R.

Proof. Let C > 0 and U > 0 be given and let u ∈ E be so that J (u) ≤ C. Deﬁne x1 = sup{x | |u(x) + 1| ≤ U and |u (x)| ≤ U}. As u ∈ E, it is clear that x1 ∈ R. Now given x2 > x1 suppose that ∀x ∈ [x1 , x2 ],

|u(x) − 1| ≥ U or |u (x)| ≥ U.

(5)

We shall give a bound on x2 − x1 in terms of C and U. Deﬁne the sets A = {x ∈ [x1 , x2 ] | |u(x) + 1| ≥ U and |u(x) − 1| ≥ U}, and B = {x ∈ [x1 , x2 ] | |u(x) + 1| < U or |u(x) − 1| < U}. It is easy to see that B is the union of intervals Ii on which |u (x)| ≥ U. Further except maybe for the ﬁrst and the last one, each of these intervals is adjacent to an interval Ji = [ci , di ] so that ∀x ∈ [ci , di ], u(x) ≥ 1 + U, or ∀x ∈ [ci , di ], u(x) ≤ −1 − U, Claim 1 – meas(A) ≤

C r ,

u (ci ) ≥ U, u (di ) ≤ −U, u (ci ) ≤ −U, u (di ) ≥ U.

where

rJ = min{f (u) | |u + 1| ≥ U and |u − 1| ≥ U}. This follows from the inequalities

f (u(x)) dx ≥ rJ meas(A).

C ≥ J (u) ≥ A

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Claim 2 – meas(Ii ) ≤ 2. On an interval I¯i = [ai , bi ], we have |u (x)| ≥ U and bi u (x) dx| ≥ U(bi − ai ). 2U ≥ |u(bi ) − u(ai )| = | ai

Claim 3 – The number n of intervals Ji is bounded : n ≤ C/ min{2U2 , rJ }. Let Ji = [ci , di ] be such that ∀t ∈ [ci , di ], u(t) ≥ 1 + U, u (ci ) ≥ U and u (di ) ≤ −U. We can write di u (x) dx| ≤ u L2 (ci ,di ) (di − ci )1/2 . 2U ≤ |u (di ) − u (ci )| = | and

ci

di

ci

[ 12 (u )2 + f (u)] dx ≥

2U2 + rJ (di − ci ) ≥ min{2U2 , rJ }. di − ci

A similar argument holds if ∀x ∈ [ci , di ], u(x) ≤ −1 − U, u (ci ) ≤ −U and u (di ) ≥ U. It follows then that di [ 12 (u )2 + f (u)] dx ≥ n min{2U2 , rJ }. C ≥ J (u) ≥ i

ci

Conclusion – We deduce from the previous claims that x2 − x1 = meas(A) + meas(B) ≤

C r

+ (C/ min{2U2 , rJ } + 2)2

and the lemma follows. For convenience we introduce the notation V+ (T, U) := {u ∈ E : |u(t) − 1| ≤ U, |u (t)| ≤ U ∀t ≥ T } where T and U are given positive numbers. Analogously we deﬁne V− (T, U) replacing −1 for 1 and t ≤ −T for t ≥ T in the above deﬁnition. Let us also set V(T, U) = V+ (T, U) ∩ V− (T, U). Lemma 5. Let C > 0 and U > 0 be given. Then there exists R > 0 such that for any u ∈ E with J (u) ≤ C, there exists v ∈ V(R, U) that satisﬁes J (v) ≤ J (u). Proof. Fix U1 such that U1 (2 +

1 α4 a3 ) ≤ min{a, U}, (4α + 2)U1 ≤ U, and (α + 2α2 + 2α3 )U21 ≤ . α 4N

Let R be given as in Lemma 4 with respect to C and U1 . Let u ∈ E be such that J (u) ≤ C. There exist x1 ≤ x2 such that |u(xi ) − (−1)i | ≤ U1 ,

|u (xi )| ≤ U1

and x2 − x1 ≤ R.

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209

Deﬁne then the function v as    −1 + z(x)   v(x) = u(x)     1 + w(x)

if x ≤ x1 , if x1 ≤ x ≤ x2 , if x ≥ x2 ,

where z and w are respectively the solutions of z iv + 4α4 z = 0,

−1 + z(x1 ) = u(x1 ), z (x1 ) = u (x1 ), z(−∞) = 0

wiv + 4α4 w = 0,

1 + w(x2 ) = u(x2 ), w (x2 ) = u (x2 ), w(+∞) = 0.

and

Next we compute z(x) = eα(x−x1 ) [z(x1 ) cos α(x − x1 ) + ( z where

(x1 ) α

− z(x1 )) sin α(x − x1 )],

|z(x1 )| ≤ U1 , |z (x1 )| ≤ U1 .

It follows that for all x ≤ x1 , we have |z(x)| ≤ U1 (2 + α1 ) ≤ min{a, U}, |z (x)| ≤ (4α + 2)U1 ≤ U and x1 [ 12 (z )2 + 2α4 z 2 ] dx = 12 [z (x1 )z (x1 ) − z (x1 )z(x1 )] K(z) := −∞

= αz (x1 )2 − 2α2 z(x1 )z (x1 ) + 2α3 z(x1 )2 ≤ (α + 2α2 + 2α3 )U21 <

α4 a3 4N .

If for any x ∈ ] − ∞, x1 ], u(x) ∈ [−1 − a, −1 + a], we compute

x1 −∞

[ 12 (u )2 + f (u)] dx = K(1 + u) ≥ K(z).

Here we used the fact that, as z is a critical point of the convex functional K(u), it is a minimum. On the other hand, if there exists x ∈ ] − ∞, x1 ] such that u(x) ∈ [−1 − a, −1 + a], there exist x3 ≤ x4 ≤ x1 so that u(x3 ) = −1 + a/2, u(x4 ) = −1 + a and ∀x ∈ [x3 , x4 ], u(x) ∈ [−1 + a/2, −1 + a]. It follows that

N (x4 − x3 ) ≥ and

x4

u (x) dx =

x3

x1

−∞

[ 12 (u )2

+ f (u)] dx ≥

x4

x3

a 2

= u(x4 ) − u(x3 ) =

[ 12 (u )2 + f (u)] dx ≥

a 2

α4 a3 4N .

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A similar computation holds for the interval [x2 , +∞[ and we can write x1 J (v) = [ 12 (z )2 + 2α4 z 2 ] dx −∞ ∞ x2 2 1 + [ 2 (u ) + f (u)] dx + [ 12 (w )2 + 2α4 w2 ] dx x2 x1 x1 [ 12 (u )2 + 2α4 (u + 1)2 ] dx ≤ −∞ ∞ x2 2 1 + [ 2 (u ) + f (u)] dx + [ 12 (u )2 + 2α4 (u − 1)2 ] dx x2

x1

= J (u). At last, it is clear that v ∈ C 1 (R), v ∈ L2 (R) and as z(−∞) = w(∞) = 0 we can write v ∈ E. Translating v if necessary, we can assume −R ≤ x1 ≤ x2 ≤ R so that v ∈ V(R, U). Lemma 6. Let a function u ∈ C 1 (R) be such that u ∈ L2 (R), u is bounded, J (u) < +∞ and there exist R > 0 and 0 R, i = 1, 2. Then u ∈ E. Proof. Clearly, if u satisﬁes the assumptions of the lemma then because of (F1)(F2)-(F3) lim inf u(x) ≤ 1 ≤ lim sup u(x). x→+∞

x→+∞

Strict inequalities are easily ruled out since u is bounded. The statement now follows from Lemma 3. Theorem 7. There exists a minimizer u of J in E which is a heteroclinic of (4) connecting ±1. Proof. Let m := inf u∈E J (u) and choose a minimizing sequence (un )n ⊂ E such that J (un ) ≤ m + 1/n, for any n ∈ N. Fix 0 0 and a sequence (vn )n ⊂ V(R, U) that satisﬁes J (vn ) ≤ J (un ) ≤ m + n1 . Estimates in Lemma 2 imply that (vn )n has a subsequence (still written (vn )n for simplicity) such that for some function v vn

Cloc (R)

→

v,

L2 (R)

vn \ v .

By Fatou’s lemma, we have J (v) ≤ m. On the other hand it is clear that |v(x) − 1| ≤ U and |v (x)| ≤ U for |x| ≥ R. It follows from Lemma 6 that v ∈ E and therefore J (v) = m. The proof is complete.

Heteroclinics for a class of fourth order conservative diﬀerential equations

3

211

Symmetric potentials with a double well

In this section we assume that f is a C 2 even function with three minima at the same level, namely we replace (F2) by the following: (F 2 )

f (u) = 0 if and only if u = 0 or u = ±1.

and we introduce (F 4)

f is even and increasing on some interval ]0, b[, b < 1.

Theorem 8. Assume that f satisﬁes (F1),(F2’), (F3) and (F4). Then there exists an odd heteroclinic of (4), with u (0) > 0, connecting ±1. Proof. As in [6] we now look for a minimizer of +∞ 1 J0 (u) = [ (u2 ) + f (u)] dx. 2 0 in the set E0 consisting of functions u ∈ C 1 ([0, +∞)) with u(0) = 0, u(+∞) = 1 and u ∈ L2 (0, +∞). A minimizer will satisfy (4) in [0, +∞) and the natural boundary condition u (0) = 0; hence its odd extension is the solution one looks for. First we note that estimates for the C 1 norm of u in terms of an upper bound of J0 (u) immediately follow from Lemma 2. In order to construct a compact minimizing sequence in E0 it suﬃces to show that, given a small number U > 0, there exists T > 0 such that u ∈ E0 may be replaced with v ∈ E0 such that J0 (v) ≤ J0 (u) and v ∈ V+ (T, U). By the argument of lemma 4, given u ∈ E0 with J0 (u) < ∞ there exists t∗ = inf{t > 0 : |u(t) − 1| ≤ U, |u (t)| ≤ U}. We replace the “tail” of u (the restriction of u to t ≥ t∗ ) with a solution of a linear problem translated to 1 as in the proof of lemma 5. Also, using arguments similar to those appearing in the proof of lemma 4, it is easy to estimate, in terms of the value C := J0 (u), the length of the interval [t , t∗ ] such that u(t ) = b/2 and u(t) ≥ b/2 if t ∈ [t , t∗ ]. Because of (F2’) we can ﬁnd γ > 0, depending only on C, such that max{u (t) : t ∈ [t , t∗ ], b/2 ≤ u(t) ≤ b} ≥ γ. Let t¯ be a point where this maximum is attained and consider the function w ∈ C 1 (R) deﬁned by   u(x) if x ≥ t¯ w(x) = (7)  u(t¯) + u (t¯)(x − t¯) if x ≤ t¯ Now the zero θ of w to the left of t¯ depends only on u (t¯) and therefore on C. If w(x) ≤ u(x) ∀x ∈ [θ, t¯] we deﬁne v(x) = w(x + θ) for x ≥ 0. If this is not the case, then ∃t˜ ∈ [θ, t¯] such that u (t˜) = max{u (t) : t ∈ [θ, t∗ ], u(t) ≥ 0} ≥ u (t¯) and 0 ≤ u(t˜) < b/2. In this case we consider w ˜ deﬁned as in (7) with t˜ instead of t¯. ˜ ˜ for x ≥ 0. In any Clearly, the zero θ of w ˜ is ≥ θ, and we deﬁne v(x) = w(x + θ) case it is obvious, because of (F4), that J0 (v) ≤ J0 (u) and we can take T = t∗ − θ.

212

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Back to the original problem

In this section we reconsider the functional F and its Euler-Lagrange equation (2). We shall assume that g is a C 1 function in R satisfying (G)

g −1 (] − ∞, 0[) =]γ− , γ+ [ where − 1 < γ− < γ+ < 1 and for some u & s < 1 we have |G(u)| ≤ s 8f (u), ∀u ∈ R, where G(u) := g(s) ds. 0

Lemma 9. Under the condition (G), there is a constant k > 0 such that ∀u ∈ E +∞ 2 u F (u) ≥ k + f (u)] dx. [ 2 −∞ Proof. Take c ∈]k, 1[ and compute +∞ 1 [ (u2 + g(u)u2 ) + f (u)] dx ≥ 2 −∞ +∞ 1 G(u) 2 G(u)2 ) ) + (f (u) − [ ((1 − c2 )u2 + (cu − )] dx 2c 8c2 −∞ 2 where we have performed integration by parts to obtain +∞ +∞ G(u)u dx = g(u)u2 dx. − −∞

Hence by our assumption F (u) ≥

−∞

+∞

−∞

[

k (1 − c2 )u2 + (1 − ( )2 )f (u)] dx. 2 c

Theorem 10. Let (F1)-(F2)-(F3)-(G) hold. Then equation (2) has a heteroclinic connecting ±1 that minimizes F on E. Proof. The proof follows the same ideas as in theorem 7. The aim is to modify a minimizing sequence (un ) for F so that it is possible to extract converging subsequences. Step 1. (un ) is bounded in C 1 (R). This is a straightforward consequence of Lemmas 9 and 2. Step 2. The statements of Lemmas 4 and 5 are true for the functional F . While this is clear with respect to lemma 4 we must make some comment on how to replace the tails of a given function u = un for a new function related to u while the value of the functional F decreases. So let us consider for instance the right tail. Given that |u(x2 ) − 1| < U and |u (x2 )| < U consider the functional +∞ 1 F2 (v) = [ (v 2 + g(v)v 2 ) + f (v)] dx 2 x2

Heteroclinics for a class of fourth order conservative diﬀerential equations

213

having as domain D2 the set of functions v ∈ C 1 [x2 , +∞[) such that v ∈ L2 (]x2 , +∞[), v(x2 ) = u(x2 ) and v (x2 ) = u (x2 ). Using integration by parts as in the proof of Lemma 8 we see that, for any function v ∈ D2 +∞ 2 v + f (v)] dx. (8) F2 (v) ≥ −G(v(x2 ))v (x2 ) + k [ 2 x2 We shall also consider the subset C2 ⊂ D2 consisting of those functions v such that |v(x) − 1| ≤ a ∀x ≥ x2 . Without loss of generality we may assume that a is so small that g(u) > 0 ∀u ∈ [1 − a, 1 + a]. a3 Claim A. If v ∈ D2 \C2 and v is bounded then F2 (v) ≥ −G(v(x2 ))v (x2 )+k 8N . Here N is an upper bound for v ∞ . In fact, as we have seen in the proof of Lemma 5, we may choose an interval [x3 , x4 ] such that x3 ≥ x2 , v(x3 ) = a/2, v(x4 ) = a and a/2 ≤ v(x) ≤ a ∀x ∈ [x3 , x4 ]. Hence the result follows as before, by using the above inequality (8). Claim B. The minimum of F2 in D2 exists and, if U is suﬃciently small, the minimum is attained at a function z ∈ C2 . Clearly, F2 is bounded below in D2 . Estimates analogous to those obtained in Lemma 2 hold. Using inequality (8) it is easily seen that by extracting convergent subsequences from a minimizing sequence the minimum is obtained. Now let C2 (δ) = {v ∈ D2 : |v(x) − 1| ≤ δ ∀x ≥ x2 }. Let also β := sup{g(u) : u ∈ [1 − a, 1 + a]}. Given 0 < δ < a there exists U > 0 such that the solution z of the linear problem v iv − βv + 4α4 (v − 1) = 0, v(x2 ) = u(x2 ), v (x2 ) = u (x2 ), v(+∞) = 1 belongs to C2 (δ). On the other hand +∞ 1 min F∈ ≤ min F2 ≤ min [ (v 2 + βv 2 ) + 2α4 (v − 1)2 ] dx = C2 C2 (δ) v∈C2 (δ) x2 2 +∞ 1 [ (z 2 + βz 2 ) + 2α4 (z − 1)2 ] dx = O(U2 ) as U → 0. 2 x2 Combining this with Claim A, Claim B follows. Step 3. The minimization procedure completed. As in theorem 7, a minimizing sequence (un ) for F is replaced by a new minimizing sequence (vn ) ∈ V(R, U) (with a convenient choice of R and U). The estimates in step 1 allow us to extract a subsequence still denoted (vn ) that converges to some function u weakly (in the same sense as in theorem 7) and C 1 -uniformly on compact intervals. To see that v is a minimizer of F it suﬃces to note that, since g(vn ) > 0 on [R, +∞[, Fatou’s lemma is applicable and +∞ +∞ g(u(x))u (x)2 dx ≤ lim inf g(vn (x))vn (x)2 dx. R

n→∞

R

The same is true on ] − ∞, −R] and of course this implies that +∞ +∞ g(u(x))u (x)2 dx ≤ lim inf g(vn (x))vn (x)2 dx. −∞

n→∞

−∞

214

P. Habets, L. Sanchez, M. Tarallo and S. Terracini

Similar arguments can be used to deal with the double well potential. However, we should rephrase (F2’) as (F 2 ) f (u) = m

There exists m such that u2 ∀x ∈] − a, a[ and f (u) = 0 if and only if u = 0 or u = ±1. 2

The next two lemmas are similar to a part of the contents of section 4 in [3]. √ Lemma 11. Let |α| < 2 β. Then there exists ∆ > 0 such that any nontrivial solution of the diﬀerential equation uiv + αu + βu = 0 changes sign in every interval of length > ∆. Proof. After rescaling we can write any solution as u = Aet sin(t+φ)+Be−t sin(t+ ψ) where A, B, φ, ψ ∈ R. We can suppose that A = 0 and B = 0. Set E(t) = |A|et , F (t) = |B|e−t and let t¯ be such that F (t¯) = G(t¯). Since max{F (t), G(t)} > min{F (t), G(t)} for t = t¯ and the points where the graphs of Aet sin(t + φ) and Be−t sin(t + ψ) touch those of ±E and ±F have abcissae that difer from π, we see that it suﬃces to choose ∆ > 3π. Lemma 12. Let T > 1 and α, β be as in the preceeding lemma. Let z be a minimizer of T 1 [ (u2 + αu2 + βu2 ) dx GT (u) := 2 0 in the subspace Z(T, δ, η) of H 2 (0, T ) consisting of those functions such that u(0) = 0, u(T ) = δ, u (T ) = η. Then given U > 0 there exist δ0 > 0 and η0 > 0 such that for all T ≥ 1, if |δ| < δ0 and |η| < η0 then |z| ≤ U in [0, T ]. Proof. First note that indeed the minimum exists, since under our assumptions there are constants c > 0, d > 0, depending on α and β only, such that we have for all u ∈ Z(T, δ, η) (9) GT (u) ≥ cu2H 2 (0,T ) − dδη. It is easy to see that the value of the minimum is O(δ 2 + η 2 ). Now if there is a constant k > 0 such that, for all choices of |δ| and |η| the minimizer z satisﬁes zC[0,T ] > k then (using lemma 1, for instance) we obtain another constant k > 0 such that zH 2 (0,T ) > k . From (9) and what has been said above we obtain a contradiction and the lemma follows. Theorem 13. Assume that f is even and satisﬁes (F1),(F2’) and (F4). Assume in addition that g is a C 2 even function and satisﬁes (G). Then there exists an odd heteroclinic of (2) connecting ±1.

Heteroclinics for a class of fourth order conservative diﬀerential equations

215

Proof. As in the preceeding case we concentrate on the details that demand a diﬀerent argument with respect to the simpler case where g ≡ 0. The main idea is to modify a minimizing sequence for +∞ 1 [ [(u2 ) + g(u)u2 ] + f (u)] dx. F0 = 2 0 in E0 so that its modiﬁed elements go from 0 to a neighborhood of 1 in bounded time. To simplify our exposition, in way similar to what we have done about f , assume in addition (G )

∃g0 < 0 such that g(u) = g0 ∀u ∈] − a, a[.

Note that the compatibility condition (G) entails √ g0 < 2 m. So ﬁx 0 U whenever t ∈]t+ , t∗ [; it is clear that t∗ − t+ is bounded in terms of an upper bound of F0 taken over the sequence. Now if u has no zero in [t+ − 1, t+ ] there is T ∈ [t+ − 1, t+ ] where 0 < u(T ) = δ < U and 0 ≤ u (T ) = η < U. Claim: If U is suﬃciently small, the minimum of F0,T = 0

T

1 [ [(u2 ) + g(u)u2 ] + f (u)] dx. 2

in Z(T, δ, η) is attained in some function z with z∞ < a and so it is the minimum of GT . Proof of the Claim: as in lemma. . . we have F0,T (u) ≥ G(δ)η + kJ0,T (u), u ∈ Z(T, δ, η) T where we have set kJ0,T = 0 [ 12 [u2 + f (u)] dx and we argue using the last summand as in the proof of Claim B in theorem 10. Suppose that U has been ﬁxed according to the Claim. We replace the restriction of u to [0, T ] with the minimizer z of F0,T , so that with the new function   z(x) if 0 ≤ x ≤ T w(x) =  u(x) if x ≥ T we clearly have F0 (w) ≤ F0 (u). Lemma 11 implies that w(t− ) = 0 for some t− ≥ T − ∆. Integration by parts again yields 0

t−

1 [ [(w2 ) + g(w)w 2 ] + f (w)] dx ≥ 0 2

216

P. Habets, L. Sanchez, M. Tarallo and S. Terracini

so that we may discard the restriction of w to [0, t− ] and the function v(x) = w(x+ t− ) has the desired properties: F0 (v) ≤ F0 (u) and it enters the U-neighborhood of (1) at a time depending only of the upper bound of the sequence and the value of U.

References 1. Jan Bouwe van den Berg, The phase plane picture for a class of fourth order conservative diﬀerential equations, J. Diﬀ. Equations,161 (2000), 110-153. 2. G. Gomper and M. Schick, Phase transitions and critical phenomena, Academic Press 1994. 3. W. D. Kalies, J. Kwapisz and R. C. A. M. VanderVorst, Homotopy classes for stable connections between Hamiltonian saddle-focus equilibria, Comm. Math. Physics 193, (1998), 337-371. 4. H. Leit˜ ao, Estrutura e Termodinˆ amica de Misturas Ternárias com Anﬁfílico, Ph. D. Thesis, Universidade de Lisboa, 1998. 5. H. Leit˜ ao and M.M. Telo da Gama, Scaling of the interfacial tension of microemulsions: A Landau theory approach, J. of Chemical Physics 108, (1998), 4189-4198. 6. W. D. Kalies and R. C. A. M. VanderVorst, Multitransition homoclinic and heteroclinic solutions of the extended Fisher-Kolmogorov equation, J. Diﬀ. Equations 131, (1996), 209-228. 7. L. A. Peletier and W. C. Troy, A topological shooting method and the existence of kinks of the extended Fisher-Kolmogorov equation, Topological Methods in Nonlin. Analysis 6, (1995), 331-355. 8. D. Smets and J. B. van den Berg, Homoclinic solutions for Swift-Hohenberg and suspension bridges type equations, preprint.

Equadiﬀ 10, August 27–31, 2001 Prague, Czech Republic

Equadiﬀ 10 CD ROM Papers, pp. 217–219

About One Property of the Generalized Liénard Diﬀerential Equations Daniela Hricišáková Department of Mathematics, University of Trenčín, Trenčín, Slovakia Email: [email protected]

Abstract. A special property for establishing boundedness and periodicity of the oscillatory properties of the solutions of equation x = F (x, x ) is deﬁned and investigated. MSC 2000. 34C10 Keywords. Liénard diﬀerential equations, oscillatory properties

In this note I will consider a certain property, called property (B), introduced in the paper [1] for the equation x = F (x, x ),

(1)

where F : R → R is a continuous function which guarantees the existence and unicity of the solutions deﬁned by the Cauchy conditions. This property, which we will later explicitely describe, gives good conditions for establishing the oscillatory properties of the solutions, mainly the boundedness and periodicity of the solutions. We will substitute the equation (1) by the system 2

x = y, y = F (x, y).

(2)

We put g(x) = −F (x, 0) and we will assume that xg(x) > 0 for

x = 0.

(3)

This guarantees that only the origin will be a singular point of the system (2) and that the trajectories of our system go clockwise. Moreover, we will assume that the

218

D. Hricišáková

function F (x, y)/y is bounded for |x| < α, |y| > β, α > 0, β > 0. This assumption guarantees the nonexistence of the vertical asymptotes. If P = (x0 , y0 ) ∈ R2 , then γ + (P ) will denote positive and γ − (P ) the negative parts of the trajectory of the system (2) going through the point P . Definition We will say that the system (2) has the property (B) if there exists a point P = (x0 , y0 ), where y0 = 0 such that γ + (P ) crosses the x-axis but γ − (P ) does not. In the paper [1] the authors prove the following theorem. Theorem 1. The system (2) has the property (B) in the positive halfplane if and only if there exists a diﬀerentiable function Φ(x) and x ¯ > 0 such that Φ(x) > 0 for x<x ¯, Φ(¯ x) = 0 and F (x, Φ(x)) ≤ Φ (x)Φ(x)

for each x < x¯.

(4)

We will consider the Liénard diﬀerential equation x = −f (x)x − g(x).

(5)

The system (2) will have the form x = y,

(6)

y = −f (x) y − g(x).

Theorem 2. Let f (x) and g(x) be continuous functions on the interval (−∞, ∞) and let be xf (x) > 0, xg(x) > 0 for

x = 0.

(7)

Moreover, let be guaranteed the existence and uniqueness of the Cauchy problem x of the system (6). Let be F (x) = 0 f (s)ds and let be F (−∞) < +∞, F (+∞) > eF (−∞) , g(x) ≤ eF (−∞) [eF (−∞) − 1] for x ≤ 0. f (x) Then the system (6) has the property (B). Proof: Respecting Theorem 1 it is suﬃcient to prove that there exists a diﬀerentiable function Φ(x) and x¯ > 0 such that Φ(x) > 0

for x < x ¯, Φ(¯ x) = 0

219

About One Property of the Generalized Liénard Diﬀerential Equations

and F (x, Φ(x)) = −f (x)Φ(x) − g(x) ≤ Φ (x)Φ(x)

for x < x ¯.

(8)

Let be x ¯ > 0 such that e−F (−∞) F (¯ x) > 0.

(9)

Put Φ1 (x) = F (¯ x) − F (x) for 0 ≤ x ≤ x¯. Then Φ1 (x) = −f (x) and from (9) we get −f (x)[F (¯ x) − F (x)] − g(x) − [F (¯ x) − F (x)](−f (x)) = −g(x) ≤ 0 for

0 ≤ x ≤ x¯.

For x ≤ 0 put Φ2 (x) = e−F (x) F (¯ x). Then x)(−f (x)). Φ2 (x) = e−F (x) F (¯ x) > 0 and Φ1 (0) = 0 = Φ2 (0). Put It is easy to see that Φ1 (0) = Φ2 (0) = F (¯ ¯ and Φ(x) = Φ2 (x) for x < 0. Setting Φ(x) = Φ2 (x) we Φ(x) = Φ1 (x) for 0 ≤ x ≤ x get − f (x)e−F (x) F (¯ x) − g(x) − e−F (x) F (¯ x)e−F (x) F (¯ x)(−f (x)) = = −f (x)e−F (x) F (¯ x)[1 − e−F (x) F (¯ x)] − g(x) ≤ ≤ −f (x)e−F (x) F (¯ x)[1 − e−F (x) F (¯ x)] − eF (−∞) [eF (−∞) − 1]f (x) = = f (x){−e−F (x) F (¯ x)[1 − e−F (x) F (¯ x)] − eF (−∞) [eF (−∞) − 1]}. (10) Using (9) we get e−F (x) F (¯ x)[e−F (x) F (¯ x) − 1] ≥ F (x)[F (x) − 1] ≥ eF (−∞) [eF (−∞) − 1]. Thus the expression in the composed brackets in (10) is nonnegative. From this it follows that Φ(x) = Φ2 (x) = e−F (x) F (¯ x) fulﬁlls (8) for x ≤ 0. This ﬁnishes the proof.

References 1. Bucci F., Villari G. Phase portrait of the system x = y, y = F (x, y), Bolletino U.M.I., 1990.

Equadiﬀ 10, August 27–31, 2001 Prague, Czech Republic

Equadiﬀ 10 CD ROM Papers, pp. 221–266

Inertial Manifolds for Nonautonomous Dynamical Systems and for Nonautonomous Evolution Equations Norbert Koksch1 and Stefan Siegmund2 1

Dep. of Mathematics, Technical University Dresden, 01062 Dresden, Germany Email: [email protected] 2 Georgia Institute of Technology, CDSNS, Atlanta, GA 30332-190, USA Email: [email protected]

Abstract. In this paper we extend an abstract approach to inertial manifolds for nonautonomous dynamical systems. Our result on the existence of inertial manifolds requires only two geometrical assumptions, called cone invariance and squeezing property, and some additional technical assumptions like boundedness or smoothing properties. In the second part of the paper we consider special nonautonomous dynamical systems, namely two-parameter semi-ﬂows. As an application of our abstract approach and for reason of comparison with known results we verify the assumptions for semilinear nonautonomous evolution equations whose linear part satisﬁes an exponential dichotomy condition and whose nonlinear part is globally bounded and globally Lipschitz. Moreover, we apply our result on parabolic evolution equation with constant selfadjoint part. So we show that our abstract approach allows to obtain the sharp conditions in the autonomous case but they are applicable for the nonautonomous case, too.

MSC 2000. 34C30, 34D45, 34G20, 35B42, 37L25

Keywords. Inertial manifolds, pullback attractor, nonautonomous dynamical systems, two-parameter semiﬂow, squeezing property, cone invariance property, spectral gap condition, exponential dichotomy condition, semilinear parabolic equation

This is the ﬁnal form of the paper.

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Introduction

Let us consider a dissipative nonlinear evolution equation of the form u˙ + Au = f (u) in a Banach space X , where A is a linear sectorial operator with compact resolvent and f is a nonlinear function. Such an evolution equation may be an ordinary diﬀerential equation (X = Rn ) or the abstract formulation of a semilinear parabolic diﬀerential equation with X as a suitable function space over the spatial domain. In the last case, A corresponds to a linear diﬀerential operator and f is a nonlinearity which may involve derivatives of lower order than A. Inertial manifolds are positively invariant, exponentially attracting, ﬁnite dimensional Lipschitz manifolds. The notion goes back to D. Henry and X. Mora [Hen81], [Mor83] and were ﬁrst introduced and studied by P. Constantin, C. Foias, B. Nicoalenko, G.R. Sell and R. Temam [FST85], [FNST85], [CFNT86] for selfadjoint A. For the construction of inertial manifolds with A being non-selfadjoint see for example [SY92] and [Tem97]. Inertial manifolds are generalizations of centerunstable manifolds and they are more convenient objects which capture the longtime behavior of dynamical systems. If such a manifold exists, then it contains the global attractor A. Usually an inertial manifold M is seeked as the graph of a suﬃciently smooth function m on P X , i.e. M = graph(m) := {x + m(x) : x ∈ P X } , where P is a ﬁnite dimensional projector. The ﬁnite dimensionality and the exponential attracting property permit the reduction of the dynamics of the inﬁnite or high dimensional equation to the dynamics of a ﬁnite or low dimensional ordinary diﬀerential equation x˙ + Ax = P f (x + m(x))

in P X

called inertial form system. A stronger reduction property is the asymptotical completeness property [Rob96]: Each trajectory of the evolution equation tends exponentially to a trajectory in the inertial manifold. There are a few ways of constructing an inertial manifold. Most of them are generalization of methods developed for the construction of unstable, center-unstable or center manifolds for ordinary diﬀerential equations. The above mentioned notion of inertial manifolds is translated and extended to more general classes of diﬀerential equations like nonautonomous diﬀerential equations, [GV97], [WF97], [LL99], retarded parabolic diﬀerential equations, [TY94], [BdMCR98], or diﬀerential equations with random or stochastic perturbations, [Chu95], [BF95], [CL99], [CS01], [DLS01]. The construction of inertial manifolds often is redone for diﬀerent classes of equations. Our aim is to separate the general structure of the construction from the technical estimates which vary from example to example. So in [KS01] we

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developed an existence result of inertial manifolds on the abstract level of nonautonomous dynamical systems and we applied it to explicit nonautonomous evolution equations under various assumptions. The main assumptions on the nonautonomous dynamical system are generalizations of the cone invariance property and the squeezing property as geometrical assumptions on the nonautonomous dynamical system. For the proof of the existence result we need some additional technical assumptions on the nonautonomous dynamical system which we called boundedness property and coercivity property. For nonautonomous semilinear evolution equations whose linear part satisﬁes an exponential dichotomy condition and these properties follows from the global boundedness of the nonlinear part and its global Lipschitz property. Whereas the global Lipschitz property is a standard assumption and which is also used to verify the cone invariance and squeezing property, in some approaches to inertial manifolds the boundedness assumption is removed or at least replaced by weaker assumptions, for example by the requirement that there is a stationary solution. For reason of completeness we repeat the essential results of [KS01]. In addition to [KS01], we will introduce another group of technical assumptions which can be veriﬁed in the case of evolution equations without boundedness assumption on the nonlinear part but by assuming of an special stationarity property and a quantiﬁed coercivity property. Moreover we give a slight extension to the case the nonautonomous dynamical system acts on a Banach space X whereas the cone invariance and squeezing property are required only with respect to the weaker norm of a larger space Y. This includes the situation of parabolic evolution equations where the smoothing action of these problems allows to use weaker assumptions on the dynamical system.

2 2.1

Nonautonomous Dynamical Systems Preliminaries

Let (X , · X ) be a Banach space. Definition 1 (Nonautonomous Dynamical System (NDS)). A nonautonomous dynamical system (NDS) on X is a cocycle ϕ over a driving system θ on a set B, i.e. (i) θ : R × B → B is a dynamical system, i.e. the family θ(t, ·) = θ(t) : B → B of self-mappings of B satisﬁes the group property θ(0) = idB ,

θ(t + s) = θ(t) ◦ θ(s)

for all t, s ∈ R. (ii) ϕ : R≥0 × B × X → X is a cocycle, i.e. the family ϕ(t, b, ·) = ϕ(t, b) : X → X of self-mappings of X satisﬁes the cocycle property ϕ(0, b) = idX ,

ϕ(t + s, b) = ϕ(t, θ(s)b) ◦ ϕ(s, b)

for all t, s ≥ 0 and b ∈ B. Moreover (t, x) -→ ϕ(t, b, x) is continuous.

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Remark 2. (i) The set B is called base and in applications it has additional structure, e.g. it is a probability space, a topological space or a compact group and the driving system has additional regularity, e.g. it is ergodic or continuous. (ii) The pair of mappings (θ, ϕ) : R≥0 × B × X → B × X ,

(t, b, x) -→ (θ(t, b), ϕ(t, b, x))

is a special semi-dynamical system a so-called skew product ﬂow (usually one requires additionally that (θ, ϕ) is continuous). If B = {b} consists of one point then the cocycle ϕ is a semi-dynamical system. (iii) We use the abbreviations θt b or θ(t)b for θ(t, b) and ϕ(t, b)x for ϕ(t, b, x). We also say that ϕ is an NDS to abbreviate the situation of Deﬁnition 1. Definition 3 (Nonautonomous Set). A family M = (M(b))b∈B of non-empty sets M(b) ⊂ X is called a nonautonomous set and M(b) is called the b-ﬁber of M or the ﬁber of M over b. We say that M is closed, open, bounded, or compact, if every ﬁber has the corresponding property. For notational convenience we use the identiﬁcation M C {(b, x) : b ∈ B, x ∈ M(b)} ⊂ B × X . Definition 4 (Invariance of Nonautonomous Set). A nonautonomous set M is called forward invariant under the NDS ϕ, if ϕ(t, b)M(b) ⊂ M(θt b) for t ≥ 0 and b ∈ B. It is called invariant, if ϕ(t, b)M(b) = M(θt b) for t ≥ 0 and b ∈ B. Definition 5 (Inertial Manifold). Let ϕ be an NDS. Then a nonautonomous set M is called (nonautonomous) inertial manifold if (i) every ﬁber M(b) is a ﬁnite-dimensional Lipschitz manifold in X of dimension N for an N ∈ N; (ii) M is invariant ; (iii) M is exponentially attracting, i.e. there exists a positive constant η such that for every b ∈ B and x ∈ M(b) there exists an x ∈ M(b) with ϕ(t, b)x − ϕ(t, b)x X ≤ Ke−ηt

for t ≥ 0 and b ∈ B

and a constant K = K(b, x, x ) > 0. The property (iii) is also called exponential tracking property or asymptotic completeness property and x or ϕ(·, b)x is said to be the asymptotic phase of x or ϕ(·, b)x, respectively. Recall that if D and A are nonempty closed sets in X , the Hausdorﬀ semi-metric d(D|A) is deﬁned by d(D|A) := sup d(x, A) , x∈D

d(x, A) := inf d(x, y) = inf x − y . y∈A

y∈A

The appropriate generalization of convergence to a nonautonomous set A is the pullback convergence deﬁned by d(ϕ(t, θ−t b)x, A(b)) → 0

for t → ∞ ,

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which was introduced in the mid 1990s in the context of random dynamical systems (see Crauel and Flandoli [CF94], Flandoli and Schmalfuss [FS96], and Schmalfuss [Sch92]) and has been used e.g. in numerical dynamics. Note that a similar idea had already been used in the 1960s by Mark Krasnoselski [Kra68] to establish the existence of solutions that exist and remain bounded on the entire time set. Now we deﬁne a handy notion (see Ludwig Arnold [Arn98, Deﬁnition 4.1.1(ii)]) excluding exponential growth of a function. Definition 6 (Temperedness). A function R : B → ]0, ∞[ is called tempered from above if for every b ∈ B lim sup t→±∞

1 log R(θt b) = 0 . |t|

Note that the following characterization holds. Corollary 7. Suppose that R : B → ]0, ∞[ is a nonautonomous variable. Then the following statements are equivalent: (i) R is tempered from above. (ii) For every ε > 0 and b ∈ B there exists a T > 0 such that R(θt b) ≤ eε|t|

for |t| ≥ T .

Definition 8 (Nonautonomous Projector). A family π = (π(b))b∈B of projectors π(b) ∈ L(X , X ) in X is called nonautonomous projector. (i) π is called tempered from above if b -→ π(b)L(X ,X ) is tempered from above. (ii) π is called N -dimensional for an N ∈ N if dim imπ(b) = N for every b ∈ B. 2.2

Inertial Manifolds for Nonautonomous Dynamical Systems

Now let (X , Y) be a pair of two Banach spaces such that X is continuously embedded in Y, X ]→ Y . Our goal is to construct an inertial manifold in X . For this we will use some assumptions with respect to the norm of X . In order to be more general, we will allow that some assumptions are required only with respect to the weaker norm of the larger space Y. To compense the diﬀerent quality of the norms we need some smoothing action of the dynamical system. Note that in many cases one can use X = Y, and for a ﬁrst reading it is good to assume X = Y and to overread the technical diﬃculties dealing with the case X = Y. Let π1 be an N -dimensional nonautonomous projector in Y. We deﬁne the complementary projector π2 (b) := idY − π1 (b)

for b ∈ B .

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Then X1 (b) := π1 (b)X

and X2 (b) := π2 (b)X ,

b∈B

deﬁne nonautonomous sets Xi consisting of complementary linear subspaces Xi (b) of X , i.e. X1 (b) ⊕ X2 (b) = X . For this fact we also write X1 ⊕ X2 = B × X . Further let Y1 (b) := π1 (b)Y

and Y2 (b) := π2 (b)Y ,

b∈B.

We assume that X1 (b) = Y1 (b)

b∈B.

We say that π1 is tempered above in X if the restriction (π1 (b)X )b∈B of π1 onto X is tempered above. We want to construct a nonautonomous inertial manifold M = (M(b))b∈B consisting of manifolds M(b) which are trivial in the sense that each of them can be described by a single chart, i.e. M(b) = graph(m(b, ·)) := {x1 + m(b, x1 ) : x1 ∈ X1 (b)} with m(b, ·) = m(b) : X1 (b) → X2 (b). For a positive constant L we introduce the nonautonomous set CL := {(b, x) ∈ B × X : π2 (b)xY ≤ Lπ1 (b)xY } . Since the ﬁbers CL (b) are cones it is called (nonautonomous) cone. The following deﬁnition is a slight generalization of that one in [KS01]. Definition 9 (Cone Invariance). The NDS ϕ satisﬁes the (nonautonomous) ˜ : ]0, ∞[ → ]0, ∞[ cone invariance property for a cone CL if there are a function L and a number T0 ≥ 0 such that ˜ ≤L L(t)

for t ≥ T0

and such that for b ∈ B and x, y ∈ X , x − y ∈ CL (b) implies ϕ(t, b)x − ϕ(t, b)y ∈ CL(t) ˜ (θt b)

for t > 0 .

Now we deﬁne a property of a cocycle ϕ which describes a kind of squeezing outside a given cone.

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Definition 10 (Squeezing Property). The NDS ϕ satisﬁes the (nonautonomous) squeezing property for a cone CL if there exist positive constants K1 , K2 and η such that for every b ∈ B, x, y ∈ X and T > 0 the identity π1 (θT b)ϕ(T, b)x = π1 (θT b)ϕ(T, b)y implies for all x ∈ X with π1 (b)x = π1 (b)x and x − y ∈ CL (b) the estimates πi (θt b)[ϕ(t, b)x − ϕ(t, b)y]Y ≤ Ki e−ηt π2 (b)[x − x ]Y ,

i = 1, 2 ,

for t ∈ [0, T ]. Remark 11. The cone invariance and squeezing property are generalization and modiﬁcations of the notion of cone invariance and squeezing property for evolution equations. A combination of both properties is sometimes called strong squeezing property, and it was ﬁrst introduced for the Kuramoto-Sivashinsky equations in [FNST85], [FNST88]. An abstract version of it was developed in [FST89], another formulation of it can be found for example in [Tem88], [FST88], [CFNT89], [Rob93], [JT96]. Essentially, a strong squeezing property states that if the diﬀerence of two solutions of the evolution equation belongs to a special cone then it remains in the cone for all further times (that is the cone invariance property); otherwise the distance between the solutions decays exponentially (that is the squeezing property). Definition 12 (Boundedness Property). The NDS ϕ satisﬁes the (nonautonomous) boundedness property if for all t ≥ 0, b ∈ B and all M1 ≥ 0 there exists a M2 ≥ 0 such that for x ∈ X with π2 (b)xX ≤ M1 the estimate π2 (θt b)ϕ(t, b)xX ≤ M2 holds. Definition 13 (Coercivity Property). The NDS ϕ satisﬁes the (nonautonomous) coercivity property if for all t ≥ 0, b ∈ B and all M3 ≥ 0 there exists an M4 ≥ 0 such that for x ∈ X with π1 (b)xX ≥ M4 the estimate π1 (θt b)ϕ(t, b)xX ≥ M3 holds. With the boundedness property we will ensure that the graph transformation mapping can be deﬁned on a complete metric space of bounded functions. The coercivity property will ensure the existence of global homeomorphisms used for the deﬁnition of the graph transformation mapping. Remark 14. As we will show later in Sec. 3.2, for evolution equations the coercivity and boundedness property of ϕ follows from the boundedness of the nonlinearity and exponential dichotomy properties of the linear part. While a global Lipschitz property of the nonlinearity is used for the cone invariance and squeezing property, too, the boundedness of the nonlinearity is an additional restriction.

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Therefore, we introduce now another group of technical assumptions which for evolution equations can be veriﬁed without boundedness assumption on the nonlinear part. Definition 15 (Stationarity Property). The NDS ϕ satisﬁes the (nonautonomous) stationarity property if there is a uniformly bounded invariant set I. The stationarity property together with the cone invariance property will allow to deﬁne the graph transformation mapping in a space of linearly bounded functions. Definition 16 (Strong Coercivity Property). The NDS ϕ satisﬁes the (nonautonomous) strong coercivity property with respect to invariant set I and the cone CL if for all b ∈ B there exist positive numbers M5 , M6 , M7 such that for x ∈ I(b) + CL (b) and all t ≥ 0 the estimate π1 (b)xX ≤ M5 eM6 t (M7 + π1 (θt b)ϕ(t, b)xX ) holds. The strong coercivity property will ensure the existence of global homeomorphisms used for the deﬁnition of the graph transformation mapping and it will be used to show the contractivity of the graph transformation mapping Remark 17. As we will show later in Sec. 3.2, for evolution equations the strong coercivity property of ϕ follows from the uniform boundedness of an invariant set I and exponential dichotomy properties of the linear part. If X = Y we need some properties to compense the weaker norm. Definition 18 (Smoothing Property). The NDS ϕ satisﬁes the smoothing property if there are function M8 , M9 : ]0, ∞[ → ]0, ∞[ such that for x, y ∈ X , b ∈ B, and t > 0 the Lipschitz estimates ϕ(t, b)x − ϕ(t, b)yX ≤ M8 (t)x − yY and π1 (b)[x − y]Y ≤ M9 (t)π1 (θt b)[ϕ(t, b)x − ϕ(t, b)y]Y

if x − y ∈ CL

hold. Remark 19. For parabolic evolution equations these smoothing property is a consequence of global Lipschitz property of the nonlinearity and the smoothing property of parabolic equations. Theorem 20 (Existence of Inertial Manifold). Let ϕ be an NDS on a Banach space X ]→ Y over a driving system θ : R × B → B on a set B and assume that ϕ satisﬁes the cone invariance and squeezing property.

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Moreover let ϕ satisfy the following technical assumptions: • ϕ possesses the coercivity and boundedness property or • ϕ possesses the strong coercivity and stationarity property with respect to the invariant set I and the cone CL with a constant M6 < η. If X = Y, we further assume that • ϕ possesses the smoothing property, and that π1 is tempered from above in X , and that there are constants M10 and M11 with π2 (b)L(X ,X ) ≤ M10 ,

π1 (b)xY ≤ M11 π1 (b)xX

for x ∈ X , b ∈ B .

Then there exists an inertial manifold M = (M(b))b∈B of ϕ with the following properties: (i) M(b) is a graph in X1 (b) ⊕ X2 (b), M(b) = {x1 + m(b, x1 ) : x1 ∈ X1 (b)} with a mapping m(b, ·) = m(b) : X1 (b) → X2 (b) which is globally Lipschitz continuous ˆ 1 − y1 X m(b, x1 ) − m(b, y1 )X ≤ Lx ˆ ≥ 0, and it satisﬁes with some L m(b, x1 ) − m(b, y1 )Y ≤ Lx1 − y1 Y with L from the cone invariance property. (ii) M is exponentially attracting in X ˆ −ηt π2 (b)x − m(b, π1 (b)x)X ϕ(t, b)x − ϕ(t, b)x X ≤ Ke for t ≥ 1, i = 1, 2 with an asymptotic phase x = x (b, x) ∈ M(b) of x and some ˆ independent of x, x , b, t, and we have K πi (θt b)[ϕ(t, b)x − ϕ(t, b)x ]Y ≤ Ki e−ηt π2 (b)x − m(b, π1 (b)x)Y for t ≥ 0, i = 1, 2 with K1 , K2 > 0 from the squeezing property. (iii) If in addition π1 is tempered from above in X , then M is pullback attracting in X , more precisely, there is a T ≥ 0 such that for each bounded set D⊂X d(ϕ(t, θ−t b)D|M(b)) ≤ e−ηt/2 d(D|M(θ−t b))

for t ≥ T .

(1)

Proof. We divide the proof into two parts. In the ﬁrst part we assume that ϕ possesses the boundedness and coercivity property. Since the proof is rather involved we split this part into four steps. In the ﬁrst step we deﬁne the graph transformation mapping. In the second step we show that it has a unique ﬁxed point m(b) which gives rise to a nonautonomous invariant set M of Lipschitz manifolds M(b). In the third step the exponential tracking property is proved and in the fourth step we investigate the pullback attractivity of M.

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In the second part we assume that ϕ satisﬁes the strong coercivity and stationarity property. There we will repeat only these parts of the proof which are changing. Part I: Let ϕ satisfy the boundedness and coercivity property. Step 1: Definition of graph transformation mapping We construct the manifolds M(b) = graph(m(b)) as the ﬁxed point of the cocycle ϕ acting on a certain class of functions g with g(b, ·) = g(b) : X1 (b) → X2 (b) ,

b∈B.

The set G of mappings of the form X1 . (b, x1 ) -→ (b, g(b, x1 )) ∈ X2 , such that g is bounded and g(b, ·) is continuous for every b ∈ B is a Banach space with the norm gG = sup g(b, x1 )X . (b,x1 )∈X1

Moreover let GL denote the subset of G containing all mappings which satisfy the global Lipschitz condition g(b, x1 ) − g(b, y1 )Y ≤ Lx1 − y1 Y for (b, x1 ), (b, y1 ) ∈ X1 with L from the cone invariance property. Note that both G and GL ⊂ G are complete metric spaces. Let T > 0 be arbitrary, but ﬁxed. We wish to deﬁne the graph transformation mapping GT : GL → G such that graph((GT g)(θT b, ·)) = ϕ(T, b)graph(g(b, ·)) and this means that x ˜ ∈ graph((GT g)(θT b, ·)) equals ϕ(T, b)x for some x ∈ graph(g(b, ·)). More precisely, for an x ˜1 ∈ X1 (θT b) we want to deﬁne (GT g)(θT b, x ˜1 ) = x ˜2 ∈ X2 (θT b) by determining an x ∈ graph(g(b, ·)) such that π1 (θT b)ϕ(T, b)x = x ˜1

and π2 (θT b)ϕ(T, b)x =: (GT g)(θT b, x ˜1 ) .

To this end we show that for each T > 0, b ∈ B, x ˜1 ∈ X1 (θT b), g ∈ GL , the boundary value problem x ∈ graph(g(b, ·)) ,

π1 (θT b)ϕ(T, b)x = x ˜1

has a unique solution x = β(T, b, x ˜1 , g). Uniqueness of a solution of (2). Assume there exist x and y with x, y ∈ graph(g(b, ·)) ,

π1 (θT b)ϕ(T, b)x = π1 (θT b)ϕ(T, b)y = x ˜1 .

(2)

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We get x − y ∈ CL (b) and the squeezing property (with x = x) implies the identity for t ∈ [0, T ] .

ϕ(t, b)x = ϕ(t, b)y

Hence x = y and there is at most one solution x = β(T, b, x ˜1 , g) of (2). Existence of a solution of (2). Note that by assumption X1 (b) = Y1 (b) for b ∈ B. Let T > 0, b ∈ B, g ∈ GL be ﬁxed and deﬁne H : X1 (b) → X1 (θT b) by H(x1 ) := π1 (θT b)ϕ(T, b)(x1 + g(b, x1 )) . By the continuity of ϕ(T, b) and g(b, ·), H is continuous. For x˜1 ∈ HX1 (b) ⊂ X1 (θT b), any x1 in the preimage H −1 (˜ x1 ) = {x1 ∈ X1 (b) : H(x1 ) = x ˜1 } gives rise to a solution x = x1 + g(b, x1 ) of the boundary value problem (2). As we have already seen, there exists at most one solution denoted by β(T, b, x ˜1 , g) and therefore the inverse H −1 of H is given by x1 ) = π1 (b)β(T, b, x ˜1 , g) H −1 (˜

on HX1 (θT b) .

Now we show the continuity of H −1 : HX1 (b) → X1 (b). Suppose, there is a ξ˜ ∈ ˜ Then there are ε > 0 HX1 (b) ⊂ X1 (θT b) such that H −1 is not continuous at ξ. and a sequence (ξ˜k )k∈N in X1 (θT b) such that ξ˜k → ξ˜0 as k → ∞ and ξk − ξ0 X ≥ ε

for all k ∈ N

(3)

where ξk := π1 (b)β(T, b, ξ˜k , g) for k = 0, 1, . . .. First, we suppose that there is a subsequence of (ξk )k∈N , denoted for shortness again by (ξk )k∈N , with ξk X → ∞ as k → ∞. Then the coercivity property of ϕ implies ξ˜k X = H(ξk )X = π1 (θT b)ϕ(T, b)(ξk + g(b, ξk ))X → ∞

as k → ∞ ,

but this contradicts ξ˜k → ξ˜0 . Therefore we have proved that (ξk )k∈N is bounded. Since X1 (b) is a ﬁnitedimensional space, there is a convergent subsequence, denoted for shortness again by (ξk )k∈N , with a limit ξ∞ = lim ξk ∈ X1 (b) . (4) k→∞

By the continuity of H, we have H(ξk ) → H(ξ∞ ). Since also H(ξk ) = ξ˜k → ξ˜0 = H(ξ0 ) we get ξ0 = ξ∞ in contradiction to (3) and (4). Therefore, H and H −1 are continuous. Now we show that H is onto, i.e. satisﬁes HX1 (b) = X1 (θT b) .

(5)

By the coercivity of ϕ, we have the norm coercivity H(ξ)X → ∞ for ξX → ∞ of H. Since X1 (b) is ﬁnite-dimensional, H is a sequentially compact mapping. By [Rhe69, Theorem 3.7], H is a homeomorphism from X1 (b) onto X1 (θT b) and hence we have (5).

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Thus we have unique solvability of (2), and we can deﬁne the graph transformation mapping GT by (GT g)(θT b, x ˜1 ) = π2 (θT b)ϕ(T, b)β(T, b, x ˜1 , g) for T > 0, b ∈ B, x ˜1 ∈ X1 (θT b), and g ∈ GL . Note that we have graph((GT g)(θT b, ·)) = ϕ(T, b)graph(g(b, ·)) . Since H −1 : X1 (θT b) → X1 (b), g(b, ·) : X1 (b) → X2 (b) and π1 (b)ϕ(T, b) : X → X are continuous, and x1 ) + g(b, H −1 (˜ x1 )) β(T, b, x ˜1 , g) = H −1 (˜ holds, the mapping (GT g)(θT b, ·) : X1 (θT b) → X2 (θT b) is also continuous. Now we show that GT gG < ∞ . Since g ∈ G there is a M1 with π2 (b)xX ≤ M1 for all b ∈ B and all x ∈ graph(g(b, ·)). By the boundedness property of ϕ there is a M2 such that π2 (θT b)ϕ(t, b)xX ≤ M2 for all b ∈ B and all x ∈ X with π2 (b)xX ≤ M1 . Let b ∈ B0 , x˜1 ∈ X1 (θT b) be arbitrary. Then β(T, b, x ˜1 , g) ∈ graph(g(b, ·)), hence π2 (b)β(T, b, x ˜1 , g)X ≤ M1 , and therefore (GT g)(θT b, x ˜1 )X = π2 (θT b)ϕ(T, b)β(T, b, x ˜1 , g)X ≤ M2 proving that GT gG < ∞, i.e. GT g ∈ G. Let T > 0, b ∈ B, x ˜1 , x ˜2 ∈ X1 (θT b), g ∈ GL be arbitrary. Since β(T, b, x ˜1 , g), β(T, b, x ˜2 , g) ∈ graph(g(b, ·)), we get ˜2 , g) ∈ CL (b) β(T, b, x ˜1 , g) − β(T, b, x

(6)

and the cone invariance property implies for T ≥ T0 a Lipschitz estimate for G g, T

˜1 ) − (GT g)(θT b, x ˜2 )Y (GT g)(θT b, x ≤ Lπ1 (θT b)(ϕ(T, b)β(T, b, x ˜1 , g) − ϕ(T, b)β(T, b, x ˜2 , g))Y = L˜ x1 − x ˜2 Y , i.e. (GT g)(θT b, ·) satisﬁes a Lipschitz condition as mapping from Y1 (b) into Y with Lipschitz constant L. Thus GT maps GL into G for every T ≥ 0, and it is self-mapping for T ≥ T0 . Moreover, using the smoothing property, for T > 0 we obtain (GT g)(θT b, x ˜1 ) − (GT g)(θT b, x ˜2 )X ≤ π2 (θT b)L(X ,X )ϕ(T, b)β(T, b, x ˜1 , g) − ϕ(T, b)β(T, b, x ˜2 , g)X ≤ M10 M8 (T )β(T, b, x ˜1 , g) − β(T, b, x ˜2 , g)Y ≤ (1 + L)M8 (T )M10 π1 (b)[β(T, b, x ˜1 , g) − β(T, b, x ˜2 , g)]Y ≤ (1 + L)M8 (T )M9 (T )M10 ˜ x1 − x ˜2 Y ˆ ≤ L˜ x1 − x˜2 X

Inertial Manifolds for Nonautonomous Dynamical Systems

233

where ˆ := (1 + L)M8 (T )M9 (T )M10 M11 . L Step 2: Unique fixed-point of graph transformation mappings Let T ≥ 0, b ∈ B, x ˜1 ∈ X1 (θT b), g, h ∈ G, and x ∈ graph(g(b, ·)), y ∈ graph(h(b, ·)) with π1 (θT b)ϕ(T, b)x = π1 (θT b)ϕ(T, b)y = x˜1 . Deﬁne

x := π1 (b)x + h(b, π1 (b)x) .

Then x − y ∈ CL , and the squeezing property implies πi (θt b)[ϕ(t, b)x − ϕ(t, b)y]Y ≤ K2 e−ηt g(b, π1 (b)x) − h(b, π1 (b)x)Y for t ∈ [0, T ]. If X = Y we obtain (GT g)(θT b, x ˜1 ) − (GT h)(θT b, x ˜1 )X ≤ K2 e−ηT g(b, π1 (b)β(T, b, x ˜1 , g)) − h(b, π1 (b)β(T, b, x ˜1 , g))X , and passing to the sup over all (θT b, x ˜1 ) ∈ X1 we get GT g − GT hG ≤ κ(T )g − hG

(7)

for all T > 0, g, h ∈ GL , where κ(T ) := K2 e−ηT . If X = Y we proceed as follows: Because of ˜i ) = π2 (θT b)ϕ(1, θT −1 b)ϕ(T − 1, b)β(T, b, x ˜i , g) (GT g)(θT b, x and using the smoothing property and the continuous embedding of X in Y we ﬁnd (GT g)(θT b, x ˜i ) − (GT g)(θT b, x ˜i )X ≤ M10 M8 (1)ϕ(T − 1, b)β(T, b, x ˜1 , g) − ϕ(T − 1, b)β(T, b, x ˜2 , g)Y ≤ κ(T )g(b, π1 (b)β(T, b, x ˜1 , g)) − h(b, π1 (b)β(T, b, x ˜1 , g))X and hence (7) with κ(T ) = (K1 + K2 )M8 (1)M10 Ce−η(T −1) for T > 1, where C is an embedding constant for the embedding from X into Y. Since η > 0 and since π2 is tempered from above in X , there is a positive T1 ≥ T0 with κ(T ) < 1 for T ≥ T1 . Thus, for T ≥ T1 , GT is a contractive selfmapping on the complete metric space GL . Now choose and ﬁx an arbitrary T˜ ≥ T1

234

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and let m denote the unique ﬁxed-point of GT in GL . We show that m is the unique ﬁxed-point of GT for every T ≥ 0. For every T ≥ 0 the mapping GT m ∈ G is uniquely determined by the graphs graph((GT m)(θT b, ·)) = ϕ(T, b)graph(m(b, ·)) ,

b∈B.

For T ≥ 0 and b ∈ B we have the identity ˜

graph((GT +T m)(θT +T˜ b, ·)) = ϕ(T + T˜, b)graph(m(b, ·)) = ϕ(T, θ ˜ b)ϕ(T˜, b)graph(m(b, ·)) T

= ϕ(T, θT˜ b)graph(m(θT˜ b, ·)) = graph((GT m)(θT +T˜ b, ·))

˜

and therefore GT m = GT +T m ∈ GL . Hence the composition GT GT m makes sense for T, T ≥ 0 and we get

graph((GT GT m)(θT +T b, ·)) = ϕ(T, θT b)graph(GT m(θT b, ·)) = ϕ(T, θT b)ϕ(T , b)graph(m(b, ·)) = ϕ(T + T , b)graph(m(b, ·))

= graph((GT +T m)(θT +T b, ·))

and therefore GT GT m = GT GT m = GT +T m for T, T ≥ 0. We get ˜

˜

GT (GT m) = GT (GT m) = GT m . ˜

Thus GT m equals the unique ﬁxed-point m of GT and we have GT m = m

for T ≥ 0 .

To prove the uniqueness of the ﬁxed-point m of GT , assume that m∗ is another ﬁxed-point. But then m and m∗ are both ﬁxed-points of GkT for every k ∈ N. Choosing k large enough such that kT ≥ T1 we know that GkT has a unique ﬁxed-point and this implies m = m∗ . Thus m is the unique mapping in GL with the invariance property ϕ(t, b)graph(m(b, ·)) = graph(m(θt b, ·))

for t ≥ 0 and b ∈ B .

We deﬁne M(b) := graph(m(b, ·)) for b ∈ B. Step 3: Existence of asymptotic phases Let b ∈ B and x ∈ X be arbitrary and let (tk )k∈N be a monotonously increasing sequence of positive real numbers tk with tk → ∞ for k → ∞. Deﬁne y := π1 (b)x + m(b, π1 (b)x) ∈ graph(m(b, ·)) and xk := β(tk , b, π1 (θtk b)ϕ(tk , b)x, m) ∈ graph(m(b, ·)) .

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We get y − xk ∈ CL (b) and the squeezing property implies for i = 1, 2, t ∈ [0, tk ] πi (θt b)[ϕ(t, b)x − ϕ(t, b)xk ]Y ≤ Ki e−ηt π2 (b)x − m(b, π1 (b)x)Y . In particular, we ﬁnd for i = 1 and t = 0 π1 (b)xk Y ≤ π1 (b)xY + π1 (b)[x − xk ]Y ≤ π1 (b)xY + K1 π2 (b)x − m(b, π1 (b)x)Y . Therefore we have proved that (π1 (b)xk )k∈N ⊂ X1 (b) = Y1 (b) is bounded. Since X1 (b) is a ﬁnite-dimensional space, there is a convergent subsequence, denoted again by (π1 (b)xk )k∈N . Since xk = π1 (b)xk + m(b, π1 (b)xk ) and m(b, ·) is continuous, also (xk )k∈N is converging to some x ∈ graph(m(b, ·)) . Then we get for i = 1, 2 πi (θt b)[ϕ(t, b)x − ϕ(t, b)x ]Y ≤ πi (θt b)[ϕ(t, b)x − ϕ(t, b)xk ]Y + πi (θt b)[ϕ(t, b)xk − ϕ(t, b)x ]Y ≤ Ki e−ηt π2 (b)x − m(b, π1 (b)x)Y + πi (θt b)[ϕ(t, b)xk − ϕ(t, b)x ]Y for all T > 0, t ∈ [0, T ] and all k ∈ N>0 with tk ≥ T . By the continuity of ϕ(t, b), and because of xk → x , the second term can be made arbitrary small for each ﬁxed t ∈ [0, T ] choosing k large enough. Therefore, πi (θt b)[ϕ(t, b)x − ϕ(t, b)x ]Y ≤ Ki e−ηt π2 (b)x − m(b, π1 (b)x)Y for t ≥ 0, i.e. x ∈ M(b) is an asymptotic phase of x if X = Y. If X = Y, then we note that the smoothing property and the continuous ˆ with embedding of X in Y implies the existence of a constant K ˆ −ηt π2 (b)x − m(b, π1 (b)x)X ϕ(t, b)x − ϕ(t, b)x X ≤ Ke

for t ≥ 1 .

Step 4: Pullback attractivity Note that with π1 also the complementary projector π2 is tempered from above. Since ϕ(t, b)x ∈ M(θt b) for every x ∈ X , t ≥ 0, b ∈ B. Step 3 implies ˆ −ηt π2 (b)[x − x ]X d(ϕ(t, b)x, M(θt b)) ≤ Ke

for t ≥ 1

ˆ Let z ∈ M(b) be arbitrary. with x = π1 (b)x + m(b, π1 (b)x) and some constant K. Then, because of x ∈ M(b), the Lipschitz property of m, and π1 (b)x = π1 (b)x , ˆ 2 (b)[z − x ]X π2 (b)[x − x ]X ≤ π2 (b)[x − z]X + Lπ ˆ 1 (b)L(X ,X ) x − zX . = π2 (b)L(X ,X ) + Lπ

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ˆ 1 (b)L(X ,X ) d(x, M(b)) . π2 (b)[x − x ]X ≤ π2 (b)L(X ,X ) + Lπ

Replacing b by θ−t b, choosing a T > 1 such that ˜ · π2 (θ−t b)L(X ,X ) + Lπ1 (θ−t b)L(X ,X ) ≤ e η2 t K

for t ≥ T

(Corollary 7), we obtain η

η

d(ϕ(t, θ−t b)x, M(b)) ≤ e− 2 t d(x, M(θ−t b)) ≤ e− 2 t d(D, M(θ−t b)) for t ≥ T proving the pullback attractivity of M with T independent of D. Part II: Let ϕ satisfy the stationarity property and strong coercivity property with respect to the invariant set I and the cone CL with a constant M6 < η. The main diﬀerence in comparison to Part I concerns the utilization of another complete metric space (a space of linearly bounded functions instead of a space of bounded functions) and therefore an appropriate proof of contraction property of the graph transformation mapping. Let G be the set mappings of the form X1 . (b, x1 ) -→ (b, g(b, x1 )) ∈ X2 , such that x1 + g(b, x1 ) ∈ I(b) + CL (b) for every (b, x1 ) ∈ X1 and that g(b, ·) is linearly bounded uniformly in b. i.e., for g there are C1 , C2 ≥ 0 with g(b, x1 )X ≤ C0 + C1 x1 Y

for (b, x1 ) ∈ X1 = Y1 .

We equipped G with the norm gG =

sup (b,x1 )∈X1

g(b, x1 )X . M7 + x1 Y

As in the proof of Theorem 20, let GL denote the subset of G containing all mappings which satisfy the global Lipschitz condition g(b, x1 ) − g(b, y1 )Y ≤ Lx1 − y1 Y for (b, x1 ), (b, y1 ) ∈ X1 with L from the cone invariance property. Note that both G and GL ⊂ G are complete metric spaces. As in the proof of Theorem 20, the strong coercivity property and the squeezing property allow to deﬁne the graph transformation mapping GT : GL → G for T > 0 by ˜1 ) = π2 (θT b)ϕ(T, b)β(T, b, x ˜1 , g) (GT g)(θT b, x ˜1 , g) ∈ I(b) + CL (b) for T > 0, b ∈ B, x˜1 ∈ X1 (θT b), and g ∈ GL , where x = β(T, b, x is the unique solution of the boundary value problem x ∈ graph(g(b, ·)) ,

˜1 . π1 (θT b)ϕ(T, b)x = x

(8)

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237

The cone invariance property implies ˜ )˜ (GT g)(θT b, x ˜1 ) − (GT g)(θT b, x ˜2 )X ≤ L(T x1 − x ˜2 X for T ≥ T0 , b ∈ B, x˜1 , x ˜2 ∈ X1 (θT b), g ∈ GL . Especially, by the stationarity property for each x ∈ I(b) with π2 (b)x = g(b, π1 (b)x) we have x ˜2 + (GT g)(θT b, x ˜2 ) ∈ ˜2 = π1 (θT b)x. Therefore I(θT b) for x (GT g)(b, x1 ) − (GT g)(b, x2 )Y ≤ Lx1 − x2 Y and (GT g)(b, x1 ) ∈ I(b) + CL for T ≥ T0 , b ∈ B, x1 , x2 ∈ X1 (b), g ∈ GL . As in Step 1 of Part I of the proof follows that GT g is a Lipschitz mapping from X1 into X2 . Thus, GT maps GL into itself for T ≥ T0 . Remains to show the contractivity of GT with respect to the norm · G of the new space G. Proceeding as in Step 2 of Part I of the proof, the squeezing property implies π2 (θt b)[ϕ(t, b)x − ϕ(t, b)y]Y ≤ K2 e−ηt g(b, π1 (b)x) − h(b, π1 (b)x)Y for t ∈ [0, T ]. We restrict us to the more complicate case X = Y. Because of (GT g)(θT b, x ˜i ) = π2 (θT b)ϕ(1, θT −1 b)ϕ(T − 1, b)β(T, b, x ˜i , g) and using the smoothing property and the continuous embedding of X in Y we ﬁnd (GT g)(θT b, x ˜i ) − (GT g)(θT b, x ˜i )X ≤ M10 M8 (1)ϕ(T − 1, b)β(T, b, x ˜1 , g) − ϕ(T − 1, b)β(T, b, x ˜2 , g)Y ≤κ ˜ (T )g(b, π1 (b)β(T, b, x ˜1 , g)) − h(b, π1 (b)β(T, b, x ˜1 , g))X with

κ ˜ (T ) = (K1 + K2 )M8 (1)M10 Ce−η(T −1)

for T > 1 where C is an embedding constant for the embedding from X into Y. Thus (GT g)(θT b, x ˜1 ) − (GT h)(θT b, x ˜1 )X M7 + π1 (θT b)˜ x1 X g(b, π1 (b)β(T, b, x ˜1 , g)) − h(b, π1 (b)β(T, b, x ˜1 , g))X ≤κ ˜ (T )k5 M7 + π1 (b)β(T, b, x ˜1 , g)X with k5 :=

˜1 , g)X M7 + π1 (b)β(T, b, x . M7 + π1 (θT b)˜ x1 X

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Using the strong coercivity property we ﬁnd k5 ≤

x1 X ) M7 + M5 eM6 T (M7 + π1 (θT b)˜ ≤ 1 + M5 eM6 T M7 + π1 (θT b)˜ x1 X

and passing to the sup over all (θT b, x ˜1 ) ∈ X1 we get GT g − GT hG ≤ κ(T )g − hG for all T > 1, g, h ∈ GL , where κ(T ) := (1 + M5 eM6 T )˜ κ(T ) = (1 + M5 eM6 T )π2 (θT b)L(X ,X )M8 (1)C(K1 + K2 )e−η(T −1) . Since η > M6 and since π2 is tempered from above in X , there is a positive T1 ≥ T0 with κ(T ) < 1 for T ≥ T1 . Thus, for T ≥ T1 , GT is a contractive self-mapping on the complete metric space GL . The rest of the proof proceeds as in the ﬁrst part of the proof.

3 3.1

Nonautonomous Evolution Equations Two-Parameter Semi-Flow

Let (X , · X ) be a Banach space. The solutions of a nonautonomous evolution equation will not generate a semi-ﬂow but a two-parameter semi-ﬂow. Definition 21 (Two-parameter Semi-Flow). A two-parameter semi-ﬂow µ on X is a continuous mapping {(t, s, x) ∈ R × R × X : t ≥ s} . (t, s, x) -→ µ(t, s, x) ∈ X which satisﬁes (i) µ(s, s, ·) = idX for s ∈ R; (ii) the two-parameter semi-ﬂow property for t ≥ τ ≥ s, x ∈ X , i.e. µ(t, τ, µ(τ, s, x)) = µ(t, s, x) . The next lemma explains how a two-parameter semi-ﬂow deﬁnes an NDS. Lemma 22 (Two-parameter Semi-Flow defines NDS). Suppose that µ is a two-parameter semi-ﬂow. Then ϕ : R≥0 × B × X → X , ϕ(t, b)x = µ(t + b, b, x) is an NDS with base B = R and driving system θ : R × B → B, θ(t)b = t + b . Moreover, for t ≥ s and x ∈ X the relation µ(t, s, x) = ϕ(t − s, s)x holds.

(9)

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Proof. θ is a dynamical system. We have ϕ(0, b) = µ(b, b, ·) = idX . We use the two-parameter semi-ﬂow property of µ to obtain for t, s ≥ 0, b ∈ B ϕ(t + s, b) = µ(t + s + b, b, ·) = µ(t + s + b, s + b, µ(s + b, b, ·)) = ϕ(t, θs b) ◦ ϕ(s, b) , proving the cocycle property of ϕ. The continuity of µ implies the continuity of (t, x) -→ ϕ(t, b)x. Now substitute t by t − s and b by s in (9) to see that µ(t, s, x) = ϕ(t − s, s)x. Translating the deﬁnitions for nonautonomous dynamical systems to two-parameter semi-ﬂows we obtain the following properties: Condition 23 (Cone Invariance Property). There are L > 0 and T0 ≥ 0 such that for τ ∈ R and x, y ∈ X , x − y ∈ CL (τ ) := {ξ : π2 (τ )ξY ≤ Lπ1 (τ )ξY } implies µ(t, τ, x) − µ(t, τ, y) ∈ CL (t)

for t ≥ τ + T0 .

Condition 24 (Squeezing Property). There exist positive constants K1 , K2 and η such that for every τ ∈ R, x, y ∈ X and T > 0 the identity π1 (τ + T )µ(τ + T, τ, x) = π1 (τ + T )µ(τ + T, τ, y) implies for all x ∈ X with π1 (τ )x = π1 (τ )x and x − y ∈ CL (τ ) the estimates πi (t) [µ(t, τ, x) − µ(t, τ, y)] Y ≤ Ki e−η(t−τ ) π2 (τ ) [x − x ] Y ,

i = 1, 2,

for t ∈ [τ, τ + T ]. Condition 25 (Boundedness Property). For all t, τ ∈ R with t ≥ τ and all M1 ≥ 0 there exists a M2 ≥ 0 such that for x ∈ X with π2 (τ )xX ≤ M1 the estimate π2 (t)µ(t, τ, x)X ≤ M2 holds. Condition 26 (Coercivity Property). For all t, τ ∈ R with t ≥ τ and all M3 ≥ 0 there exists a M4 ≥ 0 such that for x ∈ X with π1 (τ )xX ≥ M4 the estimate π1 (t)µ(t, τ, x)X ≥ M3 holds.

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Condition 27 (Stationarity Property). There is a uniformly bounded invariant set I. Condition 28 (Strong Coercivity Property). For all τ ∈ R there exist positive numbers M5 , M6 , M7 such that for x ∈ I(τ ) + CL and all t ≥ τ the estimate π1 (τ )xY ≤ M5 eM6 (t−τ ) (M7 + π1 (t)µ(t, τ, x)Y ) holds. Condition 29 (Smoothing Property). There are functions M8 , M9 : ]0, ∞[ → ]0, ∞[ such that for x, y ∈ X , τ ∈ R, and t > τ the Lipschitz estimates µ(t, τ, x) − µ(t, τ, y)X ≤ M8 (t − τ )x − yY and π1 (τ )[x − y]X ≤ M9 (t − τ )π1 (t)[µ(t, τ, x) − µ(t, τ, y)]X

if x − y ∈ CL (10)

hold. Theorem 30 (Inertial Manifold for Two-parameter Semi-Flow). Suppose that µ is a two-parameter semi-ﬂow on X and let (πi (τ ))τ ∈R ⊂ L(X ), i = 1, 2, be two families of complementary projectors π1 (τ ) and π2 (τ ). Let µ satisfy the cone invariance and squeezing property. Moreover, let • the boundedness and coercivity property or • the stationarity and strong coercivity property with respect to the invariant set I and the cone CL with M6 < η be satisﬁed. If X = Y, we further assume that µ possesses the smoothing property, that π1 is tempered from above in X , and that there are constants M10 and M11 with π2 (τ )L(X ,X ) ≤ M10 ,

π1 (τ )xY ≤ M11 π1 (τ )xX

for x ∈ X , τ ∈ R .

Then there exists an inertial manifold M = (M(τ ))τ ∈R of µ with the following properties: (i) M(τ ) is a graph in π1 (τ )X ⊕ π2 (τ )X , M(τ ) = {x1 + m(τ, x1 ) : x1 ∈ π2 (τ )X } ⊂ I(τ ) + CL with a mapping m(τ, ·) = m(τ ) : π1 (τ )X → π2 (τ )X which is globally Lipschitz continuous ˆ 1 − y1 X , m(τ, x1 ) − m(τ, y1 )X ≤ Lx ˆ and with some L, m(τ, x1 ) − m(τ, y1 )Y ≤ Lx1 − y1 Y

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with L from the cone invariance property. (ii) M is exponentially attracting, ˆ −η(t−τ ) π2 (τ )x − m(τ, π1 (τ )x)X µ(t, τ, x) − µ(t, τ, x )X ≤ Ke ˆ for t ≥ τ + 1 with an asymptotic phase x = x (τ, x) ∈ M(τ ) of x and some K, moreover πi (t)[µ(t, τ, x) − µ(t, τ, x )]Y ≤ Ki e−η(t−τ ) π2 (τ )x − m(τ, π1 (τ )x)Y for t ≥ τ , i = 1, 2, and the constants K1 , K2 > 0 from the squeezing property. Proof. By Lemma 22, the two-parameter semi-ﬂow µ deﬁnes an NDS ϕ with base B = R and driving system θ : R × B → B with θ(t)τ = t + τ , τ = b ∈ B. Now Theorem 30 follows from Theorem 20. In the next two subsections we verify the assumptions of Theorem 30 for evolution equations under the assumptions of exponential dichotomy conditions on the linear part or under the requirement that the linear part A is constant and selfadjoint such that we may use the eigenvalues of A. 3.2

Exponential Dichotomy Conditions

Let X ]→ Y ]→ Z be Banach spaces equipped with norms · X , · Y , · Z , and let (A(t))t∈R be a family of densely deﬁned linear operators A(t) on Z with domain D(A(t)) in Z. We consider a nonautonomous evolution equation x˙ + A(t)x = f (t, x)

(11)

which satisﬁes the following assumptions: (A1) Linearly A(t): • Existence of evolution operator of the linear system: Under suitable additional assumptions on X , Z, A and f (see for example [Hen81], [DKM92], [Lun95]), we may deﬁne the evolution operator Φ : {(t, s) ∈ R2 : t ≥ s} → L(Z, Z) of the linear equation x˙ + A(t)x = 0 (12) in Z as the solution of d Φ(t, s) + A(t)Φ(t, s) = 0 dt

for t > s, s ∈ R

and Φ(τ, τ ) = idZ

for τ ∈ R .

• There are constants k0 , . . . , k4 ≥ 1, β2 > β1 , γ ∈ [0, 1[, a monotonously decreasing function ψ ∈ C(R>0 , R>0 ) with ψ(t) ≤ k0 t−γ , and a family π1 = (π1 (t))t∈R of linear, invariant projectors π1 (t) : Z → Z, i.e. π1 (t)Φ(t, s) = Φ(t, s)π1 (s)

for t ≥ s,

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such that Φ(t, s)π1 (s) can be extended to a linear, bounded operator for t ∈ R with d Φ(t, s)π1 (s) + A(t)Φ(t, s)π1 (s) = 0 for t, s ∈ R dt and for t ≤ s , Φ(t, s)π1 (s)L(Y,Y) ≤ k1 e−β1 (t−s) −β2 (t−s) for t ≥ s , Φ(t, s)π2 (s)L(Y,Y) ≤ k2 e (13) for t ≤ s , Φ(t, s)π1 (s)L(Z,Y) ≤ k3 e−β1 (t−s) Φ(t, s)π2 (s)L(Z,Y) ≤ k4 ψ(t − s)e−β2 (t−s) for t > s with π2 , π2 (t) = idZ − π1 (t), as the complementary projector to π1 in Z. For the case X = Y we need the additional estimates Φ(t, s)π1 (s)L(X ,X ) ≤ k5 e−β1 (t−s) Φ(t, s)π2 (s)L(X ,X ) ≤ k5 e−β2 (t−s) Φ(t, s)π1 (s)L(Z,X ) ≤ k6 e−β1 (t−s) Φ(t, s)π2 (s)L(Z,X ) ≤ k6 (t − s)−α e−β2 (t−s) Φ(t, s)L(Y,X ) ≤ k7 (t − s)−γ e−β0 (t−s) Φ(t, s)L(Z,X ) ≤ k8 (t − s)−α e−β0 (t−s)

for for for for for for

t≤s, t≥s, t≤s, t>s, t>s, t>s

(14)

and π2 (τ )L(X ,X ) ≤ M10 ,

π1 (τ )xY ≤ M11 π1 (τ )xX

for x ∈ X , τ ∈ R . (15) and constants β0 > 0, α ∈

with nonnegative constants k5 , k6 , k7 , k8 , M10 , M11 [γ, 1[. (A2) Nonlinearity f (t, x): The nonlinear function f ∈ C(R × X , Z) is assumed to satisfy the Lipschitz inequality πi (t)[f (t, x) − f (t, y)]Z ≤ γi (π1 (t)[x − y]Y , π2 (t)[x − y]Y )

(16)

for all t ∈ R, x, y ∈ X , where γi are suitable norms on R2 . (A3) Existence of mild solutions: We have the existence and uniqueness of the mild solutions µ(·, τ, ξ) ∈ C([τ, ∞[, X ) of (11) with initial condition x(τ ) = ξ ∈ X , i.e. let µ be the continuous solution of the integral equation t Φ(t, r)f (τ, x(r)) dr for t ≥ τ . x(t) = Φ(t, τ )ξ + τ

These were the assumptions. Remark 31. Conditions like our assumptions can be found in the literature and they are standard for ordinary diﬀerential equations and for time-independent evolution equations in the non-selfadjoint case, see for example [Tem97]. For concrete examples of the realization of these assumptions we refer to Sec. 4, where we will apply our following Theorem 42 on the existence of inertial manifolds in these special situations.

Inertial Manifolds for Nonautonomous Dynamical Systems

243

Remark 32. 1. If X = Y then we choose ki + 4 = ki for i = 1, 2, 3, 4. 2. In the special cases k1 = k2 = 1, ψ = 1, γi (w) = ^i1 |w1 | + ^i2 |w2 |, the following considerations can be drastically simpliﬁed: For k1 = k2 = 1 we can ˜ If ψ = 1 we show the cone invariance property with T0 = 0 and constant L. don’t have a singularity in the integral inequalities for the estimation of solutions. If γ1 , γ2 have the above mentioned structure then we can use linear comparison problems. In order to apply Theorem 30, we have to show the cone invariance property and the squeezing property for the two-parameter semi-ﬂow µ with respect to the projector π1 . For ﬁxed r1 , r2 ≥ 0 and T ≥ 0, we deﬁne T 1 e−β1 (t−r) γ1 (w(r)) dr , (Λ w)(t) := k3 t t 2 ψ(t − r)e−β2 (t−r) γ2 (w(r)) dr + k2 e−β2 t Lw1 (0) (Λ w)(t) := k4 0

and

q(t) := k1 e−β1 (t−T ) r1 , k2 e−β2 t r2

for t ∈ [0, T ] and w ∈ C([0, T ], R2 ). Then q ∈ C([0, T ], R2≥0 ). Because of ψ(t) ≤ k0 t−γ with γ ∈ [0, 1[, Λ is an at most weakly singular integral operator from C([0, T ], R2 ) into C([0, T ], R2 ). Moreover, Λ is completely continuous. ˜ : ]0, ∞[ → ]0, ∞[, L > 0 and T0 ≥ 0 such that Lemma 33. Assume there are L ˜ )≤L L(T

for T ≥ T0

and such that ˜ )r1 v 2 (T ) ≤ L(T holds for each solution v ∈

C([0, T ], R2≥0 ) i

v i (t) ≤ (Λv) (t) + q i (t)

(17)

of for i = 1, 2, t ∈ [0, T ]

(18)

with r1 ≥ 0, r2 = 0. Then µ possesses the cone invariance property with respect ˜ L and T0 . to π with the parameters L, Proof. See [KS01] for X = Y. Lemma 34. Assume there are positive numbers L, η, K1 , K2 such that v i (t) ≤ Ki e−ηt r2

for t ∈ [0, T ]

(19)

holds for each T > 0 and each solution v ∈ C([0, T ], R2≥0 ) of (18) with r1 = 0, r2 ≥ 0. Then µ possesses the squeezing property with respect to π with the parameters L, η, K1 , K2 .

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Proof. See [KS01].

To estimate the solutions v of (18), we use the following comparison theorem for monotone iterations in ordered Banach spaces. The basic ideas and notions go back for example to [KLS89]. Let B be a Banach space and let C be an order cone in B. The order cone C induces a semi-order ≤C in B by u ≤C w

:⇐⇒

w−u∈C.

The norm in B is called semi-monotone if there is a constant c with xB ≤ cyB for each x, y ∈ B with 0 ≤C x ≤C y. The cone C is called normal if the norm in is semi-monotone, and C is called solid if C contains an open ball with positive radius. N Note that C([0, T ], RN ≥0 ) is a normal, solid cone in C([0, T ], R ). In a Banach space B with normal and solid cone C, we study the ﬁxed-point problem u = Pu + p (20) with p ∈ B and P : B → B. We assume that P is completely continuous, increasing, P u ≤C P v if u ≤C v , subadditive, P (u + v) ≤C P u + P v ,

u, v ∈ C ,

and homogeneous with respect to nonnegative factors, P (λu) = λP u

λ ∈ R≥0 , u ∈ C .

Definition 35. A function w ∈ B is called upper (lower) solution of (20) if P w + p ≤C w (w ≤C P w + p). We need the existence of a unique solution w ∈ C of (20) and an estimation of lower solutions v ∈ C of (20) by solutions or upper solutions of (20). Lemma 36. Assume that there are y ∈ intC and δ ∈ [0, 1[ with P y ≤C δy . Then there is a unique solution x∗ of (20) in C and x ≤C x∗ ≤C x holds for each lower solution x ∈ C and each upper solution x ∈ C of (20). Proof. See [KS01].

(21)

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Inertial Manifolds for Nonautonomous Dynamical Systems

In order to apply Lemma 36 to our situation, we choose B = C([0, T ], R2 ) and C = C([0, T ], R2≥0 ). Then C is a normal. The operator P = Λ is increasing and completely continuous, and p = q belongs to C. So we only have to ﬁnd a function w∗ in the interior of C with Λw∗ ≤C εw∗

with some ε ∈ [0, 1[ .

(22)

Further we can estimate the solutions v of (18) by solutions w ¯ ∈ C of Λw ¯ + q ≤C w ¯.

(23)

Lemma 37. Let t∗ ≥ 0 be ﬁxed with ψ∗ := lim ψ(t) < ∞ ,

ψ(t) > ψ∗

t→t∗

for t < t∗ .

(24)

Further let

t∗

k9 ≥ δ

ψ(r)e−δr dr + ψ∗ lim e−δt t→t∗

0

for all δ ∈ ]0, β2 − β1 [ .

Assume that there are positive numbers ρ1 < ρ2 with G(ρ1 ) = G(ρ2 ) = 0 , G(ρ)

[ρ1 ,ρ2 ]

= 0

(25)

(26)

and k1 k2 ρ1 < k9−1 ψ∗ ρ2

(27)

where G : R>0 → R is deﬁned by G(ρ) := β2 − β1 − k3 γ1 (1, ρ) − k4 k9 ρ−1 γ2 (1, ρ) . Then there are positive numbers η1 < η2 with ηi = β1 + k3 γ1 (1, ρi ) = β2 − k4 k9 ρ−1 i γ2 (1, ρi ) , and the cone invariance and squeezing property hold with L ∈ ]k1 ρ1 , k2−1 k9−1 ψ∗ ρ2 [ , k2 k9 , K2 := ρ2 K1 K1 := ρ2 ψ∗ − k2 k9 L η := η2 ,

and

−η1 t

−η2 t

+ (˜ ρ − ρ1 )ρ2 e ˜ = k1 (ρ2 − ρ˜)ρ1 e L(t) (ρ2 − ρ˜)e−η1 t + (˜ ρ − ρ1 )e−η2 t

with some ρ˜ ∈ ] max{ρ1 , k2 k9 ψ∗−1 L}, ρ2 [.

.

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Remark 38. Because of limρ→0 G(ρ) = limρ→∞ G(ρ) = −∞, the existence of ρ∗ > 0 with G(ρ∗ ) > 0 implies the existence of positive numbers ρ1 < ρ2 with (26) G(ρ1 ) = G(ρ2 ) = 0 and ρ1 < ρ∗ < ρ2 . Since G(ρ1 ) = 0 and ρ1 < ρ∗ imply ρ1 <

k4 k9 ρ−1 ∗ γ2 (1, ρ∗ ) , β2 − β1 − k3 γ1 (1, ρ∗ )

the inequality (27) holds if β2 − β1 > k3 γ1 (1, ρ∗ ) +

k1 k2 k4 k92 γ2 (1, ρ∗ ) . ψ∗ ρ∗

(28)

Since (28) implies G(ρ∗ ) > 0, condition (26) can be replaced by (28) for some ρ∗ > 0. Proof. (of Theorem 42) We show that the two-parameter semiﬂow µ generated by (11) satisﬁes the assumptions of Theorem 22. By Lemma 39 or Lemma 40 it remains to show that the cone invariance and squeezing property are satisﬁed. Step 1: Determining of Solutions of (22) and (23) In order to ﬁnd a solution w∗ of (22), ﬁrst we look for w ∈ C in the form w(t) = e−ηt (1, ρ)

(29)

w1 (t) ≥ (Λw)1 (t) + c1 e−β1 (t−T )

(30)

w2 (t) ≥ (Λw) (t) + (c2 ρ − k2 L)e−β2 t

(31)

with ρ > 0 and satisfying

and

2

for t ∈ [0, T ] with suitable positive c1 and c2 . If we assume η > β1 and η ≥ β1 + k3 γ1 (1, ρ)

(32)

then, because of ηt

1

T

e(η−β1 )(t−r) dr

e (Λw) (t) = k3 γ1 (1, ρ) t

=

k3 γ1 (1, ρ) 1 − e(η−β1 )(t−T ) , η − β1

we may choose c1 = c1 (ρ, η) :=

k3 γ1 (1, ρ) −ηT e η − β1

in order to satisfy (30). Remains to satisfy (31).

(33)

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Inertial Manifolds for Nonautonomous Dynamical Systems

Inserting (29) in (31) and dividing by ρe−ηt , we have to satisfy 1 ≥ H(t, ρ, η)

(34)

for t ∈ [0, T ] where H : [0, ∞[ ×R × R → R is deﬁned by H(t, ρ, η) :=

γ2 (1, ρ) k4 ρ

t

ψ(r)e−(β2 −η)r dr + c2 e−(β2 −η)t .

0

We choose c2 = c2 (ρ, η) := Because of

γ2 (1, ρ) k4 ψ∗ . (β2 − η)ρ

γ2 (1, ρ) k4 ψ(t) e−(β2 −η)t D1 H(t, ρ, η) = −(β2 − η)c2 + ρ

and because of the monotonicity of ψ, the function H(·, ρ, η) is maximized at t∗ . Hence we have H(t, ρ, η) ≤ H(t∗ , ρ, η) =

γ2 (1, ρ) k4 ρ

t∗

ψ(r)e−(β2 −η)r dr + (β2 − η)−1 ψ∗ e−(β2 −η)t∗

0

for all t ≥ 0. Because of (25), inequality (34) is satisﬁed if γ2 (1, ρ) k4 k9 ≤ β2 − η . ρ

(35)

Combining (32) with (35), we ﬁnd β1 + k3 γ1 (1, ρ) ≤ η ≤ β2 − k4 k9 ρ−1 γ2 (1, ρ)

(36)

as a suﬃcient condition for (30) and (31). By assumption there are positive numbers ρ1 < ρ2 with (26) and (27). Let ηi := β1 + k3 γ1 (1, ρi ) = β2 − k4 k9 ρ−1 i γ2 (1, ρi ) . Then (η1 , ρ1 ) and (η2 , ρ2 ) solve (36). Moreover, η2 > η1 . To show this, we note that η2 ≥ η1 by the monotonicity of −1 γ1 . Assuming η1 = η2 we ﬁnd γ1 (1, ρ1 ) = γ1 (1, ρ2 ) and γ2 (ρ−1 1 , 1) = γ2 (ρ2 , 1). By the convexity of the γ1 - and γ2 -balls we had γ1 (1, ρ) = γ1 (1, ρ1 ) and γ2 (ρ−1 , 1) = γ2 (ρ−1 1 , 1) for ρ ∈ [ρ1 , ρ2 ]. This would imply the constance of G on [ρ1 , ρ2 ] in contradiction to (26). Because of (27) we can choose L ∈ ]k1 ρ1 , k2−1 k9−1 ψ∗ ρ2 [ .

(37)

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Now we deﬁne wi ∈ C, i = 1, 2, by wi (t) := e−ηi t (1, ρi ) .

(38)

Then wi1 (t) ≥ (Λwi )1 (t) + and

k3 γ1 (1, ρ) −ηT −β1 (t−T ) e e η − β1

wi2 (t) ≥ (Λwi )2 (t) + ρi k9−1 ψ∗ − k2 L e−β2 t

on [0, T ] because of c1 (ρi , ηi ) =

k3 γ1 (1, ρi ) =1, ηi − β1

Because of (37) we have

c2 (ρi , ηi ) =

γ2 (1, ρi )k4 ψ∗ = k9−1 ψ∗ . (β2 − ηi )ρi

ρ2 k9−1 ψ∗ > k2 L

(39)

and inequality (22) holds for w∗ := w2 . Let now C1 ∈ [0, 1], C2 > 0 satisfy C2 C1 e−η1 T + (1 − C1 )e−η2 T ≥ k1 r1 , C2 C1 ρ1 k9−1 ψ∗ + (1 − C1 )ρ2 k9−1 ψ∗ − k2 L ≥ k2 r2 . Then w ¯ := C2 (C1 w1 + (1 − C1 )w2 ) solves ¯, Λw ¯ + q ≤C w and Lemma 36 implies ¯ v ≤C w for each solution v ∈ C of (18). Step 2: Verification of the Cone Invariance Property Because of (39) we can ﬁx ρ˜ ∈ ] max{ρ1 , k2 k9 ψ∗−1 L}, ρ2 [ . Let r2 = 0 and r1 ≥ 0. Then (40) is satisﬁed with C1 :=

ρ2 − ρ˜ , ρ2 − ρ1

C2 :=

C1

e−η1 T

k1 r1 . + (1 − C1 )e−η2 T

Thus we ﬁnd ¯ ρ, t)r1 v 2 (t) ≤ w ¯2 (t) = C2 C1 w12 (t) + (1 − C1 )w22 (t) = L(˜

(40)

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249

for t ∈ [0, T ] with −η1 t + (˜ ρ − ρ1 )ρ2 e−η2 t ¯ ρ, t) = k1 (ρ2 − ρ˜)ρ1 e L(˜ . −η T (ρ2 − ρ˜)e 1 + (˜ ρ − ρ1 )e−η2 T

Especially we have ¯ ρ, T )r1 . v 2 (T ) ≤ L(˜ The inequalities η2 > η1 > β1 imply ¯ ρ, T ) → k1 ρ1 L(˜

as T → ∞ .

Hence there are T0 ≥ 0 and L ≥ 0 with (17), if the additional inequality k1 ρ1 < L holds, which trivially follows from (37). By Lemma 33, the cone invariance property of µ as required in Theorem 30 is ˜ := L(˜ ρ, t). veriﬁed with L(t) Step 3: Verification of the Squeezing Property Now let r1 = 0 and r2 ≥ 0. Then we may choose C1 := 0 ,

k2 k9 r2 ρ2 ψ∗ − k2 k9 L

C2 :=

in order to satisfy (40). Thus we ﬁnd v(t) ≤

k2 r2 e−η2 t (1, ρ2 ) ρ2 ψ∗ − k2 k9 L

for t ∈ [0, T ] .

Hence (19) holds with η := η2 ,

K1 :=

k2 k9 , ρ2 ψ∗ − k2 k9 L

K2 := ρ2 K1

and L satisfying (37). By Lemma 34, the squeezing property of µ as required in Theorem 30 is veriﬁed. Lemma 39. Let f be globally bounded. Then two-parameter ﬂow µ possesses the boundedness property and the coercivity property. Proof. First we verify the boundedness property: By the boundedness of f there is a number F ≥ 0 with f (x)Z ≤ F

for x ∈ X .

Thus, for τ ∈ R, t ≥ τ , x ∈ X , π2 (t)µ(t, τ, x) = Φ(t, τ )π2 (τ )x +

t

Φ(t, r)π2 (r)f (r, µ(r, τ, x)) dr τ

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N. Koksch and S. Siegmund

and, by the exponential dichotomy conditions (13), π2 (t)µ(t, τ, x)X ≤ Φ(t, τ )π2 (τ )L(X ,X ) π2 (τ )xX t + Φ(t, r)π2 (r)L(Z,X ) f (r, µ(r, τ, x))X dr τ t ψ(t − r)e−β2 (t−r) dr ≤ k5 e−β2 (t−τ ) π2 (τ )xX + F k6 τ

= k5 e−β2 (t−τ ) π2 (τ )xX + F k6

t−τ

∞

0

≤ k5 e−β2 (t−τ ) π2 (τ )xX + F k6

ψ(r)e−β2 (r) dr

ψ(r)e−β2 (r) dr .

0

Thus, for any t, τ with t ≥ τ and any M1 ≥ 0 there is an M2 ≥ 0 such that for x ∈ X with π2 (τ )xX ≤ M1 we have π2 (t)µ(t, τ, x)X ≤ M2 , i.e. the twoparameter ﬂow possesses the boundedness property of µ as required in Theorem 22. Now we verify the coercivity property: For τ ∈ R, t ∈ [τ, τ + T ], x ∈ X , we have π1 (t)µ(t, τ, x) = Φ(t, τ + T )π1 (τ + T )µ(τ + T, τ, x) t + Φ(t, r)π1 (r)f (r, µ(r, τ, x)) dr τ +T

and hence π1 (τ )x = Φ(τ, τ + T )π1 (τ + T )µ(τ + T, τ, x) τ + Φ(τ, r)π1 (r)f (r, µ(r, τ, x)) dr . τ +T

The exponential dichotomy conditions (13) imply π1 (τ )xX ≤ Φ(τ, τ + T )π1 (τ + T )L(X ,X )π1 (τ + T )µ(τ + T, τ, x)X τ +T +F Φ(τ, r)π1 (r)L(Z,X ) dr τ

≤ k5 eβ1 T π1 (τ + T )µ(τ + T, τ, x)X + F k6

τ +T

e−β1 (τ −r) dr

τ

= k5 e

β1 T

F k6 β1 T π1 (τ + T )µ(τ + T, τ, x)X + (e − 1) . β1

Hence π1 (τ + T )µ(τ + T, τ, x)X ≥

1 F k6 π1 (τ )xX − (1 − e−β1 T ) k5 β1 k5

for T ≥ 0, τ ∈ R, x ∈ X which shows the coercivity property of the two-parameter ﬂow µ as required in Theorem 22.

251

Inertial Manifolds for Nonautonomous Dynamical Systems

˜ which Lemma 40. Let the cone invariance property be satisﬁed with a function L ˆ is bounded by L. Let (I(τ ))τ ∈R be an invariant set which is uniformly bounded in τ . Then the two-parameter ﬂow µ possesses the strong coercivity property with respect to CL and I with M6 < η2 . Proof. By the uniform boundedness of I there is a number M7 > 0 with π1 (t)xY ≤ M7 ,

π2 (t)xY ≤ M7

for all t ∈ R ,

x ∈ I(t) .

Let τ ∈ R and x ∈ I(τ ) + CL . The forward invariance of I and the cone invariance property imply ˜ − τ ) (M7 + π1 (t)µ(t, τ, x)Y ) π2 (t)µ(t, τ, x)Y ≤ M7 + L(t

for t > τ .

Let x1 , x2 ∈ X with x1 − x2 ∈ CL and x2 ∈ I(τ ). Let µ∆ (t) := µ(t, τ, x1 ) − µ(t, τ, x2 ). Because of π1 (t)µ(t, τ, xi ) = Φ(t, τ + T )π1 (τ + T )µ(τ + T, τ, xi ) t + Φ(t, r)π1 (r)f (r, µ(r, τ, xi )) dr , τ +T

the exponential dichotomy conditions (13) imply π1 (t)µ∆ (t)Y ≤ Φ(t, τ + T )π1 (τ + T )µ∆ (τ + T )Y τ +T + Φ(t, r)π1 (r)[f (r, µ(r, τ, x1 )) − f (r, µ(r, τ, x2 ))]Y dr t

≤ k1 e−β1 (t−τ −T ) π1 (τ + T )µ∆ (τ + T )Y τ +T + k3 e−β1 (t−r) γ1 (π1 (r)µ∆ (r)Y , π2 (r)µ∆ (r)Y ) dr ≤ k1 e

t −β1 (t−τ −T )

τ +T

+ k3

π1 (τ + T )µ(τ + T, τ, x)Y

˜ − τ ))π1 (r)µ∆ (r)Y dr . e−β1 (t−r) γ1 (1, L(r

t

Setting u(s) = π1 (τ + T − s)µ∆ (τ + T − s)Y e−β1 s

for s ∈ [0, T ] ,

we ﬁnd the Gronwall inequality s ˜ − σ))u(σ) dσ γ1 (1, L(T u(s) ≤ k1 u(0) + k3

for s ∈ [0, T ] .

0

Hence

u(s) ≤ k1 u(0) 1 + k3 = k1 u(0)ek3

s 0

s

e

k3

s τ

˜ −σ)) dσ γ1 (1,L(T

0 ˜ −σ)) dσ γ1 (1,L(T

˜ − τ )) dτ γ1 (1, L(T

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N. Koksch and S. Siegmund

and therefore π1 (τ )µ∆ (τ )Y ≤ M9 (T )π1 (τ + T )µ∆ (τ + T )Y with

˜ 6 (T ) := β1 T + k3 M

T

˜ − σ)) dσ , γ1 (1, L(T

(41)

˜

M9 (T ) := k1 eM6 (T )

0 ∗

Let ρ ∈]k1 ρ1 , ρ2 [ be a number with γ1 (1, ρ∗ ) < γ1 (1, ρ2 ). To show the existence of ρ∗ , we note that η2 = β1 + γ1 (1, ρ2 ) > η1 = β1 + γ1 (1, ρ1 ). Assuming γ1 (1, ρ) = γ1 (1, ρ2 ) for ρ ∈ [k1 ρ1 , ρ2 ], the convexity of the γ1 -balls would imply γ1 (1, ρ) = γ1 (1, ρ2 ) for |ρ| ≤ ρ2 and hence γ1 (1, ρ1 ) = γ1 (1, ρ2 ) in contradiction to η1 < η2 . Therefore γ1 (1, k1 ρ1 ) < γ1 (1, ρ2 ) and by the continuity of γ1 the existence of k1 follows. ˜ = k1 ρ1 < ˜ is monotonously decreasing with L(0) ˜ Since L = k1 ρ˜ and limt→∞ L(t) ∗ ∗ ∗ ∗ ˜ ρ < ρ2 , there is a T ≥ 0 with L(t) ≤ ρ for t ≥ T . Thus 1 T 1 ˜ ˜ − σ)) dσ M6 (T ) = β1 + k3 γ1 (1, L(T T T 0 ∗ T ≤ β1 + k3 γ1 (1, k1 ρ˜) + γ1 (1, ρ∗ ) T 1 ˜ = M6 + M5 T for T > 0 where M6 := β1 + k3 γ1 (1, ρ∗ ) < β1 + k3 γ1 (1, ρ2 ) = η2 and

˜ 5 := T ∗ γ1 (1, k1 ρ˜) . M

Hence

˜

π1 (τ )µ∆ (τ )Y ≤ k1 eM5 +M6 T π1 (τ + T )µ∆ (τ + T )Y . Thus π1 (τ )µ(τ, τ, x1 )Y ≤ π1 (τ )µ∆ (τ )Y + π1 (τ )µ(τ, τ, x2 )Y ˜

≤ M7 + k1 eM5 +M6 T π1 (τ + T )µ∆ (τ + T )Y ˜

≤ M7 + k1 eM5 +M6 T π1 (τ + T )µ(τ + T, τ, x1 )Y ˜

+ k1 eM5 +M6 T π1 (τ + T )µ(τ + T, τ, x2 )Y ˜

≤ M7 + k1 eM5 +M6 T (M7 + π1 (τ + T )µ(τ + T, τ, x1 )Y ) ≤ M5 eM6 T (M7 + π1 (τ + T )µ(τ + T, τ, x1 )Y ) with

˜

M5 := (1 + k1 )eM5 .

Inertial Manifolds for Nonautonomous Dynamical Systems

253

Lemma 41. The two-parameter ﬂow µ possesses the smoothing Lipschitz property. Proof. Let x, y ∈ X , τ ∈ R and let µ∆ (t) = µ(t, τ, x) − µ(t, τ, y). By assumption ˆ independent of t. The f (t, ·) is global Lipschitz from X to Z with some constant L exponential dichotomy conditions imply the generalized Gronwall inequality t Φ(t, s)[f (s, µ(s, τ, x)) − f (s, µ(s, τ, x))]X ds µ∆ (t)X ≤ Φ(t, τ )[x − y]X + τ t ˆ Φ(t, s)L(Z,X )µ∆ (s)X ds ≤ Φ(t, τ )L(Y,X ) x − yY + L τ t ˆ 8 (t − s)−α eβ0 (t−s) µ∆ (s)X ds ≤ k7 (t − τ )−γ eβ0 (t−τ ) x − yY + Lk τ

for µ∆ (·)X . As proved for example in [Hen81], there is a function M8 : ]0, ∞[ → ]0, ∞[ with for t > τ . µ∆ (t)X ≤ M8 (t − τ )x − yY Inequality (10) is shown in the proof of the previous Lemma as inequality (41). Now we are in a position to state the following theorem as a direct consequence of Theorem 22 and the previous lemmata: Theorem 42. Let the assumptions of Lemma 37 and the assumptions of Lemma 39 or Lemma 40 be satisﬁed. Then the claim of Theorem 22 holds for the twoparameter semi-ﬂow µ generated by (11) with η, L, K1 and K2 as given in Lemma 37. 3.3

Indefinite Quadratic Forms

Let H = Z be a Hilbert space equipped with norm | · |, and let A be a densely deﬁned linear operator on H which is selfadjoint, positive and which has compact resolvent. We consider a nonautonomous parabolic evolution equation x˙ + Ax = f (t, x)

(42)

where the nonlinear part f : R × X → H satisﬁes the following assumptions: • The Hilbert space X = D(Aα ) with norm uX := |u|α := |Aα u| is the domain of a power Aα of A with some α ∈ [0, 1[. • f (t, x) is locally Hölder continuous in t and global Lipschitz continuous in x. Then there are maximally deﬁned (classical) solutions µ(·, τ, ξ) ∈ C([τ, ∞[, X ) ∩ C 1 (]τ, ∞[, H) of (42) with initial condition x(τ ) = ξ, see [Hen81], [Mik98], and (42) generates a two-parameter semi-ﬂow µ on X .

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Let λ1 ≤ λ2 ≤ · · · denote the eigenvalues of A counted with their multiplicity and let e1 , e2 , . . . denote the corresponding eigenvectors of A. We ﬁx N ∈ N. Let π1 be the orthogonal projector from H onto span{e1 , . . . , eN } and let π2 := idH − π1 . For a ﬁxed Banach space Y with X ]→ Y ]→ Z, we introduce the quadratic forms Qρ : X → R Qρ (x) = π2 x2Y − ρ2 π1 x2Y

for x ∈ X ,

ρ > 0.

Our goal is to use inequalities Qρ (µ(t, τ, x) − µ(t, τ, y)) ≤ e−2Λ(ρ)(t−τ ) Qρ (x − y)

(43)

for t ≥ τ , x, y ∈ X and suitable ρ > 0 in order to show the cone invariance and squeezing property. Lemma 43. Under the general assumption given above, let there exist ρ1 < ρ2 , a function Λ : [ρ1 , ρ2 ] → R and a number L0 ∈ ]ρ1 , ρ2 ] with Λ(L0 ) > 0 and (43) for ρ ∈ [ρ1 , ρ2 ], t ≥ τ , x, y ∈ X . Then the two-parameter semi-ﬂow µ possesses the cone invariance property for all L ∈ [ρ1 , ρ2 ] and the squeezing property with the parameters L = ρ1 ,

η = Λ(L0 ) ,

ρ2 L 0 & K1 = & 2 , ρ2 − L20 L20 − ρ21

K 2 = ρ1 K 1 .

Proof. 1. Let ρ ∈ [ρ1 , ρ2 ] and π2 [x − y]Y ≤ ρπ1 [x − y]Y . Then Qρ (x − y) ≤ 0 and hence by assumption Qρ (µ(t, τ, x) − µ(t, τ, y)) ≤ 0, i.e., π2 [µ(t, τ, x) − µ(t, τ, y)]Y ≤ ρπ1 [µ(t, τ, x) − µ(t, τ, y)]Y

for all t ≥ τ , (44)

i.e., the cone invariance property is satisﬁed in Y for any parameter L = ρ ∈ [ρ1 , ρ2 ]. 2. Let L = ρ1 and let τ ∈ R, T ≥ 0, and x, y, z ∈ X with π1 µ(τ + T, τ, x) = π1 µ(τ + T, τ, y) and π2 [x − z]Y ≤ Lπ1 [x − y]Y . Assuming Qρ (µ(t, τ, x) − µ(t, τ, y)) < 0 for some ρ ∈ [ρ1 , ρ2 ], t ∈ [τ, τ + T ], we get a contradiction to Qρ (µ(t, τ, x) − µ(t, τ, y)) = π2 [µ(t, τ, x) − µ(t, τ, y)]2Y ≥ 0 . Hence, 0 ≤ Qρ (µ(t, τ, x) − µ(t, τ, y)) ≤ e−2Λ(ρ)(t−τ ) Qρ (x − y)

for all t ∈ [τ, τ + T ] . (45)

Using this inequality and setting µ∆ (t) := µ(t, τ, x) − µ(t, τ, y), we ﬁnd Qρ (µ∆ (t)) = π2 µ∆ (t)2Y − ρ2 π1 µ∆ (t)2Y

2 2 −2 π2 µ∆ (t)2Y − ρ22 π1 µ∆ (t)2Y = (1 − ρ2 ρ−2 2 )π2 µ∆ (t)Y + ρ ρ2 2 ≥ (1 − ρ2 ρ−2 2 )π2 µ∆ (t)Y ,

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Inertial Manifolds for Nonautonomous Dynamical Systems

i.e., π2 [µ(t, τ, x) − µ(t, τ, y)]2Y ≤

ρ22

ρ22 e−2Λ(ρ)t Qρ (x − y)ρ ∈ [ρ1 , ρ2 [, t ∈ [τ, τ + T ] . − ρ2

for all ρ ∈ [ρ1 , ρ2 [, t ∈ [τ, τ + T ]. With the ﬁrst inequality in (45) and Qρ (x − y ) = π2 [x − y]2Y − ρ2 π1 [x − y]2Y 2

≤ (π2 [y − z]Y + π2 [x − z]Y ) − ρ2 π1 [x − y]2Y 2

≤ (π2 [y − z]Y + Lπ1 [x − y]) − ρ2 π1 [x − y]2Y ≤ (1 + ε−1 )π2 [y − z]2Y + (1 + ε)L2 − ρ2 π1 [x − y]2Y =

ρ2

ρ2 π2 [y − z]2Y − L2

for ε = ρ2 L−2 − 1. Thus, for ρ = L0 , we get πi [µ(t, τ, x) − µ(t, τ, y)]Y ≤ Ki e−tη π2 [y − z]Y

for all t ∈ [τ, τ + T ] ,

i.e., the modiﬁed squeezing property is satisﬁed. Theorem 44. Let the assumptions of Lemma 43 be satisﬁed. Moreover, we assume that • f is globally bounded or • there is an bounded invariant set I. Then the claim of Theorem 22 holds for the two-parameter semi-ﬂow µ generated by (11) with η, L, K1 and K2 as given in Lemma 43. Proof. We only have to note that the exponential dichotomy conditions (13), (14) and the inequalities (15) can be satisﬁed for Φ(t, τ ) = e−A(t−τ ) . Remark 45. An inequality of the form (43) is used in [Rom94] for the special case Y = D(Aα/2 ). Assuming the Lipschitz inequality |f (x) − f (y)| ≤ ^|x − y|α

x, y ∈ X = D(Aα )

(46)

and the spectral gap condition α λN +1 − λN > ^(λα N + λN +1 ) ,

(47)

A.V. Romanov [Rom94] shows that (43) holds for ρ ∈ [h, h−1 ], Λ(ρ) = λN +1 − ^λα N +1 and with h < 1 satisfying 1 2 −2 α λN +1 − λN > ^(λα )λN +1 ) . N + (h + h 2 Thus our Theorem 44 allows to ensure the existence of inertial manifolds under the sharp spectral gap condition (47) for the Lipschitz inequality (46).

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In the following two Lemmata we verify the assumption of Lemma 43. For simplicity we restrict us to Y = H and to Lipschitz inequalities of the type |f (t, x) − f (t, y)| ≤ ν(x − y)

(48)

for all x, y ∈ D(A), t ∈ R where M

ν(x) =

! 12 di |x|2δi

i=1

with positive di , i = 1, . . . , M , and 1 γ = δ1 ∈ [0, min{α, }] , 0 ≤ δi+1 < δi 2 Let M

g1 :=

for i = 1, . . . , M − 1 .

! 12 i di λ2δ N

,

g2 :=

i=1

M

! 12 i di λ2δ N +1

i=1

We show that in this case the assumptions of Lemma 43 and hence of Theorem 44 can be satisﬁed if the spectral gap condition λN +1 − λN > g1 + g2

(49)

holds. Especially, for |f (t, x) − f (t, y)| ≤ ^ · |x − y|γ

for x, y ∈ D(A) , t ∈ R

we have the spectral gap condition λN +1 − λN > ^(λγN +1 + λγN ) . Let the auxiliary function p : R → R deﬁned by 2 p(ρ) := (λN +1 − λN ) ρ2 − ρ2 + 1 g12 + ρ2 g22 Further, let L0 :=

for ρ ∈ R .

(50)

& g1 /g2 .

Lemma 46. Let the spectral gap condition (49) be satisﬁed. Then there are uniquely determined numbers 0 < ρ1 < ρ2 with p(ρ1 ) = p(ρ2 ) = 0 ,

ρ1 < L 0 < ρ2 .

The function Λ : ]0, ∞[ → R deﬁned by Λ(ρ) := λN +1 −

ρ2 (1 + ρ2 )g22 + g12 + ρ2 g22 & & 2L 1 + ρ2 g12 + ρ2 g22

for ρ > 0

(51)

is maximized at L0 with Λ(L0 ) = λN +1 − g2 > 0 .

(52)

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Inertial Manifolds for Nonautonomous Dynamical Systems

Proof. Because of p(L0 ) =

g1 (λN +1 − λN )2 − (g1 + g2 )2 , g2

the spectral gap condition (49) implies p(L0 ) > 0. Since p is a quadratic polynomial in ρ2 and p(0) < 0, the existence and uniqueness of zeroes ρ1 , ρ2 of p in ]0, ∞[ follows. Thus p(ρ) > 0 for ρ ∈ ]ρ1 , ρ2 [ and L0 ∈ ]ρ1 , ρ2 [. We have 1 Λ(ρ) = λN +1 − g2 H(ρ) + H(ρ)−1 2 with

H(ρ) := g2

1 + ρ2 ρ−2 g12 + g22

12 ,

such that Λ has a global maximum on ]0, ∞[ at ρ with H(ρ) = 1, i.e., at ρ = L0 . Since λN > 0, the spectral gap condition (49), we have (52). Lemma 47. Let Y = H and let (48) and (49) be satisﬁed. Then Qρ (µ(t, τ, x) − µ(t, τ, y)) ≤ Qρ (x − y)e−2Λ(ρ)(t−τ ) for all x, y ∈ X , ρ ∈ [ρ1 , ρ2 ] and τ ≤ t. Proof. Let ρ ∈ [ρ1 , ρ2 ], τ < t, x, y ∈ X be ﬁxed. For shortness let µ∆ := µ(t, τ, x)− µ(t, τ, y), f∆ := f (t, µ(t, τ, x) − f (t, µ(t, τ, y)). We have , + 1 d Qρ (µ∆ ) = G −Aµ∆ + f∆ , π2 µ∆ − ρ2 π1 µ∆ 2 dt , + = − Aµ∆ , π2 µ∆ + ρ2 Aµ∆ , π1 µ∆ + G f∆ , π2 µ∆ − ρ2 π1 µ∆ and

+ , G f∆ , π2 µ∆ − ρ2 π1 µ∆ ≤ ν(µ∆ )|π2 µ∆ − ρ2 π1 µ∆ | 1 ≤ 2ε ν(µ∆ )2 + 2ε |π2 µ∆ − ρ2 π1 µ∆ |2 ε 1 2 ≤ 2 ν(π1 µ∆ ) + 2ε ν(π2 µ∆ )2 + 2ε |π2 µ∆ |2 +

1 4 2 2ε ρ |π1 µ∆ |

.

Note that

Aµ∆ , π1 µ∆ ≤ λN |π1 µ∆ |2 ,

2 − Aµ∆ , π2 µ∆ ≤ −λ1−2γ N +1 |π2 µ∆ |γ

and ν(π1 µ∆ ) ≤ g1 |π1 µ∆ | ,

ν(π2 µ∆ ) ≤ g2 λ−γ N +1 |π2 µ∆ |γ .

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With U > 0, we estimate 1 d 2 2 2 Qρ (µ∆ ) ≤ −λ1−2γ N +1 |π2 µ∆ |γ + ρ λN |π1 µ∆ | 2 dt ε ε 1 1 + ν(π1 µ∆ )2 + ν(π2 µ∆ )2 + |π2 µ∆ |2 + ρ4 |π1 µ∆ |2 2 2 2ε 2ε 1 ε ≤ |π1 µ∆ |2 ρ2 λN + ρ4 + g12 2ε 2 1 ε 2 −2γ |π2 µ∆ |2 + |π2 µ∆ |2γ −λ1−2γ N +1 + g2 λN +1 . 2ε 2 Under the assumption λN +1 >

ε 2 g , 2 2

(53)

we ﬁnd

1 d 1 ε Qρ (µ∆ ) ≤ −ΛQρ (µ∆ ) + |π1 µ∆ |2 −Λρ2 + ρ2 λN + ρ4 + g12 2 dt 2ε 2 ε 1 − λN +1 + g22 + |π2 µ∆ |2γ Λ + 2ε 2 ≤ −ΛQρ (µ∆ )

(54)

if Λ satisﬁes

ε 1 2 ε 2 −2 1 ρ + g1 ρ ≤ Λ ≤ λN +1 − − g2 . 2ε 2 2ε 2 2 This inequality is solvable with respect to Λ if and only if λN +

λN +1 − λN ≥

1 ε (1 + ρ2 ) + (g12 ρ−2 + g22 ) . 2ε 2

The right-hand side is minimized at U := with the value

1 + ρ2 ρ−2 g12 + g22 1

(55)

12 (56) 1

(1 + ρ2 ) 2 (g12 ρ−2 + g22 ) 2

such that we obtain the suﬃcient and necessary condition 1

1

λN +1 − λN ≥ (1 + ρ2 ) 2 (g12 ρ−2 + g22 ) 2 . By deﬁnition of p and by Lemma 46, this inequality holds for ρ ∈ [ρ1 , ρ2 ]. So ε 1 − g22 2ε 2 1 12 1 + ρ2 1 g12 ρ−2 + g22 2 1 = λN +1 − − g22 2 1 + ρ2 2 g12 ρ−2 + g22

Λ = λN +1 −

= Λ(ρ)

(57)

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Inertial Manifolds for Nonautonomous Dynamical Systems

solves (55) with U given by (56). Remains to show (53) with (56), i.e., we have to show ( 1 & g12 + ρ2 g22 λN +1 > ρ 1 + ρ2 g22 . 2

(58)

Indeed, (58) holds, since the inequalities (57) and λN +1 > λN > 0 imply ρ

& 1 1 + ρ2 g22 ≤ (λN +1 − λN )ρ2 g22 (g12 + ρ2 g22 )− 2 1

≤ λN +1 (g12 + ρ2 g22 )(g12 + ρ2 g22 )− 2 ( = λN +1 g12 + ρ2 g22 . Summarizing we have that Λ = Λ(ρ) satisﬁes (54). Therefore, d Qρ (µ(t, τ, x) − µ(t, τ, y)) ≤ −2Λ(ρ)Qρ (x − y) dt for all t > τ , x, y ∈ X , ρ ∈ [ρ1 , ρ2 ] such that the claim of the lemma follows. Corollary 48. Under the general assumptions of this section let f satisfy the Lipschitz inequality (48) with some γ ∈ [0, min{α, 12 }] such that the spectral gap condition (49) holds. Moreover, we assume that • f is globally bounded or • there is an bounded invariant set I. Then the claim of Theorem 22 holds for the two-parameter semi-ﬂow µ generated by (11) with η, L, K1 and K2 as given in Lemma 43.

4

Conclusion

Exponential dichotomy conditions of the form (13) are used, for example, in [Hen81], [Tem97], [BdMCR98], [LL99], [CS01]. There k3 = β1α k1 , k4 = β2α with some α ∈ [0, 1[ depending on the spaces X and Z, and ψ(t) = β2−α max{t−α , 1}, ψ(t) = β2−α t−α + 1, or ψ(t) = max{αα β2−α t−α , 1} where 00 := 1. If A is a timeindependent sectorial operator, then usually X is the domain D((A + a)α ) of the power (A + a)α of A + a with some α ∈ [0, 1[ and some a ∈ R. If X = Z then we may choose α = 0 and ψ = 1. In the special case that A is a time-independent, selfadjoint positive linear operator with compact resolvent and dense domain D(A) on the Hilbert space Z, usually one uses X = D(Aα ) with some α ∈ [0, 1[. Let π1 be the orthogonal projector from Z onto the linear subspace spanned by the N eigenvectors of A corresponding to the ﬁrst N eigenvalues λ1 ≤ · · · ≤ λN (counted with their multiplicity). Then we may choose β1 = λN , β2 = λN +1 , k1 = k2 = 1, k3 = β1α , k4 = β2α , ψ(t) := max{αα β2−α t−α , 1}, see for example [FST88, Lemma 3.1].

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In [LL99] (and with X = Z), a Lipschitz inequality of the form πi (t)[f (t, x) − f (t, y)]X ≤ ^i max{π1 (t)[x − y]X , π2 (t)[x − y]X } is utilized. This special form of a Lipschitz inequality is contained in our Lipschitz assumption with γi (w) = ^i |w|∞ and | · |∞ as the maximum norm in R2 . The standard Lipschitz inequality f (t, x) − f (t, y)Z ≤ ^x − yX in a Hilbert space Z and with orthogonal projectors π1 (t) leads to (16) with γi (w) = ^|w|2 or γi (w) = ^|w|1 , where | · |1 denotes the sum norm and | · |2 denotes the euclidean norm in R2 . In order to compare known results with ours we verify the assumptions of Theorem 42 for diﬀerent forms of Lipschitz estimates for f and for concrete functions ψ in the exponential dichotomy property. Corollary 49. Under the general assumptions in Sec. 3.2, let f satisfy (16) with weighted maximum norms γi (w) = ^i max{|w1 |, |w2 |} ,

^i > 0

for w ∈ R2 .

(59)

Let t∗ , ψ∗ and k9 with the properties as in Theorem 42. Then condition (28) and hence the claim of Theorem 42 hold if * k1 k2 k3 k4 k92 ^1 ^2 (k3 ^1 − k4 k9 ^2 )2 k3 ^1 + k4 k9 ^2 + + β2 − β1 > . (60) 2 4 ψ∗ Proof. Calculating the zeroes of G with G(ρ) = β2 − β1 − k3 ^1 max{1, ρ} − k4 k9 ^2 ρ−1 max{1, ρ}, we ﬁnd (60) as suﬃcient and necessary condition for (26), (27). Latushkin and Layton [LL99] consider −A as generator of a strongly continuous semigroup on the Banach space X = Z. Let X be the direct sum of two subspace X1 and X2 and let πi the projector from X onto Xi . Assuming exponential dichotomy conditions (13) with k1 = k2 = 1 (and k3 = k4 = 1, ψ = 1 because of X = Z) and f (0) = 0 and πi [f (x) − f (y)]X ≤ ^i max{π1 [x − y]X , π2 [x − y]X } for the time-independent nonlinearity f , they found β2 − β 1 > ^ 1 + ^ 2

(61)

as optimal spectral gap condition. They extended this result to (−A(t)) as a family of linear operators on the Banach space X = Z generating a strongly continuous

261

Inertial Manifolds for Nonautonomous Dynamical Systems

semiﬂow, see [LL99] too. Again, assuming exponential dichotomy conditions (13) with k1 = k2 = k3 = k4 = 1, ψ = 1, and the Lipschitz estimate πi (t)[f (t, x) − f (t, y)]X ≤ ^i max{π1 (t)[x − y]X , π2 (t)[x − y]X } and f (t, 0) = 0 for f , they found the spectral gap condition (61) for nonautonomous inertial manifolds. Since X = Z, we have k3 = k1 , k4 = k2 , ψ = 1. Thus we have to choose t∗ = 0 and ﬁnd k9 = ψ∗ = 1. Our condition (60) reduces to β2 − β1 > k1 ^1 + k2 ^2 , which in the special case of k1 = k2 = 1 reduces to the optimal spectral gap condition (61) found by Y. Latushkin and B. Layton, [LL99]. Corollary 50. Under the general assumptions in Sec. 3.2, let f satisfy (16) with weighted sum norms γi (w) = ^i1 |w1 | + ^i2 |w2 | ,

^i1 , ^i2 > 0

for w ∈ R2 .

(62)

Let t∗ , ψ∗ and k9 with the properties as in Theorem 42. Then condition (28) and hence the claim of Theorem 42 hold if β2 − β1 > k3 ^11 + k4 k9 ^22 +

k1 k2 k9 + ψ∗ & √ ^12 ^21 k3 k4 . k1 k2 ψ∗

(63)

Proof. Calculating the zeroes of G with G(ρ) = β2 − β1 − k3 ^11 − k3 ^12 ρ − k4 k9 ^21 ρ−1 − k4 k9 ^22 , we ﬁnd (63) as a suﬃcient and necessary condition for (26), (27). First let

ψ(t) := max{αα β2−α t−α , 1}

as in [FST88, Lemma 3.1]. Here and in the following we set 00 := 1 in order to continuously extend the expression for ψ to the limit case α = 0. We choose t∗ := αβ2−1 and hence we have ψ∗ = 1. To satisfy (25) we note that

t∗

δ 0

ψ(r)e−δr dr + ψ∗ lim e−δt = δ α αα β2−α t→t∗

δαβ2−1

−1

r−α e−r dr + e−δαβ2 .

0

The right hand side is monotonously increasing in δ > 0. Therefore, we may satisfy (25) for 0 < δ ≤ β2 − β1 ≤ β2 with α (β2 − β1 )α α k10 := α r−α e−r dr + e−α − 1 ≥ 0 , k9 := 1 + k10 . β2α 0 If k3 = k1 β1α , k4 = k2 β2α and ^11 = ^12 = ^21 = ^22 = ^, condition (63) reads & (64) β2 − β1 > k1 β1α + k2 k9 β2α + (1 + k9 k1 k2 ) β1α β2α ^ .

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Now we assume that Z is a Hilbert space, A is a time-independent, selfadjoint, positive linear operator on Z with dense domain and compact resolvent, f is a continuous mapping from R × X , X = D(Aα ), into Z satisfying a global Lipschitz condition f (t, x) − f (t, y)Z ≤ ^x − yX for x, y ∈ X . Let λ1 ≤ λ2 ≤ · · · denote the eigenvalues of A counted with their multiplicity and let π1 be the orthogonal projector from Z onto the N -dimensional subspace spanned by the ﬁrst N eigenvectors of A. Then (13) is satisﬁed with k1 = k2 = 1, β1 = λN , β2 = λN +1 , and we ﬁnd the spectral gap condition ! α/2 α/2 2 λN + λN +1 α/2 α/2 α λN +1 − λN > λN + λN +1 + k10 (λN +1 − λN ) ^ (65) α/2 λN +1 which holds if α α λN +1 − λN > 2 λα ^. N + λN +1 + k10 (λN +1 − λN )

(66)

Romanov [Rom94] showed that a spectral gap condition α λN +1 − λN > (λα N +1 + λN )^

(67)

is suﬃcient for the existence of an N -dimensional (autonomous) inertial manifold. Note that the right hand side in (66) is at most by the factor 2(1 + k10 ) worse than the right hand side in the sharp condition (67), where k10 = 0 for α = 0 and k10 ≈ 0.46 for α = 12 . For α ≤ 12 , we may apply Corollary 48 which yields the strong spectral gap condition (67), too. If α ∈ ] 12 , 1], we refer to Remark 45, which says that our approach also allows to get the sharp condition (67) in that case. Moreover, for some evolution equations it usefull to distinguish the space X = D(Aα ), in which the semiﬂow acts, from the space D(Aγ ) used in the Lipschitz inequality: One has to choose α ∈ [0, 1[ in such a way that f is a suﬃciently smooth mapping from R × X as required for the existence theory. However it is possible to satisfy and to require a Lipschitz inequality f (x) − f (y)Z ≤ ν(x − y) for x, y ∈ D(A) with γ ∈ [0, min{α, 12 }[ and some norm ν on D(Aγ ). Especially, for ν(x) = ^|x|D(Aγ ) , our Corollary 48 yields a spectral gap condition λN +1 − λN > (λγN +1 + λγN )^ which is weaker than (67) if γ < α. As an concrete application we consider a reaction-diﬀussion equation ut = uξξ + F (ξ, u, ∇u) , with

u(t, 0) = u(t, 1) = 0

|F (ξ, u, v) − F (ξ, u , v )| ≤ ^0 |u − u | + ^1 |v − v |

in Z = L2 ([0, 1]) as studied by P. Brunovský and I. Teresščák, [BT91]. Here −A is the Laplacian with Dirichlet boundary condition on [0, 1], and f (t, x)(ξ) =

Inertial Manifolds for Nonautonomous Dynamical Systems

263

F (t, ξ, x(ξ), ∇x(ξ)). For the existence theory we need α > 34 , but it is possible to √ √ choose ν(x) = 2^0 |x|+ 2^1 |x| 12 in order to cover the gradient in the nonlinearity. Corollary 48 yields the spectral gap condition √ 1 1 2 2 + λN λN +1 − λN > 2(2^0 + ^1 (λN +1 )) , i.e.

√ √ (2N + 1)π 2 > 2 2^0 + 2(2N + 1)π^1

which is weaker than the spectral gap condition found in [BT91]. Now let

ψ(t) = β2−α t−α + 1

as in [Tem97]. Then t∗ = ∞, ψ∗ = 1 and we may choose k10 := Γ (1 − α) ,

k9 := 1 + k10

to satisfy (25). In the special case ^1 = ^2 = ^, k3 = k1 β1α , k4 = k2 β2α , condition (63) reads now & (68) β2 − β1 > k1 β1α + (1 + k1 k2 (1 + k10 )) β1α β2α + k2 (1 + k10 )β2α ^ . For a Banach space Z, a time-independent, sectorial linear operator A on Z with dense domain D(A), and a time-independent, continuous mapping f from X = D(Aα ) into Z satisfying a global Lipschitz condition f (x) − f (y)Z ≤ ^x − yX where X = D((A + a)α ) with ﬁxed a ∈ R, α ∈ [0, 1[ with Gσ(A) + a > 0, and under assumption (13) with ψ(t) = β2−α t−α + 1, Temam showed ([Tem97], Theorem IX.2.1) that there are constants c1 and c2 independent of the Lipschitz constant ^ and the boundedness constant ^0 of the nonlinearity f , such that the spectral gap condition β2 − β1 ≥ c1 (^0 + ^ + ^2 )(β2α + β1α ) ,

β11−α ≥ c2 (^0 + ^)

(69)

implies the existence of an autonomous inertial manifold in the autonomous case. Note that our condition (68) is of similar form as (69) but (68) contains only known constants and is applicable for the nonautonomous case, too. Moreover, in contrast to (69), in our condition (68), the right hand side is linear in the Lipschitz constant ^. Finally let

ψ(t) = αα β2−α t−α + 1 .

Then we choose t∗ := ∞ and have ψ∗ = 1. Since ∞ t∗ ψ(τ )e−δτ dτ = δβ2−α (αα τ −α + β2α )e−δτ dτ δ 0

0

= αα δ α β2−α Γ (1 − α) + 1 ,

264

N. Koksch and S. Siegmund

for δ > 0, we may choose k10 := αα Γ (1 − α) ,

k9 := 1 +

(β2 − β1 )α k10 β2α

(70)

in order to satisfy (25) for δ ∈]0, β2 −β1 [. In the special case ^1 = ^2 = ^, k3 = k1 β1α , k4 = k2 β2α , condition (63) takes the form (64). Whilst we are yet not in a position to deal with retardation or stochastic perturbation, we try to compare our result with that one found by L. Boutet de Monvel, I.D. Chueshov and A.V. Rezounenko in [BdMCR98] and by I.D. Chueshov, M. Scheutzow in [CS01] for the special case of a semilinear parabolic equation without perturbation and without retardation. There Z is a Hilbert space, A is a time-independent, selfadjoint, positive linear operator on Z with dense domain and compact resolvent, f is a continuous mapping from R × X , X = D(Aα ), into Z satisfying a global Lipschitz condition f (t, x) − f (t, y)Z ≤ ^x − yX . Let λ1 ≤ λ2 ≤ · · · denote the eigenvalues of A counted with their multiplicity and let π1 be the orthogonal projector from Z onto the N -dimensional subspace spanned by the ﬁrst N eigenvectors of A. Chueshov and Scheutzow [CS01] found the spectral gap condition α α α ^, (71) λN +1 − λN > 2 λα N + λN +1 + α Γ (1 − α)(λN +1 − λN ) and Boutet de Monvel, Chueshov and Rezounenko found α α α λN +1 − λN ≥ 4 λα ^, N + λN +1 + α Γ (1 − α)(λN +1 − λN ) which is is little bit worse than (71). In this situation the exponential dichotomy condition (13) is satisﬁed with α k1 = k2 = 1, β1 = λN , β2 = λN +1 , k3 = λα N , k4 = λN +1 , and we ﬁnd again the spectral gap condition (65) but here with k10 given by (70). Obviously our condition (65) is a little weaker than (71). Note again that Corollary 48 would only require the spectral gap condition 67. So we have good chances to extend our result to retarded semilinear parabolic equations and, possibly, to semilinear parabolic equations with stochastic perturbation. Summarizing the examples above, one can see that at least in these examples our approach allows to get the same or weaker spectral gap conditions.

References Arn98.

L. Arnold, Random dynamical systems, Springer-Verlag, Berlin Heidelberg New York, 1998. BdMCR98. L. Boutet de Monvel, I.D. Chueshov, and A.V. Rezounenko, Inertial manifolds for retarded semilinear parabolic equations, Nonlinear Analysis 34 (1998), 907–925. BF95. A. Bensoussan and F. Flandoli, Stochastic inertial manifold., Stochastics Rep. 53 (1995), no. 1-2, 13–39.

Inertial Manifolds for Nonautonomous Dynamical Systems BT91. CF94. CFNT86.

CFNT89.

Chu95.

CL99. CS01.

DKM92.

DLS01. FNST85.

FNST88.

FS96.

FST85.

FST88. FST89.

GV97.

Hen81. JT96. KLS89.

265

P. Brunovský and I. Tereščák, Regularity of invariant manifolds, J. Dyn. Diﬀer. Equations 3 (1991), 313–337. H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Relat. Fields 100 (1994), 365–393. P. Constantin, C. Foias, B. Nicolaenko, and R. Temam, Nouveaux resultats sur les variétés inertielles pour les équations diﬀérentielles dissipatives. (New results on the inertial manifolds for dissipative diﬀerential equations), C. R. Acad. Sci., Paris, Ser. I 302 (1986), 375–378. P. Constantin, C. Foias, B. Nicolaenko, and R. Temam, Integral manifolds and inertial manifolds for dissipative partial diﬀerential equations, Applied Mathematical Sciences, vol. 70, Springer, 1989. I.D. Chueshov, Approximate inertial manifolds of exponential order for semilinear parabolic equations subjected to additive white noise, J. Dyn. Diﬀer. Equations 7 (1995), no. 4, 549–566. T. Caraballo and J.A. Langa, Tracking properties of trajectories on random attracting sets, Stochastic Anal. Appl. 17 (1999), no. 3, 339–358. I.D. Chueshov and M. Scheutzow, Inertial manifolds and forms for stochastically perturbed retarded semilinear parabolic equations, J. Dyn. Diﬀer. Equations 13 (2001), no. 2, 355–380. D. Daners and P. Koch Medina, Abstract evolution equations, periodic problems and application, Pitman Research Notes in Mathematics Series, vol. 27, Longman Scientiﬁc & Technical, Harlow, 1992. J. Duan, K. Lu, and B. Schmalfuss, Invariant manifolds for stochastic partial diﬀerential equations, manuscript (2001). C. Foias, B. Nicolaenko, G.R. Sell, and R. Temam, Variétés inertielles pour l’équation de Kuramoto-Sivashinsky. (Inertial manifolds for the KuramotoSivashinsky equation), C. R. Acad. Sci., Paris, Ser. I 301 (1985), 285–288. C. Foias, B. Nicolaenko, G.R. Sell, and R. Temam, Inertial manifolds for the Kuramoto-Sivashinsky equation and an estimate of their lowest dimension, J. Math. Pures Appl., IX. 67 (1988), no. 3, 197–226. F. Flandoli and B. Schmalfuss, Random attractors for the 3d stochastic navier stokes equation with multiplicative white noise, Stochastics and Stochastic Reports 59 (1996), 12–45. C. Foias, G.R. Sell, and R. Temam, Variétés inertielles des équations différentielles dissipatives. (Inertial manifolds for dissipative diﬀerential equations), C. R. Acad. Sci., Paris, Ser. I 301 (1985), 139–141. C. Foias, G.R. Sell, and R. Temam, Inertial manifolds for nonlinear evolutionary equations, J. Diﬀer. Equations 73 (1988), 309–353. C. Foias, G.R. Sell, and E.S. Titi, Exponential tracking and approximation of inertial manifolds for dissipative nonlinear equations, J. Dyn. Diﬀer. Equations 1 (1989), no. 2, 199–244. A.Yu. Goritskij and M.I. Vishik, Local integral manifolds for a nonautonomous parabolic equation, J. Math. Sci., New York 85 (1997), no. 6, 2428–2439. D. Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics, vol. 850, Springer, 1981. D.A. Jones and E.S. Titi, C 1 approximations of inertial manifolds for dissipative nonlinear equations, J. Diﬀer. Equations 127 (1996), no. 1, 54–86. M.A. Krasnoselski, Je.A. Lifshits, and A.V. Sobolev, Positive linear systems, the method of positive operators, Heldermann Verlag, Berlin, 1989.

266 Kra68.

KS01. LL99. Lun95.

Mik98. Mor83. Rhe69. Rob93. Rob96. Rom94.

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TY94. WF97.

N. Koksch and S. Siegmund M.A. Krasnoselski, The operator of translation along trajectories of diﬀerential equations, Translations of Mathematical Monographs, vol. 19, American Math. Soc., Providence, R.I., 1968. N. Koksch and S. Siegmund, Pullback attracting inertial manifolds for nonautonomous dynamical systems, (submitted) (2001). Y. Latushkin and B. Layton, The optimal gap condition for invariant manifolds, Discrete and Continuous Dynamical Systems 5 (1999), no. 2, 233–268. A. Lunardi, Analytic semigroups and optimal regularity in parabolic problems, Progress in Nonlinear Diﬀerential Equations and Their Applications, vol. 16, Birkhäuser, Basel, 1995. M. Miklavčič, Applied functional analysis and partial diﬀerential equations, World Scientiﬁc, Singnapur, New Jersey, London, Hong Kong, 1998. X. Mora, Finite dimensional attracting manifolds in reaction-diﬀusion equations, Contemp. Math. 17 (1983), 353–360. W.C. Rheinboldt, Local mapping relations and global implicit function theorems, Trans. Am. Math. Soc. 138 (1969), 183–198. J.C. Robinson, Inertial manifolds and the cone condition, Dyn. Syst. Appl. 2 (1993), no. 3, 311–330. J.C. Robinson, The asymptotic completeness of inertial manifolds, Nonlinearity 9 (1996), 1325–1340. A.V. Romanov, Sharp estimates of the dimension of inertial manifolds for nonlinear parabolic equations, Russ. Acad. Sci., Izv., Math. 43 (1994), no. 1, 31–47. B. Schmalfuss, The stochastic attractor of the stochastic Lorenz system, Nonlinear Dynamics: Attractor Approximation and Global Behaviour. ISAM ’92 (N. Koksch, V. Reitmann, and T. Riedrich, eds.), Technical University Dresden, 1992, pp. 27–52. G.R. Sell and Y. You, Inertial manifolds: The non-self-adjoint case, J. Diﬀer. Equations 96 (1992), no. 2, 203–255. R. Temam, Inﬁnite-dimensional dynamical systems in mechanics and physics, Applied Mathematical Sciences, vol. 68, Springer, New York, 1988. R. Temam, Inﬁnite-dimensional dynamical systems in mechanics and physics, 2nd ed., Applied Mathematical Sciences, vol. 68, Springer, New York, 1997. M. Taboado and Y. You, Invariant manifolds for retarded semilinear wave equations, J. Diﬀer. Equations 114 (1994), no. 2, 337–369. Zongxin Wang and Xianling Fan, Inertial manifolds for nonautonomous evolution equations, Adv. Math., Beijing 26 (1997), no. 2, 185–186.

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Equadiﬀ 10 CD ROM Papers, pp. 267–274

The Asymptotic Properties of the Solutions of the n-th order Neutral Diﬀerential Equations Dáša Lacková Department of Applied Mathematics and Business Informatics, Faculty of Economics of Technical University Košice, B. Němcovej 32, 040 01 Košice, Slovak Republic, Email: [email protected]

Abstract. The aim of this paper is to deduce oscillatory and asymptotic behavior of the solutions of the n-th order neutral diﬀerential equation (x(t) − px(t − τ ))(n) − q(t)x(σ(t)) = 0, where σ(t) is a delayed or advanced argument.

MSC 2000. 34C10, 34K11 Keywords. Neutral diﬀerential equation, delayed argument, advanced argument

We consider the n-th order diﬀerential equation with a deviating argument of the form (1) (x(t) − px(t − τ ))(n) − q1 (t)x(σ1 (t)) = 0, where (i) n is even, (ii) p and τ are positive numbers, (iii) q1 (t), σ1 (t) ∈ C(R+ , R+ ), q1 (t) is positive, lim σ1 (t) = ∞. y→∞

By a solution of Eq.(1) we mean a function x : [Tx , ∞) → R which satisﬁes (1) for all suﬃciently large t. Such a solution is called oscillatory if it has a sequence of zeroes tending to inﬁnity; otherwise it is called nonoscillatory. Eq.(1) is said to be oscillatory if all its solutions are oscillatory. This is the preliminary version of the paper.

268

D. Lacková

We introduce the notation Qj (t) = qj (t)

m

pi ,

where m is a positive integer, j = 1, 2.

(2)

i=0

Lemma 1. Let z(t) be an n times diﬀerentiable function on R+ of constant sign, z (n) (t) ≡ 0 on [T0 , ∞) which satisﬁes z (n) (t)z(t) ≥ 0. Then there is an integer l, 0 ≤ l ≤ n such that n + l is even and z(t)z (i) (t) > 0, i−O

(−1)

z(t)z

(i)

(t) > 0,

0 ≤ i ≤ ^, ^ ≤ i ≤ n.

(3)

Lemma 1 is a well-known lemma of Kiguradze [5]. A function z(t) satisfying (3) is said to be a function of degree l. The set of all functions of degree l is denoted by Nl . If we denote by N the set of all functions satisfying z (n) (t)z(t) ≥ 0 then the set N has the following decomposition N = N0 ∪ N2 ∪ · · · ∪ Nn . Lemma 2. Let y(t) be a positive function of degree ^, ^ ≥ 2. Then t y(t) ≥

y (O−1) (s) t1

(t − s)O−2 ds. (^ − 2)!

(4)

The proof of this lemma is immediate from integration the identity y (l−1) (t) = y (l−1) (t). Theorem 3. Assume that m is a positive integer. Let σ1 (t) < t − τ, σ1 (t) ∈ C 1 , σ1 (t) ≥ 0.

(5)

Further assume that the diﬀerential equation 1 y (n) (t) + q1 (t)y(σ1 (t) + τ ) = 0 p

(6)

is oscillatory and the diﬀerential inequality z (n) (t) − Q1 (t)z(σ1 (t)) ≥ 0

(7)

has no solution of degree 0. Then every nonoscillatory solution of Eq.(1) tends to ∞ as t → ∞. Proof. Without loss of generality let x(t) be an eventually positive solution of Eq.(1) and deﬁne z(t) = x(t) − px(t − τ ). (8)

269

The Asymptotic Properties of Neutral Diﬀerential Equations

It is easy to see that z(t) < x(t).

(9)

From Eq.(1) we have z (t) > 0 for all large t, say t ≥ t0 . Thus z (t) are monotonous, i = 0, 1, . . . , n − 1. If z(t) < 0 eventually, then we set u(t) = −z(t). In the view of (8) 1 x(t − τ ) > u(t), p that is 1 x(t) > u(t + τ ). p One gets that u(t) is a positive solution of the inequality (n)

(i)

1 u(n) (t) + q1 (t)u(σ1 (t) + τ ) ≤ 0 p and by Kusano and Naito [1] the corresponding equation 1 u(n) (t) + q1 (t)u(σ(1 t) + τ ) = 0 p has a positive solution u(t). This contradicts that (6) is oscillatory. Therefore z(t) > 0. According to Lemma 1 we have two possibilities for z (t) : (a) z (t) > 0, for t ≥ t1 ≥ t0 , (b) z (t) < 0, for t ≥ t1 . For case (a) by Lemma 1 we obtain z(t) > 0, z (t) > 0, z (t) > 0. It implies that lim z(t) = ∞ and from (9) also lim x(t) = ∞. t→∞

t→∞

For case (b) Eq.(1) can be written in the form z (n) (t) − q1 (t)x(σ1 (t)) = 0. Using (8) we have z (n) (t) − q1 (t)z(σ1 (t)) − pq1 (t)x(σ1 (t) − τ ) = 0. Repeating this procedure m-times we arrive at z (n) (t) − q1 (t)

m

pi z(σ1 (t) − iτ ) − pm+1 q1 (t)x(σ1 (t) − (m + 1)τ ) = 0.

i=1

Since z(t) is decreasing, we get z (n) (t) − q1 (t)z(σ1 (t))

m

pi ≥ 0.

i=1

In the view of (2) we have z (n) (t) − Q1 (t)z(σ1 (t)) ≥ 0.

(10)

Hence z(t) is a solution of degree 0 of the inequality (10). This is a contradiction. A @

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D. Lacková

Corollary 4. Let m be a positive integer. Further assume that (5) holds, diﬀerential equation (6) is oscillatory and there exists k ∈ {0, 1, . . . , n − 1} such that

1 lim sup k!(n − k − 1)! t→∞

t k

n−k−1

[s − σ1 (t)] [σ1 (t) − σ1 (s)]

Q1 (s)ds > 1.

(11)

σ1 (t)

Then every nonoscillatory solution of Eq.(1) tends to ∞ as t → ∞. Proof. It follows from (11) and Theorem 1 of [2] that the diﬀerential inequality (7) has no solution of degree 0. Our assertion follows from Theorem 3. A @ Let us consider the n-th order diﬀerential equation with an advanced argument of the form (12) (x(t) − px(t − τ ))(n) − q2 (t)x(σ2 (t)) = 0, where (i), (ii) hold and moreover (iv) q2 (t), σ2 (t) ∈ C(R+ , R+ ), q2 (t) is positive, lim σ2 (t) = ∞. y→∞

We introduce the notation ∞ AO (t) =



(s − t)  × (n − ^ − 1)! n−O−1

q2 (s) t

σ2 (s)

t

 (t − u)  du ds (^ − 2)! O−2

(13)

for ^ = 2, 4, . . . , n − 2. Theorem 5. Assume that m is a positive integer and σ2 (t) − mτ > t, σ2 (t) ∈ C 1 , σ2 (t) ≥ 0, 0 1

for

^ = 2, 4, . . . , n − 2

(15)

and the diﬀerential inequality z (n) (t) − Q2 (t)z(σ2 (t) − mτ ) ≥ 0

(16)

has no solution of degree n. Then every nonoscillatory solution of Eg.(12) is bounded. Proof. Without loss of generality let x(t) be an eventually positive solution of Eq.(12) and deﬁne z(t) = x(t) − px(t − τ ). (17)

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The Asymptotic Properties of Neutral Diﬀerential Equations

From Eq.(12) we have z (n) (t) > 0 for all large t, say t ≥ t0 . Thus z (i) (t) are monotonous, i = 0, 1, . . . , n − 1. If z(t) < 0 eventually, then x(t) < px(t − τ ) < p2 x(t − 2τ ) < · · · < pk x(t − kτ ) for all large t, which implies lim x(t) = 0. t→∞

If z(t) > 0, then according to a Lemma 1 we have two possibilities for z (t) : (a) z (t) > 0, for t ≥ t1 ≥ t0 , (b) z (t) < 0, for t ≥ t1 . For case (a) we have two possibilities: (i) ∃ ^ ∈ 2, 4, . . . , n − 2, such that z(t) ∈ NO , (ii) ^ = n, i.e. z(t) ∈ Nn . For case (i) Eq.(12) can be written in the form z (n) (t) = q2 (t)x(σ2 (t)). Integrating this equation from t to ∞ n − ^ times and taking Lemma 2 into account, one gets ∞ z

(O)

(t) t

(s − t)n−O−1 ds q2 (s)x(σ2 (s)) (n − ^ − 1)! 

∞

q2 (s) t

(s − t)n−O−1  × (n − ^ − 1)!

∞ q2 (s)z(σ2 (s)) t

σ2 (s)

z (O−1) (u) t1

(s − t)n−O−1 ds (n − ^ − 1)! 

(t − u)O−2  du ds (^ − 2)!

Taking into account that σ2 (t) is nondecreasing, t ≥ t1 and z (O−1) (t) is increasing, the above inequalities led to z (O) (t) ≥ z (O−1) (t)AO (t).

(18)

Integration of the identity z (O) (t) = z (O) (t) from t1 to t provides t z

(O−1)

(t)

z (O) (s)ds z (O) (t)(t − t1 ), t1

which in the view of (18) implies 1 ≥ (t − t1 )AO (t). This contradicts (15). For case (ii) Eq.(12) can be written in the form z (n) (t) − q2 (t)x(σ2 (t)) = 0.

t ≥ t1 ,

272

D. Lacková

Using (17) we have z (n) (t) − q2 (t)z(σ2 (t)) − pq2 (t)x(σ2 (t) − τ ) = 0. Repeating this procedure m-times we arrive at z (n) (t) − q2 (t)

m

pi z(σ2 (t) − iτ ) − pm+1 q2 (t)x(σ2 (t) − (m + 1)τ ) = 0.

i=1

Since z(t) is increasing, we get z

(n)

(t) − q2 (t)z(σ2 (t) − mτ )

m

pi ≥ 0.

i=1

In the view of (2) we have z (n) (t) − Q2 (t)z(σ2 (t) − mτ ) ≥ 0.

(19)

Hence z(t) is a solution of degree n of the inequality (19). This is a contradiction. For case (b) we have z(t) > 0, z (t) < 0. Hence there exists lim z(t) = c > 0.

(20)

t→∞

If x(t) is unbounded eventually, then we can deﬁne the sequence {tn } where tn → ∞ as n → ∞ as follows. Let us choose tm for every m ∈ N such that x(tm ) = max{x(s), t0 ≤ s ≤ tm }. Since x(tm − τ ) = max{x(s), t0 ≤ s ≤ tm − τ } ≤ max{x(s), t0 ≤ s ≤ tm } = x(tm ), we have z(tm ) = x(tm ) − px(tm − τ ) ≥ x(tm ) − px(tm ) = (1 − p)x(tm ). This implies lim z(t) = ∞. This contradicts (20).

A @

t→∞

Corollary 6. Let m be a positive integer. Further assume that (14) and (15) hold and there exists k ∈ {0, 1, . . . , n − 1} such that 1 lim sup k!(n − k − 1)! t→∞

σ2 (t) k

n−k−1

[σ2 (s) − σ2 (t)] [σ2 (t) − s] t

Then every nonoscillatory solution of Eq.(12) is bounded.

Q2 (s)ds > 1.

(21)

The Asymptotic Properties of Neutral Diﬀerential Equations

273

Proof. It follows from (21) and Theorem 4 of [2] that the diﬀerential inequality (16) has no solution of degree n. Our assertion follows from Theorem 5. A @

Now we want to extend our previous results to more general diﬀerential equation. So let us consider the n-th order diﬀerential equation with both arguments (advanced and delayed) of the form (x(t) − px(t − τ ))(n) − q1 (t)x(σ1 (t)) − q2 (t)x(σ2 (t)) = 0,

(22)

where (i), (ii), (iii), (iv) hold. Theorem 7. Let m be a positive integer. Further assume that (5), (14) and (15) hold, diﬀerential equality (6) is oscillatory, diﬀerential inequality (7) has no solution of degree 0 and diﬀerential inequality (16) has no solution of degree n. Then every solution of Eg.(22) is oscillatory. Proof. Without loss of generality let x(t) be an eventually positive solution of Eq.(22). Then x(t) is solution of the inequality (x(t) − px(t − τ ))(n) − q1 (t)x(σ1 (t) ≥ 0. Using the same arguments as in Theorem 3 we can prove that x(t) tends to ∞ as t → ∞. On the other hand, x(t) is also solution of the inequality (x(t) − px(t − τ ))(n) − q2 (t)x(σ2 (t) ≥ 0. Now arguing exactly as in the proof of Theorem 5 we get that x(t) is bounded. This is a contradiction. A @

In Theorem 7 of [2] Kusano has presented conditions when the functional differential equation y (n) (t) − q1 (t)y(σ1 (t)) − q2 (t)y(σ2 (t)) = 0 is oscillatory. We have extended these conditions also for the neutral diﬀerential equation of the form (22). In a paper [6] Džurina and Mihalíková have presented suﬃcient conditions for all bounded solutions of the second order neutral diﬀerential equation with a delayed argument to be oscillatory. We have extended these conditions also for the n-th order neutral diﬀerential equation involving both delayed and advanced arguments.

274

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References 1. Kusano T., Naito M., Comparison theorems for functional diﬀerential equations with deviating argument, J. Math. Soc. Japan 33 1981. 2. Kusano T., On even functional diﬀerential equations with advanced and retarded arguments, Diﬀerential Equations 45 1985, 75–84. 3. Džurina J., Mihalíková B., Oscillation criteria for second order neutral diﬀerential equations, Mathematica Bohemica 12 2000, 145–153. 4. Džurina J., Mihalíková B., Oscillations of advanced diﬀerential equations, Fasciculi Math. 25 1995, 95–103. 5. Kiguradze T. I., On the oscillation of sollutions of the equation dm u/dtm + a(t)|u|n sign u = 0, Mat. Sb. 65 1964, (in Russian). 6. Džurina J., Mihalíková B., A note of unstable neutral diﬀerential equations of the second order, Fasciculi Math. 29 1999, 17–22.

Equadiﬀ 10, August 27–31, 2001 Prague, Czech Republic

Equadiﬀ 10 CD ROM Papers, pp. 275–278

A Class of Competing Models with Discrete Delay. Julio Marín1 and Mario Cavani2 1

Departamento de Matemática, Universidad de Oriente, Cumaná 6101, Venezuela. Email: [email protected] 2 Departamento de Matemática, Universidad de Oriente, Cumaná 6101, Venezuela. Email: [email protected]

Abstract. In this paper we consider a predator-prey models with discrete time lag. The prey, is assumed to regenerate in the absence of predators by logistic growth with carring capacity K. Two competing predators feed over the prey without interference between them. We assume the functional response of the predator population in the Michaelis-Menten forms. We show that the system is pointwise dissipative and the existence of a global attractor for the solutions of the model.

MSC 2000. 39B82, 34K60

Keywords. delay diﬀerential equations, predator-prey models, point dissipative

Research supported by Consejo de Investigación, Univeridad de Oriente, Proyecto No. C.I. 5-1003-1036/01

This is the preliminary version of the paper.

276

1

J. Marín and M. Cavani

Statement of the model

In 1978, Hsu, Hubbel and Waltman in the papers [4,5], have introduced the model S (t) = γS(t)(1 −

m1 X1 (t)S(t) m2 X2 (t)S(t) S(t) )− − K a1 + S(t) a2 + S(t)

X1 (t) =

m1 X1 (t)S(t) − D1 X1 (t), a1 + S(t)

X2 (t) =

m2 X2 (t)S(t) − D2 X2 (t). a2 + S(t)

(1)

where S(t) is the number of the prey at time t, Xi (t) is the number of the ith predator at time t, It is assumed that in the absence of predation, the prey growth logistically with carring capacity K. The predators are assumed to feed on the prey with saturing functional response to prey density. Speciﬁcally, we assume that the Michaelis-Menten Kinetics describes how the predators feed on the prey. The parameter mi is the maximun birth rate of the ith predator, Di is the death rate for the ith predator, ai is the half-saturation constant for the ith predator i.e., the prey density at with the functional respose of the predator is half maximal, the parameter γ is the intrisic rate of increase, while K is the carrying capacity for the prey population. In this model it is assumed that there are no signiﬁcant time lags in the system. A more realist situation occur if considered the past history of species are considered. This is , consider that prey population growth instantaneously but the dynamic of the predators depend on the prey density in the past by mean of a discrete delay. We get the following system

S (t) = γS(t)(1 −

m1 X1 (t)S(t) m2 X2 (t)S(t) S(t) )− − K a1 + S(t) a2 + S(t)

X1 (t) =

m1 X1 (t)S(t − τ1 ) − D1 X1 (t), a1 + S(t − τ1 )

X2 (t) =

m2 X2 (t)S(t − τ2 ) − D2 X2 (t). a2 + S(t − τ2 )

(2)

with initial condition S0 (θ) = φ(θ), θ ∈ [−τ, 0], φ ∈ C([−τ, 0], R+ ) and τ = max{τ1 , τ2 }, τ1 > 0, τ2 > 0, S(0) = φ(0) ≥ 0, X10 (θ) = X10 ≥ 0, X20 (θ) = X20 ≥ 0.

Competing Models with Discrete Delay.

2

277

Main Results

We deﬁne the parameters µ1 y µ2 as follow µi =

ai D i , i = 1, 2 mi − D i

and we suppose that µ1 = µ2 . In the following result we show that the solutions of system (2) is one positive and the pointwise dissipativity is established. Theorem 1. Let 3 E = {φ = (ψ1 , ψ2 , ψ3 ) ∈ C([−τ, 0], R+ ) : ψi (θ) ≥ 0, θ ∈ [−τ, 0], i = 1, 2, 3}

then, E is positively invariant under the ﬂow induced by the system (2). Futhermore, the system (2) is point dissipative and the absorbent set; that is, the set where all the solutions eventualy enters and remains is B = [0, K] × [0, M1 ] × [0, M2 ], a1 + K + 1 a2 + K + 1 and M2 = γ . where M1 = γ m1 m2 3 Corollary 2. The systems (2) have a global attractor in C([−τ, 0], R+ ). If µ1 = µ2 the point of equilibrium of the system (2) are

EK = (K, 0, 0), E0 = (0, 0, 0), γ a1 D 1 ∗ ∗ E = (s , (K − s∗ )(a1 + s∗ ), 0), s∗ = , m1 K m1 − D 1 γ a2 D 2 (K − s∗ )(a2 + s∗ )), s∗ = E∗ = (s∗ , 0, m2 K m2 − D 2 where mi > Di , 0 < s∗ < K y 0 < s∗ < K. Lemma 3. If Xi (t) survives then 0 < µi < K. Theorem 4. – If

– E0 = (0, 0, 0) is unstable.

a1 D 1 > K, and m1 − D 1 a2 D 2 b) m2 − D2 ≤ 0 or > K, then m2 − D 2 lim S(t) = K y lim Xi (t) = 0, i = 1, 2. a) m1 − D1 ≤ 0 or

t−→∞

t−→∞

The following lemma gives us a necessary condition for the extintion of X1 and X2 . Lemma 5. If lim Xi (t) = 0, i = 1, 2, then t−→∞

mi − D i 1 , i = 1, 2. ≤ ai D i ai + K

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Lemma 6. mi − Di ≤ 0 or 0 < K < Lemma 7. 0 <

ai D i mi − D i 1 . if and only if < mi − D i ai D i ai + K

ai D i mi − D i 1 < K if and only if > . mi − D i ai D i ai + K

Lemma 8. Let 0 <

2a1 D1 a1 D 1 < K < a1 + . If m2 − D2 ≤ 0 or if m1 − D 1 m1 − D 1

a1 D 1 a2 D 2 < . Then critical point (s∗, x∗1 , 0) is local asymptotically stable, m1 − D 1 m2 − D 2 a1 D 1 γ (K − s∗ )(a1 + s∗ ). , x∗ = where s∗ = m1 − D 1 1 mK Theorem 9. Let a1 D 1 < K, and a) 0 < m1 − D 1 a2 D 2 2a1 D1 b) m2 − D2 ≤ 0 or > K. If K < a1 + , Then m2 − D 2 m1 − D 1 a1 D 1 , lim S(t) = S ∗ = m − D1 γ 1 (K − s∗ )(a1 + s∗ ), lim X1 (t) = t−→∞ Km1 lim X2 (t) = 0. t−→∞

t−→∞

References 1. Diekman, O., Van Gils, S. A., Verduyn Lunel, S. M., Walther, H. O. Delay Equations Functional-Complex and Nonlinear Analysis, Springer-Verlag, New York.(1997). 2. Freedman, H. I., So, J., Waltman, P. Coexistence in a model of competition in the chemostat incorporating discrete delays, SIAM J. Appl. Math., 49 (1989), 859-870. 3. Hale, J. K., Lunel, S. V. Introduction to Functional Diﬀerential Equations, SpringerVerlag, New York (1997). 4. Hsu, S. B., Hubbell, S. P., Waltman, P. Competing Predators, SIAM J. Appl. Math. 35 (1978), 617-625. 5. Hsu, S. B., Hubbell, S. P., Waltman, P. A contribution to the theory of competing predators, Ecol. Monogr., 48 (1978), 337-349. 6. Kuang, Y. Delay Diﬀerential Equations with Aplications in Population Dynamics, Academics Press, Boston, (1979). 7. Marín J., Cavani M. A two-predators and one-prey model with discrete delay (to appear). 8. Wolkowicz G. and Xia H., Global asymptotic behaviour of a chemostat model with discrete delays, SIAM J. Appl. Math. 57 (1981), 1019-1043.

Equadiﬀ 10, August 27–31, 2001 Prague, Czech Republic

Equadiﬀ 10 CD ROM Papers, pp. 279–287

Zero Convergent Solutions for a Class of p-Laplacian Systems Mauro Marini1 , Serena Matucci2 and Pavel Řehák3 1

2

Department of Electronic and Telecommunications, Faculty of Engineering, University of Florence, via S. Marta 3, I-50139 Florence, Italy Email: [email protected] Department of Electronic and Telecommunications, Faculty of Engineering, University of Florence, via S. Marta 3, I-50139 Florence, Italy Email: [email protected] 3 Mathematical Institute, Academy of Sciences of Czech Republic, Žižkova 22, CZ-616 62 Brno, Czech Republic Email: [email protected]

Abstract. We present some results about the existence of positive decaying solutions for a class of systems of two second order coupled nonlinear equations with p-laplacian operator, p > 1. In addition the generalized Emden-Fowler type systems are considered and necessary and suﬃcient conditions are given in order for the system to have regularly and/or strongly decaying solutions. The results here presented complete the ones in [10]. MSC 2000. 34B15, 34C11. Keywords. p-laplacian operator, regularly decaying solutions, strongly decaying solutions, ﬁxed point theorems.

1

Introduction

In this contribution we present some asymptotic results for a system of two coupled nonlinear equations with the 1-dimensional p-laplacian operator Ψp (u) = |u|p−2 u, u ∈ R, of the form [r(t)Ψp (x )] = −F (t, x, y) (S) [q(t)Ψk (y )] = G(t, x, y)

Supported by the Grants No. 201/01/P041 and No. 201/01/0079 of the Czech Grant Agency and by Italian C.N.R.

This is in the ﬁnal form and completes recent results by the authors.

280

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where we assume p, k > 1 and (i) r, q : [0, ∞) -→ (0, ∞) continuous, (ii) F, G : [0, ∞) × (0, δ] × (0, δ] -→ (0, ∞) continuous, with δ a suitably small positive constant, (iii) ∃f, g : [0, ∞) × (0, δ] -→ (0, ∞) continuous s.t. F (t, u, v) ≤ f (t, v), G(t, u, v) ≤ g(t, u), ∀(t, u, v) ∈ [0, ∞) × (0, δ] × (0, δ]. Systems of type (1) comes out in the study of the existence of radial solutions for nonlinear coupled elliptic systems with p-laplacian ∆p u = div(|∇u|p−2 ∇u) ∆p u = F (|x|, u, v) ∆k v = G(|x|, u, v) in exterior domains Ea = {x ∈ RN : |x| ≥ a}, N ≥ 2, p > 1, k > 1. Systems of partial diﬀerential equations of this form have been object of increasing interest in the last years, due to their relevance in applied sciences, especially in plasma physics, biomathematics, chemistry and, in general, in reaction-diﬀusion problems. Some relevant examples can be found in [2], [4], [5], [6], [7] and in the book [3] to which we address the interested reader. The particular case of an ordinary diﬀerential system of two coupled generalized Emden-Fowler equations of the form (p(t)|x |α−1 x ) = ϕ(t)y λ (q(t)|y |β−1 y ) = ψ(t)xµ .

(2)

has been recently studied in [8], [9], [12], with regard to existence of positive decreasing solutions, but the sign condition on the nonlinear terms is opposite to our case. The results here presented are therefore both more general and complementary to the corresponding ones in [8], [9],[12]. As both the nonlinear terms in (2) have positive sign, the dynamics which is described is completely diﬀerent from our case; further (2) seems to be more close to the case of a single equation [r(t)Ψp (x )] = f (t, x). that has been considered, for instance, in [1], [11] (see also the references therein). Here we want to present some existence results which complete the ones obtained by the authors in [10]; in particular system (1) is more general than the one considered in [10] since the nonlinear terms F and G in (1) depend on both the unknowns x and y. No assumption is done on the continuity and on the boundedness of F and G in a neighborhood of v = 0 and of u = 0 respectively, thus we can treat the regular case and the singular case at the same time, and the forced case as well. For sake of completeness we will also quote the necessary and suﬃcient conditions that can be derived for a Emden-Fowler type system, whose proofs can be found in [10]. We end this section stating some deﬁnitions.

Zero Convergent Solutions of p-Laplacian Systems

281

In this contribution we deal with the existence of decaying solutions of (1), i.e. solutions (x, y) such that x and y are eventually positive nonincreasing and x(∞) = y(∞) = 0. If (x, y) is a decaying solution of (1), then the ﬁrst quasiderivative of x, x[1] := r(t)Ψp (x (t)) is eventually negative decreasing and the ﬁrst quasiderivative of y, y [1] := q(t)Ψk (y (t)) is eventually negative increasing. Thus they admit limit as t → ∞ and −∞ ≤ x[1] (∞) < 0, −∞ < y [1] (∞) ≤ 0. A decaying solution (x, y) is called regularly decaying if x[1] is bounded and y [1] tends to a (negative) nonzero limit, strongly decaying if x[1] is bounded and y [1] tends to zero. The set of regularly decaying solutions will be denoted by DR and the set of strongly decaying solutions with DS . In the next section we will ﬁnd minimal conditions in order that (1) has solutions in the class DR and in DS , using both a topological approach (the Shauder-Tychonoﬀ ﬁxed point theorem in a Frechét space) and integral inequalities. This method allows us also to obtain an asymptotic estimate of the convergence rate of the solutions. Finally, we close this section by giving two necessary conditions that can be easily proved (see [10]); the ﬁrst one will be always assumed in the following, the second one will be assumed only in the existence results in DR : – If (1) has at least one decaying solution then the condition ∞ 1 Ψp∗ dt < ∞ r(t) 0

(H1)

is satisﬁed, where p∗ is the conjugate numbers of p, i.e. 1/p + 1/p∗ = 1. – If (1) has at least one regularly decaying solution then conditions (3) and ∞ 1 Ψk ∗ dt < ∞ (H2) q(t) 0 are satisﬁed, where k ∗ is the conjugate numbers of k.

2

Existence results in DR and in DS for (1)

Concerning the existence of solutions in the class DR , the following result holds. Theorem 1. Assume (3), (4) and 1. there exist a positive continuous functions ϕ1 on [0, ∞), a nonnegative continuous function ϕ2 on [0, ∞) and a monotone positive continuous function fˆ on (0, δ] such that for (t, u) ∈ [0, ∞) × (0, δ] f (t, u) ≤ ϕ1 (t)fˆ(u) + ϕ2 (t), ∞ ∞ ∞ 1 ϕ2 (t) dt < ∞, ϕ1 (t)fˆ Ψk ∗ dτ dt < ∞; q(τ ) 0 0 t

(5)

282

M. Marini, S. Matucci and P. Řehák

2. there exist a positive continuous functions γ1 on [0, ∞), a nonnegative continuous functions γ2 on [0, ∞) and a monotone positive continuous function gˆ on (0, δ] such that for (t, u) ∈ [0, ∞) × (0, δ]

∞ 0

g (u) + γ2 (t), g(t, u) ≤ γ1 (t)ˆ ∞ ∞ 1 γ2 (t) dt < ∞, γ1 (t)ˆ g Ψp∗ dτ dt < ∞. r(τ ) 0 t

(6)

Then (1) has solutions in the class DR . Proof. The argument is a slight generalization of the one employed in [10] in the proof of Theorem 1, in which the particular case of a nonforced system is considered. We denote by C[t0 , ∞) the locally convex space of all continuous functions deﬁned on [t0 , ∞) with the topology of uniform convergence on any compact subintervals of [t0 , ∞). Thus C[t0 , ∞) is a Fréchet space. We choose t0 ≥ 0 such that ∞ ∞ ∞ 1 1 1 ˆ ϕ2 (t) dt ≤ , ϕ1 (t)f Ψk ∗ dτ dt ≤ , 4 q(τ ) 4 t t t ∞ (7) 0∞ 0∞ 1 1 1 γ2 (t) dt ≤ , γ1 (t)ˆ g Ψp∗ dτ dt ≤ . 4 r(τ ) 4 t0 t0 t

Let

∞

Ψk ∗

m = max t0

1 q(τ )

∞

Ψp∗

dτ ; t0

1 r(τ )

dτ

and, without loss of generality, assume 2m ≤ δ. Consider the set Ω ⊂ C[t0 , ∞) × C[t0 , ∞) given by Ω = (u, v) ∈ C[t0 , ∞) × C[t0 , ∞) such that: ∞ ∞ M1 M2 Ψp∗ Ψp∗ dτ ≤ u(t) ≤ dτ, r(τ ) r(τ ) t t ∞ ∞ N1 N2 Ψk ∗ Ψk ∗ dτ ≤ v(t) ≤ dτ q(τ ) q(τ ) t t where Mi , Ni , i = 1, 2 are four suitable positive constants. On this set we deﬁne the operator T with values in C[t0 , ∞) × C[t0 , ∞), by T (u, v) = (T1 (u, v), T2 (u, v)) ∞ ∞ 1 T1 (u, v)(t) = Ψp∗ F (τ, u(τ ), v(τ )) dτ ds, M2 − r(s) t s ∞ ∞ 1 Ψk ∗ G(τ, u(τ ), v(τ )) dτ ds. T2 (u, v)(t) = N1 + q(s) t s We have to prove that T is continuous and maps Ω into a compact subset of Ω.

283

Zero Convergent Solutions of p-Laplacian Systems

First we consider the case fˆ and gˆ nondecreasing on (0, δ]; for this case let M1 = N1 = 1/2, M2 = N2 = 1. (i) T (Ω) ⊂ Ω. The positivity of F and G immediately implies, for every (u, v) ∈ Ω ∞ ∞ 1 1/2 Ψp∗ Ψk ∗ T1 (u, v)(t) ≤ ds, T2 (u, v)(t) ≥ ds. r(s) q(s) t t The assumptions on F and G, together with (5) and (7), imply that for any (u, v) ∈ Ω it holds ∞ ∞ ∞ ∞ 1 1 F (t, u(t), v(t)) dt ≤ ϕ1 (t)fˆ Ψk ∗ ϕ2 (t) dt ≤ , dτ dt + q(τ ) 2 t t t t0 ∞ 0∞ 0∞ ∞ 1 1 G(t, u(t), v(t)) dt ≤ γ1 (t)ˆ g Ψp∗ γ2 (t) dt ≤ . dτ dt + r(τ ) 2 t0 t0 t t0 These inequalities imply ∞ 1/2 ∗ T1 (u, v)(t) ≥ Ψp ds, r(s) t

T2 (u, v)(t) ≤

∞

Ψ

k∗

t

1 q(s)

ds,

and (i) follows. (ii) T (Ω) is relatively compact. Since T (Ω) ⊂ Ω, functions in T (Ω) are equibounded. The equicontinuity of functions in T (Ω) easily follows by observing that, in virtue of the above estimates, for any (u, v) ∈ Ω it holds 1 1 , 0 ≤ −(T2 (u, v)) (t) ≤ Ψk∗ . 0 ≤ −(T1 (u, v)) (t) ≤ Ψp∗ r(t) q(t) (iii) T is continuous in Ω ⊂ C[t0 , ∞) × C[t0 , ∞). Let {(un , vn )}, n ∈ N, be a sequence in Ω which uniformly converges on every compact interval I of [t0 , ∞) to (¯ u, v¯) ∈ Ω. In view of the assumption on F and of the upper bound (5), the Lebesgue dominated convergence theorem and the uniform

∞ convergence on I of the sequence {F (t, un , vn (t))} imply that the sequence t F (τ, un (τ ), vn (τ )) dτ ∞ uniformly converges to t F (τ, u¯(τ ), v¯(τ )) dτ on I. Analogously, the upper bound 0 ≤ Ψp∗

1 r(t)

1− t

∞

F (τ, un (τ ), vn (τ )) dτ

≤ Ψp∗

1 r(t)

allows us to apply again the Lebesgue dominated convergence theorem to the sequence ∞ 1 F (τ, un (τ ), vn (τ )) dτ . Ψp∗ 1− r(t) t u, v¯), that It follows that the sequence {T1 (un , vn )} uniformly converges on I to T1 (¯ is the continuity of T1 . The argument for the continuity of T2 is quite the same and (iii) is proved.

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Since (i), (ii), (iii) are satisﬁed, the Schauder-Tychonoﬀ theorem implies that the operator T has a ﬁxed point (z, w) ∈ Ω. It is easy to show that (z, w) is solution of (1) on [t0 , ∞) and (z, w) ∈ DR . In the case fˆ and gˆ nonincreasing on (0, δ] the assertion can be proved by using an argument analogous to that given in the ﬁrst part of the proof and choosing M1 = N1 = 1, M2 = N2 = 3/2. Finally, when fˆ is nondecreasing and gˆ nonincreasing on (0, δ] or vice-versa, it is suﬃcient to choose M1 = N2 = 1, M2 = 3/2, N1 = 1/2 or N1 = M2 = 1, M1 = 1/2, N2 = 3/2 respectively. The details are left to the reader. A @ Other existence results in the class DR can be found in [10] (Th. 1 and Prop. 1) in case that the nonlinearities in (1) satisfy some additional assumptions. Remark 2. The assumptions in Theorem 1 are quite general and Theorem 1 can be applied also to systems with singular forcing terms; an interesting case is when F (t, u, v) = ϕ(t)uλ1 v −λ2 and/or G(t, u, v) = ψ(t)u−µ1 v µ2 , with λ1 , µ2 ≥ 0 and λ2 , µ1 > 0. With regards to existence of solutions of (1) in the class DS , here we treat only the case in which F (t, u, v) and G(t, u, v) are both singular in a neighborhood of v = 0 and u = 0 respectively. The remaining cases, i.e. F and G both regular functions or one regular and the other singular in a neighborhood of v = 0 and u = 0 respectively, can be obtained with minor changes from the results in [10] (Th. 2, Th. 3, Prop. 2). The details are left to the reader. To state the following existence, we need a further assumption in addition to (i)-(iii). Theorem 3. Assume (iv) ∃h : [0, ∞) × (0, δ] -→ (0, ∞) continuous s.t. G(t, u, v) ≥ h(t, u), ∀(t, u, v) ∈ [0, ∞) × (0, δ] × (0, δ]. Let (3) be satisﬁed and suppose that there exist two positive continuous functions γ1 , ϕ1 on [0, ∞), two nonnegative continuous functions γ2 , ϕ2 on [0, ∞), three ˆ gˆ, fˆ on (0, δ] and two positive connondecreasing positive continuous functions h, stants 0 < N < M , such that for (t, u) ∈ [0, ∞) × (0, δ]: ˆ h(t, u) ≥ γ1 (t)h(u) + γ2 (t),

g(t, u) ≤ γ1 (t)ˆ g (u) + γ2 (t), f (t, u) ≤ ϕ1 (t)fˆ(u) + ϕ2 (t) ∞ ∞ γ2 (t) dt < ∞, ϕ2 (t) dt < ∞, 0 ∞ 0 ∞ ∞ DN (t) CM (s) ˆ Ψk ∗ ϕ1 (t)f Ψk ∗ dt < ∞, ds dt < ∞, q(t) q(s) 0 0 t where

N dτ + γ2 (s) ds r(τ ) t s ∞ ∞ M ˆ CM (t) = Ψp∗ γ1 (s)h dτ + γ2 (s) ds. r(τ ) t s

DN (t) =

∞

g γ1 (s)ˆ

∞

Ψp∗

285

Zero Convergent Solutions of p-Laplacian Systems

Then (1) has solutions in the class DS . Proof. Choose t0 ≥ 0 such that ∞ ∞ CM (s) ˆ Ψk ∗ ϕ1 (t)f ds + ϕ2 (t) dt ≤ M − N. q(s) t0 t

Let

∞

m1 = max

Ψp∗ t0

M r(t)

∞

Ψk ∗

dt, t0

DN q(t)

dt

and, without loss of generality, suppose 2m1 < δ. Deﬁne the set Ω as follows Ω = (u, v) ∈ C[t0 , ∞) × C[t0 , ∞) such that: ∞ ∞ N M Ψp∗ Ψp∗ dτ ≤ u(t) ≤ dτ, r(τ ) r(τ ) t t ∞ ∞ CM (σ) DN (σ) Ψk ∗ Ψk ∗ dσ ≤ v(t) ≤ dσ q(σ) q(σ) t t Consider the operator Tˆ : Ω -→ C[t0 , ∞) × C[t0 , ∞), T˜(u, v) = (T˜1 (u, v), T˜2 (u, v)), given by ∞ ∞ 1 T˜1 (u, v)(t) = Ψp∗ F (τ, u(τ ), v(τ )) dτ ds, M− r(s) t s ∞ ∞ 1 T˜2 (u, v)(t) = Ψk ∗ G(τ, u(τ ), v(τ )) dτ ds. q(s) s t ∞ M Clearly T˜1 (u, v)(t) ≤ t Ψp∗ r(τ ) dτ for every (u, v) ∈ Ω, t ≥ t0 . The following inequalities hold for s ≥ t0 : ∞ ∞ ∞ ϕ1 (t)fˆ(v(t)) + ϕ2 (t) dt F (t, u(t), v(t))dt ≤ f (t, v(t))dt ≤ s ∞ s ∞ s CM (s) ≤ Ψk ∗ ϕ1 (t)fˆ ds + ϕ2 (t) dt ≤ M − N ; q(s) t0 t ∞ ∞ ∞ γ1 (t)ˆ G(t, u(t), v(t))dt ≤ g(t, u(t))dt ≤ g (u(t)) + γ2 (t) dt s s ∞ s ∞ N ≤ g Ψp∗ γ1 (t)ˆ ds + γ2 (t) dt = DN (s); r(s) s t ∞ ∞ ∞ ˆ γ1 (t)h(u(t)) G(t, u(t), v(t))dt ≥ h(t, u(t))dt ≥ + γ2 (t) dt s s ∞ s ∞ M ˆ ≥ Ψp∗ γ1 (t)h ds + γ2 (t) dt = CM (s). r(s) s t From the above estimates it follows that the operator T˜ maps Ω into itself. To apply Tychonov ﬁxed point theorem, it is suﬃcient to show that T˜(Ω) is relatively

286

M. Marini, S. Matucci and P. Řehák

compact and T˜ is continuous in Ω. The argument is similar to that given in the proof of Theorem 1 and the details are left to the reader. Then the operator T˜ has at least one ﬁxed point (z, w). It is easy to show that (z, w) is solution of (1) on [t0 , ∞) and both components of the ﬁxed point are positive in each interval of [t0 , ∞): so (z, w) ∈ DS . A @ Remark 4. Unlike Theorem 1, that can be applied to both the regular case and the singular one (see Remark 2), Theorem 3 is for the singular case only. Existence results for the regular case can be found in [10] and applied to (1) with minor changes. This fact depends on the structure of the operator T˜ whose ﬁxed points are singular solutions of (1). The conditions stated in Theorem 1 and in Theorem 3 are sharp conditions for existence of solutions of (1) in DR and in DS respectively, since these conditions becomes also necessary in case of a forced Emden-Fowler type system [r(t)Ψp (x )] = −ϕ1 (t)Ψµ (y) − ϕ2 (t),

µ = 1

(8)

ν = 1,

[q(t)Ψk (y )] = γ1 (t)Ψν (x) + γ2 (t),

where we assume p, k > 1, µ, ν = 1, ϕ1 , γ1 : [0, ∞) -→ (0, ∞) continuous, ϕ2 , γ2 : [0, ∞) -→ [0, ∞) continuous, and ϕ2 , γ2 ∈ L1 (0, ∞). The following result holds, whose proof can be found in [10] Theorem 5 ([10] Th. 3, Th. 4, Prop. 3). • The Emden-Fowler forced system (8) has solutions in the class DR if and only if conditions (3), (4) and

µ−1 1 ϕ1 (t) Ψk ∗ dt < ∞ dτ q(τ ) 0 t ∞ ν−1 ∞ 1 γ1 (t) Ψp∗ dt < ∞ dτ r(τ ) 0 t

∞

∞

are satisﬁed. • The Emden-Fowler forced system (8) has solutions in the class DS if and only if conditions (3) and

∞

ϕ1 (t) 0

are satisﬁed, where 6 ∞ 1 ϑ(t) = Ψk∗ q(t) t

µ−1

∞

dt < ∞

ϑ(s) ds t

γ1 (s)

∞

Ψp∗ s

1 r(τ )

ν−1 dτ

!

7

+ γ2 (s) ds .

Comments and examples that illustrate the previous result can be found in [10].

Zero Convergent Solutions of p-Laplacian Systems

287

References 1. M. Cecchi, Z. Dosla, M. Marini, On nonoscillatory solutions of diﬀerential equations with p-Laplacian, Advances in Math. Sci. and Appl. 11 1 (2001), 419-436. 2. P. Clémen, J. Fleckinger, E. Mitidieri and F. de Thélin, Existence of positive solutions for a nonvariational quasilinear elliptic system, J. Diﬀ. Eq. 166 (2000), 455–477. 3. J.I. Diaz, Nonlinear Partial Diﬀerential Equations And Free Boundaries, Volume I: Elliptic Equations, Research Notes in Mathematics 106, Pitman Publishing Ltd., London, 1985. 4. J. Fleckinger, R. Manásevich and F. de Thélin, Global bifurcation for the ﬁrst eigenvalue for a system of p-laplacians, Math. Nachr. 182 (1996), 217–242. 5. J. Fleckinger, R. Pardo and F. de Thélin, Four parameter bifurcation for a pLaplacian system, Elect. J. Diﬀ. Eq. 2001 (2001), 1–15. 6. Y.X. Huang and J.W.H. So, On bifurcation and existence of positive solutions for a certain p-Laplacian system, Rocky Mount. J. Math. 25 (1995), 285–297. 7. D. Jiang and H. Liu, On the existence of nonnegative radial solutions for p-Laplacian elliptic systems, Ann. Polonici Math. LXXLI (1999), 19–29. 8. T. Kusano and T. Tanigawa, Existence of positive decreasing solutions to a class of second-order nonlinear diﬀerential systems, Fukuoka Univ. Sci. Rep. 29 2 (1999), 231–242. 9. T. Kusano and T. Tanigawa, Positive decreasing solutions of systems of second order singular diﬀerential equations, J. of Inequal. and Appl. 5 (2000), 581–602. 10. M. Marini, S. Matucci and P. Řehák, On decay of coupled nonlinear diﬀerential systems, to appear in Adv. Math. Sci. Appl. 11. S. Matucci, On asymptotic decaying solutions for a class of second order diﬀerential equations, Arch. Math.(Brno) 35 (1999), 275-284. 12. T. Tanigawa, Positive decreasing solutions of systems of second order quasilinear diﬀerential equations, Funk. Ekvac. 43 (2000), 361–380.

Equadiﬀ 10, August 27–31, 2001 Prague, Czech Republic

Equadiﬀ 10 CD ROM Papers, pp. 289–296

Nonoscillatory Solutions for Nonlinear Discrete Systems Mauro Marini1 , Serena Matucci2 and Pavel Řehák3 1

2

Department of Electronics and Telecommunications, Faculty of Engineering, University of Florence, via S. Marta, I-50139 Florence, Italy Email: [email protected] Department of Electronics and Telecommunications, Faculty of Engineering, University of Florence, via S. Marta, I-50139 Florence, Italy Email: [email protected] 3 Mathematical Institute, Academy of Sciences of Czech Republic, Žižkova 22, CZ-616 62 Brno, Czech Republic Email: [email protected]

Abstract. We investigate some asymptotic properties of the nonlinear forced diﬀerence system ˆk , ∆(rk Φα (∆xk )) − σϕk f (yk+1 ) = σ ϕ ∆(qk Φβ (∆yk )) − ψk g(xk+1) = ψˆk . In particular we give necessary and suﬃcient conditions for existence of the so-called regularly decaying solutions and thereby we complete the results presented in [10]. MSC 2000. 39A10, 39A11. Keywords. (singular) nonlinear diﬀerence system, asymptotic behavior, positive decreasing solutions.

1

Introduction

In [10] the authors investigated certain discrete asymptotic boundary value problems on the discrete interval [m, ∞) := {m, m + 1, ..}, m ∈ Z, associated to the

Supported by the Grants No. 201/01/P041 and No. 201/01/0079 of the Czech Grant Agency and by C.N.R. of Italy

This is the preliminary version of the paper.

290

M. Marini, S. Matucci and P. Řehák

nonlinear forced diﬀerence system ∆(rk Φα (∆xk )) − σϕk f (yk+1 ) = σ ϕˆk , ∆(qk Φβ (∆yk )) − ψk g(xk+1 ) = ψˆk .

(1)

In particular, necessary and suﬃcient conditions for the existence of the so-called strongly decaying solutions of (1) were presented. The principal aim of this contribution is to complete those results examining also the existence of the so-called regularly decaying solutions. For the deﬁnitions of these concepts see Deﬁnition 1. In system (1) we assume that {ϕk }, {ψk }, {rk }, {qk } are real positive sequences deﬁned for any k ≥ m, the forcing terms {ϕ k }, {ψk } are real nonnegative sequences p−1 deﬁned for any k ≥ m, Φp (u) = |u| sgn u with p > 1 is the one-dimensional p-laplacian operator, f, g : R+ → R+ are monotone continuous functions and σ ∈ {−1, 1}. There are several motivations for the investigation of (1): • System (1) arises in the discretization process of diﬀerential systems with p-laplacian operator. For results concerning the role of system (1) in various applications, we refer the reader to [6], [9], [12] and to the references contained therein. • The study of system (1) is motivated also by certain results obtained for scalar second order diﬀerence equations of the form ∆(rk Φα (∆xk )) = σϕk f (xk+1), which attracted a considerable attention in recent years, see, e.g., [2], [3], [4], [7], [8], [11], [15]. Other interesting contributions can be found also in the monograph [1]. • System (1) can be rewritten in the form of a fourth-order nonlinear equation of the type ∆2 (rk ∆2 uk ) + pk f (uk+2 ) = 0. For results from this point of view see for instance [13], [14]. Here we are interested in the existence of positive nonoscillatory solutions, which are asymptotically decreasing towards zero. A nonoscillatory solution of (1) is a vector sequence (x, y) = ({xk }, {yk }) satisfying (1) for k ≥ m, and such that both components {xk }, {yk } are eventually of ﬁxed sign. Definition 1. A solution (x, y) of (1) is said to be – decaying, if x, y are eventually positive decreasing and lim xk = 0 = lim yk ;

k→∞

k→∞

– regularly decaying, if it is decaying and lim rk Φα (∆xk ) = −Ax ,

k→∞

where Ax , By are positive constants;

lim qk Φβ (∆yk ) = −By ,

k→∞

Nonoscillatory Solutions for Nonlinear Discrete Systems

291

– strongly decaying, if it is decaying and limk→∞ qk Φβ (∆yk ) = 0, −Cx for σ = −1, lim rk Φα (∆xk ) = 0 for σ = 1, k→∞ where Cx is a positive constant. Note that the constant Cx cannot be zero when σ = −1, since in this case the quasidiﬀerence rk Φα (∆xk ) is eventually negative decreasing. As already mentioned at the beginning of this section, the main purpose of this paper is to complete the results presented in [10], in which the existence of strongly decaying solutions is considered. To this end we will give here necessary and sufﬁcient conditions for existence of regularly decaying solutions (see Section 2), in view of their crucial role in a variety of physical applications, as already claimed. We will also restate the main results proved in [10] in order to have a comparison with our new results (see Section 3). Note that both the regular case (i.e. with the nonlinearities f, g bounded in a right neighborhood of zero) and the singular case (i.e. with f or g unbounded in a right neighborhood of zero) will be considered. Some remarks and comments are given throughout this contribution. Finally, nontrivial prototypes of nonlinearities f, g are the one-dimensional Laplacians Φγ , Φδ , respectively, with γ, δ = 1 (i.e., with possible singularities at zero). Then (1) leads to the Emden-Fowler type system ∆(rk Φα (∆xk )) = σϕk Φγ (yk+1 ) ∆(qk Φβ (∆yk )) = ψk Φδ (xk+1 )

(2)

as a special case. Some of our main results remain valid for more general systems of the form ∆(rk α(∆xk )) = σF (k, yk+1 , xk+1 ) ∆(qk β(∆yk )) = G(k, xk+1 , yk+1 ),

(3)

where the functions α and β are monotone continuous increasing with α(0) = 0 = β(0), and F, G are positive continuous on {m, m + 1, . . . } × (0, ε] × (0, ε] with some ε > 0 and bounded with respect to the third variable.

2

Regularly Decaying Solutions

Denote with α∗ the conjugate number of α i.e. 1/α + 1/α∗ = 1, and analogously for β. The following necessary and suﬃcient condition holds: Theorem 2. System (1) has at least one regularly decaying solution if and only if ∞ ∞ 1 1 < ∞, < ∞, (4) Φα∗ Φβ∗ rk qk k=m

∞ k=m

ϕˆk < ∞,

k=m ∞

k=m

ψˆk < ∞,

(5)

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and there exist A > 0, B > 0 such that   ∞ ∞ A  < ∞, ϕk f  Φβ∗ qj k=m j=k+1   ∞ ∞ B  < ∞. ψk g  Φα∗ rj k=m

(6)

(7)

j=k+1

In addition, if σ = 1, then this solution is decreasing for any k ≥ m. Proof. The “if part”. Choose an integer T ≥ m such that     ∞ ∞ ∞ ∞ 1  1  ϕk f  Φβ∗ ψk g  Φα∗ < 1/2, < 1/2. qj rj k=T

j=k+1

k=T

j=k+1

Denote with ^∞ T the Banach space of all bounded sequences deﬁned for k ≥ T , endowed with the topology of the supremum norm, and consider the set Ω ⊂ ∞ ^∞ T × ^T given by   Ω=



∞

∞ (u, v) = ({uk }, {vk }) ∈ ^∞ T × ^T :

Φα∗

j=k+1

≤ uk ≤

∞

Φα∗

j=k+1

M2 rj

,

∞ j=k+1

Φβ∗

N1 qj

M1 rj

≤ vk ≤

≤ ∞

Φβ∗

j=k+1

 N2  , qj 

where M1 , N1 , M2 , N2 are suitable positive constants which will be determined ∞ later. Consider the operator T : Ω → ^∞ T × ^T deﬁned by T (u, v) = (T1 (v), T2 (u)) = ({(T1 (v))k }, {(T2 (u))k }), where



 ∞ Mτ (σ) σ (T1 (v))k = Φα∗  + [ϕi f (vi+1 ) + ϕˆi ] , rj rj i=j j=k   ∞ ∞ N1 1 ψi g(ui+1 ) + ψˆi  Φβ∗  + (T2 (u))k = qj qj i=j ∞

j=k

with τ (σ) = 1 for σ = 1 and τ (σ) = 2 for σ = −1. In order to show that T has a ﬁxed point in Ω, for the Schauder-Tychonoﬀ ﬁxed point theorem it is suﬃcient ∞ to verify that: (i) Ω is a nonempty, closed and convex subset of ^∞ T × ^T , (ii) T (Ω) ⊆ Ω, (iii) T (Ω) is relatively compact, (iv) T is continuous in Ω.

Nonoscillatory Solutions for Nonlinear Discrete Systems

293

The validity of (i) is obvious, and furthermore it is not diﬃcult to show that T maps Ω into itself, i.e. (ii) holds. To show it, if f, g are nondecreasing it is suﬃcient to choose M1 = N1 = 1/2 and M2 = N2 = 1. If f, g are nonincreasing we choose M1 = N1 = 1, M2 = N2 = 3/2, and ﬁnally, when f is nondecreasing and g is nonincreasing, or vice versa, it is suﬃcient to choose M1 = N2 = 1, M2 = 3/2, N1 = 1/2, or N1 = M2 = 1, M1 = 1/2, N2 = 3/2, respectively. (iii) To show that T (Ω) is relatively compact it is suﬃcient to prove that T (Ω) ∞ is uniformly Cauchy in the topology of ^∞ T × ^T by [4, Theorem 3.3], i.e. for any ε > 0 there exists N ≥ T such that for any k, l ≥ N it holds |(T1 (v))k − (T1 (v))l | < ε

and |(T2 (u))k − (T2 (u))l | < ε

for (u, v) ∈ Ω. The details are left to the reader. (iv) The continuity of T in Ω can be proved using a similar argument to that in the proof of [10, Theorem 1], namely using the discrete analogue of the Lebesgue dominated convergence theorem, since the series occuring in the deﬁnition of the operator T are totally convergent. Thus the Schauder ﬁxed point theorem can be applied and the operator T has a ﬁxed point (x, y) ∈ Ω. It is easy to see that (x, y) is a regularly decaying solution of (1) for k ≥ T . To show that this solution can be extended to the left in a decreasing manner for σ = 1 we use the fact that system (1) is actually a recurrence relation. We proceed in the same way as it is done in the proof of [10, Theorem2]. The “only if part”. Let (x, y) be a regularly decaying solution of (1). Then there exist positive constants M1 , M2 , N1 , N2 and T ≥ m such that ∞ ∞ M1 M2 ≤ x Φ ≤ Φ α∗ k α∗ j=k j=k rj rj , (8) ∞ ∞ N1 N2 ≤ y ≤ Φ k β∗ j=k Φβ∗ j=k qj qj for k ≥ T . Assume f, g nondecreasing; the remaining cases can be treated similarly. Summing twice both equations in (1) from k to ∞ we get ∞ ∞ xk = σ j=k Φα∗ r1j i=j (ϕi f (yi+1 ) + ϕˆi ) , (9) ∞ ∞ yk = j=k Φβ∗ q1j i=j (ψi g(xi+1 ) + ψˆi ) . Now (8) and (9) imply the conditions (4), (5), (6) and (7), that leads to the assertion. A @ Remark 3. Note that the necessary and suﬃcient conditions in Theorem 2 are the same for both cases σ = ±1 in spite of the fact that diﬀerent sign condition causes a diﬀerent dynamical behavior as regards other types of solutions. Remark 4. Using similar arguments to those in the above proof, we are able to prove the “if part” for more general system (3), where the ﬁrst condition in (4)

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M. Marini, S. Matucci and P. Řehák

∞ is replaced by k=m α−1 (1/rj ) < ∞, α−1 being the inverse of the function α. Instead of (6) we suppose the existence of certain “upper” functions ϕ, ¯ F¯ such ¯ ¯ ¯ F satisfy that F (k, u, v) ≤ ϕ¯k F (u) on {m, m + 1, . . . } × (0, ε] × (0, ε], where ϕ, ∞ ∞ 1 ϕ¯k F¯ β −1 < ∞, qj k=m

j=k+1

β −1 being the inverse of β. The latter condition in (4) and (7) would be rearranged in a similar way. Obviously, a necessary condition in the above sense cannot be stated in this case. Remark 5. Closer examination of the proof enables us to obtain an asymptotic estimate for regularly ∞ decaying solutions. Indeed, it is not diﬃcult ∞ to see that xk is asymptotic to j=k Φα∗ (1/rj ), while yk is asymptotic to j=k Φβ∗ (1/qj ). Remark 6. Finally note that, besides the regular case, the statement of Theorem 2 includes the singular one as well, i.e. when f, g are unbounded in a right neigborhood of zero. It should be emphasized that we do not require any additional conditions to treat this case.

3

Strongly Decaying Solutions

For sake of completeness in this section we recall some results that were proved in [10] in order to have “complementary” statements. We start with a necessary and suﬃcient criterion guaranteeing the existence of strongly decaying solutions of system (1) with σ = −1. Theorem 7 ([10], Theorem 1). System (1) with σ = −1 has strongly decaying solutions if and only if condition (5) and ∞ 1 Φα∗ <∞ rk k=m

hold, and there exists a constant A > 0 such that ∞ ∞ ϕk f ωj (A) < ∞, k=m

where

ωk (A) = Φβ∗

(10)

j=k+1

∞ ∞ 1 A ˆ Φα∗ ψj g + ψj < ∞. qk rj i=j+1 j=k

Remark 8. In the contrast to the existence of regularly decaying solutions, the ﬁrst condition in (4) is not necessary in Theorem 7. On the other hand, if the ﬁrst condition in (4) holds, then condition (10) can be relaxed to simpler (but only suﬃcient) conditions (6), (7). Thus we have analogous suﬃcient conditions guaranteeing the existence of both regular and strongly decaying solutions of (1) with σ = −1. See Remark 13 for additional information concerning the case σ = 1.

295

Nonoscillatory Solutions for Nonlinear Discrete Systems

Remark 9. The proof of Theorem 7 gives an asymptotic estimate for strongly decaying solutions. However, in the contrast to the case in Section 2, we have an ∞asymptotic estimate only for the ﬁrst component. In fact, xk is asymptotic to j=k Φα∗ (1/rj ). In the case of system (1) with σ = 1, the following existence result holds. Theorem 10 ([10], Theorem 2). Let f, g be nondecreasing. Suppose that at least one of the forcing terms {ϕˆk }, {ψˆk } is eventually nontrivial.

(11)

∞ ∞ ∞ 1 1 ˆ (ϕj + ϕˆj ) < ∞, Φβ∗ (ψj + ψj ) < ∞, rk qk

(12)

If ∞

Φα∗

k=m

j=k

k=m

j=k

then system (1) with σ = 1 has at least one strongly decaying solution that is decreasing for any k ≥ m. Remark 11. Observe that condition (4) is not necessary. On the ∞other hand, The ϕ and orem 10 requires the convergence of the series ∞ j j=m j=m ψj . When at least one of the series the following criterion.

∞ j=m

ϕj and

∞ j=m

ψj diverges, we can state

Theorem 12 ([10], Theorem 3). Let f, g be nondecreasing. If (11), (4), (5), (6), (7) hold, then system (1) with σ = 1 has at least one strongly decaying solution that is decreasing for any k ≥ m. Remark 13. If f, g are nondecreasing and (11) holds, then the conditions of Theorem 2 are suﬃcient for the existence of both regularly and strongly decaying solution of (1) with σ = 1 and with σ = −1 as well. Remark 14. In the contrast to Theorems 2 and 7, the presence of forcing terms plays an important role in Theorems 10 and 12 for proving the existence of strongly decaying solutions. On the other hand, as it is shown in [10], there exist discrete nonforced systems having strongly decaying solutions. The problem of ﬁnding sufﬁcient conditions for nonforced discrete systems to have singular solutions remains open; in the continuous case this problem can be solved using the concept of singular solution (see [12]) that however has no discrete counterpart.

References 1. R. P. Agarwal, Diﬀerence Equations and Inequalities, 2nd edition, Pure Appl. Math. 228, Marcel Dekker, New York, 2000. 2. M. Cecchi, Z. Došlá and M. Marini, Positive decreasing solutions of quasilinear difference equations, Comp. Math. Appl 42 (2001), 1401–1410.

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3. M. Cecchi, Z. Došlá and M. Marini, Unbounded solutions of quasilinear diﬀerence equations, to appear in Comp. Math. Appl. 4. S. S. Cheng and W. T. Patula, An existence theorem for a nonlinear diﬀerence equation, Nonlinear Analysis 20 (1993), 193–203. 5. P. Clément, J. Fleckinger, E. Mitidieri and F. de Thélin, Existence of positive solutions for a nonvariational quasilinear elliptic system, J. Diﬀ. Eq. 166 (2000), 455–477. 6. J. I. Díaz, Nonlinear Partial Diﬀerential Equations and Free Boundaries, Volume I: Elliptic equations, Research Notes in Math. 106, Pitman Publ. Ltd., London, 1985. 7. J. R. Graef and E. Thandapani, Oscillation of two-dimensional diﬀerence systems, Comp. Ath. Appl. 36 (1999), 157–165. 8. B. Liu and S. S. Cheng, Positive solutions of second order nonlinear diﬀerence equations, J. Math. Anal. Appl. 204 (1996), 482–493. 9. M. Marini, S. Matucci and P. Řehák, On decay of coupled nonlinear systems, to appear in Adv. Math. Sci. Appl. 10. M. Marini, S. Matucci and P. Řehák, Strongly decaying solutions of nonlinear forced discrete systems, to appear in Proc. of ICDEA 2001, Augsburg. 11. R. Medina, Nonoscillatory solutions for the one-dimensional p-laplacian, J. Comp. Applied Math. 98 (1998), 27–33. 12. T. Tanigawa, Positive decreasing solutions of systems of second order quasilinear diﬀerential equations, Funkc. Ekvac. 43 (2000), 361–380. 13. E. Thandapani and I. M. Arockiasamy, Some oscillation and non-oscillation theorems for fourth order diﬀerence equations, Zeitricht für Analysis und ihre Anwend. 19 (2000), 863–872. 14. J. Yan and B. Liu, Oscillatory and asymptotic behaviour of fourth order nonlinear diﬀerence equations, Acta Math. Sinica 13 (1997), 105–115. 15. P. J. Y. Wong and R. P. Agarwal, Oscillation and monotone solutions of second order quasilinear diﬀerence equations, Funk. Ekv. 39 (1996), 491–517.

Equadiﬀ 10, August 27–31, 2001 Prague, Czech Republic

Equadiﬀ 10 CD ROM Papers, pp. 297–321

Mixed-hybrid Model of the Fracture Flow Jiří Maryška1, Otto Severýn2 and Martin Vohralík3 1

Department of Modelling of Processes, Faculty of Mechatronics and Interdisciplinary Engineering Studies, Technical University of Liberec, Hálkova 6, 461 17 Liberec 1, Czech Republic Email: [email protected] 2 Department of Modelling of Processes, Faculty of Mechatronics and Interdisciplinary Engineering Studies, Technical University of Liberec, Hálkova 6, 461 17 Liberec 1, Czech Republic Email: [email protected] 3 Department of Mathematics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Trojanova 13, 120 00 Prague 2, Czech Republic Email: [email protected]

Abstract. A stochastic discrete fracture network model of Darcy’s underground water ﬂow in disrupted rock massifs is introduced. Lowest order Raviart–Thomas mixed-hybrid FEM discretization is chosen, and it is properly imbedded in both the framework of mixed and mixed-hybrid FEM for the special conditions of the ﬂow through connected system of 2-D polygons in 3-D. Model problems are tested and generation and triangulation of fracture networks for real situations is described. MSC 2000. 76M10, 65C20, 65N50, 65N15, 35J25 Keywords. Fracture ﬂow, stochastic discrete fracture network model, mixed and mixed-hybrid ﬁnite element method

1

Introduction

We consider a steady saturated Darcy’s law governed ﬂow of an incompressible ﬂuid through a system of 2-D polygons placed in the 3-D space and connected under certain conditions into one network. This may simulate underground water ﬂow through natural geological disruptions of a rock massif, fractures, e.g. for the purposes of ﬁnding suitable nuclear waste repositories. Note that intersection of This is the preliminary version of the paper.

298

J. Maryška, O. Severýn and M. Vohralík

three or more triangles through one edge in the discretization is possible owing to the special geometrical situation, see Fig 11. We use the lowest order RaviartThomas mixed-hybrid ﬁnite element approximation, and we study its existence and uniqueness, existence and uniqueness of appropriate weak solutions, and the approximation error in the framework of the mixed FEM and the mixed-hybrid FEM, see [10], [8] respectively. For technical details of the following, see [14]. The outline of the contribution is as follows: in Section 2, we state the mathematical-physical formulation, in Section 3, we give appropriate function spaces, in Section 4, we state the weak primal formulation and verify its existence and uniqueness, which we will need in Sections 5 and 6 in order to prove the existence and uniqueness of weak and discrete solutions for the mixed/mixed hybrid ﬁnite element methods. The theoretical estimates are conﬁrmed by the carried out numerical examples, see Section 7. Description of stochastic discrete fracture networks generation and discretization for simulation of real situations is given in Section 8.

2

Mathematical-physical Formulation

We suppose that we have S =

αO \ ∂S ,

(1)

O∈L

where αO is an opened 2-D polygon placed in a 3-D Euclidean space; we call αO as a fracture. We denote as L the index set of fractures, |L| is the number (ﬁnite) of considered fractures. We suppose that all closures of these polygons are connected into one “fracture network”, the connection is possible only an edge, not a through point. Moreover, we require that if αi αj = ∅ then αi αj ⊂ ∂αi ∂αj , i.e. the connection is possible only through fracture boundaries (we state this requirement in order to be able to deﬁne correct function spaces). The situation is schematically viewed in Figure 4. Let us have a 2-D orthogonal coordinate system in each polygon αO . We are looking for the fracture ﬂow velocity u (2-D vector in each αO ), which is the solution of the following problem u = −K ∇p + ∇z in S , (2) ∇·u= q p = pD

in ΛD ,

in S ,

u · n − σ(p − pD ) = uN

(3) in

ΛN ,

(4)

where all variables are expressed in local coordinates of appropriate αO and also the diﬀerentiation is always expressed towards these local coordinates. The equation (2) is Darcy’s law, (3) is the mass balance equation and (4) is the expression of

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Mixed-hybrid Model of the Fracture Flow

appropriate boundary conditions. The variable p denotes the modiﬁed ﬂuid presp sure p (p = Pg ), g is the gravitational acceleration constant, ` is the ﬂuid density, q represents stationary sources/sinks density and z is the elevation, positive upward taken vertical 3-D coordinate expressed in appropriate local coordinates. We require the second tensor K to be symmetric and uniformly positive rank 2 2 deﬁnite on each αO , i.e. i,j=1 Kij (x)ηi ηj ≥ β i=1 ηi2 , β > 0 for any (η1 , η2 ) ∈ 2 R and almost all x ∈ αO \ ∂S, for all ^ ∈ L, and pose also the requirement ΛD ∩ ΛN = ∅ , ΛD ∪ ΛN = ∂S , ΛD = ∅.

3

Function Spaces

1 We start from L2 (αO ), u0,α = ( α u2 dS) 2 and L2 (αO ) = L2 (αO ) × L2 (αO ) in order to introduce < L2 (S) ≡ L2 (αO ) , L2 (S) ≡ L2 (S) × L2 (S) . (5) O∈L

We begin with classical Sobolev space H 1 (αO ) of scalar functions with square integrable weak derivatives, H 1 (αO ) = {ϕ ∈ L2 (αO ); ∇ϕ ∈ L2 (αO )}, ϕ1,α = 1 ( α [ϕ2 + ∇ϕ · ∇ϕ] dS) 2 , so as to introduce H 1 (S) ≡ {v ∈ L2 (S) ; v|α ∈ H 1 (αO ) ∀^ ∈ L , (v|αi )|f = (v|αj )|f ∀ f = αi αj , i, j ∈ L } . 1

(6)

1

We then have the spaces H 2 (∂S) and H − 2 (∂S) and the surjective continuous trace 1 operator γ : H 1 (S) → H 2 (∂S) as in the standard “planar” case. We further can 1 1 (S) = {ϕ ∈ H 1 (S) ; γϕ = 0 on ΛD }, H 2 (ΛN ) = { µ : ΛN → R; ∃ ϕ ∈ deﬁne HD 1 1 1 1 (S), µ = γϕ}, H − 2 (ΛN ) = { ψ ∈ H − 2 (∂S); ψ, ϕ∂S = 0 ∀ϕ ∈ HN (S)} and, HD 1 1 1 2 for pd ∈ H (ΛD ), HD,∗ (S) = {ϕ ∈ H (S) ; γϕ = pD on ΛD } . We denote as H(div, αO ) the Hilbert space of vector functions with square integrable weak divergences, H(div, αO ) = {v ∈ L2 (αO ); ∇·v ∈ L2 (αO )}, vH(div,α ) = 1 2 (v20,α + ∇ · v0,α ) 2 . We can deﬁne now H(div, S) ≡ {v ∈ L2 (S) ; v|α ∈ H(div, αO ) ∀^ ∈ L , i∈If v|αi · ni , ϕi = 0 ∀f such that |If | ≥ 2 , If = {i ∈ L ; f ⊂ ∂αi } , ∀ϕi ∈

1 H∂α i \f

(7) .

We have the surjective continuous normal trace operator ζ : u ∈ H(div, S) → 1 u · n ∈ H − 2 (∂S) as in the standard “planar” case. We further deﬁne the space 1 H0,N (div, S) = {u ∈ H(div, S) ; u · n, ϕ∂S = 0 ∀ϕ ∈ HD (S)}. The norms on the spaces deﬁned by (5), (6), (7) are given as

2·,S

=

|L| O=1

2

·,α .

(8)

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Remark 1. Note that deﬁnitions (5), (6), (7) are essential. The system S, however consisting of plane polygons, is not plane by oneself. Moreover, one edge can be common to three or more polygons αO creating the system S. Hence the deﬁnition of the space H 1 (S) of scalar continuous functions and especially of H(div, S) of normal trace vector continuous functions, expressing the mass balance condition at each inner edge. Note also that these deﬁnitions coincide with the classical ones for at most two polygons intersecting through one edge, cf. [11, Theorem 1.3.] We now turn to function spaces necessary for the mixed-hybrid formulation. Let us thus suppose a triangulation of the system S; we require that the triangulation in each fracture incorporates the intersections with other fractures. We deﬁne an index set Jh to number the elements of the triangulation Th , |Jh | denotes the number of elements. We deﬁne two sets of edges, Λh = ∪e∈Th ∂e ,

Λh,D = ∪e∈Th ∂e \ ΛD ,

(9)

and on Λh,D , we set 1

1 H 2 (Λh,D ) = {µ : Λh,D → R ; ∃ϕ ∈ HD (S), µe = γh ϕe

∀e ∈ Th } ,

(10)

1 where γh is the trace mapping of functions from HD (S) on the edges structure 1 2 Λh,D . For the functions from H (Λh,D ), we have the norm

|µ| 12 ,Λh,D =

inf

1 (S) ϕ∈HD

{|ϕ|1,S ; µe = γh ϕe

∀e ∈ Th } .

Due to ΛD = ∅, | · | 12 ,Λh,D is a norm on Λh,D equivalent with · 12 ,Λh,D . This deﬁnition will be suitable later in order to obtain the inf–sup condition. The standard hybrid-divergence space H(div, Th ) without any continuity requirements has the form H(div, Th ) = {v ∈ L2 (S) ; ∇ · ve ∈ L2 (e) ∀e ∈ Th } , |L| |Jh | 2 2 2 ∇ · v0,e . with the norm vH(div,Th ) = O=1 v0,α + O=1

4

(11)

Existence and Uniqueness of the Primal Formulation

1 Let us suppose that we have a p˜ such that p˜ ∈ HD,∗ (S).

Definition 2. As a primal weak solution of the steady saturated fracture ﬂow problem described by (2) - (4) on the system S, we understand a function p = 1 (S), such that p0 + p˜, p0 ∈ HD (K∇p0 , ∇ϕ)0,S + σp0 , ϕΛN = (q, ϕ)0,S + σpD − uN , ϕΛN − −(K∇z, ∇ϕ)0,S − (K∇˜ p, ∇ϕ)0,S − σ p˜, ϕΛN

1 ∀ϕ ∈ HD (S) . 1

(12)

Our general requirements are Kij ∈ L∞ (S), q ∈ L2 (S), pD ∈ H 2 (ΛD ∪ ΛN ), 1 uN ∈ H − 2 (ΛN ) and σ ∈ L∞ (ΛN ), 0 ≤ σ ≤ σM .

301

Mixed-hybrid Model of the Fracture Flow

Theorem 3. The problem (12) has a unique solution. Proof. Considering standard techniques (triangle inequality, bound |Kij (x)| ≤ K, Schwarz’s inequality, positive deﬁniteness of the tensor K on each αO , estimates for Sobolev spaces H 1 (αO ), ΛD = ∅) and the deﬁnition of all norms like in (8), the only crucial point is the use of Hölder’s inequality 1 1 ( ni=1 a2i ) 2 ( ni=1 b2i ) 2 ≥ ni=1 ai bi , ai , bi ∈ R . (13) The term z1,S is surely ﬁnite, since in each fracture, the 3-D z coordinate is expressed in its local coordinates as a linear function, and ∇z is thus a constant vector in each αO , ^ ∈ L. We have that the left hand side of (12) represents a 1 1 1 (S) × HD (S), coercive on HD (S), and since the bilinear continuous form on HD 1 right hand side of (12) represents a linear continuous functional on HD (S), the assertion is assured by the Lax–Milgram lemma.

5

Mixed FEM and Subsequent Hybridization

Although the possibility to use the mixed ﬁnite element method is not apparent at the ﬁrst sight because of the special geometrical situation treated, we will see that it is possible due to the special function spaces introduced in Section 3 and special approximation spaces introduced in this section, considering also the existence and uniqueness of the primal formulation. We take σ = 0, i.e. the problem (2) - (4) with only Neumann boundary conditions. 5.1

Weak Mixed Solution

The tensor of the fracture hydraulic conductivity K is positive deﬁnite on each αO , and therefore there exists, on each αO , a positive deﬁnite tensor A = K−1 which ˜ that u ˜ · n = uN characterizes the medium resistance. Let us now consider such u on ΛN in appropriate sense. Definition 4. As a weak mixed solution of the steady saturated fracture ﬂow ˜ , u0 ∈ problem described by (2) – (4), we understand functions u = u0 + u H0,N (div, S), and p ∈ L2 (S) satisfying (Au0 , v)0,S − (∇ · v, p)0,S = − v · n, pD ΛD + (∇ · v, z)0,S − − v · n, z∂S − (A˜ u, v)0,S

(14)

∀ v ∈ H0,N (div, S) ,

˜ , φ)0,S −(∇ · u0 , φ)0,S = −(q, φ)0,S + (∇ · u

∀φ ∈ L2 (S) . 1 2

(15)

Our general requirements are Aij ∈ L∞ (S), q ∈ L2 (S), pD ∈ H (ΛD ) and uN ∈ 1 H − 2 (ΛN ). Theorem 5. The problem (14), (15) has a unique solution.

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Proof. Let us denote a(u, v) = (Au, v)0,S =

|L| O=1

b(v, φ) = (∇ · v, φ)0,S =

Au · v dS ,

α

|L| O=1

∇ · v φ dS ,

α

f (v) and g(φ) the functionals on the right hand sides of (14), (15) respectively, and V = H0,N (div, S), Φ = L2 (S), and W = {v ∈ V ; b(v, φ) = 0 ∀φ ∈ Φ}. It is easy then to show that the form a(·, ·) is a bilinear continuous form on V × V, moreover coercive on W, the form b(·, ·) is a bilinear continuous form on V × Φ satisfying the inf–sup (Babuška–Brezzi) condition b(v, φ) ≥ k1 φ∈Φ v∈V vH(div,S) φ(0,S) inf sup

(16)

with k1 > 0. Further, the functional f (·) is a linear continuous functional on V and the functional g(·) is a linear continuous functional on Φ. The proofs are direct applications of standard techniques; essential is the existence and uniqueness of the primal solution and again Hölder’s inequality (13), see [14] for details. Using [11, Theorem 10.1.] (originally Brezzi 1974), we have the assertion. 5.2

Mixed Finite Element Approximation

We deﬁne a 3-dimensional space RT0 (e) of vector functions linear on a given element e with the basis vie , i ∈ {1, 2, 3}, where e e e e e x − α11 e e x − α21 e e x − α31 v1 = k1 , v2 = k2 , v3 = k3 . y − αe12 y − αe22 y − αe32 Concerning its dual basis, we state classically Nje , j = 1, 2, 3, Nje (uh ) = f e uh · j nej dl, where each functional Nje expresses the ﬂux through one edge for uh ∈ RT0 (e); we have Nje (vie ) = δij after appropriate choice of αe11 − αe32 , k1e − k3e . The local interpolation operator is then given by πe (u) =

3

Nie (u)vie

∀ u ∈ (H 1 (e))2 .

(17)

i=1

We start from the Raviart–Thomas space RT0−1 (Th ) of on each element linear vector functions without any continuity requirements, RT0−1 (Th ) ≡ {v ∈ L2 (S) ; v|e ∈ RT0 (e) ∀e ∈ Th } , to deﬁne the necessary “continuity assuring” space RT00 (Th ) by RT00 (Th ) ≡ {v ∈ RT0−1 (Th ) ; i∈If v|ei · nf,∂ei = 0 ∀f such that |If | ≥ 2 , If = {i ∈ Jh ; f ⊂ ∂ei } = RT0−1 (Th ) ∩ H(div, S) .

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Mixed-hybrid Model of the Fracture Flow

We set furthermore RT00,N (Th ) ≡ {v ∈ RT00 (Th ) ; v · n = 0 in ΛN } = RT0−1 (Th ) ∩ H0,N (div, S) and 0 M−1 (Th ) ≡ {φ ∈ L2 (S) ; φ|e ∈ M 0 (e)

∀e ∈ Th } ,

0

where M (e) is the space of scalar functions constant on a given element e. Looking for the basis, appropriate dual basis, and global interpolation operator for RT00 (Th ), we have the following deﬁnitions and lemmas: We set Nh = {N1 , N2 , . . . , NINh } as the dual basis of RT00 (Th ), where for each border edge f , we have one functional Nf deﬁned by Nf (uh ) = f uh |e ·n∂e dl, and for each inner edge f common to elements e1 , e2 , . . . , eIf , we have If −1 functionals given by 1 1 uh |e1 · n∂e1 dl − uh |ej+1 · n∂ej+1 dl , j = 1, . . . , If − 1 . Nf,j (uh ) = If f If f Lemma 6. For all uh ∈ RT00 (Th ), from Nj (uh ) = 0 ∀ j = 1, . . . , INh follows that uh = 0. Proof. Let us suppose that Nj (uh ) = 0 ∀ j = 1, . . . , INh . From the deﬁnition of the functionals Nf on border edges, we have f uh |e · n∂e dl = 0 for all border edges f . Concerning the deﬁnition of the functionals Nf,j on inner edges and the condition Nf,j (uh ) = 0, we have f uh |e1 · n∂e1 dl = f uh |ej · n∂ej dl for all j = 2, . . . , If . If Considering the equality i=1 u | · n∂ei dl = 0 characterizing the continuity f h ei of the functions from RT00 (Th ), we come to f uh |e · n∂e dl = 0 for all f ∈ Λh and all e, f ⊂ e. Since RT00 (Th ) ⊂ RT0−1 (Th ) and all coeﬃcients for the local interpolants on each e ∈ Th are equal to zero, uh = 0 follows. We set Vh = {v1 , v2 , . . . , vINh }, where for each border edge f , we have one base function vf deﬁned by vf = vfe with vfe being the local base function appropriate to the element e and its edge f , and for each inner edge f common to elements e1 , e2 , . . . , eIf , we have If − 1 base functions given by vf,i =

If

e

vfek − (If − 1)vfi+1

,

i = 1, . . . , If − 1 .

k=1, k=i+1

Lemma 7. For the bases Nh and Vh , Nj (vi ) = δij , i, j = 1, . . . , INh holds. Proof. We have from the deﬁnition of base functions of RT0 (e) that Nf (vf ) = 1 for all border edges f , and simply Nf (v) = 0 for all v ∈ Vh , v = vf . Concerning the inner edges, we easily come to Nf,j (vg ) = 0 for all j = 1, . . . , If − 1, f an inner edge, g border edge, and to Nf,j (vg,i ) = 0 for all j = 1, . . . , If −1, i = 1, . . . , Ig −1, f an inner edge, g another inner edge. We have 1 1 1 1 e vfe1 · n∂e1 dl − v j+1 · n∂ej+1 dl = − =0 Nf,j (vf,i ) = If f If f f If If

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for i = j, and Nf,j (vf,i ) =

1 If

f

vfe1 · n∂e1 dl − =

1 If

f

e

−(If − 1)vfj+1 · n∂ej+1 dl =

1 1 + (If − 1) = 1 If If

for i = j, i, j = 1, . . . , If − 1, f an inner edge. Thus the proof is completed. We introduce ﬁrst a space smoother than H(div, S), corresponding to the classical (H 1 (S))2 , H(grad, S) = {v ∈ L2 (S) ; v|α ∈ (H 1 (αO ))2 i∈If v|αi · nf,∂αi = 0

∀^ ∈ L , (18)

∀f such that |If | ≥ 2 , If = {i ∈ L ; f ⊂ ∂αi } , in order to set the global interpolation operator INh

πh (u) =

Ni (u)vi

∀ u ∈ H(grad, S) .

(19)

i=1

Lemma 8. Concerning the local and global interpolation operators given by (17), (19) respectively, we have their equality on each element, i.e. πh (u)|e = πe (u|e )

∀ e ∈ Th , ∀ u ∈ H(grad, S) .

Proof. As the the base functions vi , i = 1, . . . , INh of RT00 (Th ) are combined from the base functions vje on each element, we only have to verify that the coeﬃcients by vje are the same. By border edges, the situation is apparent – the coeﬃcients for both local and global interpolation operators are given by f u|e · n∂e dl. For the inner edges, we have f −1 I

i=1

f −1 I 1 1 Nf,i (u)vf,i = u|e1 · n∂e1 dl − u|ei+1 · n∂ei+1 dl · If f If f ej i=1 If

k=1, k=i+1

−

1 If

1 u|e1 · n∂e1 dl − If f ej i=1, i=j−1 1 e · n∂ei+1 dl · vfj − (1 − δj1 ) u|e1 · n∂e1 dl − If f 1 e u|ej · n∂ej dl · (If − 1)vfj − If f e

vfek − (If − 1)vfi+1

f

u|ei+1

=

If −1

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Mixed-hybrid Model of the Fracture Flow

using the deﬁnition of Nf,i and vf,i for an inner edge f , i = 1, . . . , If − 1, j = e 1, . . . , If . Considering now only the coeﬃcients by vfj , we come to If −1 1 1 u|e1 · n∂e1 dl − u|ei+1 · n∂ei+1 dl = I I i=1 f f i=1 f f 1 1 u|e1 · n∂e1 dl = u|e1 · n∂e1 dl = (If − 1) + If If f f

If −1

If for j = 1, using the normal trace continuity of u expressed by i=1 f u|ei · n∂ei dl = 0. Similarly, we have 1 1 1 u|e1 · n∂e1 dl + u|e1 · n∂e1 dl + u|ej · n∂ej dl − (If − 2) If f If f If f 1 1 −(If − 1) u|e1 · n∂e1 dl + (If − 1) u|ej · n∂ej dl = u|ej · n∂ej dl If f If f f for j ≥ 2, and thus the proof is completed. Lemma 9. Even for the considered special function spaces and their ﬁnite dimensional subspaces, we have div

H(grad, S) −→ L2 (S)    >πh >Ph ,

(20)

div

0 RT00 (Th ) −→ M−1 (Th )

i.e. the commutativity diagram property, where πh is the global interpolation op0 erator deﬁned by (19), and Ph is the L2 (S)-orthogonal projection onto M−1 (Th ). Proof. The proof is easy using the previous lemma and the validity of the commutativity diagram property for the local interpolation operator, see e.g. [9, Section 3.4.2]. Definition 10. As the lowest order Raviart–Thomas mixed approximation of the the problem (14), (15), we understand functions u0,h ∈ RT00,N (Th ) and ph ∈ 0 (Th ) satisfying M−1 (Au0,h , vh )0,S − (∇ · vh , ph )0,S = − vh · n, pD ΛD + (∇ · vh , z)0,S − u, vh )0,S ∀ vh ∈ RT00,N (Th ) , − vh · n, z∂S − (A˜

˜ , φh )0,S −(∇ · u0,h , φh )0,S = −(q, φh )0,S + (∇ · u

0 ∀φh ∈ M−1 (Th ) .

Theorem 11. The problem (21), (22) has a unique solution.

(21)

(22)

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J. Maryška, O. Severýn and M. Vohralík

0 Proof. Let us denote Vh = RT00,N (Th ), Φh = M−1 (Th ) and Wh = {vh ∈ Vh ; b(vh , φh ) = 0 ∀φh ∈ Φh }. As Vh ⊂ V, Φh ⊂ Φ, the (bi)linearity and continuity of the forms a(·, ·), b(·, ·) and of the functionals f (·) and g(·) from Theorem 5 holds also on Vh and Φh . Since Wh ⊂ W, also the coercivity of the form a(·, ·) is assured. For the discrete inf–sup condition, essential is the commutativity diagram property expressed by (20), at least for the case ΛN = ∅. For the case ΛN = ∅, a special procedure has to be carried out, cf. [11], pages 587-588.

5.3

Error Estimates

If the solution (u0 , p) ∈ V×Φ of (14), (15) is such that (u0 , p) ∈ H(grad, S)×H 1 (S) and ∇ · u0 ∈ H 1 (S) and if (u0,h , ph ) ∈ Vh × Φh is the solution of (21), (22), then u0 − u0,h H(div,S) + p − ph 0,S ≤ Ch(|p|1,S + |u0 |1,S + |∇ · u0 |1,S ) ,

(23)

where the constant C does not depend on h and |ϕ|1,S = ∇ϕ0,S , |u|21,S = 2 2 i=1 |ui |1,S (see [11], Theorem 13.2). 5.4

Hybridization of the Mixed Method

If f ∈ Λh , we deﬁne ﬁrst the space M 0 (f ) of functions constant on this edge and ﬁnally, on the edges structure Λh,D deﬁned by (9), 0 (Λh,D ) ≡ {µh : Λh → R ; µh |f ∈ M 0 (f ) ∀f ∈ Λh , M−1

µh |f = 0 ∀f ∈ ΛD } . RT0−1 (Th ),

It now follows immediately that if vh ∈ only if

vh · n, λh ∂e ∩Λh,D = 0

(24) then vh ∈

RT00,N (Th )

if and

0 ∀λh ∈ M−1 (Λh,D ) ,

e∈Th

which allows us to state the hybrid version of the lowest order Raviart–Thomas mixed method: Definition 12. As the lowest order Raviart–Thomas mixed-hybrid approximation of the the problem (14), (15), we understand functions u0,h ∈ RT0−1 (Th ), 0 0 ph ∈ M−1 (Th ), and λh ∈ M−1 (Λh,D ) satisfying {(Au0,h , vh )0,e − (∇ · vh , ph )0,e + vh · n, λh ∂e ∩Λh,D } = e∈Th

=

{− vh · n, pD ∂e ∩ΛD + (∇ · vh , z)0,e − vh · n, z∂e − (A˜ u, vh )0,e }

e∈Th

∀ vh ∈ RT0−1 (Th ) , −

e∈Th

(∇ · u0,h , φh )0,e = −

(25)

˜ , φh )0,e } {(q, φh )0,e − (∇ · u

e∈Th 0 ∀φh ∈ M−1 (Th ) ,

(26)

307

Mixed-hybrid Model of the Fracture Flow

u0,h · n, µh ∂e ∩Λh,D =

e∈Th

{ uN , µh ∂e∩ΛN − ˜ u · n, µh ∂e ∩Λh,D }

e∈Th 0 ∀µh ∈ M−1 (Λh,D ) .

(27)

Due to the previously mentioned, the triple u0,h , ph , λh surely exist and is unique, u0,h and ph are moreover at the same time the unique solutions of (21), (22); moreover, the multiplier λh is an approximation of the trace of p on all edges from Λh,D . Consequently, all error estimates from Section 5.3 are valid also for the mixed-hybrid solution triple u0,h , ph , λh and we have the following theorem: Theorem 13. The problem (25) – (27) has a unique solution.

6

Mixed-hybrid FEM

We will use in this part the mixed-hybrid ﬁnite element method as introduced by [8] to treat our problem of steady saturated fracture ﬂow. In fact, its use is more straightforward in the given case, since it handles even the case of one edge belonging to three or more triangles without any changes of formulation – through additional summation equation assuring the mass balance condition on inner edges. 6.1

Weak Mixed-hybrid Solution

We deﬁne the space 1

WD (Th ) = H(div, Th ) × L2 (S) × H 2 (Λh,D )

(28)

with the norm w2WD (Th ) = v2H(div,Th ) + φ20,S + |µ|21 ,Λh,D . On WD (Th ) × 2 WD (Th ), we deﬁne a bilinear form B, w) = (Au, v)0,e − (∇ · v, p)0,e + v · n, λ∂e∩Λh,D − B(w, (29) e∈Th

−(∇ · u, φ)0,e + u · n, µ∂e∩Λh,D − σλ, µ∂e∩ΛN

and on WD (Th ) a linear functional Q, Q(w) = − v · n, pD ∂e∩ΛD + (∇ · v, z)0,e − v · n, z∂e − e∈Th

−(q, φ)0,e + uN − σpD , µ∂e∩ΛN

(30)

,

= (u, p, λ) ∈ WD (Th ) and w = (v, φ, µ) ∈ WD (Th ). where w Definition 14. As a weak mixed-hybrid solution of the steady saturated fracture = (u, p, λ) ∈ ﬂow problem described by (2) – (4), we understand a function w WD (Th ) satisfying w) = Q(w) B(w,

∀ w = (v, φ, µ) ∈ WD (Th ) .

(31)

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J. Maryška, O. Severýn and M. Vohralík 1

Our general requirements are Aij (x) ∈ L∞ (S), q ∈ L2 (S), pD ∈ H 2 (ΛD ∪ ΛN ), 1 uN ∈ H − 2 (ΛN ) and σ ∈ L∞ (ΛN ), 0 ≤ σ ≤ σM . Theorem 15. The problem (31) has a unique solution. Proof. The problem (31) is the same as originally studied by Oden and Lee in [8], except of incorporation of Newton boundary condition and occurrence of the vertical coordinate z. Using the existence and uniqueness of the primal solution even in our geometrical situation, there are no complications to show that the form B(·, ·) is a bilinear continuous form on WD (Th ) × WD (Th ), satisfying the inf–sup condition of the type (16), and that the functional Q(·) is a linear continuous functional on WD (Th ). The existence and uniqueness follows by the theory of Babuška, see [1, Theorem 2.1]. 6.2

Mixed-hybrid Finite Element Approximation

0 We equip the space M−1 (Λh,D ) of edge-wise constant functions deﬁned by (24) with the norm |µh |2∗ 1 ,∂e , (32) |µh |2∗ 1 ,Λh,D = 2

2

e∈Th 1

where |µh |∗ 12 ,∂e = |µ∗h | 12 ,∂e with µ∗h ∈ H 2 (∂e), µ∗h edge-wise linear and satisfying µ dl = ∂ei µ∗h dl, i ∈ {1, 2, 3}, where ∂ei are the sides of the triangle e. One ∂ei h can show that such deﬁnition is correct. We can now deﬁne the ﬁnite-dimensional approximation space h 0 0 (Th ) = RT0−1 (Th ) × M−1 (Th ) × M−1 (Λh,D ) WD

equipped with the norm wh 2Wh (Th ) = vh 2H(div,Th ) + φh 20,S + |µh |2∗ 1 ,Λh,D ,

h wh = (vh , φh , µh ) ∈ WD (Th ).

2

D

Definition 16. As the lowest order Raviart–Thomas mixed-hybrid approxima h = (uh , ph , λh ) ∈ tion of the the problem (31), we understand a function w h (Th ) satisfying WD h , wh ) = Q(wh ) B(w

h ∀ wh = (vh , φh , µh ) ∈ WD (Th ) .

(33)

Theorem 17. The problem (33) has a unique solution. Proof. Using the deﬁnition of the norm (32), it is rather elaborate to show that h h the form B(·, ·) is a bilinear continuous form also on WD (Th )×WD (Th ), satisfying the discrete inf–sup condition, and that the functional Q(·) is a linear continuous h (Th ). Special qualities of the space RT0 (e) are used while functional also on WD 1 replacing edge-wise constant µh by its edge-wise linear counterpart µ∗h ∈ H 2 (∂e) in the proofs.

309

Mixed-hybrid Model of the Fracture Flow

0 Remark 18. The deﬁnition of the norm (32) for µh ∈ M−1 (Λh,D ) is necessary. h h The form B(·, ·) is not a continuous form on W (T ) ×W (T ) equipped with the h h D D h (Th ) as norm µh 20,Λh,D = f ∈Λh,D µh 20,f for µh . If we improve the norm on WD 2 2 2 2 2 wh Wh (Th ) = vh H(div,Th ) + vh · n0,Λh + φh 0,S + µh 0,Λh,D , the form B(·, ·) D gets continuous, but the discrete inf–sup condition is not satisﬁed with a constant independent of h, all owing to the mixture of the spaces RT0 (e) approximating 1 0 (∂e) of edgeH(div, e) with generally only u·n ∈ H − 2 (∂e), u ∈ H(div, e), and M−1 2 wise constant functions belonging generally only to L (∂e). By the deﬁnition (32), 1 we “skip” into necessary H 2 (∂e).

6.3

Error Estimates

We shall notice ﬁrst that we are dealing with a nonconform approximation. We 1 0 have approximated the space H 2 (Λh,D ) given by (10) by the space M−1 (Λh,D ) given by (24). We set X = v ∈ H(div, Th ) ; v · n ∈ L2 (Λh,D ) × φ ∈ L2 (S) × 1 0 × µ + µh ; µ ∈ H 2 (Λh,D ) , µh ∈ M−1 (Λh,D ) , w2X = v2H(div,Th ) + φ20,S + |µ + µh |2∗ 1 ,Λ 2

h,D

, and after some tedious manipu-

lations, we come to the bilinearity and continuity (with the constant C1X ) of the h (Th ). The error estimate is consequently of the form form B(·, ·) on X × WD

−w h X ≤ w X wh )| C 1 |Q(wh ) − B(w, − wh X + d ≤ 1 + 1d w sup , inf h wh X h (T ) C2 wh ∈WD (Th ) C2 wh ∈WD h = (u, p, λ) ∈ WD (Th ) X being the unique weak mixed-hybrid solution deﬁned w h h = (uh , ph , λh ) ∈ WD (Th ) the mixed-hybrid approximation deﬁned by (31), w d by (33), and C2 being the constant from the discrete inf–sup condition. However, due to the restriction v · n ∈ L2 (Λh,D ) (necessary to give the sense to the term wh ) from the nonconform error estimate), we come to B(w, | e∈Th { u · n, µh ∂e\∂S + wh )| |Q(wh ) − B(w, sup = sup wh X 0 (Λ µh ∈M−1 wh ∈Wh (Th ) h,D ) D

+ u · n − uN , µh ∂e∩ΛN + σ(pD − λ), µh ∂e∩ΛN }| =0 |µh |∗ 12 ,Λh,D and thus the ﬁnal error estimate has the form CX ( −w h X ≤ 1 + 1d Ch (|u|1,S + |∇ · u|1,S )2 + p22,S , w C2

(34)

for u ∈ H(grad, S), ∇ · u ∈ H 1 (S) and p ∈ H 2 (S) except of the previous requirements, the constant C independent of h. This means that we have the O(h) accuracy, however not for the edge-wise constant µh , but for its extension µ∗h .

310

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Model Problems

We consider two simple model problems in this section. The ﬁrst model problem is a model problem on a system S, where at most two polygons intersect through one edge, i.e. almost classical planar case; we call this as special geometrical situation. In the second model problem, one edge is common to four polygons; we call this situation, the kernel of what has been investigated in this contribution, as a general geometrical situation. All the computations were done in double precision on a personal computer, the resulting symmetric indeﬁnite systems of linear equations were solved by the solver GI8 of the Institute of Computer Science, Academy of Sciences of the Czech Republic, see [6]. This is based on the sequential elimination onto a system with Schur’s complement and subsequent solution of this system by a specially preconditioned conjugate gradients method. The solver accuracy was set to 10−8 .

z

y

L

5

a

y

1

A

L

x1

4

1 0.5

0

L

1

x

y

2

B L

6

-0.25

a L

1

3

x 2=x 1

2

X L

2

-1

Fig. 1. Model Problem I – Special Geometrical Situation

311

Mixed-hybrid Model of the Fracture Flow

7.1

Model Problem I – Special Geometrical Situation

The problem with the system S viewed in Figure 1 is as follows: S = α1 α2 \ ∂S , u = − ∇p + ∇z in S , ∇ · u = 0 in p=0

in Λ1

,

S,

p=0

in Λ2

u · n = 0 in Λ3 , u · n = 0 in Λ4 πx1 + S · A in Λ5 , p = S · y1 p = sin 2X sinh π(A+B) 2X

in Λ6 .

The exact solutions in α1 can be easily found as π(y + B) πx 1 1 sinh + S · y1 , pα1 = sin 2X 2X πx1 π(y1 + B) π cos sinh , uα1 = − 2X 2X 2X πx π(y + B) π 1 1 − sin cosh − S − ∇zαy 1 , 2X 2X 2X where ∇zα1 = (0, ∇zαy 1 ), S + ∇zαy 1 = ∇zαy 2 , and we can see them in Figures 2 and 3. Notice the occurrence of the term S assuring the continuity of the velocity ﬁeld because of diﬀerent z gradients in α1 and α2 . The following table gives pressure, velocity, and pressure trace errors in the fracture α1 . There is the O(h) convergence in pressure, velocity, and pressure 1 trace in | · |∗ 12 ,Λh,D norm, but only O(h 2 ) in pressure trace in · 0,Λh,D norm. N triangles p − ph 0,S u − uh H(div,Th ) λ − λh 0,Λh,D |λ − λh |∗ 1 ,Λh,D 2 2 8×2 0.4481 1.2236 1.4984 1.2236 4 32×2 0.2212 0.6262 1.0564 0.6262 8 128×2 0.1102 0.3150 0.7509 0.3150 16 512×2 0.0550 0.1577 0.5332 0.1577 32 2048×2 0.0275 0.0789 0.3779 0.0789 64 8192×2 0.0138 0.0394 0.2676 0.0394 128 32768×2 0.0069 0.0197 0.1893 0.0197 256 131072×2 0.0034 0.0099 0.1339 0.0099

Table 1. Pressure, Velocity, and Pressure Trace Errors in α1 for the Special Geometrical Situation, Model Problem I

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6 5 4 3 2 1 0 1.5

1 1 0.5 0.5 0

0

Fig. 2. Exact Pressure Graph in Fractures α1 and α2 , Model Problems I and II

1.4

1.2

α1

1

0.8

0.6

α

2

0.4

0.2

0 0

0.2

0.4

0.6

0.8

1

Fig. 3. Exact Velocity Graph in Fractures α1 and α2 , Model Problems I and II.

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Mixed-hybrid Model of the Fracture Flow

x 3=x 1

z

y

8 3

B

5

1

y

1

7

A

9

x 1=x 4

4

1

0.5 0

x

y

2

6

-0.25

B

y

3

12

B

10

1

x 2=x 1 2

4

3

X

2

11

-1

y

4

Fig. 4. Model Problem II – General Geometrical Situation

7.2

Model Problem II – General Geometrical Situation

The system S for this model problem is viewed in Figure 4. We have S = α1 α2 α3 α4 \ ∂S , u = − ∇p + ∇z in S , ∇ · u = 0 in

S,

p = 0 in Λ7 , p = 0 in Λ8 u · n = 0 in Λ9 , u · n = 0 in Λ10 π(B+B) 4 sinh in Λ11 , p = 0 in Λ12 , p = sin πx 2X 2X and boundary conditions on Λ1 – Λ6 as for the model problem I. The exact solutions in α1 and α2 stay the same.

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We give approximation errors in the ﬁrst fracture α1 . As the exact solutions coincide for the model problems I a II, we can compare tables 1 and 2, i.e. special (almost classical, planar) and general (the situation characteristic for the investigated problem of the fracture ﬂow) geometrical situations. A slight diﬀerence appears only for rough triangulations and disappears for the increasing N. Thus, demonstrated for these two model problems, the existence of triangulation edges common to three and more triangles does not inﬂuence the error, resp. convergence order. This was veriﬁed also for the pressure, velocity and pressure trace error distributions, as we can see in Figures 5, 6 and 7.

N triangles p − ph 0,S u − uh H(div,Th ) λ − λh 0,Λh,D |λ − λh |∗ 1 ,Λh,D 2 2 8×4 0.4445 1.2247 1.4973 1.2247 4 32×4 0.2212 0.6263 1.0562 0.6263 8 128×4 0.1102 0.3150 0.7509 0.3150 16 512×4 0.0550 0.1577 0.5332 0.1577 32 2048×4 0.0275 0.0789 0.3779 0.0789 64 8192×4 0.0138 0.0394 0.2676 0.0394 128 32768×4 0.0069 0.0197 0.1893 0.0197 256 131072×4 0.0034 0.0099 0.1339 0.0099

Table 2. Pressure, Velocity, and Pressure Trace Errors in α1 for the General Geometrical Situation, Model Problem II

−4

Pressure Error

x 10

0.5

9

0.4

8

y1

7 0.3 6 0.2

5 4

0.1

3 0

0

0.1

0.2

0.3

0.4

0.5 x1

0.6

0.7

0.8

0.9

1

Fig. 5. Distribution of the Pressure Error in α1 , Model Problems I and II

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Mixed-hybrid Model of the Fracture Flow

−3

Velocity Error

x 10 2.8

0.5

2.6 2.4

0.4

y1

2.2 2

0.3

1.8 0.2

1.6 1.4

0.1

0

1.2 1 0

0.1

0.2

0.3

0.4

0.5 x1

0.6

0.7

0.8

0.9

1

Fig. 6. Distribution of the Velocity Error in α1 , Model Problems I and II

−3

Pressure Trace Error

x 10

0.5

14 12

0.4

y1

10 0.3

8 6

0.2

4 0.1 2 0

0

0.1

0.2

0.3

0.4

0.5 x1

0.6

0.7

0.8

0.9

1

Fig. 7. Distribution of the “L2 ” Pr. Trace Error in α1 , Model Problems I and II

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Fig. 8. Generated Fracture Network on a 5 × 5 × 8 m Domain

8

Simulation of a Real Situation: Stochastic Discrete Fracture Network Generation and Discretization

We describe in this section the preparation of the fracture network and its triangulation in real situations. 8.1

Stochastic Discrete Fracture Network Generation

In order to generate stochastic discrete fracture networks, an original software called Fracture Network Generator was developed. Each fracture (geological 3-D object) is in the generator approximated by a ﬂat circle disk characterized by its middle coordinates, radius, orientation, possibly hydraulic conductivity or aperture distribution and roughness. Fractures are divided into four sets: fractures in fracture zones, deterministically measured single fractures, hydraulically important fractures and other (common) fractures. Fractures are further supposed to be divided into three types according to their mean orientation in 3-D cartesian coordinates [0,0,1], [0,1,0] or [1,0,0]. Each combination of set and type, except of

Mixed-hybrid Model of the Fracture Flow

317

deterministically measured single fractures, is treated as an independent statistical population. Number of fractures and spacing determines fracture frequency, deﬁned as amount of fractures per one depth meter in each part of the simulated domain. Fracture lengths are supposed as lognormally distributed, i.e. with the probability density function (p.d.f.) f (x) = σ√12πx exp(−(ln x−µ)2 /2σ 2 ). Concerning orientations, fractures are supposed to have the Fisher-von-Mises distribution k exp(k cos α) sin α of angles α between fracture normal vectors and f (α) = exp(k) vectors of mean orientations. In order to validate methods used for the statistical description and generating algorithms, χ2 tests for each simulated variable were carried out, see [12]; we only have to put an emphasis on strictly distinguishing between real statistical characteristics and by “exploration boreholes” measured distributions, the latter of them being aﬀected by a selective eﬀect. Indeed, while drilling borehole, we have a bigger chance to intersect a larger fracture than a smaller one. We can see an example of a generated network in Figure 8. 8.2

Final Triangular Mesh Construction

Discretization of approximating circle disks into triangle elements has occurred as a crucial point. Although many algorithms solving the discretization of a given 2-D domain are known, only few of them are able to involve pre-deﬁned interface lines, intersections of approximating circles in our case. The searched algorithm should be in addition capable to simplify the given geometrical situation (to save computer storage and avoid numerical faults and ill-conditioned resulting matrix), i.e. it can be only approximate; obtaining the initial mesh is also suﬃcient. In the Fracture Network Generator, an originally developed discretization algorithm is implemented. It contains of a preliminary phase and of an algorithm for triangulation of an arbitrary polygonally bounded domain with pre-deﬁned interface lines (triangulation algorithm).

Fig. 9. 2-D Geometrical Simpliﬁcations

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Fig. 10. Final Discretization of a Circle Disk

In the preliminary phase, identiﬁcation and various geometrical simpliﬁcations are made. Close, almost parallel fractures are removed from the fracture population or equivalently replaced. 2-D geometrical simpliﬁcations (moving and stretching intersections in fracture planes) are carried out before the start of the actual triangulation algorithm; we can see an example of these simpliﬁcations in Figure 9. Using these 2-D geometrical simpliﬁcations has naturally non-trivial 3-D consequences - if the simpliﬁcations are used, then the real geometrical correspondence vanishes. It is then replaced by extra connectivity information. The actual triangulation algorithm is based on combining the Domain Decomposition Conception, expressing that the domain is split into two parts along an intersection whenever possible, and adapted classical triangulation algorithm, Advancing Front Method. Many user’s setting inﬂuencing incorporating the geometrical simpliﬁcations and thus the precision/complexity of the ﬁnal triangulation are possible. An example of ﬁnal triangulation of a random circle disk is in Fig. 10. An on-element aperture distribution function is used after the discretization in order to assign to each triangle element an imaginary aperture. It is derived again from the Fisher-von-Mises distribution p.d.f., but depends also on the size of the given fracture and emplacement of the element inside the fracture. Based on the aperture and on a parameter describing roughness of the fracture walls, the element hydraulic conductivity can be later set, in order to simulate the “channeling eﬀect” (the ﬂow throughout a natural fracture is not evenly distributed throughout the whole fracture plane because of its non-constant aperture – channels of ﬂow

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319

occur). The fracture is, however, still supposed as planar. Data ﬁles with complete information about elements in the ﬁnal triangular network (emplacement, aperture, roughness, hydraulic conductivity and connection to other elements) are the ﬁnal results of the Fracture Network Generator. We can see the ﬁnal triangular mesh of the network from Fig. 8 in Fig. 11. A colour gradiation is used in order to denote elements apertures.

Fig. 11. Triangulation of the Network from Figure 8

9

Conclusion

In the submitted text, a discrete fracture network model of the fracture ﬂow with existence and numerical analysis, examples of model problems, and description of application on real situations was introduced. The ﬁnite element method for numerical approximation was chosen so as to achieve the biggest possible accuracy. The mixed-hybrid approximation was chosen because it, unlike the primal approximation, assures mass balance on each element and its data structures are

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though suitable for subsequent ﬁnite volume contaminant transport models, which are perspectively our main goal. In this approximation, the velocity is moreover computed directly and has good continuity rank, on the other hand, slightly bigger resulting matrix is the only drawback. The mixed-hybrid weak formulation is much more straightforward in the given case, when three or more triangles can intersect through one edge in the discretization; it expresses directly the mass balance condition on such edges. However, the lowest order Raviart-Thomas mixed-hybrid approximation is a nonconform approximation to the weak mixed-hybrid solution, which causes diﬃculties while investigating its existence, uniqueness, and approximation error. Under the construction of special function spaces, mixed weak solution and approximation were established too, and their existence and uniqueness were shown using classical techniques.

Acknowledgement This paper came to existence in framework of the project of the Grant Agency of the Czech Republic No. 205/00/0480. The last of the authors was also partially supported by the project No. J04/98:210000010 “Application of Mathematics in Technical Sciences” of the Government of the Czech Republic.

References 1. Babuška I., Error-Bounds for Finite Element Method, Numerische Mathematik 16 (1971), 322–333. 2. Cacas M. C., Ledoux E., De Marsily G., Modelling Fracture Flow With a Stochastic Discrete Fracture Network: Calibration and Validation the Flow and the Transport Model, Water Resources Research 26, No 3, 1990, 479–489. 3. Characterization and Evaluation of Sites for Deep Geological Disposal of Radioactive Waste in Fractured Rocks–Proceedings from The 3rd Äspö International Seminar, Oskarshamn, June 10–12, SKB Technical Report 98-10, Stockholm, 1998. 4. Kaasschieter E. F., Huijben A. J. M., Mixed-hybrid Finite Elements and Streamline Computation for the Potential Flow Problem, Report PN 90-02-A, TNO Institute of Applied Geoscience, Delft, 1990. 5. Maryška J., Rozložník M., Tůma M., Mixed-hybrid Finite Element Approximation of the Potential Fluid Flow Problem, Journal of Computational and Applied Mathematics 63 (1995), 383–392. 6. Maryška J., Rozložník M., Tůma M., Schur Complement Reduction in the Mixedhybrid Finite Element Approximation of Darcy’s Law: Rounding Error Analysis, Technical Report TR-98-06, Swiss Center for Scientiﬁc Computing, Swiss Federal Institute of Technology, Zurich, 1998, 1–15. 7. Maryška J., Severýn O., Vohralík M., Mixed-hybrid Model of the Fracture Flow, Proceedings of international conference ALGORITMY 2000, Slovak Technical University, Bratislava, 2000, 75–84.

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8. Oden J. T., Lee J. K., Dual-mixed Hybrid Finite Element Method for Second-order Elliptic Problems, in: Mathematical Aspects of Finite Element Methods (I. Galligani and E. Magenes, eds.), Lecture Notes in Math. Vol 606, Springer Verlag, Berlin, 1977, 275–291. 9. Quarteroni A., Valli A., Numerical Approximation of Partial Diﬀerential Equations, Springer-Verlag Berlin Heidelberg, Berlin, 1994. 10. Raviart P. A., Thomas J. M., A Mixed Finite Element Method for Second-order Elliptic Problems, in: Mathematical Aspects of Finite Element Methods (I. Galligani and E. Magenes, eds.), Lecture Notes in Math. Vol 606, Springer Verlag, Berlin, 1977, 292–315. 11. Roberts J. E., Thomas J. M., Mixed and Hybrid Methods in: Handbook of Numerical Analysis, vol. II, Finite Element Methods, Part 1 (P. G. Ciarlet and J. L. Lions, eds.), Elsevier Science Publishers B.V. (North-Holland), Amsterdam, 1991, 523–639. 12. Vohralík M., Mixed-hybrid Model of the Fracture Flow, Diploma Thesis, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Prague, 2000. 13. Vohralík M., Modelling the Fracture Flow, Proceedings of international conference SIMONA 2000, Technical University of Liberec, Liberec, 2000, 58–68. 14. Vohralík M., Existence- and Error Analysis of the Mixed-hybrid Model of the Fracture Flow, Technical Report MATH-NM-06-2001, Dresden University of Technology, Dresden, 2001.

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Lp -estimates for solutions of Dirichlet and Neumann problems to heat equation in the wedge with edge of arbitrary codimension

Alexander I. Nazarov Faculty of Mathematics and Mechanics, St. Petersburg State University, Bibliotechnaya pl. 2, Stary Peterhof, 198904 St. Petersburg, Russia, Email: [email protected]

Abstract. Coercive estimates in anisotropic weighted Lp -spaces are obtained for solutions of the Dirichlet and Neumann problems to the heat equation in the wedge with arbitrary codimensional edge (in particular, in the cone). MSC 2000. 35B45, 35K05, 35R20 Keywords. coercive estimates, heat equation, weighted spaces

Denote x = (x , x ) the point in Rn , x ∈ Rm , x ∈ Rn−m (2 ≤ m ≤ n). Let Km (ω) = {x : x /|x | ∈ ω} be a cone in Rm , cutting a domain ω ⊂ S1 with a smooth boundary. In the case m < n we denote by Km (ω) = Km (ω) × Rn−m the wedge in Rn with (n − m)-dimensional edge (if m = n we set Km (ω) = Km (ω)). We introduce the weighted spaces Lp,(µ) (Km ) with the norm up,(µ),Km = u · |x |µ p,Km ,

µ ∈ R,

( · p stands for the standard norm in Lp ). We also introduce two scales of anisotropic weighted spaces: Lp,q,(µ) (Km × [0, T ]) = Lq [0, T ] −→ Lp,(µ) (Km )

This work was partially supported by Russian Fund for Fundamental Research, grant no. 99-01-00684.

A note on results published in [2].

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with the norm

% % up,q,(µ) = %u(·, t)p,(µ),Km %q,[0,T ] ; " p,q,(µ) (Km × [0, T ]) = Lp,(µ) Km −→ Lq ([0, T ]) L

with the norm

% % |||u|||p,q,(µ) = %u(x, ·)q,[0,T ] %p,(µ),Km .

Let us consider the Dirichlet and Neumann initial-boundary value problems for the heat equation in Km (ω):   ut − 8u = f (x, t), x ∈ Km (ω), t > 0 (D) (1)  u = 0, ut=0 = 0; x∈∂Km (ω) (N )

  ut − 8u = f (x, t), x ∈ Km (ω),  ∂u ut=0 = 0 ∂n x∈∂Km (ω) = 0,

t>0 (1 )

(n stands for the unit outward normal). Theorem 1. Let ΛD be the ﬁrst eigenvalue of the Dirichlet problem to the Beltrami-Laplacian in ω: − 8 U = ΛD U in ω, U ∂ω = 0. Let λD be the positive root of the equation λ2 + (m − 2) · λ − ΛD = 0. Let p, q ∈ ]1, +∞[, and 2−

m m − λD < µ < m − + λD . p p

Then a solution of (1) satisﬁes the inequalities ut p,q,(µ) + D(Du)p,q,(µ) + u · |x |−2 p,q,(µ) ≤ C f p,q,(µ) ,

(2)

|||ut |||p,q,(µ) + |||D(Du)|||p,q,(µ) + |||u · |x |−2 |||p,q,(µ) ≤ C |||f |||p,q,(µ) .

(3)

Theorem 2. Let ΛN be the ﬁrst nonzero eigenvalue of the Neumann problem to the Beltrami-Laplacian in ω: ∂ U = 0. − 8 U = ΛN U in ω, ∂n ∂ω

Lp -estimates for solutions of Dirichlet and Neumann problems

325

Let λN be the positive root of the equation λ2 + (m − 2) · λ − ΛN = 0. Let p, q ∈ ]1, +∞[, and 2−

m m − min{λN , 2} < µ < m − . p p

Then a solution of (1’) satisﬁes the inequalities ut p,q,(µ) + D(Du)p,q,(µ) ≤ C f p,q,(µ) ,

(2 )

|||ut |||p,q,(µ) + |||D(Du)|||p,q,(µ) ≤ C |||f |||p,q,(µ) ,

(3 )

Remark 1. In (2), (2’), (3), (3’) C does not depend on T . Remark 2. For m = 2 , p = q the results of Theorems 1 and 2 were established in [1]. All the details and closed results are contained in [2].

References 1. V. A. Solonnikov, Lp -estimates for solutions of the heat equation in a dihedral angle, to appear in Rendiconti di matematica. 2. A. I. Nazarov, Lp -estimates for a solution to the Dirichlet problem and to the Neumann problem for the heat equation in a wedge with edge of arbitrary codimension , (in Russian) Probl. Mat. Anal., No 22, (2001), 126–159; English transl. in J. Math. Sciences 106, No 3 (2001), 2989–3014.

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A multiplicity theorem for the Neumann problem for Nonlinear Hemivariational Inequalities Nikolaos S. Papageorgiou1 and George Smyrlis2 1

2

Department of Mathematics, National Technical University, Zographou Campus, Athens 15780, Greece Email: [email protected] Department of Mathematics, National Technical University, Zographou Campus, Athens 15780, Greece Email: [email protected]

Abstract. In this paper we study a nonlinear hemivariational inequality driven by the p-Laplacian with Neumann boundary conditions. We prove a multiplicity theorem which produces at least three distinct solutions. We employ a Landesman-Lazer type condition and our approach is based on the nonsmooth critical point theory for locally Lipschitz functions.

MSC 2000. 35J20, 35J85 Keywords. Hemivariational inequality, Neumann problem, locally Lipschitz function, Clarke subdiﬀerential, nonsmooth Cerami condition, nonsmooth critical point theory, multiplicity theorem, p-Laplacian.

1

Introduction

Hemivariational inequalities are a new type of variational expressions, which arise in physical and engineering problems, when we deal with nonsmooth, nonconvex energy functionals. Generally speaking, mechanical problems involving nonmonotone, possibly multivalued stress-strain laws or boundary conditions derived by nonconvex superpotentials, lead to hemivariational inequalities. For concrete applications we refer to the books of Naniewicz -Panagiotopoulos [22] and Panagiotopoulos [23]. Hemivariational inequalitites have intrinsinc mathematical interest as a new form of variational expression. They include as a particular case problems with discontinuities. In the last ﬁve years hemivariational inequalities This is the ﬁnal form of the paper.

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have been studied from a mathematical viewpoint primarily for semilinear Dirichlet problems. We refer to the works of Goeleven-Motreanu-Panagiotopoulos [12], Motreanu-Panagiotopoulos [21] and the references therein.Quasilinear Dirichlet problems were studied recently by Gasinski-Papageorgiou [8],[9], [10], [11]. The study of the Neumann problem is lagging behind. In the past Neumann problems with a C 1 -energy functional (i.e. continuous forcing term) were studied by Mawhin-Ward-Willem [20], Drabek-Tersian [7] (semilinear problems) and Huang [16], Arcoya-Orsina [2] and Hu-Papageorgiou [13] (quasilinear problems). The only work on the Neumann problem with a discontinuous forcing term is that of CostaGoncalves [6], where the right hand side of the semilinear equation is independent of z ∈ Z, it is bounded and it has mean value zero. The aim of this paper is to prove a multiplicity result for a quasilinear hemivariational inequality with Neumann boundary condition, using conditions of Landesman-Lazer type. Similar conditions were employed by Goeleven -MotreanuPanagiotopoulos [12] (semilinear Dirichlet problems) and by Arcoya-Orsina [2] (quasilinear Neumann problems with a C 1 -potential function). In [12], the approach is degree theoretic and the authors make a rather restrictive hypothesis, namely they assume that there exists a continuous map W : L2 (Z) −→ L2 (Z) such that W (x)(z) ∈ ∂j(z, x(z)), a.e. on Z (here ∂j(z, ·) denotes the subdiﬀerential in the sense of Clarke). Given that the Clarke’s subdiﬀerential is only strong -to-weak upper semicontinuous, we realize that this is a quite restrictive hypothesis. In Arcoya-Orsina [2] the approach is variational. However, we think that there is a gap in the proof of the existence theorem (theorem 3). Namely, the claim (p. 1631) that the proof of lemma 2.1 extends to the Neumann problem is not precise, since we no longer can appeal to Poincaré’s inequality. A more careful analysis is needed. Our approach is variational and it is based on the nonsmooth critical point theory for locally Lipschitz functionals as this was formulated by Chang [4] and extended recently by Kourogenis-Papageorgiou [18].

2

Mathematical background

Let X be a Banach space and X ∗ its dual. A function ϕ : X −→ R is said to be locally Lipschitz, if for every x ∈ X there exists a neighborhood U of x and a constant kU such that |ϕ(z) − ϕ(y)| ≤ kU z − y, for all z, y ∈ U. Recall that if ψ : X −→ R = R ∪ {+∞} is proper, convex and lower semicontinuous, it is locally Lipschitz in the interior of its eﬀective domain domψ = {x ∈ X : ψ(x) < +∞}. Given x, h ∈ X we can deﬁne the generalized directional derivative of ϕ at x in the direction h by ϕ(x + th) − ϕ(x ) . t t↓0

ϕ0 (x; h) = lim sup x →x,

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Multiple solutions

It is easy to see that h -→ ϕ0 (x; h) is sublinear continuous, so by the Hahn-Banach theorem it is the support function of a nonempty, convex and w∗ -compact set

∂ϕ(x) = x∗ ∈ X ∗ : (x∗ , h) ≤ ϕ0 (x; h), for all h ∈ X . The set ∂ϕ(x) is called the generalized (Clarke) subdiﬀerential of ϕ at x ∈ X.If ϕ is also convex, Clarke’s subdiﬀerential coincides with the subdiﬀerential in the sense of convex analysis. If ϕ, ψ are both locally Lipschitz functions, then for all x ∈ X and all λ ∈ R, we have ∂(ϕ + ψ)(x) ⊆ ∂ϕ(x) + ∂ψ(x) and ∂(λϕ)(x) = λ∂ϕ(x). If ϕ ∈ C 1 (X), then ∂ϕ(x) = {ϕ (x)}. This fact makes the nonsmooth critical point theory an extension of the smooth theory. A point x ∈ X is said to be a critical point of ϕ, if 0 ∈ ∂ϕ(x), i.e. ϕ0 (x; h) ≥ 0, for all h ∈ X. Evidently, if x ∈ X is a local extremum of ϕ, then 0 ∈ ∂ϕ(x). It is well known that the smooth critical point theory uses a compactness-type condition known as the Palais-Smale condition or the more general Cerami condition. In the present nonsmooth setting this condition takes the form:“ We say that ϕ satisﬁes the nonsmooth Cc -condition, if every sequence {xn }n≥1 ⊆ X such that ϕ(xn ) → c and (1 + xn )m(xn ) → 0, as n → +∞, has a strongly convergent subsequence”. Here m(x) = inf{x∗ : x∗ ∈ ∂ϕ(x)}, for all x ∈ X. Let Z ⊆ RN be a bounded open domain with a C 1 -boundary. We consider 1,p the following direct sum decomposition : W 1,p (Z) = Rp⊕ V, V = {v ∈W (Z) : Dvp v(z)dz = 0}, 2 ≤ p < ∞. Let λ1 = inf : v ∈ V, v = 0 . We can vpp Z show that λ1 > 0 and that it is the ﬁrst nonzero eigenfunction of the p-Laplacian −∆p x = −div (Dxp−2 Dx) with Neumann boundary condition (note that λ0 = 0 is also an eigenvalue).

3

Multiplicity Theorem

We study the following quasilinear hemivariational inequality:   −div (||Dx(z)||p−2 Dx(z)) ∈ ∂j(z, x(z)) a.e. on Z       ∂x   (z) = 0, on Γ.  ∂np

  

(1)

∂x (z) = ||Dx(z)||p−2 (Dx(z), n(z))RN , with n(z) the outward normal to ∂np Γ (= the boundary of Z) and 2 ≤ p < ∞. Our hypotheses on the nonsmooth potential j(z, x) are the following: H(j) : j : Z × R −→ R is a function such that j(·, 0) ∈ L1 (Z) and

Here

(i) for all x ∈ R, z -→ j(z, x) is measurable; (ii) for almost z ∈ Z, x -→ j(z, x) is locally Lipschitz; (iii) for almost z ∈ Z, all x ∈ R and all u∗ ∈ ∂j(z, x) we have 1 1 |u∗ | ≤ a(z) + c|x|r−1 , with a ∈ Lq (Z) + = 1 , c > 0, 1 ≤ r < p; p q

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(iv) for almost z ∈ Z and all x ∈ R, pj(z, x) ≤ λ1 |x|p (v) there exist two functions j± ∈ L1 (Z) such that

lim j(z, x) = j± (z), uni-

x→±∞

formly for almost all z ∈ Z (vi) there exist c− < 0 < c+ such that j(z, c+ )dz, j(z, c− )dz > j± (z)dz > 0. Z

Z

Z

We consider the energy functional ϕ : W (Z) −→ R deﬁned by 1 p j(z, x(z))dz. ϕ(x) = Dxp − p Z 1,p

We know that ϕ is locally Lipschitz (see Hu-Papageorgiou [15], p.313). Proposition 1. If hypotheses H(j) hold, then ϕ satisﬁes the nonsmooth Cc j± (z)dz . condition for c = − Z

Proof. Let {xn }n≥1 ⊆ W 1,p (Z) be a sequence such that j± (z)dz and (1 + xn )m(xn ) → 0, ϕ(xn ) → c, with c = −

as n → ∞.

Z

We will show that {xn }n≥1 ⊆ W 1,p (Z) is bounded. Suppose not. Then by passing xn , to a subsequence if necessary, we may assume that xn → ∞. Let yn = xn n ≥ 1. We may assume that w

yn −→ y in W 1,p (Z), yn → y in Lp (Z), a.e. on Z and |yn (z)| ≤ k(z), a.e. on Z, for all n ≥ 1, with k ∈ Lp (Z). From the choice of the sequence {xn }n≥1 we have that 1 Dxn pp − j(z, xn (z))dz ≤ M1 , for all n ≥ 1, with M1 > 0. p Z Dividing by xn p we obtain 1 j(z, xn (z)) M1 Dyn pp − dz ≤ . p p xn xn p Z

(2)

Using Lebourg’s mean value theorem (see Clarke [5], p.41), we see that for almost all z ∈ Z we can ﬁnd u∗λ ∈ ∂j(z, λx) with 0 < λ < 1, such that |j(z, x) − j(z, 0)| = |u∗λ x| =⇒ |j(z, x)| ≤ |j(z, 0)| + (a(z) + c|x|r−1 )|x| ≤ a1 (z) + c1 |x|r , a1 ∈ L1 (Z), c1 > 0

331

Multiple solutions

(see hypothesis H(j)(iii) and recall that j(·, 0) ∈ L1 (Z)). So we obtain j(z, xn (z)) a1 (z) |yn (z)|r dz ≤ dz+ c dz → 0, as n → ∞( since r < p). 1 p p p−r xn Z Z xn Z xn Thus by passing to the limit in (2) and using the weak lower semicontinuity of the norm functional, we obtain Dypp = 0 =⇒ y = ξ ∈ R. Note that yn → ξ in W 1,p (Z) and because yn = 1, for n ≥ 1, we infer that ξ = 0. We may assume without loss of generality that ξ > 0. Then xn (z) → +∞, a.e. on Z. Also let x∗n ∈ ∂ϕ(xn ), n ≥ 1, such that m(xn ) = x∗n . The existence of such an element follows from the fact that ∂ϕ(xn ) is w-compact and x∗ -→ x∗ is weakly lower semicontinuous on W 1,p (Z)∗ . From the choice of the sequence {xn }n≥1 ⊆ W 1,p (Z) we have | x∗n , xn | ≤ εn , with εn ↓ 0 and ·, · being the duality brackets for the pair W 1,p (Z), W 1,p (Z)∗ . Let A : W 1,p (Z) → W 1,p (Z)∗ be the nonlinear operator deﬁned by Dx(z)p−2 (Dx(z), Dy(z))RN dz, for all x, y ∈ W 1,p (Z).

A(x), y = Z

It is easy to check that A is demicontinuous, monotone, thus it is maximal (see Hu-Papageorgiou [14], p. 309). For every n ≥ 1, we have x∗n = A(xn ) − u∗n , with 1,p deﬁned by u∗n ∈ ∂J(x n ), where J : W (Z) −→ R is the integral functional p j(z, x(z))dz. If J1 : L (Z) −→ R is deﬁned by J1 (x) = j(z, x(z))dz, J(x) = Z

Z

then J = J1 |W 1,p (Z) and both are locally Lipschitz. Moreover, from Chang [4] q = {u∗ ∈ Lq (Z) : u∗ (z) ∈ (theorem 2.2) we have that ∂J(x) ⊆ ∂J1 (x) = S∂j(·,x(·)) ∂j(z, x(z)) a.e. on Z} ⊆ Lq (Z) (see Clarke [5], p. 83). So we have

x∗n , xn = Dxn pp − u∗n (z)xn (z)dz ≤ εn . Z

We consider the direct -sum decomposition W 1,p (Z) = R ⊕ V , with V = {v ∈ v(z)dz = 0}. So we can write xn = xn + x ˆn with xn ∈ R and W 1,p (Z) : Z

x ˆn ∈ V, n ≥ 1 and we we have

u∗n (z)xn (z)dz ≤ εn .

Dˆ xn pp −

(3)

Z

From the deﬁnition of the Clarke subdiﬀerential (see section 2) we have u∗n (z)xn (z) ≤ j 0 (z, xn (z)) = lim sup vn →xn (z)

j(z, vn + ε xn (z)) − j(z, vn ) . ε

ε↓0

Recall that for almost all z ∈ Z, xn (z) → +∞ as n → ∞. So vn → +∞ as n → ∞. Hence by virtue of hypothesis H(j) (v), given ε > 0 we can ﬁnd n0 (ε) ≥ 1 such

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that for all n ≥ n0 and all z ∈ Z \ N1 , |N1 | = 0 (| · | being the Lebesgue measure on RN ), we have j+ (z) − ε2 ≤ j(z, vn + εxn (z)) ≤ j+ (z) + ε2 and j+ (z) − ε2 ≤ j(z, vn ) ≤ j+ (z) + ε2 . So for all n ≥ n0 and all z ∈ Z \ N1 , we have u∗n (z)xn (z) ≤

2ε2 j+ (z) + ε2 − j+ (z) + ε2 = = 2ε ε ε

and

j+ (z) − ε2 − j+ (z) − ε2 −2ε2 = = −2ε. ε ε Therefore for all n ≥ n0 and all z ∈ Z \ N1 , we have |u∗n (z)xn (z)| ≤ ε, hence u∗n (z)x n (z) → 0 as n → ∞ uniformly for almost all z ∈ Z and so it is fulﬁlled u∗n (z)xn (z) ≥

u∗n (z)xn (z)dz → 0 as n → ∞. Thus if we pass to the limit as n → ∞ in

that Z

(3), we obtain Dˆ xn pp → 0 as n → ∞. This by virtue of the Poincaré-Wirtinger inequality (see Hu-Papageorgiou [15], p. 866) implies that x ˆn → 0 in W 1,p (Z) as n → ∞. xn (z)), j(z, xn + x ˆn (z)) − Let Γn (z) = {(v ∗ , λ) ∈ R × (0, 1) : v ∗ ∈ ∂j(z, xn + λˆ j(z, xn ) = v ∗ xn (z). From the Lebourg mean value theorem, we have that for almost all z ∈ Z, Γn (z) = ∅. By redeﬁning Γn on the exceptional Lebesgue-null set (setting Γn to be equal to {0}, for example), we may assume without any loss of generality that Γn (z) = ∅ for all z ∈ Z. We claim that for every h ∈ R, the xn (z); h) is measurable. Indeed, from the deﬁnition function (z, λ) -→ j 0 (z, xn + λˆ of the generalized directional derivative, j 0 (z, xn + λˆ xn (z); h) equals to inf

m≥1

j(z, xn + λˆ xn (z) + r + sh) − j(z, xn + λˆ xn (z) + r) . s r,s ∈ Q ∩ (−1/m, 1/m) sup

But j(z, x) is jointly measurable (see Hu-Papageorgiou [14], p.142). So it follows xn (z); h) is measurable. Let Sn (z, λ) = ∂j(z, xn +λˆ xn (z)) that (z, λ) -→ j 0 (z, xn +λˆ and {hm }m≥1 ⊆ R be a countable dense set. Because j 0 (z, xn + λˆ xn (z); ·) is continuous, we have GrSn = {(z, λ, u) ∈ Z × (0, 1) × R : u ∈ Sn (z, λ)} ? = {(z, λ, u) ∈ Z × (0, 1) × R : uhm ≤ j 0 (z, xn + λˆ xn (z); hm )}, m≥1

so GrΓn = GrSn {(z, v ∗ , λ) ∈ Z × R × (0, 1) : j(z, xn + xˆn (z)) − j(z, xn ) = ∗ v xˆn (z)} ∈ L × B(R) × B(0, 1), with L being the Lebesgue σ-ﬁeld of Z. Invoking the Yankov-von-Neumann-Aumann selection theorem (see Hu-Papageorgiou [14], p. 158), we obtain measurable functions vn∗ : Z −→ R and λn : Z −→ (0, 1) such that (vn∗ (z), λn (z)) ∈ Γn (z) a.e. on Z. Therefore we have j(z, xn + xˆn (z)) − j(z, xn ) = vn∗ (z)ˆ xn (z), vn∗ (z) ∈ ∂j(z, xn + λˆ xn (z)), a.e. on Z, for all n ≥ 1. Thus we can write that 1 p ∗ ϕ(xn ) = Dxn p − vn (z)ˆ xn (z)dz − j(z, xn )dz. p Z Z

(4)

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Multiple solutions

Arguing as before, we can show that

vn∗ (z)ˆ xn (z)dz → 0 as n → ∞ (note

Z

xn (z)}n≥1 is bounded for that xn → +∞ since xn (z) → +∞ a.e. on Z and {ˆ almost all z ∈ Z (recall that x ˆn → 0 in W 1,p (Z))). Also we know that Dxn p = Dˆ xn p → 0 as n → ∞. So by passing to the limit as n → ∞ in (4), we obtain that c = −

j+ (z)dz, a contradiction. Similarly, if we assume that ξ < 0, we reach the contradiction that c = − j− (z)dz. Z

Z

w

Therefore {xn }n≥1 ⊆ W 1,p (Z) is bounded and so we may assume that xn −→ x ∗ in W 1,p (Z) and xn → x in Lp (Z). We have | xn , xn − x| ≤ εn , with εn ↓ 0, which u∗n (xn − x)dz ≤ εn , n ≥ 1.

implies that A(xn ), xn − x − Z

∗ r q By virtue of hypothesis H(j) (iii), {un }n≥1 ⊆ L (Z) ⊆ L (Z) is bounded u∗n (xn − x)dz → 0. It follows that (1/r + 1/r = 1 and r < p). So Z

lim sup A(xn ), xn − x ≤ 0 and because A is maximal monotone, it is generalized pseudomonotone (see Hu-Papageorgiou [14], p. 365) and so A(xn ), xn → w

A(x), x, hence Dxn p → Dxp . Because Dxn −→ Dx in Lp (Z, RN ) and the latter is a uniformly convex Banach space, it follows that Dxn → Dx in Lp (Z, RN ) (see Hu-Papageorgiou[14], p. 28) and so xn → x in W 1,p (Z). Proposition 2. If hypotheses H(j) hold, then ϕ is bounded below and ϕ |V ≥ 0. Proof. By virtue of hypothesis H(j) (v), we can ﬁnd N2 ⊆ Z Lebesgue-null set and M2 > 0 such that |j(z, x) − j+ (z)| ≤ 1, for all z ∈ Z \ N2 and all x ≥ M2 and |j(z, x) − j− (z)| ≤ 1, for all z ∈ Z \ N2 and all x ≤ −M2 . Also because |j(z, x)| ≤ a1 (z) + c1 |x|r a.e. on Z with a1 ∈ L1 (Z), c1 > 0 (see the proof of proposition 1) we have that for almost all z ∈ Z \ N2 and all |x| ≤ M2 , |j(z, x)| ≤ a2 (z) with a2 ∈ L1 (Z). So for all x ∈ W 1,p (Z) we have 1 p j(z, x(z))dz ϕ(x) = Dxp − p Z ≥− j(z, x(z))dz − j(z, x(z))dz − −

{x(z)≥M2 }

{|x(z)|≤M2 }

{x(z)≤−M2 }

j(z, x(z))dz ≥

≥ −j+ 1 − j− 1 − 2|Z| − a2 1 , which implies that ϕ is bounded from below.

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For v ∈ V , recall that Dvpp ≥ λ1 vpp (see section 2). So using hypothesis H(j) (iv) for all v ∈ V we have 1 1 λ1 p ϕ(v) = Dvp − j(z, v(z))dz ≥ Dvpp − vpp ≥ 0. p p p Z Using these two auxiliary results we can prove the following multiplicity theorem for problem (1). Theorem 3. If hypotheses H(j) hold, then problem (1) has at least three distinct solutions. Proof. We introduce the following open subsets of W 1,p (Z) U ± = {x = ±η + v : η > 0, v ∈ V }. Let m± = inf [ϕ(x) : x ∈ U ± ] > −∞ (see proposition 2). Also let ϕ± : W 1,p (Z) −→ R = R ∪ {+∞} be deﬁned by   ϕ(x), ϕ± (x) =



if x ∈ U

+∞,

±

otherwise

Both functions ϕ± are lower semicontinuous and bounded below. In what follows we shall work with ϕ+ but a similar analysis can be conducted using ϕ− . 1 Invoking theorem 1.1 of Zhong [25] with ε = , n ≥ 1, we generate a sequence n {xn }n≥1 ⊆ U + such that ϕ(xn ) ↓ m+ and 1 xn − yn n , ϕ+ (xn ) ≤ ϕ+ (y) + 1 + xn − =⇒

for all y ∈ W 1,p (Z),

1 xn − yn n ≤ ϕ+ (y) − ϕ+ (xn ), 1 + xn

for all y ∈ W 1,p (Z).

Let u ∈ W 1,p (Z) and set y = xn + tu, t > 0. Since U + is open for t ∈ (0, δ) we have y ∈ U + . So 1 u ϕ(xn + tu) − ϕ(xn ) n ≤ , 1 + xn t −

1 u ≤ ϕ0 (xn ; u), =⇒ n 1 + xn −

t ∈ (0, δ), (recall ϕ+ |U + = ϕ+ ),

(see section 2), for all u ∈ W 1,p (Z).

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Multiple solutions

1 + xn 0 ϕ (xn ; u), n ≥ 1. Evidently, ϑn is continuous, 1 n sublinear ( hence ϑn (0) = 0 ) and for all u ∈ W 1,p (Z) we have −u ≤ ϑn (u). So we can apply lemma 1.3 of Szulkin [24] to obtain yn∗ ∈ W 1,p (Z)∗ such that yn∗ ≤ 1 1 n y ∗ we and yn∗ , u ≤ ϑn (u), for all u ∈ W 1,p (Z) and all n ≥ 1. If x∗n = 1 + xn n have x∗n , u ≤ ϕ0 (xn ; u) for all u ∈ W 1,p (Z), n ≥ 1. So xn ∈ ∂ϕ(xn ) for all n ≥ 1. We have Introduce ϑn (u) =

(1 + xn )m(xn ) ≤ (1 + xn )x∗n ≤

1 −→ 0, n

as n → ∞.

Recall that ϕ(xn ) ↓ m+ and m+ ≤

j(z, c+ )dz < Z

j+ dz < 0 (see hyZ

pothesis H(j) (vi)). From proposition 1 we know that ϕ satisﬁes the nonsmooth Cm+ -condition. So by passing to a subsequence if necessary, we may assume that xn → y1 in W 1,p (Z). We have ϕ(xn ) → ϕ(y1 ) = m+ < 0. If y1 ∈ bd U + = V , then ϕ(y1 ) = m+ ≥ 0 (see proposition 2), a contradiction. Therefore y1 ∈ U + and it follows that y1 is a local minimum of ϕ, hence 0 ∈ ∂ϕ(y1 ). With a similar argument we obtain y2 ∈ U − such that 0 ∈ ∂ϕ(y2 ) and of course y1 = y2 = 0. Finally because of hypothesis H(j) (vi) and the fact that ϕ |V ≥ 0, we can apply the nonsmooth Saddle Point Theorem (see Kourogenis-Papageorgiou [18]) and produce y3 ∈ W 1,p (Z) such that ϕ(y3 ) = c ≥ 0 − 0 ∈ ∂ϕ(y3 ). Now let y = yk , k = 1, 2, 3. We have 0 ∈ ∂ϕ(y) and so A(y) = u∗ ,

j± (z)dz > m± and Z

for some u∗ ∈ ∂J(x) ⊆ Lq (Z)

(5)

(see the proof of proposition 1). Let ψ ∈ C0∞ (Z). Since −div(Dyp−2 Dy) ∈ W −1,q (Z) = W01,p (Z)∗ (see for example Adams [1], p. 50), by integration by parts we obtain

A(y), ψ = −div(Dyp−2 Dy), ψ = u∗ ψ dz = u∗ , ψ. Z

But C0∞ (Z) is dense in W01,p (Z), so we obtain −div(Dy(z)p−2 Dy(z)) = u∗ (z) ∈ ∂j(z, y(z)),

a.e. on Z.

(6)

Also from the “quasilinear ” Green’s identity (see Kenmochi [17], Casas-Fernandez [3] or Hu-Papageorgiou [15], p. 867), for every v ∈ W 1,p (Z) we have @ A ∂x (−div(Dyp−2 Dy)) v dz + Dyp−2 (Dy, Dv)RN dz = , γ(v) ∂np Z Z Γ

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with ·, ·Γ being the duality brackets for the pair W 1/q, p (Γ ), W −1/q, q (Γ ) and γ : W 1,p (Z) −→ Lp (Γ ) is the trace operator. From (5) and (6) we obtain @ A ∂x −u∗ v dz + A(y), v = , γ(v) . 0= np Z Γ But γ W 1,p (Z) = W 1/q, p (Γ ) (see Kufner-John- Fučik [19], p. 338). So it follows ∂x that = 0. Therefore y1 , y2 , y3 are distinct solutions of (1). ∂np As a simple example of a function which satisﬁes hypotheses H(j), consider the following locally Lipschitz nonsmooth potential j(x) (for simplicity we drop the z- dependence):  λ1   |x|p , if |x| ≤ 1   p j(x) = , with p a < λ1 , a > 0,   a λ1   − a, if |x| ≥ 1 + x2 p  λ1 |x|p−2 x, if |x| < 1       =⇒ ∂j(x) = [−2a, λ1 ], if |x| = 1       − 2a , if |x| > 1. x3 Clearly, j± =

λ1 λ1 − a > 0 and so we can have c± = ±1. Then j(c± ) = > p p

λ1 − a = j± . Also clearly pj(x) ≤ λ1 |x|p for all x ∈ R and ﬁnally for all x ∈ R p and all u∗ ∈ ∂j(x), we have |u∗ | ≤ M0 , M0 > 0. So hypotheses H(j) are satisﬁed.

References 1. R.Adams, Sobolev Spaces Academic Press, New York, (1975). 2. D. Arcoya-L.Orsina, Landesman-Lazer conditions and quasilinear elliptic equations, Nonlin. Anal. 28(1997), 1623-1632. 3. E.Casas-L. Fernandez, A Green’s formula for quasilinear elliptic operators, J.Math.Anal.Appl. 142(1989), 62-73. 4. K.-C. Chang, Variational methods for nondiﬀerentiable functionals and their applications to partial diﬀerential equations, J.Math.Anal.Appl. 80(1981), 102-129. 5. F.H.Clarke, Optimization and Nonsmooth Analysis, Wiley, New York (1983). 6. D.Costa-J.V.A.Goncalves, Critical point theory for nondiﬀerntiable functionals and applications, J.Math. Anal.Appl. 153(1990), 470-485. 7. P. Drabek-S. Tersian, Characterization of the range of Neumann problem for semilinear elliptic equations, Nonlin. Anal. 11(1987), 733-739.

Multiple solutions

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8. L.Gasinski-N.S.Papageorgiou, Nonlinear hemivariational inequalities at resonance, Bull. Austr.Math. Soc. 60(1999), 353-364. 9. L.Gasinski-N.S.Papageorgiou, An existence theorem for nonlinear hemivariational inequalities at resonance, Bull. Austr.Math. Soc. 63(2001), 1-14. 10. L.Gasinski-N.S.Papageorgiou, Multiple solutions for nonlinear hemivariational inequalities near resonance, Funkcialaj Ekvaciaj 43(2000), 271-284. 11. L.Gasinski-N.S.Papageorgiou, Solutions and multiple solutions for quasilinear hemivariational inequalities at resonance, Proc.Royal Soc. Edinburg(Math) 131A(2001), 1-21. 12. D.Goeleven-D.Motreanu-P.Panagiotopoulos, Eigenvalue problems for variationalhemivariational inequalities at resonance, Nonlin. Anal. 33(1998), 161-180. 13. S.Hu-N.S.Papageorgiou, Nonlinear elliptic problems of Neumann type, Rendiconti Circolo Matem. di Palermo L(2001), 47-66. 14. S.Hu-N.S.Papageorgiou, Handbook of Multivalued Analysis, Volume I: Theory Kluwer, Dordrecht, The Netherlands(1997). 15. S.Hu-N.S.Papageorgiou, Handbook of Multivalued Analysis, Volume II: Applications Kluwer, Dordrecht, The Netherlands(2000). 16. Y.X.Huang, On eigenvalue problems of the p-Laplacian with Neumann boundary conditions, Proc.AMS 109(1990), 177-184. 17. N.Kenmochi, Pseudomonotone operators and nonlinear elliptic boundary value problems, J.Math.Soc. Japan 27 (1975), 121-149. 18. N.Kourogenis-N.S.Papageorgiou, Nonsmooth critical point theory and nonlinear elliptic equations at resonance, J.Austr.Math.Soc.(SerA) 69(2000), 245-271. 19. A.Kufner-O.John-S.Fu˘cik, Function Spaces, Noordhoﬀ Publ.Co., Leyden, The Netherlands(1997). 20. J.Mawhin-D. Ward-M.Willem, Variational methods and semilinear elliptic equations, Arch. Rat. Mech.Anal. 95(1986),269-277. 21. D.Motreanu-P.Panagiotopoulos, A minimax approach to the eigenvalue problem of hemivariational inequalities and applications, Appl.Anal. 58(1995), 53-76. 22. Z.Naniewicz-P.Panagiotopoulos, Mathematical Theory of Hemivariational Inequalities and Applications, Marcel Dekker, New York(1995). 23. P.Panagiotopoulos, Hemivariational Inequalities. Applications to Mechanics and Engineering, Springer-Verlag, New York(1993). 24. A.Szulkin, Minimax principles for lower semicontinuous functions and applications to nonlinear boundary value problems, Ann.Inst.H.Poincaré, Analyse Non Lineaire 3(1986), 77-109. 25. C.K.Zhong, On Ekeland’s variational principle and a minimax theorem, J.Math.Anal.Appl. 205(1997), 239-250.

Equadiﬀ 10, August 27–31, 2001 Prague, Czech Republic

Equadiﬀ 10 CD ROM Papers, pp. 339–347

Logistic Equation on Measure Chains Zdeněk Pospíšil Department of Mathematics, Faculty of Science, Masaryk University, Janáčkovo nám 2a, 662 95 Brno, Czech Republic, Email: [email protected]

Abstract. The dynamic equation (14) unifying both the Verhulst diﬀerential and the Pielou diﬀerence logistic equations is derived. Some application of it is brieﬂy discussed. MSC 2000. 34A99, 39A12, 92D25 Keywords. Logistic equation, continuous and discrete population growth

1

Introduction

The logistic equation

x (1) x = rx 1 − K introduced by Pierre Fran¸cois Verhulst in 1838 [8] has became a useful tool in modeling of a population growth. Here, x denotes a “size” of population (number of individuals, population density, biomass etc.), r an intrinsic growth rate and K a carrying capacity of environment. The equation serves as a “standard equation” in population dynamics and it uses to be generalized in various directions. Among these generalizations, discrete analogies of equation (1) play an important role — such diﬀerence equations represents models of populations with non-overlapping generations. There are several discrete logistic equations. The Euler discretization of equation (1) with the step 1 gives the equation xk , (2) xk+1 = `xk 1 − K which was utilized e.g. by Maynard Smith [5]. This equation has one great disadvantage: if the population size is greater than K, it become negative in the next

Research supported by grant 201/99/0295 of the Czech Grant Agency

This is the preliminary version of the paper.

340

Z. Pospíšil

time step. This is why, another discrete population models were constructed. The most popular one is the May [4] equation xk . (3) xk+1 = xk exp ρ 1 − K A diﬀerent discrete equation xk+1 =

`xk 1 + P−1 K xk

(4)

was proposed by Pielou [6]. Now, the equation arises — which equation among (2), (3) and (4) is a “correct” discrete analogy of equation (1)? By the “correctness”, I mean the following property: there exists a general equation such that equation (1) and a diﬀerence equation are particular cases of it. Recently, a powerful tool for uniﬁcation of diﬀerential and diﬀerence equations was discovered by Stefan Hilger [2]. It is the calculus on a special structure — measure chains or time scales. (The two concepts were used as synonyms in the original Hilger’s papers. Now, the words “measure chain” denote an abstract structure while the notion “time scale” stands for a particular subset of reals.) The uniﬁcation of discrete and continuous models of population growth has much more than a theoretical (or aesthetic) signiﬁcance. An evolution of population may be neither completely continuous nor completely discrete. A certain species may evolve in a continuous way during a period of favorable conditions. But such a period may be interrupted by an event or by a season of bad conditions and the size of population “jumps” to a diﬀerent value after such event or season. As a typical examples, we can consider an insect population taken for a pest in agriculture and hence chemically destroyed in regular or irregular time intervals, or a population of mites which reproduces itself with several generations during spring and summer times and only some fertilized females survive winter season. A uniﬁed equation — neither continuous nor discrete but possessing properties of the both cases — should describe such populations, too. The derivation of uniﬁed equation is the main result of the contribution. In the next section, the notion of measure chain is brieﬂy described, the dynamic equations and the exponential function on measure chain are reminded. This section is intended mainly as an “advertisement” of the theory. The calculus on measure chain (time scales) is described in details in the monographs [1] and [3]. The last section contains the announced result.

2

Measure chains

Measure chain is a set T satisfying the following axioms. Axiom 1 (Chain) T is totally ordered set. This means that there is a relation ≤ on T which is reﬂexive (t ≤ t for all t ∈ T), antisymmetric (s ≤ t and t ≤ s ⇒ s = t), transitive (r ≤ s and s ≤ t ⇒ r ≤ t) and total (s ≤ t or t ≤ s for all t ∈ T).

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As usual, s < t means s ≤ t and s = t, s > t means t < s, and so on. The open, close and half-open intervals of T are deﬁned by [r, s] = {t ∈ T : r ≤ t ≤ s}, ]r, s[ = {t ∈ T : r < t < s}, ]r, s] = {t ∈ T : r < t ≤ s},

[r, s[ = {t ∈ T : r ≤ t < s}.

The order topology is generated by the open intervals of T. An open interval of T containing t ∈ T is called neighborhood of t. Axiom 2 (Conditionally complete chain) Each nonempty subset of T, which is bounded above, has a least upper bound (supremum). Consequently, each nonempty subset of T, which is bounded below, has a greatest lower bound (inﬁmum). The forward and backward jump operators are mappings σ, ρ : T → T deﬁned by σ(t) = inf{s ∈ T : s > t}, ρ(t) = sup{s ∈ T : s < t}. Via these two operators, points in T can be classiﬁed with respect to their right and left order neighborhood: t ∈ T is called right-dense, right-scattered, left-dense or left-scattered, if σ(t) = t, σ(t) > t, ρ(t) = t or ρ(t) < t, respectively; t ∈ T is called dense if it is right-dense or left-dense. Axiom 3 (Growth calibration) There exists a mapping µ : T × T → R with the following properties – (Cocycle property) For all r, s, t ∈ T we have µ(r, s) + µ(s, t) = µ(r, t). – (Strong isotony) For all r, s ∈ T we have the implication r > s ⇒ µ(r, s) > 0. – (Continuity) µ is continuous with respect to the product order topology. The function d(r, s) := |µ(r, s)| is a metric on T which generates the order topology. Because of the conditionally completeness with respect to the ordering, T is also complete with respect to the metric. The graininess function µ∗ : T → {x ∈ R : x ≥ 0} is deﬁned as µ∗ (t) = µ(σ(t), t). Any closed subset of R with the natural ordering and with growth calibration µ deﬁned by µ(r, s) = r − s is an example for a measure chain; it is called a time scale. If T = R, then σ(t) = ρ(t) = t and µ∗ (t) = 0 for each t ∈ T; if T = Z, then σ(t) = t + 1, ρ(t) = t − 1 and µ∗ (t) = 1 for each t ∈ T. It is remarkable that there is no algebraic structure (e.g. group or ring structure) on T. A function f : T → R is called rd-continuous (right-dense-continuous), if it is continuous in each right dense instant and has a left-sided limit in each instant of T, which is at the same time right-scattered and left dense. A function f : T → R is called regressive provided 1 + µ∗ (t)f (t) = 0 for all # T \ {m}, if T has a left-scattered maximum m κ t∈T = . T, otherwise

342

Z. Pospíšil

In the subsequent text, for function f : T → R, we will abbreviate f ◦ σ by f σ . A function f : T → R is called (delta) diﬀerentiable at t ∈ T, with (delta) derivative f ∆ (t) ∈ R, if for each ε ∈ R, ε > 0, there is a neighborhood Ω of t such that σ f (t) − f (s) − f ∆ (t)µ(σ(t), s) ≤ ε|µ(σ(t), s)| for all s ∈ Ω. It follows that f σ (t) = f (t) + µ∗ (t)f ∆ (t)

(5)

for each t ∈ T where the derivative exists, see e.g. [3, Theorem 1.2.2]. Let f, g : T → R be functions, which are diﬀerentiable at t ∈ T and let c ∈ R. Then the following hold ∆ (6) (cf (t)) = cf ∆ (t), ∆

(f (t) ± g(t)) = f ∆ (t) ± g ∆ (t), ∆

(f (t)g(t)) = f ∆ (t)g(t) + f σ (t)g ∆ (t),

1 f (t)

∆ =−

f ∆ (t) f σ (t)f (t)

if f (t) = 0 = f σ (t).

(7) (8)

(9)

Let τ ∈ Tκ . If g : Tκ → R is an rd-continuous function, then there exists exactly one function f : T → R with the properties: f ∆ (t) = g(t) for all t ∈ T

and f (τ ) = 0,

see e.g. [3, Theorem 1.4.4]. According to the corresponding notation in real analysis we write t g(s)∆s = f (t). τ

Applying (5), we obtain the useful formula t

σ(t) g(s)∆s = g(s)∆s − µ∗ (t)g(t).

τ

(10)

τ

The basic concepts concerning measure chains and the two important special cases of it — i.e. the sets of reals R and integers Z — are summarized in table 1. ∞ Example 1. Let {ak }∞ k=0 be an increasing sequence of reals and let {dk }k=1 be a sequence of positive reals. Let T = {(k, τ ) ∈ N0 × R : ak ≤ τ ≤ ak+1 }. Let us deﬁne the lexicographic ordering on T, i.e.

(k, ξ) ≤ (l, η)

⇔

k < l or (k = l and ξ ≤ η)

343

Logistic Equation on Measure Chains general T forward jump operator σ(t) backward jump operator ρ(t) growth calibration µ(r, s) graininess function µ∗ (t) “shift” f σ (t) rd-continuous f regressive f derivative f ∆ (t) t integral f (s)∆s τ

T=R t t r−s 0 f (t) continuous f any f f (t) t f (s)ds

T=Z t+1 t−1 r−s 1 f (t + 1) any f f (t) = −1 ∆f (t) = f (t + 1) − f (t) t−1 f (k)

τ

k=τ

Table 1. Two most important special cases of measure chains k

and a mapping µ : T × T → R as µ ((k, ξ), (l, η)) = ξ − η + convention:

m i=n

n−1

αi = −

αi for m < n − 1 and

i=m+1

n−1

di (we use the

i=l+1

αi = 0). Then T is a

i=n

measure chain with growth calibration µ. The forward jump operator σ and the graininess function µ∗ are given by   (k, τ ), 0, if τ = ak+1 if τ = ak+1 σ(k, τ ) = , µ∗ (k, t) = . (k + 1, ak+1 ), if τ = ak+1 dk+1 , if τ = ak+1 This measure chain can underlay a model of a process whose continuous evolution is usually interrupted by an event (impulse, catastrophe etc.) of “size” di at time instants ai , i = 1, 2, . . . . In such a case, (k, τ ) denotes a time instant with “absolute time distance” τ from beginning a0 after k occurrences of the interrupting events. (This example was introduced in [7], but it contained an awkward misprint in the graininess function there.) Let f : T → R be a function. Then  ∂f (k, τ )/∂τ , if τ = aka+1 and ∂f (k, τ )/∂τ exists f ∆ (k, τ ) = [f (k + 1, τ ) − f (k, τ )] /dk+1 , if τ = ak+1 , (k,τ )

f (l, s)∆(l, s) = (0,a0 )

k−1

ai+1

i=0 a i

τ

f (i, σ)dσ +

f (i, σ)dσ + ak

[τ ]

di+1 f (i, ai+1 ).

i=1

A dynamic equation is an equation of the form x∆ = f (t, x), where the mapping f : T × R → R. Let p : T → R. The dynamic equation x∆ = p(t)x

(11)

344

Z. Pospíšil

is called a (ﬁrst order) linear (dynamic) equation. If function p is regressive and rd-continuous, then the initial value problem (11), x(t0 ) = x0 admits exactly one solution, see e.g. [1, Theorems 2.33, 2.35]. Let p : T → R be an rd-continuous regressive function. The (generalized) exponential function ep (t, τ ) is deﬁned to be the unique solution of the initial value problem x∆ = p(t)x, x(τ ) = 1. Clearly, the derivative of the exponential function is given by e∆ p (t, τ ) = p(t)ep (t, τ ).

(12)

If T = R, p ≡ 1 and τ = 0, then the solution of initial value problem x = x, x(0) = 1 is x(t) = exp(t). This observation justiﬁes the terminology. If T = R, p is arbitrary continuous function and τ = R, then t ep (t, τ ) = exp

p(s)ds. τ

Example 2. Let T be the time scale introduced in Example 1 and suppose that ˜ the sequence {dk }∞ k=1 is bounded above, i.e. d = inf {1/dk : k = 1, 2, . . . } > 0. ˜ Further, let β > 0, δ ∈]0, d[ and put  β, if τ = ak+1 . r(k, τ ) = −δ, if τ = ak+1 Then the function r is rd-continuous and regressive. Now, one can easily verify that k er (k, τ ), (0, a0 ) = exp(−βa0 ) − δ di exp(−βai ) exp(βτ ). i=1

In particular, if ai = i for i ∈ N0 and di = 1 for i ∈ N then 1 − exp(−βk) exp(βτ ). er (k, τ ), (0, 0) = 1 + δ 1 − exp(β)

3

The equation

The initial value problem for the nonautonomous logistic ordinary diﬀerential equation x , x(t0 ) = x0 x = r(t)x 1 − K(t) has the solution t exp r(τ )dτ t0

x(t) = x0 1 + x0

t t0

r(τ ) K(τ )

τ . exp r(s)ds dτ t0

345

Logistic Equation on Measure Chains

Consequently, a dynamic equation possessing the Verhulst equation (1) as a particular case, should have the solution er (t, t0 )

x(t) = x0 1 + x0

t t0

.

(13)

r(τ ) K(τ ) er (τ, t0 )∆τ

By (6)–(9) and (12) the delta derivative of the previous equality is t e∆ r (t, t0 ) 1 + x0 x∆ (t) = x0

t 1 + x0 t0

t0

r(τ ) K(τ ) er (τ, t0 )∆τ

r(τ ) K(τ ) er (τ, t0 )∆τ

t er (t, t0 ) r(t) 1 + x0 = x0

t0

1 + x0

t t0

r(t) − er (t, t0 )x0 K(t) er (t, t0 )

t0

r(τ ) K(τ ) er (τ, t0 )∆τ

r(τ ) K(τ ) er (τ, t0 )∆τ

1 + x0

t r(t) 1 + x0 = x(t)

=

σ(t) r(τ ) 1 + x0 e (τ, t )∆τ 0 K(τ ) r

t0

σ(t) t0

r(t) − x0 K(t) er (t, t0 ) r(τ ) K(τ ) er (τ, t0 )∆τ

r(τ ) K(τ ) er (τ, t0 )∆τ

1 + x0

σ(t) t0

−

=

x0 K(t) er (t, t0 )

r(τ ) K(τ ) er (τ, t0 )∆τ

Now, formulae (10), (12) and (5) yield   x∆ (t) = r(t)x(t) 1 −

 r(t) x0 µ∗ (t) K(t) er (t, t0 ) +

1 + x0  x0  = r(t)x(t) 1 − K(t) 

σ(t) t0

x0 K(t) er (t, t0 ) 

=

r(τ ) K(τ ) er (τ, t0 )∆τ ∗



µ (t)e∆ r (t, t0 ) + er (t, t0 ) σ(t) r(τ ) 1 + x0 K(τ ) er (τ, t0 )∆τ t0

 =

eσr (t, t0 )

 =

x0  = r(t)x(t) 1 − K(t) 1 + x0

σ(t) t0



r(τ ) K(τ ) er (τ, t0 )∆τ

xσ (t) x(t) + µ∗ (t)x∆ (t) = r(t)x(t) 1 − = r(t)x(t) 1 − = K(t) K(t) r(t)x(t) ∆ x(t) x (t). = r(t)x(t) 1 − − µ∗ (t) K(t) K(t)

346

Z. Pospíšil

Consequently,

1 x r(t)x 1 − K(t) . = r(t) ∗ x 1 + µ (t) K(t)

x∆

(14)

Hence, the dynamic equation possessing the solution given by (13) has the form (14). If T = R then µ∗ (t) = 0, x∆ = x and we get the equation 1 x = r(t)x 1 − x . K(t) If T = Z then µ∗ (t) = 1, x∆ = x(t + 1) − x(t). Denoting as usual, xk = x(k), rk = r(k), Kk = K(k), we get the equation 1 xk rk xk 1 − (1 + rk )xk Kk + xk = . xk+1 = rk rk 1+ xk 1+ xk Kk Kk The obtained results show that the Verhulst (1) and the Pielou (4) logistic equations are particular cases of dynamic equation (14). Remark 3. The fact that the equations (1) and (4) are in certain sense the same is not striking at all. The both equations are Riccati ones. Example 4. Let T be the time scale introduced in Example 1 with a0 = 0, ai = i, di = 1 for each i ∈ N,  β, τ = k + 1 r(k, τ ) = where β > 0, δ ∈ [0, 1[. −δ, τ = k + 1, The initial value problem x∆ =

r(k, τ )x(1 − x) , 1 + µ∗ (k, τ )r(k, τ )x

x(0, 0) = x0 ∈]0, 1[

can serve to model an evolution of population with intrinsic growth rate β which is regularly exterminated by a chemical compound of eﬃciency δ. Its solution can be evaluated using the results of Examples 1 and 2.

References 1. Bohner M., Peterson A., Dynamic equations on time scales: an introduction with applications, Boston-Basel-Berlin: Birkhäuser, 2001.

Logistic Equation on Measure Chains

347

2. Hilger S., Analysis on measure chains — a uniﬁed approach to continuous and discrete calculus, Results Math., 18, 1990, 18–56. 3. Kaymak¸calan B., Laksmikantham V., Sivasundaram S., Dynamic systems on measure chains, volume 370 of Mathematics and its Applications, Dordrecht: Kluwer Academic Publishers, 1996. 4. May R. M., Biological populations obeying diﬀerence equations: stable points, stable cycles and chaos, J. Theor. Biol., 51, No 2, 1975, 511-524. 5. Maynard Smith J., Mathematical ideas in biology, Cambridge: Cambridge Univ. Press., 1968. 6. Pielou E. C., An introduction to mathematical ecology, NY-London-Sydney-Toronto: Willey-Interscience, 1969. 7. Pospíšil Z., Hyperbolic sine and cosine functions on measure chains, Nonlin. Anal. TMA, 47, No 2, 2001, 861–872. 8. Verhulst P. F., Notice sur le loi que la population suit dans son accroissement, Corr. Math. et Phys., 10, 1838, 113–121.

Equadiﬀ 10, August 27–31, 2001 Prague, Czech Republic

Equadiﬀ 10 CD ROM Papers, pp. 349–356

Boundary stabilization of the Schrödinger equation in almost star-shaped domain Salah-Eddine Rebiai Department of Mathematics, Faculty of Sciences, Batna University, 05000 Batna, Algeria, Email: [email protected]

Abstract. The question of uniformly stabilizing the solution of the Schrödinger equation yt − i∆y = 0 in Ω × (0, ∞) (Ω is a bounded domain of Rn ) sub∂y = F (y, yt ) ject to boundary conditions y = 0 on Γ0 × (0, ∞) and ∂ν on Γ1 × (0, ∞),(Γ0 , Γ1 ) being a partition of the boundary, is studied. We shall show that if {Ω, Γ0 , Γ1 } is almost star-shaped, then a suitable choice of F leads to exponential energy decay. Moreover exponential decay rate estimates will be obtained. The approach adopted is based on multipliers technique.

MSC 2000. 35B45, 35J10, 35B05, 93C20, 93D15 Keywords. Schrödinger equation, boundary stabilization, exponential stability, exponential decay rates estimates, multipliers technique

1

Introduction

Let Ω be an open bounded domain in Rn with suﬃciently smooth boundary Γ . Assume that Γ consists of two parts Γ0 and Γ1 satisfying Γ0 ∩ Γ1 = ∅

(1)

We set Q = Ω × (0, +∞), Σ0 = Γ0 × (0, +∞), Σ1 = Γ1 × (0, +∞). Let a and l be two nonnegative functions of class C 1 such that Γ0 = ∅ or a = 0 This is the preliminary version of the paper.

(2)

350

S.-E. Rebiai

Consider the problem yt − i∆y = 0

in Q

y(x, 0) = y0 in Ω y=0 on Σ0 ∂y + ay + lyt = 0 ∂ν

(3) (4) (5)

on Σ1

(6)

dy and ν is the unit normal of Γ pointing towards the exterior of where yt = dt Ω. The natural space for initial data is

V = ϕ ∈ H 1 (Ω) : ϕ = 0 on Γ0 When Γ0 has non-empty interior in Γ, by Poincaré’s inequality, we have ϕL2 (Ω) ≤ β ∇ϕ(L2 (Ω))n ,

∀ϕ ∈ V

(7)

In view of this inequality, we shall consider in V the norm induced by the inner product (ϕ, ψ)V = G

Ω

∇ϕ.∇ψdΩ

Associated with each solution of (3)–(6) is its total energy at time t; E(t) =

Ω

2

|∇y| dΩ +

Γ1

2

a |y| dΓ

A simple calculation shows, at least formally, that d dt E(t)

=−

Γ1

l |yt |2 dΓ ≤ 0

hence E(t) is nonincreasing. The question we are interested in is the following: under what conditions can we establish the exponential decay of the energy and if possible obtain explicit decay rate estimates. An aﬃrmative answer to the above question has been given by Machtyngier and Zuazua [3] under the following assumptions: (H1)- {Ω, Γ0 , Γ1 } is “star-complemented-star-shaped”, scss for short (see [1]). This means that there exists a point x0 ∈ Rn such that -(x − x0 ).ν(x) ≤ 0 on Γ0 (Γ0 is star-complemented with respect to x0 ) -(x − x0 ).ν(x) ≥ 0 on Γ1 (Γ1 is star-shaped with respect to x0 ) (H2)- a ≡ 0 and l = (x − x0 ).ν(x) The aim of this paper is to extend the result of Machtyngier and Zuazua, in two ways: ﬁrst by replacing the scss domains by a larger class of domains known

Boundary stabilization of the Schrödinger equation

351

as almost star shaped domains, second by replacing the boundary feedback (H2) by a more general one with a = 0. The rest of the paper is organized as follows. In Section 2, we recall the notion of almost star-shaped domains. In Section 3, we state and prove the boundary stabilization theorem.

2

Almost star-shaped domains

Definition 1. {Ω, Γ0 , Γ1 } is an almost star-shaped domain if there exists ϕ ∈ C 2 (Ω) such that ∆ϕ = 1

in Ω

(8)

λ1 (ϕ) = Inf {λ1 (x), x ∈ Ω} > 0 ∂ϕ ≤0 on Γ0 ∂ν ∂ϕ ≥0 on Γ1 ∂ν

(9) (10) (11)

where λ1 (x) is the smallest eigenvalue of the real symmetric squared matrix D2 ϕ(x). The simplest example is the case where {Ω, Γ0 , Γ1 } is a scss domain. The function ϕ is then given by ϕ(x) =

1 2n

|x − x0 |2

Remark 2. We refer to [4] and [5] for other examples and further details.

3

The boundary stabilization theorem

We ﬁrst state the following existence and regularity theorem for the system (3)–(6)

Theorem 3. Assume (1) and (2). 1- Given y0 ∈ V, the problem (3)–(6) has a unique weak solution y ∈ C((0, +∞), V ) ∩ C 1 ((0, +∞), L2 (Ω)) 2- If y0 satisﬁes the stronger conditions y0 ∈ H 2 (Ω) ∩ V ∂y0 + ay0 + il∆y0 = 0 on Γ1 ∂ν

(12) (13)

352

S.-E. Rebiai

Then the solution y has the stronger regularity property y ∈ L∞ ((0, +∞), H 2 (Ω)), yt ∈ L∞ ((0, +∞), V )

To prove this theorem, semigroups approach can be adopted (see [2]). Our main result is as follows. Theorem 4. Let {Ω, Γ0 , Γ1 } be an almost star shaped-domain, and choose D 1 ∂ϕ 2 ∂ϕ a= and l = (14) 2 ∂ν 3 ∂ν 8 ∇ϕ∞ Then for every y0 ∈ V, the energy corresponding to the weak solution of (3)–(6) satisﬁes the estimate ∀t ≥ 0,

λ1 (ϕ) with ω = √ ( 6 ∇ϕ∞ + β) ∇ϕ∞

E(t) ≤ E(0)e1−ωt

This theorem will be only proved for smooth initial data. The general case follows by a standard density argument. To proceed the following preliminary results are needed. Lemma 5. Given y0 verifying (12)–(13). Then the strong solution of (3)–(6) satisﬁes 2 ∀0 ≤ S ≤ T < +∞, E(S) − E(T ) = l |yt | dΣ (15) Σ1

Proof. We multiply both sides of (3) by yt and we integrate by parts over Ω We obtain (yt − i∆y)yt dΩ 0= Ω = (|yt |2 − iyt ∆y)dΩ Ω 2 2 = |yt | dΩ + i ∇y.∇yt dΩ + i (ayyt + l |yt | )dΣ Ω

It follows that

Ω

Γ1

G( ∇y.∇yt dΩ + Ω

Γ1

2

ayyt dΣ) = −

l |yt | dΣ Γ1

d But the left-hand side of the above equality is precisely E(t). Hence the desired dt result.

353

Boundary stabilization of the Schrödinger equation

Lemma 6. Given 0 ≤ S < T. Then the following identity holds true for any strong solution of (3)–(6):

∂ϕ 2 (ay + lyt )∇ϕ.∇ydΣ − G (|∇y| + iyyt ) dΣ − ∂ν Σ1 Σ 1 ∂y ∂ϕ dΣ = LX + 2G (ay + lyt )ydΣ + (D2 ϕ∇y).∇ydQ Σ1 Σ0 ∂ν ∂ν Q

−2 G

(16) where

T

X=

y∇ϕ.∇ydΩ Ω

S

Proof. (i)- We multiply both sides of (3) by ∇ϕ.∇y and integrate over Q. We obtain the following identity (see the Appendix): ∂y ∂ϕ 2 ∂ϕ ∇ϕ.∇ydΣ − +L dΣ + 2G |∇y| yyt ∂ν ∂ν Σ ∂ν Σ Σ ∂y G y∆ϕdΣ = LX + 2G (D2 ϕ∇y)∇ydQ Σ ∂ν Q

(17)

(ii)- We now use the boundary conditions (5) and (6). Thus ∂y ∂y On Γ0 : y = yt = 0; |∇y| = ; ∇ϕ.∇y = (∇ϕ.ν) ∂ν ∂ν

(18)

Therefore using (6) and (18) in the left-hand side of(17), we ﬁnd the sought- after identity for y satisfying (3)–(6).

Lemma 7. Assume that a and l are deﬁned by (14). Then for any initial data verifying (12) and (13), we have:

Σ0

∂y 2 ∂ϕ ) − 2G ∂ν ∂ν

∂ϕ 2 (|∇y| + iyyt ) dΣ − ∂ν √ 1 2 2 (ay + lyt )ydΣ ≤ 4 6 ∇ϕ∞ (E(S) − E(T )) − a |y| dΣ (19) G 4 Σ1 Σ1 (

Proof. Set α = Then

1 8 ∇ϕ2∞

(ay + lyt )∇ϕ.∇ydΣ −

Σ1

Σ1

( and λ =

2 3

354

S.-E. Rebiai

∂ϕ 2 ∂ϕ 2 − G(iyyt ) − a |y| − G(lyyt ) −2G(ay + lyt )∇ϕ.∇y − |∇y| ∂ν ∂ν ∂ϕ ∂ϕ 2 2 2 2 2 2 ∇ϕ∞ (α2 |y| + λ2 |yt | ) + |∇y| − |∇y| + ≤ ∂ν ∂ν λ2 + 1 ∂ϕ α ∂ϕ 2 ∂ϕ λ2 + 1 2 2 2 |yt | + |y| ≤ 2 ∇ϕ∞ λ2 + |yt | + 2α ∂ν 2 ∂ν ∂ν 2α ∂ϕ 1 2 ∇ϕ2∞ α − α |y|2 ∂ν 2 The estimate (19) follows now from the particular choice of the coeﬃcients α and λ given by (14), from the identity (15) and (8)–(11). Lemma 8. |X| ≤ 2βE(S) Proof. Using (7), we have y∇ϕ.∇ydΩ ≤ y(t) 2 L (Ω) ∇ϕ.∇y(t)L2 (Ω) Ω

≤ β ∇ϕ∞ y(t)2V Thus |X| ≤ 2β ∇ϕ∞ E(S)

4

Proof of Theorem 4

Applying Lemmas 7 and 8, we deduce from the identity (16), the inequality √ 1 2 2 2 2G (D ϕ∇y).∇ydQ ≤ (4 6 ∇ϕ∞ + 2β ∇ϕ∞ )E(S) + a |y| dΣ 4 Q Σ1 Using (9), we get

T

2λ1 (ϕ) S

√ 2 E(t)dt ≤ (4 6 ∇ϕ∞ + 2β ∇ϕ∞ )E(S)

Letting T → +∞, we obtain for every ﬁxed S ∈ R+ , the estimate

+∞

E(t)dt ≤ S

1 E(S) ω

355

Boundary stabilization of the Schrödinger equation

where

λ1 (ϕ) ω= √ ( 6 ∇ϕ∞ + β) ∇ϕ∞

The conclusion of the theorem follows by applying a Gronwall-type inequality as in [2].

5

Appendix Proof of (17)

We multiply both sides of (3) by ∇ϕ.∇y and integrate by parts over Q. Term yt ∇ϕ.∇y Integrating by parts in t and using the identity h.∇ψdΩ = hψ.νdΓ − ψ∇hdΩ Ω

Γ

(A1)

Ω

we obtain

yt ∇ϕ.∇ydQ = Q

T yt ∇ϕ.∇ydΩ

Q

−

S

yyt ∇ϕ.νdΣ− Σ

∆y∇ϕ.∇ydQ − i

i Q

Adding −i

Q

∆y∇ϕ.∇ydQ to both sides of (A2) yields

∆y∇ϕ.∇ydQ = LX − L

2G Q

Term

Q

y∆y∆ϕdQ (A2) Q

yyt ∇ϕ.νdΣ − G

Σ

y∆y∆ϕdQ

y∆y∆ϕdQ

Using Green’s ﬁrst theorem and the identity (20), we ﬁnd ∂y 2 y∆y∆ϕdQ = |∇y| ∆ϕdQ − y∇y.∇(∆ϕ)dQ y∆ϕdΣ − Q Σ ∂ν Q Q Term

Q

(A3)

Q

(A4)

∆y∇ϕ.∇ydQ

We use Green’s ﬁrst theorem and the identity (20). We obtain ∂y ∇ϕ.∇ydΣ − (D2 ϕ∇y).∇ydQ− ∆y∇ϕ.∇ydQ = Q Σ ∂ν Q 1 1 2 2 |∇y| ∇ϕ.νdΣ + |∇y| ∆ϕdQ 2 Σ 2 Q Inserting (A4) and (A5) into (A3), results in (17).

(A5)

356

S.-E. Rebiai

References 1. Chen G., Energy decay estimates and exact boundary value controllability for the wave equation in a bounded domain, J.Math. Pures Appl., 58, No 9, 1979, 249–274. 2. Komornick V., Exact Controllability and Stabilization. The Multiplier Method (Masson, Paris, 1994). 3. Machtyngier E., Zuazua E. Stabilization of the Schrödinger equation, Portugaliae Mathematica, 51, No 2, 1994, 243–256. 4. Martinez P., Stabilisation de syst`emes distribués semilinéaires: Domaines presque étoilés et inégalités intégrales généralisées, Thesis, University Louis Pasteur, Strasburg, France 1998. 5. Martinez P., Boundary stabilization of the wave equation in almost star-shaped domains, SIAM J.Control, 37, No 3, 1999, 673–694.

Equadiﬀ 10, August 27–31, 2001 Prague, Czech Republic

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Convergence, via summability, of formal power series solutions to a certain class of completely integrable Pfaﬃan systems Javier Sanz Department of Mathematical Analysis, University of Valladolid c/ Prado de la Magdalena s/n, 47005 Valladolid, Spain Email: [email protected]

Abstract. A theory of k-summability in a direction (k = (k1 , k2 , . . . , kn )) has been put forward for formal power series of several complex variables. It involves the study of multidimensional Laplace and Borel transforms, and their eﬀect on series (resp. functions) subject to Gevrey-like bounds (resp. admitting Gevrey strongly asymptotic expansion). As an example of the application of this tool to some questions in PDE’s, a new proof is given of a result of R. Gérard and Y. Sibuya stating the convergence of the formal power series solutions to certain completely integrable Pfaﬃan systems.

MSC 2000. 35C10, 35C20, 40C15

Keywords. Formal solutions, summability, Pfaﬃan systems

1

Introduction

The theory of k-summability in one variable was developed by Ramis [10,11], with the aim of building up analytic solutions to ODE’s in sectors, departing from a formal power series solution which, in fact, will asymptotically represent the former. The introduction by Ecalle [4,5] of a more powerful tool, multisummability, led Braaksma [3] to prove that every formal power series solution to a nonlinear system of ODE’s at an irregular singular point is multisummable, which allows to compute actual solutions from formal ones. A complete treatment of this subject can be found in the books by Balser [1,2]. The main tools for summability theory This is an overview article.

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are Gevrey asymptotics and (formal and analytic) Laplace-Borel transforms, while multisummability is a kind of “recurrent summability”. As a particular case, consider the system of nonlinear ODE’s z k+1

df = ϕ0 (z) + A(z)f + f α ϕα (z). dz |α|≥2

where f , ϕ0 and ϕα are p-vectors, and A is a p × p matrix. Suppose all data are holomorphic in D = {|z| < r} and the series in the right member uniformly converge on compact subsets of D × {f < ρ}. If k is a positive integer, ϕ0 (0) = 0 and A(0) is invertible, then the system has a unique formal power series solution f = n∈N an z n that may certainly diverge, though it is always k-summable. Regarding the case of two variables, let us consider the Pfaﬃan system z1k1 +1

∂f = ϕ0 (z1 , z2 ) + A(z1 , z2 )f + f α ϕα (z1 , z2 ), ∂z1 |α|≥2

∂f = ψ0 (z1 , z2 ) + B(z1 , z2 )f + f α ψα (z1 , z2 ). z2k2 +1 ∂z2

(1)

|α|≥2

where f , ϕ0 , ϕα , ψ0 and ψα are p-vectors, and A, B are p× p matrices. Suppose all data are holomorphic in D = {|z1 | + |z2 | < r} and the series in the right members uniformly converge on compact subsets of D × {f < ρ}. If k1 and k2 are positive integers, ϕ0 (0, 0) = ψ0 (0, 0) = 0, A(0, 0) and B(0, 0) are invertible and the system is completely integrable, then the system has a unique formal power series solution f =

anm z1n z2m ,

anm ∈ Cp .

n,m∈N

Gérard and Sibuya [6] proved that f is convergent. This fact was not readily accepted, so Sibuya has given several proofs of this result. In one of them [14] the point of view of summability is adopted, by considering one of the variables as a parameter and summing the series in the other variable “uniformly” with respect to the parameter. However, it seems desirable to ﬁnd a summation method that treats all variables equally and at the same time, as it has been explicitly pointed out by Sibuya [14] and Balser [2, Chapter 13]. Our deﬁnition of summability in a direction for series of several variables relies on (i) the concept of (Gevrey) strongly asymptotically developable functions (see Majima [8,9] and Haraoka [7]), that shares the usual stability properties –in particular, with respect to diﬀerentiation– of H. Poincaré’s deﬁnition in one variable, and (ii) the deﬁnition and study of multidimensional Laplace and Borel transforms. As an application of this theory, we provide a new proof of the result by Gérard and Sibuya. For a detailed treatment of all these topics, see [13].

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2

Notation

For k = (k1 , k2 ) ∈ (0, ∞)2 , A = (A1 , A2 ) ∈ (0, ∞)2 and α = (α1 , α2 ) ∈ N2 , we put Γ (1 + α/k) = Γ (1 + α1 /k1 )Γ (1 + α2 /k2 ),

α2 1 Aα = Aα 1 A2 ,

where the Γ s on the right stand for the Gamma function. Consider, for j = 1, 2, an open sector (on the Riemann surface of the Logarithm) with vertex at the origin, Sj = S(dj , θj , ρj ) = { z = reiϕ : 0 < r < ρj , |ϕ − dj | < θj /2 }, where dj ∈ R, θj > 0 and ρj ∈ (0, ∞] are the bisecting direction, the width and E2 the radius of Sj , respectively. The polysector j=1 Sj ⊂ R2 will be denoted by S = S(d, θ, ρ), where d = (d1 , d2 ), θ = (θ1 , θ2 ) and ρ = (ρ1 , ρ2 ). In case ρj = +∞ for j = 1, 2, we write S = S(d, θ). E A polysector T = 2j=1 T (dj , θj , ρj ) on R2 is a bounded proper subpolysector of S = S(d, θ, ρ), and we write T 7 S, if for j = 1, 2 we have ρj < ρj (so that ρj < +∞) and [dj − ρj /2, dj + ρj /2] ⊂ (dj − ρj /2, dj + ρj /2).

3

k-summability

of power series in two variables

Let us begin stating the basic deﬁnitions and results about Gevrey asymptotics in the sense of Majima. Definition 1. A function f : S = S1 × S2 ⊂ R2 → C is Gevrey strongly asymptotically developable of order k (f ∈ Ak (S)) if there exists a family TA(f ) = {hm , gn , anm : n, m ∈ N}, where hm (resp. gn ) is a holomorphic function from S1 (resp. S2 ) to C and anm ∈ C, n, m ∈ N, such that the following holds: if we deﬁne the approximate function of order α = (n, m) ∈ N2 by Appα (f )(z) =

n−1 j=0

gj (z2 )z1j +

m−1 O=0

j=n−1, O=m−1

ajO z1j z2O ,

hO (z1 )z2O −

j,O=0

then for every T 7 S, there exist CT > 0, AT ∈ (0, ∞)2 such that for every α ∈ N2 , α |f (z) − Appα (f )(z)| ≤ CT Γ (1 + α/k)Aα z ∈ T. T |z| , FA(f ) := anm z1n z2m is the (formal) series of strongly asymptotic expansion (n,m)∈N2

of f . Whenever f ∈ Ak (S) one has FA(f ) is Gevrey of order k (FA(f ) ∈ C[[z]]k ), i.e., there exist C > 0, A = (A1 , A2 ) ∈ (0, ∞)2 such that for every α = (α1 , α2 ) ∈ N2 , |aα | ≤ C Γ (1 + α/k)Aα . Ak (S) and C[[z]]k are diﬀerential algebras, and the map FA : Ak (S) → C[[z]]k is a homomorphism. The next result will be crucial for our purposes.

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Proposition 2 (Watson’s lemma [12,13]). Let S = S(d1 , θ1 , ρ1 )×S(d2 , θ2 , ρ2 ) be a polysector such that θj > π/kj for j = 1, 2. Then, the mapping FA is injective. We are now ready to introduce a concept of summability for formal power series in two variables. Definition 3. A formal power series f = α∈N2 aα z α is said to be k-summable π in direction d ∈ R2 if there exist a polysector S = S(d, θ, ρ), with θ > , and a k function f ∈ Ak (S) such that FA(f ) = f. In this case, we write f = Sk,d (f), and call f the k-sum of f in direction d. It seems natural to ask whether the analogue to Fubini’s theorem holds in this context. The fact is that basic results remain unaltered when one considers: 1. (Strong) asymptotic expansions for functions with values in a complex Banach space (E, · ), and 2. k-summability of formal power series in one variable with coeﬃcients in E. Suppose S = S1 × S2 and B ∈ (0, ∞)2 . Although Ak (S, E) (with the expected meaning) is not a Banach space, we can consider its subspace Wk,B (S, E) := {f : S → E : f k,B :=

sup z∈S, α∈N2

Dα f (z) < ∞}. α! Γ (1 + α/k)B α

Lemma 4. (Wk,B (S, E), · k,B ) is a Banach space, and the restriction to any T 7 S of the functions in Ak (S, E) provides elements of Wk,B (T, E) for some suitable B = B(T ). Also, the map f ∈ Wk,B (S, E) → f ∗ ∈ Wk1 ,B1 (S1 , Wk2 ,B2 (S2 , E)) deﬁned for every z1 ∈ S1 by f ∗ (z1 ) = f (z1 , ·) is an isomorphism. These facts together lead to the following deﬁnition and result. ∞ Definition 5. A formal power series f = n,m=0 anm z1n z2m is iteratively k-summable in direction d (in a certain order, but this will turn out to be irrelevant) if, when we write f =

∞ n=0

gn z1n ,

where gn =

∞

anm z2m ,

m=0

the following hold: (i) Every gn is k2 -summable in direction d2 , and the sum gn belongs to Ak2 (S2 ), where the sector S2 = S2 (d2 , θ2 , ρ2 ) does not depend on n. (ii) There exist T2 = T2 (d2 , ϕ2 , r2 ) 7 S2 , with ϕ2 > π/k2 , and B2 (T2 ) > 0 such ∞ that gn ∈ Wk2 ,B2 (T2 ) for every n ∈ N, and the series g = n=0 gn z1n (with coeﬃcients in that Banach space) is k1 -summable in direction d1 .

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Proposition 6. A formal power series f is k-summable in direction d if and only if it is iteratively k-summable in direction d (in any order). It will also be interesting to consider the case that convergence appears in some of the summation steps. Let S1 = S(d1 , θ1 , ρ1 ) be a sector, D2 a disk around 0 and k1 > 0. Definition 7. The space A(k1 ,∞) (S1 × D2 ) consists of the holomorphic functions f : S1 × D2 → C for which there exists a family TA(f ) = {hm , gn , anm : n, m ∈ N}, where hm (resp. gn ) is a holomorphic function from S1 (resp. D2 ) to C and anm ∈ C, n, m ∈ N,such that, if we deﬁne Appα (f ) as before, then for every T1 7 S1 and every compact K2 ⊂ D2 there exist C > 0 and A ∈ (0, ∞)2 (both depending on T1 and K2 ) such that for every α = (n, m) ∈ N2 and z ∈ T1 × K2 , |f (z) − Appα (f )(z)| ≤ C Γ (1 + n/k1 )Aα |z|α . Definition 8. A formal power series f is (k1 , ∞)-summable in direction d1 ∈ R if there exist S1 = S(d1 , θ1 , ρ1 ), with θ1 > π/k1 , a disk D2 and a function f ∈ A(k1 ,∞) (S1 × D2 ) such that FA(f ) = f. For a complex Banach space E, a disk D and B > 0, we consider the space W∞,B (D, E) := {f : D → E : f ∞,B :=

f (p) (z) < ∞}. p! B p z∈D, p∈N sup

(W∞,B (D, E), · ∞,B ) is a Banach space, and it plays a role in the characterization, similar to that in Proposition 6, of (k1 , ∞)-summability in terms of iterative summability.

4

New proof of Gérard-Sibuya’s result

The information in the paper of Sibuya [14] can be seen to imply that the formal ∞ solution to (1), f = n,m=0 anm z1n z2m , is: (a) (iteratively) (k1 , ∞)-summable in every direction d1 except for those in a ﬁnite set E1 , and (b) (iteratively) (∞, k2 )-summable in every direction d2 except for those in a ﬁnite set E2 . ∞ By (b), hm := n=0 anm z1n converges in a disk D1 for every m, with sum hm . / E1 in such a way that the sectors By (a), we can choose directions d11 , . . . , dO1 ∈ 1 ⊂ D1 around 0 (of Tj where the sums Sk1 ,dj hm are deﬁned cover a whole disk D 1 course, these sums glue together and agree with hm );also, there exists Aj > 0 such ∞ that hm ∈ Wk1 ,Aj (Tj ) for every m, and the series m=0 hm z2m , with coeﬃcients in Wk1 ,Ad (Td ), converges. Since sup |hm (z1 )| ≤ sup Sk1 ,d hm k1 ,Ad ,

1 z1 ∈D

d∈F1

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∞ we see that m=0 hm z2m converges in a disk D2 when its coeﬃcients are considered 1 )) of holomorphic functions in D 1 (with the compact open in the space H(D 1 )) topology). In order to conclude, it suﬃces to take into account that H(D2 , H(D and H(D1 × D2 ) are isomorphic.

References 1. W. Balser, From divergent power series to analytic functions. Theory and application of multisummable power series, Lecture Notes in Math. 1582, Springer-Verlag, Berlin, 1994. 2. W. Balser, Formal power series and linear systems of meromorphic ordinary diﬀerential equations, Springer-Verlag New York, Inc., 2000. 3. B. L. J. Braaksma, Multisummability of formal power series solutions of nonlinear meromorphic diﬀerential equations, Ann. Inst. Fourier (Grenoble) 42 (1992), 517– 540. 4. J. Ecalle, Les fonctions résurgentes I, II, Publ. Math. Orsay, Univ. Paris XI, Orsay, 1981; III, Publ. Math. Orsay, Univ. Paris XI, Orsay, 1985. 5. J. Ecalle, Introduction à l’Accélération et à ses Applications, Travaux en Cours, Hermann, Paris, 1993. 6. R. Gérard, Y. Sibuya, Étude de certains syst`emes de Pfaﬀ avec singularités, in: Lecture Notes in Math. 172, Springer, Berlin, 1979, 131–288. 7. Y. Haraoka, Theorems of Sibuya-Malgrange type for Gevrey functions of several variables, Funkcial. Ekvac. 32 (1989), 365–388. 8. H. Majima, Analogues of Cartan’s Decomposition Theorem in Asymptotic Analysis, Funkcial. Ekvac. 26 (1983), 131–154. 9. H. Majima, Asymptotic Analysis for Integrable Connections with Irregular Singular Points, Lecture Notes in Math. 1075, Springer, Berlin, 1984. 10. J.P. Ramis, Dévissage Gevrey, Astérisque 59–60 (1978), 173–204. 11. J.P. Ramis, Les séries k-sommables et leurs applications, in: Lecture Notes in Phys. 126, Springer-Verlag, Berlin, 1980, 178-199. 12. J. Sanz, Linear continuous extension operators for Gevrey classes on polysectors, to appear in Glasgow Math. J. 13. J. Sanz, Summability in a direction of formal power series in several variables, to appear in Asymptotic Anal. 14. Y. Sibuya, Convergence of formal solutions of meromorphic diﬀerential equations containing parameters, Funkcial. Ekvac. 37 (1994), 395–400.

Equadiﬀ 10, August 27–31, 2001 Prague, Czech Republic

Equadiﬀ 10 CD ROM Papers, pp. 363–377

Computation and Continuation of Quasiperiodic Solutions Frank Schilder1 and Werner Vogt2 1

Frank Schilder, TU Ilmenau, Fakultät Mat.-Nat., Postfach 100565, 98693 Ilmenau, Deutschland Email: [email protected] 2 Werner Vogt, TU Ilmenau, Fakultät Mat.-Nat., Postfach 100565, 98693 Ilmenau, Deutschland Email: [email protected]

Abstract. We consider periodically forced ODEs which exhibit quasiperiodic oscillations. These oscillations are investigated by an approximation and continuation of the associated invariant torus with respect to free system parameters. For the invariant torus we derive an uncomplicated invariance equation whereby we do not require the system to be partitioned or an a-priori-coordinate transformation to be applied. This equation is solved by semidiscretisation methods where Fourier-Galerkin methods especially in the case of periodically forced ”weakly nonlinear” ODEs lead to low dimensional autonomous systems which can be treated by standard algorithms. Also in the general case it turns out that this approach allows an eﬃcient computation and continuation of quasiperiodic solutions. A number of problems has been analysed successfully and an example is given in this paper. MSC 2000. 65P30, 65L10, 37M20 Keywords. Dynamical Systems, Quasiperiodic oscillations, Numerical methods

1

Introduction

We consider periodically forced ordinary diﬀerential equations (ODEs) of order n≥2 dx = f (x, t) , f : Rn × R -→ Rn (1) dt This is the preliminary version of the paper.

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under the following assumptions: 1. f ∈ C r (Rn × R) with r ≥ 1 suﬃciently large. f is 2π−periodic with respect to t: f (x, t + 2π) = f (x, t) ∀(x, t) ∈ Rn × R. 2. System (1) has a locally unique quasiperiodic solution y ∈ C 1 (R) with p (2 ≤ p ≤ n) rationally independent basic frequencies ω1 = 1, ω2 , . . . , ωp . These p frequencies form a rationally independent (incommensurate) frequency basis Ω = (ω1 , ω2 , . . . , ωp ) with ω1 = 1 because of the 2π−periodicity of f . The following methods can be applied to autonomous systems as well, but then the basic frequency ω1 is also unknown and an additional phase condition must be introduced. During the last 15 years very diﬀerent approaches have been developed for approximating quasiperiodic solutions and invariant tori. The method of invariance equations tries to compute the parametrisation of an invariant torus by solving the quasilinear partial diﬀerential equations. While [4], [5], [6], [1], [2], [16], [10] use special forms of diﬀerence methods, in [12], [9], [3] multidimensional Fourier methods are applied. The drawback of these approaches is that an a-priori transformation from Cartesian to radius-angle coordinates (u, θ) is required and in most applications such a global parametrisation is neither possible nor numerically feasible. As an alternative approach, [8], [14], [15], [11], [9] consider a suitable Poincaré map P and try to compute an invariant manifold of P as a solution of a functional equation. The performance of these methods strongly depends on discretisation and interpolation techniques and also on the stability of solutions. For special 2nd order systems and systems with small perturbations the following additional methods are widely used in enineering: – – – –

the averaging method of Krylov and Bogoljubov, the method of amplitudes of Van der Pol, the harmonic balance method and generalizations of C. Hayashi and multiscale methods.

In principle, they ”reduce the quasiperiodicity” of the solutions by transforming periodic solutions into equilibria and, if possible, quasiperiodic to periodic solutions. Unfortunalety they are restricted to special ODEs and do not extend to the general case of system (1). The aim of our approach is a numerical approximation of quasiperiodic solutions x(t) = u(Ωt) of the original system (1) without using a-priori transformations into radius-angle coordinates (u, θ). By means of a suitable torus system, we can analyse quasiperiodic solutions x(t, λ) and their corresponding torus solutions u(θ, λ) depending on parameters λ ∈ Rm by methods for periodic solutions. So we are able to use existing continuation methods and methods for bifurcation analysis.

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2

Transformation into the torus equation

For the quasiperiodic solution y of (1) the representation y(t) = u(Ωt) = u(t, ω2 t, . . . , ωp t) with the associated torus function u = u(θ) : Tp → Rn is used. u is assumed to be continously diﬀerentiable and 2π−periodic in every variable θi , i = 1, 2, . . . , p. Inserting this formulation for y into (1) yields ω1

p ∂u ∂u (Ωt) + ωj (Ωt) = f (u(Ωt), t) , ∂θ1 ∂θ j j=2

(2)

which by using the vector-valued function g : R → Rn p ∂u ∂u g(t) = (Ωt) + ωj (Ωt) − f (u(Ωt), t) , ∂θ1 ∂θ j j=2

(3)

becomes equivalent to the equation ∀t ∈ R .

g(t) = 0

(4)

According to assumptions 1 and 2, g ∈ C(R, Rn ) is also quasiperiodic with basic frequencies ω1 = 1, ω2 , . . . , ωp . Its associated torus function G : Tp → Rn with g(t) = G(Ωt) = G(t, ω2 t, . . . , ωp t) is deﬁned by G(θ) =

p ∂u ∂u (θ) + ωj (θ) − f (u(θ), θ1 ) . ∂θ1 ∂θ j j=2

(5)

For G ∈ C(Tp , Rn ) the range of the quasiperiodic function g(t) = G(ωt) is dense in the range of the torus function G(θ) , θ ∈ Tp (see [12], p.10). With scalar product and norm in Cn

x, y =

n

xj yj ,

|x| = x, x2 = 2

j=1

n

|xj |2

j=1

the identity sup |g(t)| = maxp |G(θ)| t∈R

θ∈T

(6)

holds ([12], p.11). As a consequence it follows that g(t) = 0 ∀t ∈ R

⇐⇒

G(θ) = 0 ∀θ ∈ Tp

(7)

which leads to in the invariance equation (the torus system) on Tp p ∂u ∂u (θ) + ωj (θ) = f (u(θ), θ1 ) . ∂θ1 ∂θ j j=2

(8)

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Any solution u of this system yields a quasiperiodic solution x(t) = u(Ωt) of g(t) = 0 , t ∈ R. Note that (8) is a special case of the general invariance equation of an invariant p-torus u(θ) p

Ψj (u(θ), θ)

j=1

∂u (θ) = f (u(θ), θ). ∂θj

(9)

In our approach, the basic frequencies ωj for j > 1 are unknowns and may be determined by appropriate extensions to system (8). For simplicity we consider the 2-dimensional case, but all the ideas can also be generalized to p-tori. In the case p = 2 semidiscretisation methods with respect to θ1 may be applied, for example: – Fourier-Galerkin, – ﬁnite diﬀerences or – collocation. System (8) becomes thereby transformed into an autonomous system of ordinary diﬀerential equations for functions uk (θ2 ) : T1 → Rn , k ∈ Zk ⊂ Z. We get an initial value for the frequency ω2 at a Neimark-Sacker bifurcation point (nonresonant case) of a periodic solution when a quasiperiodic solution is born. The ﬁrst two approaches are discussed in more detail here.

3

Semidiscretisation by Fourier-Galerkin methods

We deﬁne the nonlinear operator F : H 1 → H 0 as F (u) ≡

∂u ∂u + ω2 − f (u, θ1 ) , ∂θ1 ∂θ2

(10)

where H s = H s (T2 ), s = 0, 1 are the Sobolew spaces of torus functions F : T2 → Cn with generalized derivatives Dα F ∈ L2 (T2 ) up to order s. Especially let H 0 = L2 (T2 ). Scalar product and norm in H s are deﬁned by

Dα F (θ), Dα G(θ) dθ1 dθ2 , (F, G)s = 0≤|α|≤s T2

||F ||2s = (F, F )s =

|Dα F (θ)|2 dθ1 dθ2

0≤|α|≤s T2

With (10) we obtain a zero problem which is equivalent to G(θ) = 0 ∀θ ∈ T2 in operator form F (u) = 0 , u ∈ H s (T2 ). (11) For simplicity we now replace t = θ1 because of ω1 = 1 and θ = θ2 with frequency ω = ω2 . Let ϕi (t), i = −N, . . . , −1, 0, 1, . . . , N , be an orthonormal

Computation and Continuation of Quasiperiodic Solutions

367

system of a linear subspace of L2 (T1 ) for which N

ϕ˙ k (t) =

ckj ϕj (t)

(12)

j=−N

holds. In case of the trigonometric functions 1 ϕk (t) = √ eikt , 2π

k = −N, −N + 1, . . . , N

the constant matrix C = (ckj ) is especially C = diag(−iN, . . . , −2i, −i, 0, i, 2i, . . . , iN ) . n

We discretise u(t, θ) by projection onto the subspace HN = (span{ϕk , |k| ≤ N }) with projector N PN u(t, θ) = uk (θ) · ϕk (t) (13) k=−N

where the Fourier coeﬃcients are

uk (θ) =

u(t, θ)ϕk (t) dt.

(14)

T1

Inserting uN ∈ HN into (10) and applying PN to (11) yields the Galerkin or spectral system (15) PN F (uN ) = 0 , uN ∈ HN . This Galerkin procedure can be deﬁned component-wise by introducing the vector function ϕ(t) = (ϕ−N (t), ..., ϕ0 (t), ϕ1 (t), ..., ϕN (t))T and the matrix function U (θ) = (u−N (θ), ..., u−1 (θ), u0 (θ), u1 (θ), ..., uN (θ)) . Then (13) in the representation uN (t, θ) = U (θ)ϕ(t) ,

(t, θ) ∈ T2

(16)

can be inserted into (10) by using (12) F (uN (t, θ)) = U (θ)Cϕ(t) + ωU (θ)ϕ(t) − f (U (θ)ϕ(t), t) . We now expand f (U (θ)ϕ(t), t) in (17) into a θ−dependent Fourier series f (U (θ)ϕ(t), t) = Γ (U (θ))ϕ(t) + RN (θ, t) ,

(17)

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where RN (θ, t) is the remainder for |k| > N and the coeﬃcients are Γ (U (θ)) = (γjk ) ∈ Cn×(2N +1) . Applying the scalar product (14) to (17) in L2 (T1 ) yields the component-wise representation ω · uil (θ) + (U (θ) · C)il − γil (U (θ)) = 0 ,

i = 1(1)n, |l| ≤ N .

In vector notation, the spectral system (Galerkin system) is now ω · U (θ) + U (θ) · C = Γ (U (θ)) .

(18)

If we consider the Fourier series for N → ∞, then the periodic solutions of spectral system (18) will obviously yield quasiperiodic solutions of the original system (1). The following theorem can be proven (see [13]): Theorem 1. With assumptions 1 and 2 it holds for N → ∞: (i) U (θ) is a 2π−periodic solution of system (18) if and only if u(t) = U (ωt)ϕ(t) is a quasiperiodic solution of the original system (1). (ii) A = (akl ), k = 1 . . . n, l ∈ Z, is an equilibrium point of system (18) with A · C = Γ (A)

(19)

iﬀ u(t) = Aϕ(t) is 2π−periodic solution of the original system (1). The Galerkin system (18) is an autonomous system with n(2N + 1) equations. For harmonically forced ”weakly nonlinear” systems which frequently appear in electrical engineering we already achieve in practice good approximations for small N = 1, 2, 3. Applying the transformation to the independent variable θ = ωτ

with U (θ) = U (ωτ ) = Y (τ ) ,

to (18) we can eliminate the unknown parameter ω and obtain the spectral system Y (τ ) = Γ (Y (τ )) − Y (τ ) · C

(20)

for periodic solutions Y (τ ) with unknown period T . This standard problem can now be solved by software tools for periodic oscillations and is an eﬃcient way to compute and continue quasiperiodic solutions. With such an approximation at hand the invariant closed curves γ1 and γ2 of the two Poincaré sections P1 and P2 of a quasiperiodic solution can easily be calculated. Using (13) uN (t, θ) =

N

uk (θ) · ϕk (t)

(21)

k=−N

we get immediately the approximations: N 1 γ1N (θ) = √ uk (θ) 2π k=−N N 1 γ2N (t) = √ uk (0) · ϕk (t) 2π k=−N

for t = 2πm, m ∈ N and for θ = T m, m ∈ N.

(22)

(23)

Computation and Continuation of Quasiperiodic Solutions

4

369

Semidiscretisation by ﬁnite diﬀerence methods

Let the nonlinear operator F : B 1 → B 0 with ∂u ∂u + ω2 − f (u, θ1 ) , ∂θ1 ∂θ2

F (u) ≡

(24)

now act in Banach spaces B k = C k (T2 ), k = 0, 1 . The zero problem G(θ) = 0 on T2 is then in operator form F (u) = 0 ,

u ∈ C 1 (T2 ) .

(25)

If this problem is semidisretised by ﬁnite diﬀerences on θ1 the resulting ODE system could also be treated by standard software. Here we use our own continuation methods and therefore we ﬁrst linearise (24) and then discretise the linear systems. Applying Newton’s method to (25) yields the linearised problem F (uk ) v k = F (uk ) uk+1 = uk − v k

(26)

where the Frechét derivative is F (uk ) =

∂ ∂ + ω2 − fx (uk , θ1 ) . ∂θ1 ∂θ2

(27)

fx is the Jacobian of f with respect to x. Then we have to solve linear systems of the form ∂v ∂v + ω2 − A(θ1 , θ2 ) · v = r(θ1 , θ2 ). (28) ∂θ1 ∂θ2 with quasiperiodic vector function r and matrix function A. Replacing t = θ1 , θ = θ2 and ω = ω2 , we deﬁne a mesh in t by 2π GN = ti | ti = i · h; i = 0, 1, . . . , N ; h = N and discretise v(t, θ) on GN by v(ti , θ) = vi (θ),

v0 (θ) ≡ vN (θ),

i = 0, 1, . . . , N.

Approximation of the derivative with respect to t by ﬁnite diﬀerences l ∂v 1 (ti , θ) ≈ {DN v}i = cj · vi+j (θ) . ∂t h

(29)

j=−k

and insertion into (28) leads to ωvi +

l 1 cj · vi+j − Ai (θ) · vi = ri (θ) h j=−k

(30)

370

F. Schilder and W. Vogt

where Ai (θ) = A(ti , θ) and ri (θ) = r(ti , θ). Using ω = 2π T and isolating vi yields the cyclic periodically forced linear ODE system for the Newton corrections

vi =

l T 1 cj · vi+j + ri (θ) Ai (θ) · vi − 2π h

(31)

j=−k

with unknown period T . This linear system can now be solved by standard methods for periodic solutions. Again, we can easily obtain approximations to the invariant closed curves γ1 and γ2 by: γ1 (θ) = u(0, θ) ≈ u0 (θ) γ2 (ti ) = u(ti , 0) ≈ ui (0)

5

(32) (33)

Application to an electrical circuit

As an example we study a dynamical system given by H. Kawakami and T. Yoshinaga in [17]. The Duﬃng-type system of order 3 is given by x˙ 1 = x2 1 x˙ 2 = −k1 x2 − (x21 + 3x23 )x1 + B cos t 8 1 x˙ 3 = − k2 (3x21 + x23 )x3 + B0 , 8

(34)

which describes a resonant electric circuit with two saturable inductors. The period of the Poincaré map is T = 2π and we explore the system for the parameter values B0 = 0.03, B = 0.22, k2 = 0.05 and k1 ∈ [0.04, 0.15]. By numerical integration (”brute force” method), a bifurcation of the 2π−periodic solution into an invariant torus can be observed at k1∗ ≈ 0.1214. A stable quasiperiodic solution arises for smaller values k1 < k1∗ , which is continued in ﬁgure 1. Obviously a cascade of period doublings with respect to one basic frequency (torus doublings) arises and ﬁnally a strange attractor can be seen. The 2π−stroboscopic Poincaré map is displayed by bold dots. 1. Solution via spectral system. We choose a truncated Fourier series of order 1 x1 (t) = y1 (ωt) + y2 (ωt) sin(t) + y3 (ωt) cos(t) x2 (t) = y4 (ωt) + y5 (ωt) sin(t) + y6 (ωt) cos(t) x3 (t) = y7 (ωt) + y8 (ωt) sin(t) + y9 (ωt) cos(t) with the real functions   y1 (ωt) y2 (ωt) y3 (ωt) Y (ωt) =  y4 (ωt) y5 (ωt) y6 (ωt)  y7 (ωt) y8 (ωt) y9 (ωt)



 1 and ϕ(t) =  sin(t)  . cos(t)

371

Computation and Continuation of Quasiperiodic Solutions

1.451

1.327

1.202 1.40

. . . . .. . . .. . . ... . .. ... .......... . .. . . .... .. . . ... . ... ...... ....... ... .. .. . .. ... ......... .. ... .. . . . . ... . .. .. . .. . . .. .. ... . ... .. ... .... .. . ..... . . .. . . .. .................. .... .. .. ... . .. ... . . .. . . . .. . ... .. .. . ... .. ... . .. . . . ... .. . . . . . ... . . .. .. ... ... .... ... .. . .. ....... . ... .. .. ..... .. .. ...... .... .. .... . . .. . .. . ... .. .. . . . ... . .. . . .. . . . . ... . ... .. .... . ... .... .. .. ... ...... ......................................................................... ............ ... . . .. ...... ... ............ . . .. ... .. . . . . .. . . . . .. .. .. .. . . . . . .. . .. . . .. ..... . . ....... ..... . .... ... ........... ... .. ... ..... ............. ... .....⊕ .. ..⊕.⊕ .. ... . .. . . . .. . . . . . . . . ... .. ... .... . . ... . ....... .. . ..... ... . ...... . .⊕........⊕ . .. . . . ...... ....... ... ... . .... .⊕ .....⊕ ....⊕ ....⊕ ......... . ... . . . .. . ....... . .... ........ ... ⊕ ....... . . ... ⊕ ... . . .. .. . . . . . .. . . ..... ... ... .. . . . . . ... .. . . . ........... ⊕ . . . .. . .. . . ... .........⊕ . . . . . . ......... ... . ... . .. . .⊕ .. ... ......... ....... ................. .. ...... ....... ........ ..... ... ...... ..... . .... . ..... . .... ....... ...... .... ........ .... . . .. . .... .. .. . . ... ...........⊕ . .. . . .. . . ... . . . . ... .. .. .. . .... . . . . . . .... .. . . ... ... ... . . . .. .. . ... .. . .. . .. . . ... ... . ..... ........... .. .. ... ⊕ . .⊕ . . . . . . . . . . . . ........ ⊕ . ...... . .. .... .. ........ . . ..... ... .... . .. .. .. .. . . . ..... . . .. .. . .............. .. . . . . . . . . .. . ... ... . . . ... ...... . . . .... . .... .⊕⊕ .. . ... . .. .. . . .. . . ... . ... . . ............ ⊕ . . . .. . .. . . . . . . .... . . .. . .. . . . . .. . . ..... ... . . ⊕ . . . . . ... .......⊕.... . . ....... . . .... . . .... ... .. . .... .. .. ... ..... .................... . . .. . . . . . . ... . ... . .. .. ... ...... . . .. . ... . . . . . . . .. .. . .. .. . ... . .. ... . . . .. .. .. .. ........ . ..... ... . .. .. .⊕ . . . .. . .. . .. . . ... ......... . ..⊕ . . .. .... . .. . . . .. . . . . . ... . .. . .. . . . . . . . . . . . . .. . .. ..... .... . ....... . . . . . .. .... .. . .. ...⊕ . . . . . . . . . . . . . . . . ⊕ . . . . . . . . ⊕. . . . .. . . . ... . . . .. . .. . . . . . . . . .. . . . .. . .... .. .. . . . . . .. .. . . . . . . . . . ... ... ... ..... .. .... .... .... . .. . .... . . .... . . . .. ... . ......... .. . .. . .. . . . . . . . . .. . . . . .. .. ......... . . .. ... .. .. .. ⊕. . . . . .. .. .. . . . . ........ .. ...⊕ . . . . .. .. .. .. .. . .. . . .. . . . .. . . .. . . . . . . ⊕. . . . . . .. . . ...... .. . . . ..... . . . . . .. . . .. . .... . . .... .. . .. ⊕ . . . . . . . . . . . . . . . ... . .. ... . . .. . . .. ... .... ... ... .. .. . . .. .. .. .. . .. . . . . . . ... .. . .. . . . . . . . .. ... . .. . . . .⊕ . .. . . . . .. . . .. .... . .. . .. .. . . . . .. ..... .. .. ⊕ . . . .. . . . . . .. . . . . .......... .... .... . .⊕ . . .. . ...... .... . .⊕. . . . . . . .. . . . .. .... . .. . ... ....⊕ .... . .. .. .. ... . . ... ... . ....... . . . .. . . .. .. . ... . .. .. ... . . . . ....... . . . .... . . . ... . .. . .. ..... . .... . . .⊕. . .. . . . . . . . . .. . . . .. . . . . . ... ..... . . .. .... .. .⊕ . . . .... . . . . .. .. . . . . . . .. . .. . . .. .. .... . ... ... . . ... . .. .. . . . . . . . . . .. . .. . .. . ..... .... . . . ... . . . . . . . . ... .. ... ⊕ . . . . .. .... .. .. .. ...... . . . . .. .. .. ... . ⊕... ... . . ... .. .... ......... .... . . . . . .. . . . . . .. . . . . .. . .. . ⊕ . . . . . . .. .. . . .. ... ....... . . . . . . . . . . . ... ... ... .... .. . . . . .. . . . . . .. . .... .. ..... . . ... . .... ⊕ . .. . . . . . .. . ..... . . . ⊕. .. . .. . . . .. . . . . ... . .. . ... . . .. ..... .. .. . .. . . . ... .. . . . .. ..... .. . . . . .. .. . .. . . . .. . .. . . .. ... ... . . .. .. . . ... . . . . . . . . .. . . . . . . . . . . . . . . .. .. . . .... .. . .. . .⊕. . . . .. . . . . . . . . ....... ..... .. .. .. ... .. .. .... ⊕.. . ... .. . ..... .. .. ....... . . . . . . . . . . . . . . . . . . . . . .. .... . . . . .. .. ... . . . . . . . . .. .. .. .. . . . . .. . . . . . .. .. . . .. .. . .... ........ .. . ..... .. .. .⊕ . . .. . . . .. . . . . . . . . . . . . .. .. .. .. ...... .... . . .. . .. ... . . . ... . .. . ⊕ .. .. .. .. .⊕ . . ... . .. . ... ... . . . .... . . . . . . . . . . . . . . . ..... . . . .. .. .. . .. . . . . .. . .. . ⊕. . . .. . ... .. . .. ... ..... .. . . . . . . . . . .. .. . .. .. .. .... .. . . . . .. . .. . .. . .... . . . ... .. ⊕.. . . . . . . . . . . . . . ....... . .... ... .. . . ... . .... . . . . . ⊕ . . .. .. .. .. . . .. ..... .. . . . . . . . . . . . .. .. . .... . . . . . .. . . . . . . ⊕ ... . ... . . . . .. . ... . . . .. . . . . . . . . . . . .... ..... ... .. . . .. ... . . . . . .. . . .... . .. . ... . ... . .. . .. .. ... ... .... .... .. .. .. . . . . ... .. . . . . .. .... . . . . . .. . . ... . . . .. . . . . . . . . . .. ⊕ .. .. . .. . ⊕ .. . . . . . . . ... .. ... . . . .⊕. .. .. . .. . .. .. .. .. ... ...... . .. .. ... . . ...... .... ..... .. ..... . ... .. . . ... . ⊕.. ... .. . .... . . .... . ..... ... ........... . . . . . . .. . . . . .. .. ... ... ..... .... . .. . . . . . .. . . . .. . ...... .... . . .. ... . ..⊕. . . . . . .. . ... .. .. .. ... . ... .. .... . ... .. . .. .. ... . . .. . ... .. .... . . ... . .... .. . .... ...... . . . .. . . .. . . .. .. . . . . . .. .. .. . . . . .. ... ... . ..... ..... . .... . . .. .. ... . .. ... .. .. .. . . . . ... . ⊕ . ... . . . .. ... . ... . ... .. . .. . .. .... .. .. . .. . .. . . ⊕. . . . . ...... . .. ... . .... .. .. . . .. . .. . .. ... .. .... . . . ..... . ... .... . .... ...... .. ... . . .............. . . . . . .... . . ... .. . .... .. . .. . .. . .. ........ . . ... . . . . . .. . ... .. .. ...... ....... .. . . .. . . ⊕. .. . . . .. .......... ........ .. ...... ... ..... .. ...... . ...... . ....... . ...... . ..... .... .⊕ . . . . .. . . . . .. . . . . .... .. .. .. .. . . . ⊕. . . . .. ... ... .... .. . .. ... .. . . .. . . .. ... . .. ....... ... .... ... .....⊕ .. ......... ... .... .... .... . .... ... .. . .. ... . .. .... . .......... .... ........ . ... .. .. .. . . .. ... .... . ... . . . .. . . ... . .. . .......... . . . . . . . . .. . .. .. . . . . . . ⊕ . ..... ......... .... . . . .. . . . . . . . . . . . . . . ... . . ⊕ . . . . . . . .. ... . ....... . . . ........ ......... . .. ... . .... ... ............. .. .. ... .. . . .... .. .............. ..... ... .. . ... . .. .. .....⊕ . . . . . . . . ⊕ . . . . . . . . . .. . . . . . . . . .. .. . . ... . . . . . . . . . . . . . . . . . .. ... . . .. . ... ..... .. . . .. . . ........... . . . . ...... .. ....... .... ... .. .... .⊕ ..... ... . . . . . .. . .. . . ... . . . .. . . .. . .. . ⊕. . . .. . . ... . . .. .. .. . .... . .. ... .... ... ... ............... ...... . ... .. .... ......... ... .... ..... . . . .. . ... ⊕ ......... . . .. . . . .. . . .. . . . .. . . .. . ⊕ ... . . .... . ... . ... . . . ....⊕ . .. . . . . . . .. . . ... . . .. .. . ... . . . ... . . .. ....... ....... ....... .............................. ......... ..... .. . .. . .. . . . ... ..⊕. ... . . . .. . .. . . . .. .. . . . . ... . ⊕ . ⊕. .. . ..... . ... . .. .. . ... .. .... .. .... .. ...... ... ... . . .. . . .. . . . ..... . ... . ... . . . . . . . ⊕ . . . . . ⊕.. ⊕ .. . ... . ... .. ... .. . ... .. . . .. . ...... . . . . . . .. . ⊕... ⊕. . . ... . .. . . . . .. ... ... .. . ...⊕ .. ⊕ . . ⊕. . . ... .... .... . . . . . .. ........ . ... ... .. . . ... .. ...⊕. .. .⊕. ⊕ . . . . . . .. .. .. .. .. .. .... ⊕ ... ...⊕... .. .. . .

1.46

1.25

1.51 0.00 1.04

-1.51 0.00

1.65

-1.40

k1 = 0.09

1.457

1.191

0.926 1.81

-2.1 -1.81

k1 = 0 05

0.00

1.8 0.0

-1.8 -1.65

k1 = 0.06

... .. ....................................... ............. ......... ....... .. ............ ...... ............ ........... ...... .. . . ............................. .. .. .. .. .. . . .... ... ... . .. .... .. . .... . ... .. . . . ... ... .... ....... ........ ... ... .. ... ...... .. ........ ..... . .. .. . .... .. . .. . .. .. . . .. .. .. . . ... . . ... ... .. . ... .................. ................. . . ....... ......⊕ ............ ..............................⊕ . .. ....⊕ . ....⊕ .. . .. ... . . . ... ....... .... .............................................. ............. . ... . .....⊕ . .....⊕ ... ⊕ ....................... ............... ................. ...... ........ ..............⊕ ....⊕ . ...⊕ ......⊕ .............⊕ .. ....... ... ....... .... ...... ... ...... .. ........... ⊕...⊕ .. . ............................................................. ............................................. ................................. ..... . . ...... .... ........ .........⊕ . ......... . ..........⊕..⊕ . ...⊕ ...... .... .... ........ ... .. .... ... . ⊕ . .⊕.. .. ... ... ............. ............ ........... ........ ............... ⊕............................................... .....................⊕ ...... .... ..... ... ................... . .. .. ...... ...... . ... ...... ......... .. .. .......⊕ . ...................... ........................ ............... .... ..... ........................................⊕ . ...⊕ . .......... .......... ..... .... .. .......... ........................... .... ................. .... ...... ......... ............................................... . ... .......... ...... .......... ...... ..⊕ ....... .. .......⊕ . ....... . ... .... ................⊕ .⊕ . ........ ..................⊕ .⊕ . ....⊕ .....⊕ .. ...⊕ ...⊕ . . .............. ....................................................... ............. ...... .................... ... ...... ... ..... ............................................. . ...... ...⊕ ... .. . ..........⊕ ⊕ .. ............................ .⊕ ..⊕.......⊕ .....⊕ . ...⊕ .... . ........⊕ ..........................................⊕ ⊕ ..... ....... . . .......... . ....................... ...... .. ..................................... ............... . ⊕ . .. ................................................................. . ........... ................⊕ . ..... . . . ⊕. ....⊕ ......... .......⊕ ..................... .............⊕ ..... ........ ... ............... ........⊕ ....... . .. . .... . ........⊕ .............................. . . . ..... .. .... . ............. .⊕ ... .......................................................... .....................................⊕ .. ⊕ ⊕ . . . . . . .. ........⊕ ...... . ............⊕ .... ... .... ... .. .... . .. ..... .. ........ ........ ............... ............... ............................ .... ............... . .... .... ... .... ... ... .. . ......... .............. .... ........ .....⊕ ...⊕ ...... ........ ..... .......... ... .. . ......... ..... .... ........⊕ . ⊕ . . . ... .. ...... ..........⊕ . . . ⊕ . . . ... . ...⊕ .. ... . ...............⊕ . . . . . . . . . . . . . . . . . . . . . . . ⊕ . . . . . . . . . . . ........... .....⊕ . . .. ... ..... ......................... . . . .... .............. ..⊕......... ........................ . .. . . . .⊕ .⊕ . ..... . . .. . .. .......... ........ ................ . . . .... ......... .......... ........ ....... . ....... ...... ........... ........ . . ............ ..... . .. ... . ... .. ..⊕ .... ...... .... ... . ... ................. ... .. .. .. ............. .......⊕ ... .. .........⊕ .⊕ .. ... ........ ....................... .............................⊕ ... ........... . . ... . . . .. . . ..... ..... . .. ..... .... ... ............. ....⊕ . .. . .. . ...... ..........................⊕ . . ..... . .. . . ... .......⊕ . . . . . . . . . . . . . . . . . . . . ..... ... .................⊕ . . . . . . ... . . . . ⊕ . . . . . . ⊕ . . . . . . .. ... . . . ........ ...... .. . .. . ... .. . ... ... .. . . .. ..... . ............. .......... ⊕. .... .. .... . . . .. .. . . . . .......... ................ ...........................⊕ .............. ......... .. .⊕ . .⊕ ..............................⊕ .. ...... . ... . .. ... .. . . . .. .. . ..... ... . .... ...... .. ...... ............. ........ ...... ....... .............. .. ...... ..... .⊕ ... ... ..⊕ .. ... ..⊕...... ... . ..... . ..... .. . . ..... . .. ........ ... ...........⊕ .................⊕ ............... ... ............................ . ... .. ... ....... .. ..... ... .......... . . . . . . . . . . . . .. . . . . . . . ........ ..............⊕ . . . . . .. . . . . . . . . . ⊕ . . . . . . . . . . . ⊕ . .. . . . . . . . . . . . . . . . . . . ⊕ . . . . . . . . . . . . ... .. . ... .. .. . ......... ................... ...... .................... ................ ..........⊕ ..... .. .. ... ... . . .... .... .. .. .. ...... ....... .......... ...⊕ .. .. .... ....................... ...... ............ .. .. ... .⊕ . . .. . . ... ... ..... ... .. ...................... ..... ............. ..... ........... ..... ................................... . .... . .. . ...... ........ .⊕ ...... .... ......⊕ ⊕ ...... .. .⊕ ................. ........ ........... ............... ..⊕ ........... ........... . ...... . .. . .. ...... ..... . .... .. .... ...... ...... ................... ............................................... ....... ........ ............................................... ..........⊕ ....... .⊕ . . .. . ... ..... ... .............................................. . . .. ....... ...........⊕ . . . . . . . . . . . . . . . . ⊕ . . . . . . ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..... ..................... . .. ...... .................... . ...... ..⊕ . . .. .. .. .. ... ...... . .... .⊕ ..................... ..⊕ ... ............. ....................... ............................................................... ............... ......................................................... ...............................⊕ ..... ⊕ ............. ........................ ................... ................................................. ....... . . . .... ... .. .. ⊕ ................................ ..... .⊕ ......... . .. . . ........................ ....... ...⊕ .⊕ . ........... . ........ ..... .. ..... . ...................................... .. ................⊕ ..⊕ ..........⊕ . ... .... ....⊕....... .... .⊕ .⊕ .....................................................................................⊕ .. .......................................................................... .................................................................................. ........ ............ .⊕ ..⊕ . ... . .... ⊕ . . . .... . . . . .. ... ..... ⊕ ....⊕ .......... . .... . . ..... .. ...... ..⊕ ⊕ .... ......... .⊕ . ................ ......... ..............⊕ ....................... .......⊕ .......... .⊕ .. . .. ...... ........... ................. ... ..⊕ .......⊕ ......⊕ .............. .. ... .............. .. ...................................... ... .. ⊕ . ........ .. . . . . ⊕ ........................................................... ............⊕ . ................. ... ........ ......⊕ .⊕ ...⊕ .⊕ ..................... ............ . ... .... ............ ... ...... . ..⊕ ...⊕ ..........................⊕ . ...................................................................... ........ ..⊕ ...⊕ ..⊕ ...⊕ ........... ... ........................................... .............................. ..⊕ ...⊕ ............................................. .......................................⊕ ........ ....................... ...........⊕ ...⊕ ....⊕ .....⊕ ⊕ . ... . ... . ... .... .......... ....⊕ ........ ......... . . . . . . . . . . ................. ....⊕ . . . . . . . . .. .... ....... .. ⊕ . ⊕ . . . . . ⊕ . . . . . . .. . . . . . . . . . . . . . . ⊕ . . . . . . . . . . . . . . . . ⊕ . . . . . . . . . . . . . . . . ⊕ . . . . . . . . . . . . . . . . . . . . . . ⊕ ⊕ . . ⊕ . . ..... .. . . . . . . ... .... .. . .. ... .. .. . . ... ..... . .. . .... . .... . . .. ... . . . . . . ... . . . . . .. . . .. ..... .....⊕ .⊕ .. ⊕ ...⊕ . .⊕ ....... .................................... ...... .... .....⊕ ..........⊕ ..... .. ..... .. ...⊕....... ... . ...⊕ ......... . ..... . . .. .. . .... . . . . . .. . . . . .. . . ........ .. .. ⊕ ....................... . . . .. . .... .......... ................ ....... ............................... ..........................⊕ ⊕ . ...................................... ............ ................. . ..... .......⊕ ... . .. ... . . . . . .. .. . .. .. . ... . . . .. .. . .. . . . . .. . .. . .. .. . . . . . ... ...................... .. .. .... .................... .⊕ . . . .. . . . .⊕.. ....... ... .. .. ⊕ . . . . .. . ... . . ⊕ . . . . ⊕ .......... .............. . . . . . . . . . . . . .. . . . . ... ....... ..... ... . ... ..... .... ... .. . . . .... ........... ...... ...... .... . . .. .... ........ ........ .⊕ . ....... . ... . . .. .. . ... .... .... . . ...... .......... ......... . .. . ... . .... ⊕ .....⊕..... . .......... ...... . ...... .... ........... ......... ...⊕ . . ... ... ... .....⊕ .. . .. .. ... .. ........................................ ...... ... . ....... .. ...... ... ................................⊕ .... ................⊕ .... .. .... . ..... .. . .. . . . . .... .. .. .. .. ... ....... . . . . . .⊕ . ... ....................... ............ .... .............................................. .. .. ... ....... ................. ............ ........ . ....... ................. .................... ................................ ..⊕ ......⊕ . . . . . .... .. .. . .. ... .... .. . ... ............. .................⊕.... .... ... .. ... ⊕ . ... ........⊕ .. .. .... ... ..... .................. .. ..... ....... ......... ............................................... . . . . .. . . ... . ... . .. . ... . ......... .............. .. ⊕ ...⊕ ...... . .. ... . ⊕ ... .. .. ⊕ ..⊕ ... .. ....................................................... ............... ......... .... ................ ...... ....................................⊕ ..... ...... .. . . ... . ... .. .. ........ ...... ................... ........... .... ....... . ... .. .............. .⊕ ........... ... ..... .. .... .... . . .. .. ........ . .... ⊕ . .... ..... .... ⊕ ... ............⊕ ....⊕ . . ........................... ....... . ........ .. ... .. ... ..... ... ... .. . ..........⊕ .. ... .. .......... .... .................. . . . . . . . . . . . . . . . . . . . . ⊕ . . . . . . . . . .. . . ... . ⊕ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .... .. . . . . . . . . . . . . . . . ... . ...... ....... . .... .. . .. ....... ..... ........ .... ........ . ...... . . ..... .. . . .. ... . .... .. ........... . ........... . .. ..... ....... ..... . .. . ... .. . ⊕ ... .. .⊕ .. ................................................ ......... ..... .. ........ .. ...⊕ ........ .......⊕ ...⊕....⊕ .... . ......... ....... .. ...... .. .... . . . . . .......⊕.... ...... ...... ........... ............... .... ... .⊕ .. .⊕ ....... .... ⊕ ....... .. ... . ..... ....⊕ ....⊕ ..⊕ . ....⊕ ............................................................ ...................................................... .... ............. .................⊕ . . ...................................⊕ .⊕ .⊕ ...... ... ... ....... . . ....... .... ........................ ... . ..... .......⊕ . . . ⊕ . . ... . . . . . . ⊕ . . . . . . . . . . . . . . . . . . . . ⊕ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ⊕ . . . . . .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . ⊕.. ... . . .. ..⊕ .. . ......... ... ...... ...... ..... . . .. ... .... .. . . ... .... .. .... ....... .⊕ . ... . . ... ⊕ ..⊕.....⊕ ..⊕ . ....... .... ..... ........................................... .............. .................................... ........ .......... ...... ..................... . . . .. .. . ............... . . . ....⊕...... .. ...... ...... ....... ....⊕ .... . .. .. ........⊕ .................. .... ..... . .... .... ......... ............. ..................................... .. ......... ........ ..... .........................⊕ ..............⊕ .. ⊕ ........ ...⊕ . . . ... .... ..... ...... . ...... . ....... .. ...... ..... . ..... ....... .... ................. ..⊕ ..............⊕ ... .. . . ...⊕ . . . . . . . ... . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...... ...... .. . .............. . . . . . . . . . . . . . ⊕ . . . . . . . . ....⊕ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ........ ..... ... . ... . . .. ⊕.. .. . ............ .. ....... ... ... .. ..⊕..... ..... . .. ...... .. . .... .. .... ... ........................⊕ .. ..... ...... . . . .... . .. .. . .. ....... . ............. . . .. .. ...... ..... ............................ .. . ....... .... . .... .. .. . .. ....... .. .... . ..... . .⊕ . . .. . . ⊕ ... .... .... .. ...... ...... .. .... .. ..⊕ . . . ... ...... ............. ....... . .............. .......⊕ .⊕ ...... .... . . ....... ..... . .. .. ... . .. .... . .. ..... ............ . .. ... . ....... .. ......... .... .. . ... .... .... .. . . ..... .. ⊕ . . . .. . . ...⊕ . . ....... . .. .. .... ....... ............. ..................... .⊕ .. . . ...... ............................ ................................ ................ ....................... .⊕ . ..... . . . . ... . .... ... .⊕.⊕ ..... ................. . ......... ............. . ........ . ....⊕. .⊕ .. .. ..................... ..... .. . .... ... ............⊕ ⊕..⊕. . . .... .. ...... ... .. .......... .... .. ...... ..... ....⊕ . . ...... ... . ..... .. .... . ........ ...... .. ............ ................⊕ ... ..... .. . . . . . . . .... ... ..⊕ ... . .. ... .. . ....⊕ . .⊕......⊕ .. ..... .. ..... ......... ... ....... ...... . ... .. . .... . . .. ..... ... . .. . ..... .. .. . .. .. ... ........... ......⊕ ................ ... .................. ........ ......... . ....... . ..... .... ...... .... . .⊕ .. ..⊕ . ....⊕. . ..⊕ ..... .. ...... .⊕ ........⊕ ....⊕ ..........⊕ . . . . . . ⊕ . . . . . . . . . . . . . . . . ... . .. ... .. .. . .. .⊕ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. .. . ... .. .. ... .. . . . . .. . . ... . .... ....... .⊕...... .. .. ................. ......................... ........ ........ .... ....... ...... ................. ......... ......... .. ..... ............ ....⊕ . . . . . ..⊕ . . . . . . . . .. . .. .. .. . . .......... ...... .. . .. . .. ⊕ .. . .......... ....⊕ .. ... ... .... .. . .. . . ... .... ... .... .... ... ... ... .. ........ ...... ... .... .. .. . . .. . ... ⊕. ... .. .. . .. . .. .. .. .. .. . . .. .. . .............. . .. ............... ...... ........ .. .. ..... ... .... .. ... ......... ⊕ ... ... . .. ... ... ... ... . .. . . . ... ... ...... ........ .... . . . .. . . . . .. ⊕ .. . .. . . ... . .. .... .... ....... .......... .... ..... . ..... .. .. ... .... . .. ... ............ .. .. . .. .. ... .. ... . ... ⊕ .. . ⊕ ..... .. .. ... . ... .. .. . .. . .. .... .. . . ... .. ............ . . . . . ... . .. . . .. ......................... ..... .. ............ ........ . ... . . .. . . ..... ...... .. .... . ...⊕ ..⊕ ... .... . .. . .. .. ⊕...⊕ . . . .. . ... .. ... . .. .. . . . . . . . . ... .... ... . ... . . . ... ... . .......... ...... ........ . .. .. ........ ... . .. .. . .. .. .. ... ... .. .. ..... ... . ... ... ........ .... . . . . . . . . .. . ⊕ .⊕ . . . ........ ........ .. . .... ..... . . .. . . ... . . .. . . .... . ... . ... . ..... ........ . ...... .. .........⊕ ....... . ... ... . ........ ....................... . . . .. . . .. . .. .. ...... .............. . . . . . . . . . . . . .⊕ . . . . . . .. . ... ⊕ ......... . .. ..... ........ .. .... .. ...... .. . ... . .. ⊕ . . .⊕ .. . . ................ ...... .................. ....................... . . . . . . . . . .. . . . ... . ... .. . . . . . . .. . .... .. ... .. . ... .. . ... . ⊕ . . . ... . . .. ... . . . . ... . ... ... ......⊕ .⊕ . . ..... ... .. . ...... . ...... ... .⊕. ... ... .... . ... ... ....... .. ⊕.. ⊕. .. . .. . .. . . . . . ... . . . ⊕. ...⊕.. ⊕ ... ... .... ..... ... . .. . ... ... .. .. .. . .. . .... .⊕ .... .. .. ...... ....... . ... ... ... .... . . .... .. ................. . . .. . . .. . ..⊕ ... ⊕ .⊕ . ⊕ . . . . . . . ⊕ . . .. ..⊕ . . . . . . . . .... ...... ..... .. .. ... .... ..... . .... . ..... . .... .. .... .. .... ... ⊕ . . . . .. ...... .... . .... . ... .. .. . . . ... .... . .. . ... ..... .... .. .. ... .. . .. . .. .. . .. . .. . . . ... ... ... ... . ... .. .

0.00

. . ...... .................................................. .......... ........ . ..... ....... . .... ... ........ ............... ...... .. . .. .... .... ..... ... .... . .... ........ .......... .................................... .......... .......... ............................ ... ....................................................................... ...... ........... .... ........... .... ........ ....... . .. ....... ......... .. . ........... ........... ...................................................................... ... ................. ...... .... ....... . . ... ...... ...................................................⊕ ....⊕ . . . ..⊕ ....... ....... ......... ..⊕⊕ ....... .⊕ .. ............ ....⊕ ..⊕ .... ....⊕ ....⊕ .....⊕ ............... ............. ⊕ ...⊕ ............................................. .... ...................................................................................................................................................... ... ...⊕ .⊕ . ....... .⊕ ... .. ..... ...................... .⊕ ................ .... .........⊕ ⊕...⊕ ...⊕ ...... .................................⊕ .⊕..⊕ ...... ⊕ .⊕ .... ......... ...... ...... ....... ..... ...... .. ..... ........ .......... ............................. ... ..... .. .......... .................... ....... ........................................ .......... ....... .... ....⊕ ⊕ . .⊕... .. ... .. .. . ............. ...................⊕ ...... .... ........... ...... .. ........ ...... ...... .. ....... .. ...... ... .. ... .. .... ................ .⊕⊕ ........ ..... . .... .. . . ....... ...... ....................... .....⊕ ... . ..... .. .. ..⊕ ⊕.⊕.. .. ... .. . ... ........ .. . . . . . . . . . . . .... .. ..... .. ⊕ ......... ... .. . .... ... .. .... .. ..... .... ...... ........ ................ .⊕ .....⊕ .. . . .. ... .. ... .... .... .......................... . ...⊕ ........... . ... . .. ⊕ ............ ...... ..... ....... .......... ......... ............................................................. ... .......... .. .. . ...⊕ ⊕... . . .. . . . . . . .. ........................................ ....⊕ .. ⊕.. .... ... . ....... .. ...... . .. ..... . . .... ... ... ....... ..... . . . ... . .. . .. . .. . ... ... ... . ..... ......... ... . ..⊕ ⊕. ... . .. .. . . .. . .. .. .. ...... .. .............. ... ... .. .... .. .... . .... ... ... ... .. .......... ................ ⊕ . ... ... .... .... ...... .. ..... ..... ......... .... ............ ............. ...⊕ . . .. . . ⊕ . . . . . ..⊕........ ..... ..... . .... . ... . .... . ............. . . ................ .... .. ..⊕ . ...... ... .. . . . . . . . . .. . . . . .... ........ .... .... ..... .... . .. ....... ... ... . .. .... ............. . . . .. .. ..⊕ .... ..... .. . .. ...⊕ ... .... ... ... ..... . ..... .. ..... ... .... .. .... ⊕... . ... .. . .... . .... . . .... . ................................ ....... .......... ...........⊕ ..... .⊕⊕ ....... .... .. ... .. . ... .. . .. . . . .. . .... . ..... .................. .... .. .. .... ...⊕ .. ... ...... .... ........ .... ..... .... ......... ... ......... .. . ...... ... ....... .......... .. .⊕ ..⊕ .. ........ . . . . . . . . . . . . . . . . . . . . . . ........ .... .. .....⊕ . ..... . . . . . . . . . .. .. .... ..... ⊕ .. . . . . . . ⊕.... .. ... .. ... . ..... ... . .... .. ... ..... .... ............. . .. .. . . . . .... ................ . .. . . .. ... . ... ... . .... ... . . ........... . .. .. ... . .... . .. . . . . .. .. ............................... ...... ............ ...... ....... ......⊕ .⊕ .. . ..... ... . ... .. ... ....... .. .. .. ... ..... ............ ⊕ .... . .. . .. . ⊕ ... .. . . . . . . . . .. . .. .. . . ... ......... . ... .. . .. . ... .. ... . . . ..... ......... .... .... ...... . ....... ... ........ . ......⊕ . . . . . . ... .......... ....... ..⊕ ... . . ⊕.... . .... . .. .. ... . .. ... .. . ........... . . . ⊕ . . . . . . . . . . .. . . . . . . . .. . .......... ...... ......... ...... .. .. .. . ...... ..... ... ........ .....⊕ . ....................... .................................................. .⊕ . ............ ... . .. .. .... . ... ... . .. . . ..... .... . ............ .. . . ... . . ⊕ ...⊕ . .. ... ..... .... ....... ..... .. ... ........ ............ . . . . . . . . .. ... .. . . . . . . ...... ... ⊕ . .. .... ... .... . .. ............. ................... ............. ... ..... .. ........ ..... ... ...⊕ ... ... . .. ⊕ ...... . . . . . ⊕ . . . . . . . . . . . . . . . . ... . . . . ............ ..... .... ⊕ . . . . . .. . . . . . . . .. . .. ⊕ . . .. .. ... . . .. ... .. . ........ . .. . .. .. . .. . . .. . ... . .. . .. ........... ⊕ ... .. . . ⊕.... ... . . .. . ... .. . .. ... . . .. ......................................................................... ................ ................ ..⊕ ... . .. .. . .. .. . .. . . . .. . .. ....... .......... ....... ....... ........ .............. ......... ....................... .... ⊕ ...... ⊕ .. .. .. . .................................. ............. ....................................... .........................⊕ .......... . . . .. . .. .. .. . . . . ...... .... . . ........... ... . ...⊕ . . . . . . . . . . . . . . . . . . ⊕ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... . . . . . . . . . . . . .⊕. . . . . ... .................... .....................................................................................................................⊕ .⊕ ..⊕ ..... . . . . .. . . .. . .. .. .. . .................. ...................... ............................................................................................... ............... . .. .. .. .... ... .. .. ...... .. ........ .... .. . ... . . . . .. .. .. . .. ...⊕ ...... .. .... . . ⊕ ........ . ..... . . . . . .. .. .. .. .... ⊕......... .. ................................................................................................................................................................................................ . ⊕.... .. ................................................................................................................................ .................. ........ ...... ..... ..... . ....... .. ......⊕ ⊕ . . . ... .......... ... ... ...⊕ .. . ... . .. .. .. .. .. . . . . ... .. . .. . . . . ... . . . . . . ... . . . . . . . . . . . . . ..... ......... . ... ⊕. .. .. . . .. ... .. .. . . . ... ........................ . . . . . . . .. .. .. ⊕ .................... ..... ........... ........ .... ......................... ........... . . . . . . . ..⊕.. . .. . . ................................................................................................ .......... ...... .................. .... .. ...... .... ......... .... ...... .⊕ ..⊕.⊕ .......... .. .. . . .. . ....... ...... ......... ... ... . ... ..... ...... ......................... ..................................................... ........... .. ...... ....... .... ⊕ ..⊕ ......... .......... ..................................⊕ .⊕ . . . . . .............................................................⊕ ⊕ .... .... . . . . ⊕ ....⊕ .⊕ ..... .. . . . . .. ... ......... ............................................... .......................................................................................... ............................................................. ..⊕ ..⊕ ..⊕ .....⊕ .⊕ .....⊕ .⊕ .⊕ . ........................ ......⊕ ...⊕ ...⊕ ..⊕ .. ...⊕ ... . . . ..... ⊕ ...⊕ .⊕ .⊕ .⊕ . ..... . ... . . . . . . . .. . . . . . . .. .. .. .⊕ ....................... ............................................... . ..... .. .... .................. ....⊕ ⊕ .⊕ .⊕ .. .. . .. .. . .....................................................................................⊕ .⊕ ....⊕ ..⊕ ..⊕ ⊕ ..⊕ ..⊕ ...⊕ ...⊕ .....⊕ ..⊕ ...⊕ ⊕ ....⊕ . . . . .. . . . .. .. .. .. ... . .. .. . . .. . .... .. ... .......... .... ...⊕ ...⊕ ..⊕ ..⊕ .... ... ..... ... ⊕ ............................... ...... ..... ... . . ..... ... .. .. ... .. ........... ..⊕ ...⊕ ...⊕ .......⊕ . .... .................... ......... ................. ............. ... ....... ..... ............ ........... ...... ...................... .... ........ . . . . . . ... . ... ... . .. . ... . .. . . .... .... . . ......⊕ .. . .. .. . .. .... . ⊕ . . . ... . . . . . .. .. . . ... . . ... .. .. ..... . .. ....... .. . . . . . . . . . . . . . . . .⊕ ...⊕ .. ...... . .. . .. ...... ...... .... .. . ... ..... ..... ... ........ .... ....... ..... ....... ... . ....⊕ ... ⊕ ⊕. . . .. . ... . ... . ... . . . . . . . . .. .. . ....⊕ .....⊕ ...... ....... ..... . ..... .. .... .... .. ... .......... .. ................ .. . . . .. . . . . . .. . . .. .. . . .. . .... .... ... . . . ... .............. ...... ...... .... .. ..⊕ . . . . . .... .. . . . .. . . ... . . . . . .. . . . . .. .. .. .. ... .. . ..... .... .. .. . . .. . . . ... ... . ... .... . .. .. . . . . .. . ....... ⊕ ....⊕ .......... ........... ..... . .. .. .. .. ⊕ . . .. . ........ .... .... ......................... . . . .. .. . . . .. . . . .. ... . . . . . . . . . .. .. . .. . ... . .. ... ........ .... ..... . ... ..⊕... .. .. .. . . .. . . . . . .. . ... . ... .. . . . .. . . . ..............⊕ .. ...... .... ................ . . . . . .. .. . . .. . . . .. .. . . .. .. ... ..... . . ..... .⊕ .. . . ..... .. .. . .. ... ⊕ .⊕... . .. . .... . ... . .. . .... . .... .... .. .... . ... . ................... .⊕ . .. ....... ...................... .. . .. . ... . . . . . . . . .... . . . . .. . . .. .. .. . . . .. .. .... ....... ... . .... . . . . .. . ⊕ . . ............. . . .. . . . . . . . .. . . . .. . . ... .. . .. ⊕ . . .. . . . . . . . . . . . ... . ... .......... .. . .. . . .. ⊕ ⊕. . . .. . .. .. . .. .. . .. ... . ... .. . .. . ... . . .. . . .............................. ..... .⊕ . ... . . . . .. .. . . . .. .. . . . . .. . . . . .. ... .. . . .. .. . .... ..... ................ .... .⊕ ............. .... . ... .. . .⊕ . ... .. ... ... .. . .. . . . . .. . .. .. . ................................. .......................⊕ .... . .. . . .. . .. . . .. . . . .. . . . . .. ... .. . .. ... . ............ .⊕ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ⊕ . . . . . ............... . ... . ... . ⊕ . . .. . . . . .. .. .. . . .. . . . . . ... .. . . . .. ........ . . ... .. ...... .. .......... .......⊕ . . . . . . . . . . .. ... ... . . . .. . ... ... ... . .... ....... . . . .. .. .⊕ . . ..⊕ . .. . .. .. .. . ... . .. . .. . .. . .......... ..... ..... ... ......... ... ..... .... ....................⊕ ....... ........ ...... . ... . . .. .. ... . .. .. . .. ..... ...... . .. .... .. . . .... .. .. ..... ..... .. . .... ... . ⊕ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . .... . . ⊕ . . .. . . . . . . . . . . . . . . . . .............. . .. . .. . . .. ... . . ... .. . .. . .. ....... .. . ....... ... .. . .. . .. . .. .. .... ... .. . .... ... ........ .. ⊕ ........ ....... . .... . ... . . . .⊕ .. ... ... ... ... . ... . .. .. .. ....... ............ .. . ... .. ......... .... .. ... .... . .. .... ....⊕ ..... . .... . .. ... .... . . .. . . ... .. .. . .... ............ ....... . .⊕ .... .. .. .. . ... . ...⊕ .... . .... ...... . .... . ... ..... ... .. . .... ... . .. . ... .................. . . .. .. . .. .. . .. ... . .. ... .... .. .... .. ............ ........... ............. .......... .... ..... .............. . ... ...... ...............⊕ .. .... . ... . . ...⊕ .⊕ ........ .... .. . . . . .. . . ... . . . . ........ .. ... ... ... . ..... ...... .... ..... . .. ... . . . ..... ..... . .... ..⊕ .... . ... .. . . .⊕ . ........................ ..... .... ... .... .... ..... ...... ..... ... ................ .... ...... .. ... . . ..⊕ . ... . ..... ..... . ..... . ................................. ........... ........ ... .............................................. . ... ... . .. ........ ...... ......... ......⊕ . . .. .. . ..... . . .⊕ ⊕ . ....... . .. ..... ..... ..... . ... ... ...⊕ ............ ............. .... ... .... ... . .. .... ... . .. . . . ..... .. ..... ⊕ .⊕ ...... ... .. . ..... .. . ...... .. ................... ... . .. ..... . . ... . ... . .. . . .. . .. . ..... . .. ....... . ......... . ..... ...................⊕ . . . .⊕ .. ........ ..................... ................................ .. ..... . .. ... . ...... .. ... ...... ..... .......................... ⊕ ..... . . ... .. .. . . . . . ... . . .. ... . . .. . .. . ... . . . . .. .⊕ ....... . .... . ............ . .. . .. ....... .....⊕ . . ⊕ . . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . ⊕ . . . . .. . . . . . . . . . . . . . . .⊕ . .. . ... . . ... .. ...... ......................... . ... . . ... . ... ... . . .. . .. . . . . . ... . .. .. . . .. . .. . . . . ... .⊕ ... ... ............ .... ..... ...... .... .. ...... ... .. ... ...⊕ ⊕..⊕ .. . . . . . .. . . . ........ .. ... ...... ............... .... ......... ⊕.... . . .... . . . ... . . . .. . . .. . ...... ................................... ........ .. .. . .. .. ........... .. .. . ... .. ... ..⊕ . ⊕ . .. . ... .. ..... .. .. .. . ... ... . ... . . .. .... .. . . .. . ⊕...⊕ .⊕. .⊕. . . . . . . . .. . ... .. . . ... ⊕. .⊕ ..... .... .... ....... ... . ... .. ... . .. . . ...... .⊕ . ⊕. .. .. .. .... .......... ......... ...... . ....... ...... . .... . ..... . .. . .. .. .. ⊕ .. ⊕. .⊕ .. ........... . . ..... ... ... . .. . . .. .. . . . .... ...⊕ .. . . .. ⊕..⊕ ... .⊕ .. . .. . .. . . ...... .. .. . . ... ..... . ... ....... .... .. . ... ...... . ...⊕ . ..⊕..⊕ . . ..⊕....⊕⊕ . . ..... ..................... ...... . ...... ....... ... ....... ....... . ...... . ..... . .... . .

1.44

1.16

2.1 0.0

0.88 1.9

. . . . . . . . . .. . . . .... . . . . . . . . . . .. . . . . . . . .. .. .. . . . .. . . . . . .. ....... . . . . . ... . . .. . . . . . . . . . . ... . . ... ... . . . . ... .. . ........ ..... .... ..... .... .. .. . ....... .... ... .. .. .. . ........ ...... . ... ... ..... .. .. .... . .... .. .. . . .. .. .. .... .... . ... ..... .... . . . . . . . .... . . ....... . .. ..... ... .... ..... ... ........ .................. ......... ..... .. . .... . .... ... . ... ..... . ....... ........ ...... . .............. ............... .... . . .. ... .... ... . .. . . . . .. ..... . . ... .. ... . . . .. . . .. ..... .... ..... .. ... . ..... ...... .. .. ......... .... .... ..... ... .......... . . ... ........ .. .. . ... ... .... ........... .. . ... .. . . ...... . . ... .................... .... ...... .... .......... ...... ................. . .. .. ... .. ..... ... . .. . ... .. ........ .... ......... . ... . . ..... .. .... ................. .. .... ... .. . .. . .. . ... .. ..... . . ...... ....... ......... ......... ... ...... ................... ........ ...... .. ......... .... .... . . .. .. . . ..... .. ... . . . . . . .. . ...... . . . . . . .. . . . . . . .... . . ... . ... .. . . ... .... . ..... ..... . .... . . .. ..... ......................... .. .. .... ........................ ..... .......... . . . ... . .. . .. .. ........... ... . ... . ...... . . .. . .. . ... . . .. .. . . . . . ... . .... .... .... .. ... .. .. .. . ...... ....... . . ... .. .. . .. .. .. .... . .... ... . ... . ... . ... .. .. . ...... ............. . . .... .. ... .................. .. .. . .. . . ..... . . . .. . .. ........ . .... ..... .. ........ . .. . . . . .... ..... ........ . ... ... .. . . .... .... ... .... ........ . ... . .. . ... . .... .. . ... .. .... . .... ..... . .... . ... .................... ... . .. . . . .. . .. . ... . .... ...... ...... . ........ . . . . . . . . . . . . . . . . ... . . .. . . . .. . . . . .. . . . . ............... ........ .. . . . . . . . . . . ...... .. . .. ... . .... ...... .. .. .. . . . .. ... . .. .. ... . . . . . ... . .... .. . ....... . .... . ... .................. . .............. . ......... .. ..... ... . . . . . .. . . . .. . .. . . . .. .......... . . . .. ... .. ... .. .. .. .. . . . . . . . . . .. ... . .. .. . .... . . ..... . .. . ... . .. . .... . . .... . .. . . . . . . . . . . .. . .. . . . . . . ... . . ... . .... ....... .. . .. .. . . .... . . . . .. .. . . . . . . . .. .. . . . . . . .. .. .. .. . ... . . .... .. . .. .. . ... . . .. . . . . . .... .. . .. .... ....................... .. .... ..................... . ............. .... .... . .. .. . . .. . . .. . .. .. . . .. . . .. . . .. . .. .... . . . . . .. . .. .. ... .. . . . . .. . . . . . . ... . .. . . . . . . . ..... . .. . .. ...... ....... .... ........ . . . . . . . ... ... ........ . . .. ... . .. ... ... . ..... ... .. .... .. ... .. ...... ...... ... . ....... ......... . .... .. .. ..... ... ... . .. .. .. ..... ... ...... . .. .. .. . .. ... .. . ... ..... .. .... .... ......... .. . .. . ..... .......... . ... .... . . . . . . .. . . . . . . . . . .. . . . . . . . . . . . . . . .... . . . . . . . . . . ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ........ .......... .. ..... . . .. .... . .... .. ... ... .... ......... .... .... . .... . ...... .. .. ... .. .. . . . . . ... . . .. .. ..... .. ... .. . . . . . . ... . . .... . . .. . .. .. ... ... ... .. . . . . . . . . . . .. . ... .. .. . . . . .. ... .. .. . .. ...... .. . . . .... . . . . . . .. .. .. .. . . . . . . ... . .. . . ... . . . . ...... .. ... .... ....... ............... .. .. .. .............. ..... ......... ... ....... ........ ........... . ...... ....... .. ........ . .. ...... .... . . ....... .. ....... ........ .... . . . . .. .. .. .. .. ......... . ... ... ........ . . . . ..... .. .. ......................... ....... .... . ....... ..... . .. ... . ...... .... ..... ..... . ... .... . ..... .. ... ..................... ............... ....... ......... ....... ............ ... ... . . . . . . . . .. ...... ... ... ... . . ...... ... . .. .. .. . .... .... .. . .. . . . ... .. . .. .. . . . ... . . . . .. . .... . .. . . .... ... .... .. ... ... . ....... . ... . ............................... ....... ...... . . . .... ....... .. ..... .. ... .. .... .. .. .. .. .. .. .............. ................ ..... ......... .... .. .. .. ... ..... .... ....... ........ . ....... ... ... .................. ......... ................ .. .. ...... . . ..... .. ... .. ... . ..... ...... ............. . . . . . ... .... . .. . . . .. .. . .. . . .. . ......... . . . . . . .. .. . . .. . .. .... ...... .. ..... .. . ........ .. .. .. .. .. .. . . ... . . . . ... . . .. .... ... ... .. ... ..... .. . .. . .... ...... ...... .............. .. .......... ................ .......... ...... ............. . ... ....... ..... ....... ............... .. ... .. ... ........ . . . . . . ... . . . . .. . .. .. . .. .. . . . .. . . .. . .. . . . . .. . .. ... . .. ....... .. ........ . .. . . . . . .. .. . . . .. .. . . . . . . . .. . ... . ... ............... .. ........ . . . . . . .. . .... . . ....... . . . .. ...... .. .. . .. ..... .. . .. ... ... . .... ... .... .. .... ...... ... . . . .. .. . . . .. .. . . ... .. .. . . .. ... . ... ... . . . .. ... . .. . ... . .. . ... . ... . . .... .. .. . . ... .. . . . ........... .... . . .. . . . . .... . ... . . . .. ......... .. ....... . ... . . . . .... .. . . ... ........ . . ... .. ... ... .. . . .. ... . . . . . .. . .. . . . .. ........... . . . . . .. . .. . . . . . .. . . .. . . .. . . . . .. . .. . ...... . ..... ..... . ... .. . .. .. ... ..... . ... . .. . .. . . ..... .. ... . . . .. ........ . . . ... . . ...... . ... ....... .. . ..... ... . . .. .. . . . . .. .. ......... . ... . . . .. .. .. ...... . ........... .... .. . . . .. . . . . .. . .. . ...... . .. .. . .. ......... . ....... ..... ... . .. ... .. . ............... .... ..... ... . ... .. .... ... .. ... .. .. .. . . . .. . . . . .. . . . .... .. . .. . . .. . . .. . . . . . . ... . . .... . . .... .. ... .... . . . .. . .. .. .. . .. ..... .. . . . . . . . . .... .... . .. .. ... .. . .. . ... .. . .. . ....... . .. . ... .. ........ ... ... . ... . ............ ...... .... . . . . .. .. . . .. . ... . . .. . . . . . . . . .. . ... .... . ... . . ... .. . .. . . ... ... . . .. . . . ... .. . ..... . . .. .. . . . . . . . . . . ... . . . . . . .... .. .. . . . . ........ ......... . . .. . . ... ... . . . . . . . ... . . . .. . ... . .. . . .... .. ..... ..... . .. .. . . .. . . . . . ... . ... ..... ..... . ... .. . . ..... . .... ... . .... .. . . . .. . ... .. . ... .. . . . .. .. .. . . .. .. ... .. .. . . . . ..... ... ... ... ......... .... ..... .. .... . .... . .. .. . .. .. ... .... ....... . . . . ... ... . . . . . . . . . .. . . . . . . . . . . .. .. . . . . .. . . . .. . .. . .. .. . .. . .. . . . ... . . . . . . .. .......... ... ... . . . .. .. .... ........ . . . ........... ... . ..... .. . . .. ..... . .. . .. ..... . . . .. .. . ... .. . . . ... .. . .. .. ... . .. . . .... . .. .. ... . .. .. . .. . . .. . . . ... . .. . ... .. .. . .. .. .. . . . .. . .. . .. .. .. ... . . .. .. ..... . . . .... .. . .. . . . . ......... . .... .. . . . . . .. .... .. ... . .... . . . . . .. . . . . .. . . .. . . .. . .. ...... . . . . . . . . . .... .. . . ....... . . . . . . . .. . . . . . . . . . . . .. . . . . .. ... . . .. .. . . . . .. ... . .. . . . . .... .... . . . . .. . . . . . . . . . . . . . . .. . .. .. .. .. . .. . . . . . . . . .. . . . . . . . . . ... . . .. . . . . . . . . .. . .. . . . . . ... . . . . . . . . . .. . .. . . . . . .. . .. . . . . . . . . . . . . .

-0.0

k1 = 0 043

Fig. 1. So ut on scenar o of system (34)

-2.1 -1.9

2.2 0.0

372

F. Schilder and W. Vogt

As the right hand sides of (34) are polynomials in x1 , x2 , x3 , we can use a computer algebra system to generate the spectral system. With Maple 5.1 we obtain the following 9-dimensional spectral system with symbolic parameters B0 , B, k1 and k2 (The dots denote derivatives to τ = ωt).

y˙ 1 = y4 y˙ 2 = y5 + y3 y˙ 3 = y6 − y2 y˙ 4 = −0.1875 y1y8 2 − k1 y4 − 0.1875 y1y9 2 − 0.1875 y1y2 2 − 0.375 y1y7 2 − 0.375 y2y7 y8 − 0.375 y3y7 y9 − 0.1875 y1y3 2 − 0.125 y13 y˙ 5 = −0.375 y2y1 2 − 0.375 y2y7 2 − 0.75 y1y7 y8 − 0.28125 y2y8 2 + y6 − 0.09375 y2y3 2 − 0.09375 y2y9 2 − k1 y5 − 0.09375 y23 − 0.1875 y3y9 y8 y˙ 6 = −0.375 y3y7 2 − 0.375 y3y1 2 − 0.28125 y3y9 2 − 0.09375 y3y8 2 − 0.1875 y2y9 y8 + B − y5 − 0.09375 y33 − k1 y6 − 0.75 y1y7 y9 − 0.09375 y3y2 2 y˙ 7 = −0.375 k2 y9 y1 y3 − 0.1875 k2 y7 y2 2 − 0.1875 k2 y9 2 y7 − 0.1875 k2 y7 y3 2 − 0.375 k2 y7 y1 2 − 0.125 k2 y7 3 + B0 − 0.1875 k2 y8 2 y7 − 0.375 k2 y8 y1 y2 y˙ 8 = −0.375 k2 y8 y1 2 − 0.09375 k2 y8 3 − 0.75 k2 y7 y1 y2 + y9 − 0.1875 k2 y9 y3 y2 − 0.09375 k2 y8 y9 2 − 0.09375 k2 y8 y3 2 − 0.375 k2 y8 y7 2 − 0.28125 k2 y8 y2 2 y˙ 9 = −0.75 k2 y7 y1 y3 − 0.09375 k2 y9 3 − 0.09375 k2 y8 2 y9 − 0.09375 k2 y9 y2 2 − 0.375 k2 y9 y7 2 − 0.1875 k2 y8 y3 y2 − y8 − 0.375 k2 y9 y1 2 − 0.28125 k2 y9 y3 2 .

This autonomous system can now be analysed by the continuation and bifurcation code AUTO 97 of E.J.Doedel et al. [7]. Some of the results are displayed in ﬁgures 2 and 3. In ﬁgure 2 the spectral system is displayed for parameter k1 ∈ [0.025, 0.20]. Hopf bifurcations arise at k1 ≈ 0.1315 and 0.1281 (labels 2 and 3). The branch arising at label 3 for k1 ≈ 0.1281 has been followed. A cascade of period doublings (labels 7, 13, 15) and further bifurcations occur. Figure 3 displays periodic orbits of the spectral system. Obviously a sequence of period doublings arises. For k1 = 0.043 a phase portrait is given together with its Poincaré map. An interpretation of the periodic orbits of the spectral system in connection with the quasiperiodic solutions of the original system can be found in table 1.

373

Computation and Continuation of Quasiperiodic Solutions

2.20 14 2.15

13 12

8 16

2.10

17

11 15

9 7

2.05

10

4

6

5 2

2.00

1

3

1.95

1.90 0.025

0.050

0.075

0.100

0.125

0.150

0.175

0.200

Fig. 2. Bifurcation diagram of the spectral system

1.25

1.5

1.00

1.0

0.75 0.5 0.50 0.0 0.25 -0.5 0.00

13

-0.50 -1.50

-1.0

6

-0.25

-1.25

-1.00

-0.75

-0.50

-0.25

0.00

-1.5 -2.0

k1 = 0.09

1.445

1.0

0.5 16 1.087

-0.5

-1.0

-1.5 -2.0

0.730

-1.5

-1.0

-0.5

0.0

k1 = 0.05

-1.0

-0.5

0.0

0.5

1.0

k1 = 0.06

1.5

0.0

-1.5

0.5

1.0

2.26

. . . . . ... .. .⊕ . . . . . .... . . . . .. . . . . . . .. . . . .... .. .. . . .. . . . . . . . . . . . ... . . . . . . . . . ... .. . . .... . .⊕ . .... ... ⊕ . .. . . .⊕ . .. . .⊕ ... .. . . .. . .. . . . . ... .. ..⊕ ... ...... . . ..⊕ .. . . . .. .. . ..⊕... .⊕. . ..⊕ .. . . . . . ... ⊕ . . .... .... ......... ...... .. ... ......... ........................... ... .... .. ... .. . . .. . . .. ... . ..⊕ . ⊕ .. . .. ⊕ . ..⊕ . .. ... .... .....⊕ .. ⊕ . ... ... . .. ..... .. .. .. ..... .. ... . ... ... .... .... .......... .. . ..... . .. . . .. . .. . .. .... . .⊕ . .. .. ..... ....⊕⊕ . . .⊕ . . . ..... ..⊕ . . .. ....⊕.......... ....⊕.⊕ . .. . .. . . . ... . ..... . . . . . . . ⊕ . .⊕ . .. . . .⊕ . .......⊕ . ....⊕.......⊕ . .⊕ ⊕... ⊕ . ............ . ....... ... . ................. .... ............ ........ .... .. .. . . . . .. . . . .. ... . .. . . . . . . . . .. ⊕ . . .. .... ..... . .... .. . . .. . . . ... . . .. ... .. .. .. . . . .... ..⊕ ... ...... .⊕.. ⊕ . . ... ⊕ . . ... .... ....⊕ . . ⊕.. . . ... . ...⊕ ..⊕ . .... .⊕ .. ..... ..⊕⊕ .... .... ...⊕ .. .. .. .. . . .. .. .. .. . .. ... . ... .. . .. . . . . . . . ............ . .... ..................⊕ .. . .. ....⊕ . . .. ... .. .....⊕ . . . . . . . . . . . . . . . ⊕ .. . . ⊕ . . . . . . . . . . . . ⊕ .... .⊕ . . . ... .. . ... .. ........ . ....... ............. . ... . . . ... . ⊕ ... ....⊕ . . . . . . ... .. . ..... .. . .. . . .... . .⊕. . . . . . . . .... .. . .. ... .....⊕ .. .⊕.. ..... ⊕ .. .... . ......... . ........... .. ... . . ⊕ .. .... .. .... . . . . ............⊕ . . .....⊕ .. . . . .. ... .. ....... ...⊕ . .⊕ . ⊕.. . .. . .... . .. . . .. . .⊕ . . . . .. . . . ⊕ . . ⊕. .... ... ........... ..............⊕ . . . . .⊕ . .. ...⊕ .⊕ .......⊕ ⊕ .. ........ . ... . ... ........ . . ... .. . . . .. ... .. ... . . . .. .. . ⊕ .... .. ... ... . . . . . ⊕..... ... .. .⊕. ... ⊕ . . . .. . . ....... ...........⊕ ........... .. ..⊕ .⊕ . . . .... .... ... .. .... . . . .. ⊕ . . .⊕ ⊕ . . . . . . . . . .. . . ... ..⊕ . . ⊕ . . ⊕ . . . . . . . . . . . . . . . . .. . .. .. ... . . .......... ... . ....... . . ⊕ ... ... ... . . . . . .. . . .. . . . . .. ... .⊕ . . .. .. . . . .. ..... . . ⊕... . .... ⊕ . . . ... . ⊕ . .. . . .⊕. . .... ...... .. ⊕ . . .. ⊕.. .. . .. . .... .. . .. .. . .. . . . . . .⊕ ⊕ ⊕ . . . .... . ...........⊕ .................⊕ . . . . . .. . . ... ⊕ .................⊕ ...... . ........... ..... ....... ........ ...... .... .. .............. ...... ....... .. .. . . ... .. . . . .. .. . ........⊕ . . . . ... . ...... . ... ⊕ . . ...⊕ .......... ⊕ .. .. .. ... .. .. .⊕ .. ⊕ . .. ⊕ .. .⊕ .⊕ ... .. . ... .... .... ...... .................. ....... .... .. . . .... . . . .. . . . . . ....... .. . ............................⊕ ..⊕ . ...... ... ... . ........ .⊕ . . ..........⊕ .. ..⊕ . . . .. .. .⊕. . .. . ..⊕ .. ... .⊕ .... ...⊕ ..⊕ .. ..... . ... ⊕ ........ . .... .. . .. . .. . . . . . . . . . . . . . . ⊕ . . . . . . . . .. . .. . ⊕ . . . . . . . . . . . . . . . . . . . . . . . . ..... ... ... .. . .................⊕ . . . . . . . . ⊕ .⊕ . . ⊕ . ... . . .. ..⊕ . . . .. .. . ... .. . . ... ... .... . ..... . . . . . .. ⊕ . . ..... .. .....⊕.... ........ ⊕ . .⊕ .. . . . ...........⊕ . . . .... . . .... .⊕ .. . . .. ......⊕ . .... . ... .... .. .... .... .. .... .. ... . . . . . . . .. . . . . .. .. . ..⊕ ... .⊕ .. .. . ⊕ .....⊕ . ..⊕ .⊕ .. ........ .......⊕ . .... ......⊕ . ..... .... .⊕ ..... .⊕....... ....⊕ ⊕ . . . .. .. .. . ⊕ . . . . ⊕ . ... .. ....... ....⊕ . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . ⊕. . .. . . ... . ... ... ..... .⊕ . .. .... .. . .. . . ... . . . ...... .. . . . . . . .. . . . . . . ... .. . . ... .. . .. . .. .. . .. . .......⊕ . . .. . .. . ... . .. .... .. .....⊕ ... . . . . . . .... .... .. ⊕ .. ... ..⊕ .. ...⊕.. . ... . . . ....⊕ . ... . . ⊕.. ..⊕ . .. .. ⊕ . . . . .. . .. .. .. . . ... . ... .. .. .. ......⊕ . .... . . ⊕ .. . . ... .⊕...... . ⊕ . ..... . . .. . ⊕ . ........ ... .. .. .⊕.. .. . . ..... ........... . .... . .. .⊕. . .. .. . . . ... .. . ... . . . . . . . ... . .. . . .⊕ .. . ⊕ . .. ...⊕ .⊕. . . . . . .. . . . . . ⊕....... ... .... . . . . . . . .. .. ⊕ .. .. . .. . ... ⊕. .......... ... . . .. .. . . . ..⊕ ⊕ .. . .. . . . .. ... . .⊕ . . ..⊕ . . .. .. ... .... .. ... ... ........ ..... . ..⊕... ... . ⊕ ....... ⊕ . . . ... . .. . . . . .. . . . . .. .... .. . . . . . .... .. .. ... .. . . ⊕ . . . ..... .. . . .. .. . .. .. .. . . . . .. ... . ... . . .. .. . . .. . . . ⊕⊕ .. . . . .. . .. .. . .⊕ . ...... ................. .. . . .... .⊕.. . .. . . . . . .. ⊕. . . . . . .. . . . . . . . . . . . . . . . ⊕ . . ⊕ . . . . . . . . . . ⊕ . . . . . . . . . . . . . . .. . . . . . . . .. . . .. . . . . .. . ..... . . . .. . ... ... . . ... . . . .. ⊕.. ... ... .. ... ....... .. .. .... . ........ . . .... .. . .............. ..⊕ .. . . .. .. . . . . . . . . .⊕....... . . . . ... ⊕ .⊕ . . .. . ....⊕ . ... ... . .. . . .. .... ... . . .... . . . . ... . .. . .. .. . . . . . . . . . .. .. ... . . .. ⊕⊕. . . .. .. ...⊕.... .. ..... ... . .... . ... ...... . . .. ... .. ⊕ .. . . . .. . . ....⊕.. . .... ...⊕. .... .... . . .. .. . ⊕ . ⊕. . .. . .... ...... .. . . .. . . . . .. .. . . .⊕ . . ......⊕ . . . . .. . . .. ....... .. . .. .. .. . ⊕ .... ... . . . . . . .... . .. ...... .. ⊕⊕.. .... ......... ... ... ..... . . .... . . . . . . . . . . . . . . . . . . . . . ⊕ . . . . . . . . . . . . ... . . . .. . .. .... . ... . . .. . .⊕ . . . .... . . . . . ...... . . . . .. . . . . ⊕ .. . .. ... ... .. ... .⊕.. . . .. . . ..... .. ⊕ . . . . . . .. .. . . . . . . . . ..... . . . .... . .. . . . .. .. . .. ⊕ ...... .. ..... .⊕ . ⊕. . . .. .. ... .. .. ... .. . . .. .⊕ . . . . . . . . . . . . ... . . .⊕. .. . . . ..⊕ . ..⊕ . . .... .. . .. . . . . .. . . . . . . . . .. . . . . . ... . ... .⊕. .... ⊕. . . ... . . . .. ... .. . . .. ...... .. . .. . .. .. ... .. .. .. . .... . . . . . . . . . . ⊕. . . . . . . ... .⊕ . . . . . .⊕ . . . . . . . . . . . . . . .. ...⊕ . . . . . . . . . . . . .. .. . . .. . .. . .. . .. . . . . .. . . . ... .. ... . . .. . .... .⊕... .. . .. . ... . .. . ⊕.. ... .⊕ . . . .. . . . . . . . . . .. . . .. .. .. . .. . . .. .. .. . . ⊕ . . . ⊕ . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... . . . . . . . . . .... .. . ......⊕ . . ... . . . . . ... . . . . . . . . . . .. . ... . .. . . . . .... . .. ......... . .. . . . . ⊕ .. . . . . . .. . .. . . . . . . .. . . .. . .. ... ... ⊕...⊕ .. .. . . .... . ⊕ . . . . . . . . . . .. . .... . . . .. . . . . . ⊕ . . . .. . . . . . . .. . . . . . .. . . . . . . .. . . .... . .... ⊕. .. . . . . . . . . . . .. . . . . . .. . . . . ⊕. . . . . . . . . . . . . . .... . . . . . . . . . . . .. . . . . . .. . . . . . . . . . . . .⊕ . . . . ... . . .. .. .. . . . . .⊕ . ⊕. . . . ... .. . . .. . .. . . . . . . . . ⊕. . . . . . . . . .

0.01

k1 = 0.043

Fig. 3. Periodic orbits of the spectral system.

-2.38 -2.25

2.39 0.01

374

F. Schilder and W. Vogt No. of label spectral system

original system

1 3

stable equilibrium Hopf bifurcation

stable periodic solution torus bifurcation

6 7

stable periodic solution stable invariant torus period doubling torus doubeling

12 13

unstable 2-per. solution stable inv. double torus period doubling torus doubling

16 15

unstable 4-per. solution stable inv. 4-fold torus period doubling torus doubling

—

strange attractor

strange attractor

Table 1. Interpretation of bifurcation diagram in ﬁgure 2.

2. Solution via ﬁnite diﬀerences. For semidiscretisation we used the central diﬀerence formula of 4th order: ∂vi 1 ≈ (vi−2 − 8vi−1 + 8vi+1 − vi+2 ) . ∂t 12h

(35)

In this example the resulting linear diﬀerential equations (31) were solved by the same ﬁnite diﬀerence method. The simple torus was continued on a 20 times 20 (t, θ)-grid and the double-torus on a 20 times 40 grid. Using standard continuation techniques we obtained the bifurcation diagram in ﬁgure 4 with the following special points: bifurcation bifurcation point type k1∗ ∈ [0.12, 0.1225] torus bifurcation k1∗ ∈ [0.08, 0.0825] torus-ﬂip bifurcation

emerging solution type asymptotically stable 2-torus asymptotically stable doubled 2-torus

Figure 5 shows approximations of the invariant torus arising at k1∗ ∈ [0.12, 0.1225] and ﬁgure 6 shows cross-sections of the doubled invariant torus for diﬀerent parameter values.

375

Computation and Continuation of Quasiperiodic Solutions

Bifurkation diagramm of the system by Kawakami 1.76

1.75

|| x || L

2

1.74

1.73

1.72

1.71 0.04

0.06

0.08

0.1

0.12

0.14

0.16

k1

Fig. 4. Bifurcation diagram obtained by the ﬁnite diﬀerence method.

1.5

1.5

1.4

1.4

1.3

1.3

1.2

1.2

1.1

1.1

1

1 2

2

1 -2

1

0 -1

0

1

-2

-1 2

0 -1

0

1

-2

k1 = 0.1225

-1 2

-2

k1 = 0.12

1.5

1.5

1.4

1.4

1.3

1.3

1.2

1.2

1.1

1.1

1

1 2

2

1 -2

0 -1

0

1

-1 2

-2

k1 = 0.1175

1 -2

0 -1

0

1

-1 2

-2

k1 = 0.04

Fig. 5. Approximations of the emerging invariant 2-torus.

376

F. Schilder and W. Vogt

1.5

1.5

1.4

1.4

1.3

1.3

1.2

1.2

1.1

1.1

1

1 2

2

1 -2

1

0 -1

0

1

-2

-1 2

0 -1

0

1

-2

k1 = 0.0825

-1 2

-2

k1 = 0.08

1.5

1.5

1.4

1.4

1.3

1.3

1.2

1.2

1.1

1.1

1

1 2

2

1 -2

0 -1

0

1

-1 2

-2

k1 = 0.075

1 -2

0 -1

0

1

-1 2

-2

k1 = 0.04

Fig. 6. Approximations of the cross-sections γ1 and γ2 of the emerging invariant doubled 2-torus.

Computation and Continuation of Quasiperiodic Solutions

377

References 1. Bernet, K.: Ein Beitrag zur numerischen Approximation und Verfolgung von Toruslösungen parameterabhängiger dynamischer Systeme. TU Ilmenau, Shaker Verlag 1996. 2. Bernet, K.; Vogt, W.: Anwendung ﬁniter Diﬀerenzenverfahren zur direkten Bestimmung invarianter Tori. ZAMM 74 (1994), No. 6, T577-T579. 3. Chen, Y.; Leung, A.: Bifurcation and Chaos in Engineering. Springer, London 1998. 4. Dieci,L.; Lorenz,J.; Russell,R.D.: Numerical Calculation of Invariant Tori. SIAM J. Sci. Stat. Comput. 12 (1991), 607–647. 5. Dieci,L.; Lorenz,J.: Block M-Matrices and Computation of Invariant Tori. SIAM J. Sci. Stat. Comput. 13 (1992), 885–903. 6. Dieci,L.; Bader,G.: On Approximating Invariant Tori : Block Iterations and Multigrid Methods for the Associated Systems. Preprint Nr. 658, University Heidelberg, 1992. 7. Doedel, E.J.; Champneys, A.R.; Fairgrieve, Th.F.; Kuznetsov Y.A.; Sandstede, B.; Wang, X.: AUTO 97: Continuation and Bifurcation Software for Ordinary Diﬀerential Equations (with HomCont), 1997 8. Kevrekidis, I.G., Aris, R., Schmidt, L.D., Pelikan, S.: Numerical Computations of Invariant Circles of Maps. Physica 16 D (1985), 243-251. ¨ 9. Mingyou, H.; KŘpper, T.; Masbaum, N.: Computation of Invariant Tori by the Fourier Methods. SIAM J. Sci. Comput. 18 (1997), 918–942. 10. Moore, G.: Computation and Parametrisation of Invariant Curves and Tori. SIAM J. Numer. Anal., 33 (1996), 2333–2358. 11. Nedwal, F. Eine direkte Methode zur numerischen Approximation geschlossener Invarianzkurven von Poincaré–Abbildungen, Diplomarbeit TU Ilmenau, 1995. 12. Samoilenko, A.M.: Elements of the Mathematical Theory of Multi-Frequency Oscillations. Kluwer Academic Publishers, Dordrecht u.a. 1991. 13. Schilder, F.; Vogt,W.: Eine Spektralmethode zur Verfolgung und Analyse periodischer und quasiperiodischer Lösungen. Preprint No. M 27/00, TU Ilmenau 2000. 14. Van Veldhuizen,M.: A new algorithm for the numerical approximation of an invariant curve. SIAM J. Sci. Stat. Comput. 8 (1987), 951–962. 15. Van Veldhuizen,M.: Convergence Results for invariant curve algorithms. Math. Comp. 51 (1988), No. 184, 677–697. 16. Vogt, W.; Bernet, K.: A Shooting Method for Invariant Tori. Preprint No. M 3/95, TU Ilmenau 1995. 17. Yoshinaga, T. and Kawakami, H.: Bifurcations and Chaotic States in Forced Oscillatory Circuits containing Saturable Inductors. In: Carroll, T.; Pecora, L. Nonlinear Dynamics In Circuits. World Scientiﬁc Publishing, 1995.

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Equadiﬀ 10 CD ROM Papers, pp. 379–389

Non-Uniqueness of Solution to Quasi-1D Compressible Euler Equations Pavel Šolín1 and Karel Segeth2 1

Institute of Industrial Mathematics, Johannes Kepler University, Altenbergerstrasse 69, A-4040 Linz, Austria Email: [email protected] 2 Mathematical Institute of the Academy of Sciences, Žitná 25, 11567 Prague, Czech Republic Email: [email protected]

Abstract. Our study deals with the non-unique behaviour of the almostunidirectional ﬂow of compressible inviscid gases. This phenomenon is not unknown to engineers who deal with compressible ﬂows in pipes, ducts, tubes or nozzles of nontrivial geometries, however it is not easy to ﬁnd an exact mathematical analysis in the engineering literature. Our aim is to ﬁll this gap by providing a suﬃcient and exact mathematical insight into this phenomenon based on the analysis and numerical solution of the quasione-dimensional and three-dimensional compressible Euler equations. In the former case, we show the non-uniqueness of solution analytically. Further we mention some useful properties of sonic points which are a potential source of the non-uniqueness. Finally, we illustrate the presence of the nonuniqueness also in the three-dimensional model, using an axisymmetric ﬁnite volume scheme. Both analytical and numerical examples are presented.

MSC 2000. 35L65, 35L67, 76N10, 76N15, 76R10

Keywords. non-uniqueness, inviscid gas ﬂow, compressible Euler equations, quasi-one-dimensional, axisymmetric three-dimensional, ﬁnite volume method

This research was in part supported by the Grant No. 201/01/1200 of the Grant Agency of the Czech Republic.

This is the preliminary version of the paper.

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Introduction

The question of uniqueness of the exact as well as numerical solution is always very important if a nonlinear problem is solved. The theory of almost unidirectional inviscid ﬂow of perfect gases in tubes, ducts, pipes and nozzles is undoubtedly a relatively well-explored discipline (see e.g. [3,4,5,7,6,9,12,14,15]), which deserves a permanent industrial attention. Nevertheless, only a surprisingly few remarks can be found in the literature on the non-unique behavior that these ﬂows can exhibit at sonic and transonic regimes in nontrivial axisymmetric geometries. It is our aim to provide some analytical and numerical insight into this phenomenon in the present paper, which is based on the article [10]. As compared with [10], we are more brief concerning the derivation of the model, the recursive algorithm for the construction of all exact solutions to our problem and the axi-symmetric ﬁnite volume method. On the other hand, we explain in more detail the analytical example where non-uniqueness of the stationary quasi-one-dimensional compressible Euler equations is rooted because this is a source of non-unique behaviour of almost-unidirectional ﬂow of gases, which can be observed also in more dimensions.

2

Quasi-one-dimensional model

In this section we will analyze a basic model of almost unidirectional inviscid compressible ﬂow, which at the same time oﬀers a suﬃciently complex description of the ﬂow and is suﬃciently transparent to have an analytical solution. Obviously, for real-life industrial simulations, advanced quasi-one-dimensional models (such as, e.g., the Fanno model discussed in [4,5,6,9] including wall drag and turbulence eﬀects are recommended. Let us consider a bounded interval I = [xa , xb ] ⊂ (−∞, ∞), real constants 0 < Rmin ≤ Rmax and a bounded real function r(x) : I → [Rmin , Rmax ] describing the radius of an axisymmetric pipe or nozzle. For simplicity, we suppose that r(x) is once continuously diﬀerentiable in (xa , xb ) and continuous in I, but the results presented in this paper are valid also for r(x) continuous and only piecewise smooth. We deﬁne a varying cross-section a(x) = πr2 (x) for x ∈ I. We consider the standard stationary quasi-one-dimensional compressible Euler equations for perfect gases (see, e.g., [3,4,7,10,12,14,15]). Due to their hyperbolicity and the assumed smoothness of the radius r, discontinuous but piecewise smooth solutions can be expected. Along the smooth parts of the solutions, the partial diﬀerential equations can be rewritten into the following system of three non-linear algebraic equations a(x)`(x)u(x) = m, (1) 1 κp(x) + u2 (x) = h, (κ − 1)`(x) 2

(2)

Non-Uniqueness of Solution to Quasi-1D Compressible Euler Equations

p(x) =s `κ (x)

381

(3)

expressing the conservation of mass m, enthalpy h and entropy S (where S = cv lns + const., cv being the speciﬁc heat of the gas), respectively. Here `(x), u(x), p(x), e(x) mean the density, velocity, pressure and total energy density, respectively, and κ ∈ (1, 3) is a real constant. Derivation of (1) and (2) is straightforward, and a detailed derivation of (3) and further properties of the entropy S can be found, e.g., in [1,3,4,5,7,9,15]. In the quasi-one-dimensional case, all discontinuities are necessarily shocks (simultaneous discontinuities in all `, u and p) as contact discontinuities are not relevant (see, e.g., [15]). Rankine-Hugoniot conditions, which can be derived from the weak formulation of the compressible Euler equations, imply that m, h are conserved also at shocks (see, e.g., [15]). It is known that the entropy S is not conserved at shocks where it rises discontinuously obeying the Rankine-Hugoniot relation 2κM (x− )2 − κ + 1 . (4) p(x+ ) = p(x− ) κ+1 Here x ∈ I and p(x+ ), p(x− ), M (x− ) mean the downstream pressure limit and upstream pressure and Mach number limits at the shock, respectively. For the sake of completeness, let us recall the speed of sound c(x) and the Mach number M (x) deﬁned by & (5) c(x) = κp(x)/`(x), M (x) = |u(x)|/c(x), respectively. Flow is called subsonic where M (x) < 1, sonic where M (x) = 1 and isentropic where the quantity s from (3) is conserved. Lemma 1. In inviscid compressible ﬂow, shocks cannot occur in subsonic or sonic ﬂow regions. Flow leaves a shock always at subsonic regime. The entropy S and the quantity s deﬁned in (3) are discontinuous at shocks and always increasing with respect to the ﬂow direction. Both of these quantities stay conserved except for shocks. Proof. See any basic book on ﬂuid mechanics, e.g. [1]. Almost always we are interested in a subsonic inlet. These considerations lead us to a mathematically exact formulation of the problem of our interest: Problem 2. Let I, r, a be as described & above. Consider boundary data `a > 0, ua > 0, pa > 0 such that Ma = ua / κpa /`a ≤ 1, and pb such that pa ≥ pb > 0. Let us put m = a(xa )`a ua , h = κpa /((κ − 1)`a ) + u2a /2 according to (1), (2), respectively. For a ﬁnite set D ⊂ (xa , xb ) (corresponding to shocks), partitioning (xa , xb ) into a ﬁnite number of non-overlapping open intervals I1 , I2 , . . . , Id (ordered from the left to the right), we consider a sequence of constants pa /`κa = s1 < s2 < . . . < sd . The set D and real functions `, u, p deﬁned in I \ D are sought such

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that 1. 2. 3. 4. 5.

`, u, p are bounded, positive and smooth in I \ D; `, u, p satisfy (1), (2) in I \ D with the constants m, h, respectively; `, p satisfy (3) in I \ D with p/`κ = sk in Ik for all 1 ≤ k ≤ d; `, u, p satisfy (4) at all x ∈ D; `(xa ) = `a , u(xa ) = ua , p(xa ) = pa , p(xb ) = pb .

Remark 3. If Problem 2 has a solution, the value of ua is determined (except for a few degenerate situations) by `a , pa and pb as described, e.g., in [3,12]. Despite the fact that the prescription of ua seems to be an unnecessary additional restricting condition, it allows us not to deal with its computation here and improves the clarity of further considerations. In the next section, we are going to present a simple problem where the nonuniqueness of solution can be shown analytically in a constructive way.

3

Example of a problem with a non-unique solution

Lemma 4. If Problem 2 has a solution `, u, p, there is a unique pair of values `b , ub > 0 such that `(xb ) = `b , u(xb ) = ub . If pb = pa it is `b = `a , ub = ua . Proof. Let us assume that a solution to Problem 2 exists. Equations (1), (2), expressing the conservation of m, h in I \ D, yield a quadratic equation for u(xb ) = ub . It is easy to see that this equation has always one positive and one negative root. The negative one is meaningless with respect to (1). Relation (1) yields also the density `b . The rest is easy to see. Theorem 5. Let the radius r be as described in Section 2 and moreover satisfy r(xa ) = r(xb ) = r0 > 0, r(x)&> r0 for all x ∈ (xa , xb ). Let the boundary data of Problem 2 satisfy Ma = ua / κpa /`a = 1, pb = pa . Then, Problem 2 has exactly two solutions in I, both of them smooth in (xa , xb ). Proof. According to Lemma 4, a solution of Problem 2 must satisfy `(xb ) = `a . As pa /`κa = p(xb )/`κ (xb ), Lemma 1 yields that D = ∅. Thus, relation (4) is not relevant. Putting (1) and (3) into (2), we obtain κs κ−1 m2 ` −h=0 (x) + 2 κ−1 2a (x)`2 (x)

(6)

for all x ∈ I. For an x ∈ I, equation (6) can be written in the form fρ (`(x)) = 0,

(7)

with the implicit function fρ (ξ) =

κs κ−1 m2 ξ + 2 −h κ−1 2a (x)ξ 2

(8)

Non-Uniqueness of Solution to Quasi-1D Compressible Euler Equations f ρ

f ρ

A)

f ρ

B)

ρ

ρ

383

C)

ρ

Fig. 1. Function fρ ; situations A) with no real root, B) exactly one real root, C) two diﬀerent real roots. deﬁned for all ξ ∈ (0, ∞). The function fρ depicted in the Fig. 1 1 is smooth and achieves its only minimum at ξmin = m2 /(κa2 s) κ+1 . The deriva tive fρ is negative in (0, ξmin ) and positive in (ξmin , ∞). It is fρ (ξmin ) = 0 both for xa and xb . For all x ∈ I, the value of fρ (ξmin ) is a decreasing function of the cross-section a(x). Thus, the equation (7) has exactly one positive root `1 (x) = `2 (x) = `a for x = xa , exactly two positive roots 0 < `1 (x) < `2 (x) for all x ∈ (xa , xb ) and exactly one positive root `1 (x) = `2 (x) = `a for x = xb . The implicit function theorem immediately yields that the functions `1 (x), `2 (x) are smooth in (xa , xb ). Putting the solutions `1 (x), `2 (x) into the equation (1), we obtain two diﬀerent positive smooth solutions u1 (x), u2 (x) for the velocity. Finally, using (3), we obtain the corresponding solutions p1 (x), p2 (x) for pressure. 3.1

Properties of sonic points

As we have seen in the previous paragraph, sonic points (points x ∈ I such that M (x) = 1) play an important role in the existence of non-unique solutions to Problem 2. It is our aim to mention some of their further useful properties in this paragraph. Lemma 6. Let D, `, u, p solve Problem 2. Let x ∈ I \D such that M (x) = 1. Then the solution of Problem 2 at x has the unique form

1/2 (κ − 1)`(x) h − u2 (x)/2 2h(κ − 1) m , p(x) = . , `(x) = u(x) = κ+1 a(x)u(x) κ (9) Proof. Immediately from (1), (2) using (5). Lemma 7. Let D, `, u, p solve Problem 2. Let x1 , x2 ∈ I \ D, x1 < x2 , M (x1 ) = M (x2 ) = 1 and a(x1 ) = a(x2 ). Then `, u, p are continuous in [x1 , x2 ]. Proof. Using Lemma 6 with a(x1 ) = a(x2 ), we obtain `(x1 ) = `(x2 ), p(x1 ) = p(x2 ). Thus, p(x1 )/`κ (x1 ) = p(x2 )/`κ (x2 ). Lemma 1 implies the continuity of ˜ ∈ D, x1 < x ˜ < x2 . `, u, p in [x1 , x2 ]. This obviously means that there is no x

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Lemma 8. Let D, `, u, p solve Problem 2. Let x1 , x2 ∈ I \ D, x1 < x2 , M (x1 ) = M (x2 ) = 1 and a(x1 ) = a(x2 ). Then none of `, u, p can be continuous in [x1 , x2 ]. Moreover, necessarily it is a(x1 ) < a(x2 ). Proof. Lemma 6 with a(x1 ) = a(x2 ) yields that `(x1 ) = `(x2 ), p(x1 ) = p(x2 ). This and conservation of m, h in I \ D yield that p(x1 )/`κ (x1 ) = p(x2 )/`κ (x2 ). Lemma 1 implies that p(x1 )/`κ (x1 ) < p(x2 )/`κ (x2 ). Relation (9) yields that this is only possible if a(x1 ) < a(x2 ). Lemma 9. Let D, `, u, p solve Problem 2. Let x1 , x2 ∈ I \ D, x1 < x2 , and let `, u, p be continuous in [x1 , x2 ]. a) If M (x1 ) < 1 and r(x) is decreasing in [x1 , x2 ] then M (x) is increasing in [x1 , x2 ], but the relation M (x) < 1 is preserved in [x1 , x2 ). b) If M (x1 ) < 1 and r(x) is increasing in [x1 , x2 ] then M (x) is decreasing in [x1 , x2 ], and obviously M (x) < 1 in [x1 , x2 ]. c) If M (x1 ) > 1 and r(x) is decreasing in [x1 , x2 ] then M (x) is decreasing in [x1 , x2 ], but the relation M (x) > 1 is preserved in [x1 , x2 ). d) If M (x1 ) > 1 and r(x) is increasing in [x1 , x2 ] then M (x) is increasing in [x1 , x2 ], and obviously M (x) > 1 in [x1 , x2 ]. Proof. We put s1 = p(x1 )/`κ (x1 ). Let x ∈ (x1 , x2 ). For `, u, p continuous, relations (1), (2) and (3) with (5) yield u2 (x) =

2(κ − 1)h , 2/M 2 (x) + κ − 1

(10)

m , a(x)u(x)

(11)

`(x) =

s1 =

aκ−1 (x)(2(κ − 1)h)

κ+1 2

κmκ−1 M 2 (x) [2/M 2 (x) + κ − 1]

κ+1 2

.

(12)

Relation (12) can be written as κ+1 M 2 (x) 2/M 2 (x) + κ − 1 2

1 aκ−1 (x)

=

κ+1 2

(2(κ − 1)h) κmκ−1 s1

= const.

(13)

We consider (13) as an implicit function for the Mach number M . Analysis of the shape of its solution (with respect to the monotone of r supposed in a) to d)) yields the monotone behavior of M . This analysis also yields that in a) and c), the existence of an x ∈ (x1 , x2 ) such that M (x) = 1 is contradictory to (13). Corollary 10. Without loss of generality, we can assume that M (xa ) = 1 in the case that the radius r does not have a local minimum at xa (namely, using the implicit function fP from the proof of Theorem 5, it can be shown that there would be no solution to Problem 2). Thus, Lemma 9 excludes all possibilities for the existence of sonic points except for such x ∈ I where the radius r achieves a local minimum.

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385

Corollary 10 together with Lemma 6 will play an important role in the construction of multiple solutions as we will be able to evaluate solution of Problem 2 at these points in the pipe or nozzle a priori.

4

Example: a multiple nozzle

In this section we deal with non-unique solutions to Problem 2 corresponding to a nontrivial function r. We present analytical results obtained with a recursive algorithm described in [10] as well as numerical results obtained by a suitable axisymmetric ﬁnite volume scheme for three-dimensional compressible Euler equations derived in [10]. The function # + 0.0265, x ∈ [xa , 0.05], − cos(10π(x−0.05)) 50 r(x) = (14) − sin(10πx) + x/100 + 1/100, x ∈ [0.05, xb ] 250 is considered in the interval I given by xa = −0.05 m and xb = 0.75 m as shown in Fig. 2.

Fig. 2. Geometry of the multiple nozzle.

Boundary conditions are chosen as pa = 60000 Pa, θa = 368.16 K (where θa is the inlet temperature) and pb = 13000 Pa. Density `a is evaluated directly from the perfect gas state equation. For this set of boundary data, the nozzle works in the Laval regime with M (0.05) = 1 and the value of the subsonic inlet velocity is computed as ua = 5.42 m/s. The interval I is divided equidistantly into Nelem = 2000 ﬁnite volumes. In Figures 3 to 6, we show four diﬀerent analytical solutions to Problem 2. In Figures 7 to 10, steady state results of the corresponding axisymmetric computation are shown. Let us remark that we need one more boundary condition at the subsonic inlet in the axisymmetric case in comparison with the quasi-one-dimensional one. The reason is that the number of incoming characteristics in the axisymmetric case is greater. To be compatible with the quasi-one-dimensional case, we prescribe in addition to an inlet density `a and pressure pa also zero radial component of the subsonic inlet velocity ur = 0. In agreement with the quasi-one-dimensional case, pressure pb is prescribed at the subsonic outlet. At last, we compare the analytical results with the numerical ones (corresponding to axial cutlines of the axisymmetric geometry) in Figures 11 to 14. Here analytical solutions are depicted by

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solid lines and the ﬁnite volume cutlines by dashed ones. A structured triangular grid with 12 000 ﬁnite volumes was used for the axisymmetric ﬁnite volume computation. 4.1

Analytical quasi-1D solutions

4 3.5 3 2.5 2 1.5 1 0.5 0 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Fig. 3. Mach number, analytical solution with one shock. 3.5 3 2.5 2 1.5 1 0.5 0 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Fig. 4. Mach number, analytical solution with two shocks. 3 2.5 2 1.5 1 0.5 0 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Fig. 5. Mach number, analytical solution with three shocks. 2.5 2 1.5 1 0.5 0 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Fig. 6. Mach number, analytical solution with four shocks.

Non-Uniqueness of Solution to Quasi-1D Compressible Euler Equations

4.2

Axi-symmetric finite volume solutions

Fig. 7. Mach number color map, FV solution with one shock.

Fig. 8. Mach number color map, FV solution with two shocks.

Fig. 9. Mach number color map, FV solution with three shocks.

Fig. 10. Mach number color map, FV solution with four shocks.

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Comparison of quasi-1D and axi-symmetric solutions

4 3.5 3 2.5 2 1.5 1 0.5 0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Fig. 11. Mach number, analytical and FV solutions with one shock. 3.5 3 2.5 2 1.5 1 0.5 0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Fig. 12. Mach number, analytical and FV solutions with two shocks. 3 2.5 2 1.5 1 0.5 0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Fig. 13. Mach number, analytical and FV solutions with three shocks. 2.5 2 1.5 1 0.5 0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Fig. 14. Mach number, analytical and FV solutions with four shocks.

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References 1. M. Feistauer, Mathematical Methods in Fluid Dynamics (Longman Scientiﬁc & Technical, Harlow, 1993). 2. J. Felcman and P. Šolín, Construction of the Osher-Solomon scheme for 3D Euler equations, East-West J. Numer. Math. 6 (1998) 43–64. 3. C. Hirsch, Numerical Computation of Internal and External Flows, Vol. 2 (J. Wiley and Sons, 1990). 4. D. D. Knight, Inviscid Compressible Flow, in: The Handbook of Fluid Dynamics (CRC, 1998). 5. L. D. Landau, E. M. Lifschitz, Fluid Mechanics (Pergamon Press, 1959). 6. H. Ockendon, J. R. Ockendon, The Fanno model for turbulent compressible ﬂow, Preprint, OCIAM, Mathematical Institute, Oxford University, 2001. 7. H. Ockendon, A. B. Taylor, Inviscid Fluid Flow (Springer-Verlag, 1983). 8. S. Osher, F. Solomon, Upwind diﬀerence schemes for hyperbolic systems of conservation laws, Math. Comp. 38 (1982) 339–374. 9. A. H. Shapiro, The Dynamics and Thermodynamics of Compressible Fluid Flow, Vol. 1 (The Ronald Press Co., New York, 1953). 10. P. Šolín, K. Segeth, Non-uniqueness of almost unidirectional inviscid compressible ﬂow, submitted to J. Comput. Phys. . 11. J. L. Steger, R. F. Warming, Flux vector splitting of the inviscid gas dynamics equations with applications to ﬁnite diﬀerence methods, J. Comput. Phys. 40 (1981) 263–293. 12. E. Truckenbrodt, Fluidmechanik, Band 2 (Springer-Verlag, Berlin-Heidelberg-New York, 1980). 13. G. Vijayasundaram, Transonic ﬂow simulation using upstream centered scheme of Godunov type in ﬁnite elements, J. Comput. Phys. 63 (1986) 416 – 433. 14. A. J. Ward-Smith, Internal Fluid Flow, in: The Fluid Dynamics of Flow in Pipes and Ducts (Clarendon Press, Oxford, 1980). 15. P. Wesseling, Principles of Computational Fluid Dynamics (Springer-Verlag, 2000).

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Some Remarks On the Terminal Value Problem In Hereditary Setting Marko Švec Dept. of Mathematics, Faculty of Education, Komenský University, Moskovská 5, 811 01 Bratislava, Slovakia Email: [email protected]

Abstract. The terminal value problem (TVP) is investigated. Conditions for its solvability are examined.

MSC 2000. 34C10

Keywords. diﬀerential equations, terminal value problem

1

Introductory remarks

In the last 50 years the theory of diﬀerential equations with delayed argument has been widely developped, mainly for the initial value problems. The boundary value problems for such diﬀerential equations were more or less omitted. In the paper [1] from 1975 I considered two boundary value problems: For the diﬀerential equation x˙ = f (t, xt ), x ∈ Rn , t ∈ [t0 , T ),

(1)

1. given X1 ∈ Rn it is to prove the existence of the initial conditions (t0 , ϕ) such that the equation (1) has a solution existing on the interval [t0 , T ) with the property limt→T − = X1 ; 2. let X0 , X1 be two points of the space Rn . It is to prove the existence of a solution x(t) of (1) such that x(t0 ) = X0 , limt→T − = X1 . I have assumed that:

This is the ﬁnal form of the paper.

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Rn is the n-dimensional euclidian space with the norm |.|. If x(t) : [t0 − h, T ) → Rn , then x(t) = x(t + s), s ∈ [−h, 0] for t ∈ [t0 , T ). C = C([−h, 0], Rn ) denotes the Banach space of all continuous functions ϕ(t) : [−h, 0], Rn with the norm ϕ(t) = max |ϕ(t)| t∈[−h,0]

and C 0 = C 0 ([−h, 0], Rn ) denotes the Banach space of all ϕ(t) ∈ C with ϕ(0) = 0. B = B([t0 , T ), Rn ) is the Banach space of all functions u(t) continuous and bounded with the norm u(t)B = sup |u(t)|. t∈[t0 ,T )

B0 is the Banach space of all functions u(t) ∈ B such that u(t0 ) = 0 and Bϕ is the metric space of all functions z(t) such that z(t) = ϕ(t − t0 ), t ∈ [t0 − h, t0 ], z(t) = u(t) ∈ B0 for t ∈ [t0 , T ) with the distance ρ(z1 , z2 ) = u1 − u2 B , where z1 (t) = u1 (t), z2 (t) = u2 (t) for t ∈ [t0 , T) and z1 (t) = z2 (t) = ϕ(t − t0 ) for t ∈ [t0 − h, t0 ]. The following assumptions are made: H1 The function f (t, ϕ) is deﬁned and continuous on [t0 , t) × C and T |f (t, 0)|dt = K < ∞. t0

H2 There exists a function β(t) deﬁned and continuous on [t0 , T ) such that |f (t, ϕ1 ) − f (t, ϕ2 )| < β(t)ϕ1 − ϕ2 forall ϕ1 , ϕ2 ∈ C T and t ∈ [t0 , T ) and t0 β(t)dt = k < 1. It is proven: Lemma 1. Let H1 and H2 be satisﬁed. Let X1 ∈ Rn and ϕ ∈ C0 be given. Then for a given z ∈ Bϕ there exists a unique X0 ∈ Rn such that

T

X0 +

f (s, X0 + zs )ds = X1 . t0

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Some Remarks On the Terminal Value . . .

Theorem 1. Let H1 and H2 be satisﬁed and let be K < 1/2. Let X1 ∈ Rn and ϕ ∈ C 0 be given. Then there exists a unique X0∗ ∈ Rn such that the solution x(t, t0 , X ∗ + ϕ) of (1) exists on [t0 , T ) and lim x(t, t0 , X ∗ + ϕ) = X1 .

t→T −

From this theorem follows immediately Corollary 1. Let H1 and H2 with k < 1/2 be satisﬁed. Let X1 ∈ Rn be given. Then for each ϕ ∈ C 0 there exists one and only one point X0∗ (ϕ) and the solution x(t, t0 , X ∗ + ϕ) of (1) on [t0 , T ) such that lim x(t, t0 , X0∗ + ϕ) = X1 .

t→T −

It is evident that for a given T the value T0 may be chosen such that the requirement k < 1/2 from Theorem 1 will be satisﬁed. Lemma 2. Let H1 and H2 be satisﬁed. Let ϕ ∈ C 0 and X0 ∈ Rn be given. Then for every z ∈ Bϕ there exists unique X1 ∈ R1 such that T X0 + f (s, X0 + zs )ds = X1 . t0

Theorem 2. Let H1 and H2 be satisﬁed. Let be ψ ∈ C. Then the solution x(t) of (1) given through the initial conditions (t0 , ψ) exists on [t0 , T ), is unique and limt→T − x(t) = X1 ∈ Rn . Lemma 3. Let H1 and H2 be satisﬁed. Let ψ1 , ψ2 ∈ C and x(t, t0 , ψ1 ), x(t, t0 , ψ2 ) be the corresponding solutions of (1). Then t xt (t0 , ψ1 ) − xt (t0 , ψ2 ) ≤ exp[ β(s)ds ]ψ1 − ψ2 t0

for t ∈ [t0 , T ). Other problem which is to solve is the following: (P) Let be given X0 , X1 . It is to ﬁnd ϕ ∈ C 0 such that there exists the solution x(t) = x(t, t0 , X0 + ϕ) of (1) satisfying the boundary conditions x(t0 ) = X0 ,

lim x(t) = X1 .

t→T −

The following two theorems give us some information about the solvability of the theorem (P). Theorem 4. Let H1 and H2 be satisﬁed. Let X0 ∈ Rn and |X0 | + K = 0. a , where a ≥ 1. Then for the solution Let ϕ ∈ C 0 be such that ϕ < K 1−k x(t, t0 , X0 + ϕ) of (1) holds |x(t)| < [|X0 | + K]

a , t ∈ [t0 , T ). 1−k

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From Theorem 4 we get Theorem 5. Let the assumptions of Theorem 4 be satisﬁed. If X1 ∈ Rn , |X1 | > a [|X0 | + K] 1−k , then there is no solution of the problem (P) for ϕ ∈ C 0 , ϕ < a K 1−k . It seems to be evident that the problem (P) can have, in general, no solution. The question which has not been answered till today is: What is the weakest condition to add to H1 and H2 , (k < 1/2) to guarantee the solvability of the problem (P)? Remark: In my paper [1] mentioned at the beginning of this contribution the solutions are from space C 1 . The authors Marcello Ragni and Paola Rubbioni from the University of Perugia in the paper [2] have considered this problem and have enlarged the class of the solutions to the AC (Absolute Continuous Functions). So, in the Carathéorody setting they give an existence theorem and have obtained the classical results on terminal value problems in the functional and also in the nonfunctional case. I will give a short review of this paper: Let be a ∈ R+ ∪ {+∞}, I = (−∞, a), t, s, ∈ I ∪ {a}, Is = (−∞, s), Its = [t, s), t < s. Let L∞ loc (I) be the space of essentially bounded measurable functions, C(I) the subspace of continuous functions on I taking values on Rn . Let w be a subset of C(I) endowed with an inner operator T : w → w such that T x · χI0a = x · χI0a and such that for every x, Φ ∈ W with x(0) = Φ(0) we get T x · χI0 = T Φ · χI0 ; in other words, we assume that w is endowed with a map τ : w(0) → wI0 such that, put x(t), t ∈ I0a for every x ∈ w , T x(t) = τ (x(0)) (t), t ∈ I0 the element T x belongs to w. A recurring hypothesis will be: (τ ) τ : w(0) → wI0 is a continuous map. Remark: If (τ ) holds, T is continuous.

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Some Remarks On the Terminal Value . . .

Condition (C’): for every ﬁxed (y, x) ∈ Rn × w the function f (•, y, x) is measurable in the set {t ∈ I : (t, y, x) ∈ D × w}, Condition (CV): C holds and, for a. e. t ∈ I, for every sequence (yn )n in w(t) converging to y0 ∈ w(t), for every sequence (xn )n in w and x0 ∈ w with dIt (xn , x0 ) → 0 we have f (t, yn , xn ) → f (t, y0 , x0 ).

2

Statement of the problem (Terminal Value Problem)

Let be f : D × w → Rn . We have to solve the problem: Fixed y ∈ Rn , determine an impact - time p ∈ (0, a] and a function x ∈ w (absolutely continuous on I0a ) such that   x (t) = f (t, x(t), x), a. e. x ∈ I0p , limt→p− x(t) = y, (TVP)  x|I0 = τ (x(0)). The following theorem of the existence of solutions of problem (TVP) is founded on an admissibility condition on the mark y and the impact - time p with respect to the other data of the problem. Definition: we will say that a couple P¯ = (¯ p, y¯ ∈ (0, a] × Rn is admissible for problem (TVP) if (A) there exist two functions u ¯, v¯ ∈ L1 (I0p , Rn ) such that (A1 ) V¯Iop¯ ⊂ wIop¯ , v (t) ≤ f (t, x(t), T x) ≤ −u(t) for a. e. t ∈ Iop¯ and for every x ∈ V¯ ∩ w, (A2 ) −¯ where p¯ p¯ ¯ V = {x ∈ C(I) : y¯ + u ¯(s) ds ≤ x(t) ≤ y¯ + v¯(s) ds for every t ∈ I0p¯}. t

t

Theorem 6: Suppose that w ⊂ C(I) satisﬁes the property (τ ) and let f : D×w → Rn be a given function satisfying condition (CV). If P¯ ≡ (¯ p, y¯) ∈ (0, a] × Rn is an admissible couple for (TVP), then there exists a solution of (TVP). p¯ u(s)|, |¯ v (s)|} ds for every t ∈ Iop¯, and consider the Proof: Put M (t) = t max{|¯ set G = {x ∈ V¯Iop¯ : w(x, Iop¯) ≤ w(M, Iop¯)}. The set is trivially nonempty and it is a compact and convex set in C(Iop¯). Moreover, G ⊂ WIop¯ . Consider the functional F : G → C(Iop¯) deﬁned by

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F x(t) = y −

p¯

f (s, x(s), T x) ds for every t ∈ Iop¯. t

From (A) we get F (G) ⊂ G. Besides, F is continuous functional. Continuity follows from Lebesque dominated convergence theorem. So we can apply the SchauderTichonov ﬁxed point theorem. Thus there exists x ∈ w such that p¯ x(t)=¯ y − t f (s, x(s), x) ds in Iop¯, x|Io =τ (x(0)) in I0 . So x(t) is the solution we were looking for. Now it is to be answered: – how many solutions there exist; – which are the conditions to guarantee the unicity; – if there is a set of all solutions, which properties has this set?

References 1. Švec M., Some Properties of Functional Diﬀerential Equations, Bolletino U.M.I. (4) II Supl. fasc. 3, 1975, 467–477. 2. Ragni M., Rubbioni P., On Terminal Value Problem in Hereditary setting, Atti Se. Mat. Fis. Univ. Modena, XLIV, 1996, 427–442.

Equadiﬀ 10, August 27–31, 2001 Prague, Czech Republic

Equadiﬀ 10 CD ROM Papers, pp. 397–412

Regularity of Minimizers in Optimal Control Delﬁm F. Marado Torres Department of Mathematics, University of Aveiro 3810-193 Aveiro, Portugal Email: [email protected]

Abstract. We consider the Lagrange problem of optimal control with unrestricted controls – given a Lagrangian L, a dynamical equation x(t) ˙ = ϕ (t, x(t), u(t)), and boundary conditions x(a) = xa , x(b) = xb ∈ Rn , ﬁnd a control u(·) ∈ L1 ([a, b]; Rr ) such that the corresponding trajectory x(·) ∈ W1, 1 ([a, b]; Rn ) of the dynamical equation satisﬁes the boundary conditions, and the pair (x(·), u(·)) minimizes the functional J[x(·), u(·)] := b L (t, x(t), u(t)) dt. We address the question: under what conditions we a can assure optimal controls are bounded? This question is related to the one of Lipschitzian regularity of optimal trajectories, and the answer to it is crucial for closing the gap between the conditions arising in the existence theory and necessary optimality conditions. Rewriting the Lagrange problem in parametric form, we obtain a relation between the applicability conditions of the Pontryagin maximum principle to the later problem and the Lipschitzian regularity conditions for the original problem. Under the standard hypotheses of coercivity of the existence theory, the conditions imply that the optimal controls are essentially bounded, assuring the applicability of the classical necessary optimality conditions like the Pontryagin maximum principle. The result extends previous Lipschitzian regularity results to cover optimal control problems with general nonlinear dynamics. MSC 2000. 49J15, 49N60 Keywords. optimal control, Pontryagin maximum principle, boundedness of minimizers, nonlinear dynamics, Lipschitzian regularity

1

Introduction

We establish Lipschitzian regularity conditions for the minimizing trajectories of optimal control problems. Lipschitzian regularity has a number of important implications. For example in control engineering applications, where optimal strategies This is the preliminary version of the paper.

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are implemented by computer, the choice of discretization and numerical procedures depends on minimizer regularity [3,33]. Lipschitzian regularity of optimal trajectories also precludes occurrence of the undesirable Lavrentiev phenomenon [25,6,23,22] and provides the validity of known necessary optimality conditions under hypotheses of existence theory [11]. The techniques of the existence theory use compactness arguments which require to work with measurable control functions from Lp , 1 ≤ p < ∞ [5]. On the other hand, standard necessary conditions for optimality, such as the classical Pontryagin maximum principle [27], put certain restrictions on the optimal controls – namely, a priori assumption that they are essentially bounded. Examples are known, even for polynomial Lagrangians and linear dynamics [3], for which optimal controls predicted by the existence theory are unbounded and fail to satisfy the Pontryagin maximum principle [12]. If we are able to assure that a minimizer (˜ x(t), u ˜(t)), a ≤ t ≤ b, of our problem is such that u ˜(·) is essentially bounded, then the solutions can be identiﬁed via the Pontryagin maximum principle. As far as ϕ(t, x˜(t), u ˜(t)) is bounded, it also follows that the optimal trajectory x ˜(·) is Lipschitzian. Similarly, the Hamiltonian adjoint ˜ of the Pontryagin maximum principle turn out to be Lipschitzian multipliers ψ(·) either. Thus, regularity theory justiﬁes searching for minimizers among extremals and establishes a weaker form of the maximum principle in which the Hamiltonian adjoint multipliers are not required to be absolutely continuous but merely Lipschitzian. The study of Lipschitzian regularity conditions has received few attention when compared with existence theory or necessary conditions, which have been well studied since the ﬁfties and sixties. The question of Lipschitzian regularity, for the general Lagrange problem of optimal control, seems diﬃcult, and attention have been on particular dynamics. Most part of results in this direction refers to problems of the calculus of variations. For a survey see [9, Ch. 2] or [34, Ch. 11]. Less is known for the Lagrange problem of optimal control. Problems whose dynamics is linear and time invariant – ϕ(x, u) = Ax + Bu – were addressed in [14] and recently the case of control-aﬃne dynamics – ϕ(t, x, u) = f (t, x) + g(t, x) u – was studied [28]. For a survey see [29]. Results for general nonlinear dynamics, which is nonlinear both in state and control variables, are lacking. To deal with the problem we make use of an idea of time reparameterization that proved to be useful in many diﬀerent contexts – see e.g. [19, Sec. 10], [16,17], [20, Lec. 13], [5, p. 46], [2], [1], [10], [24, Ch. 5], [21, p. 29], and [32]. Examples which possess minimizers according to the existence theory and to which our results are applicable while previously known Lipschitzian regularity conditions fail are provided.

2

Formulation of Problems (P), (Pτ ) and (Pτ [w(·)])

We are interested in the study of Lipschitzian regularity conditions for the Lagrange problem of optimal control with arbitrary boundary conditions. For that is enough to consider the case when the boundary conditions are ﬁxed: x(a) = A and x(b) = B. Indeed, if x ˜(·) is a minimizing trajectory for a Lagrange problem with

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any other kind of boundary conditions, then x ˜(·) is also a minimizing trajectory for the corresponding ﬁxed boundary problem with A = x ˜(a) and B = x ˜(b). The data for our problem is then  a, b ∈ R (a < b)  A, B ∈ Rn  (1)   L : R × Rn × Rr −→ R ϕ : R × Rn × Rr −→ Rn . We assume L(·, ·, ·), ϕ(·, ·, ·) ∈ C, and ϕ(·, ·, u), ϕ(·, ·, u) ∈ C 1 . (Smoothness hypotheses on L and ϕ can be weakened, as is discussed later in connection with the Pontryagin maximum principle.) The Lagrange problem of optimal control is deﬁned as follows. Problem (P).

b

L (t, x(t), u(t)) dt −→ min

I [x(·), u(·)] = a



x(·) ∈ W1, 1 ([a, b]; Rn ) , u(·) ∈ L1 ([a, b]; Rr )  x(t) ˙ = ϕ (t, x(t), u(t)) , a.e. t ∈ [a, b] 

(2)

x(a) = A, x(b) = B.

The overdot denotes diﬀerentiation with respect to t, while the prime will be used in the sequel to denote diﬀerentiation in order to τ . To derive conditions assuring that the optimal controls u ˜(·) of problem (P ) are essentially bounded, u˜(·) ∈ L∞ , two auxiliary problems, deﬁned with the same data (1), will be used. Problem (Pτ ).

b

L (t(τ ), z(τ ), w(τ )) v(τ ) dτ −→ min

J [t(·), z(·), v(·), w(·)] = a



t(·) ∈ W1, ∞ ([a, b]; R) , z(·) ∈ W1, 1 ([a, b]; Rn )

 v(·) ∈ L ([a, b]; [0.5, 1.5]) , w(·) ∈ L ([a, b]; Rr ) ∞ 1   #  t (τ ) = v(τ )    z (τ ) = ϕ (t(τ ), z(τ ), w(τ )) v(τ )    t(a) = a, t(b) = b; z(a) = A, z(b) = B.

(3)

Remark 1. The fact that the control variable v(·) takes on its values in the set [0.5, 1.5], guarantees that t(τ ) has an inverse function τ (t).

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The following problem is the same as problem (Pτ ) except that w(·) ∈ L1 ([a, b]; Rr ) is ﬁxed and the functional is to be minimized only over t(·), z(·) (the state variables) and v(·) (the control variable). Problem (Pτ [w(·)]).

b

F (τ, t(τ ), z(τ ), v(τ )) dτ −→ min

K [t(·), z(·), v(·)] = a



t(·) ∈ W1, ∞ ([a, b]; R) , z(·) ∈ W1, 1 ([a, b]; Rn )  v(·) ∈ L∞ ([a, b]; [0.5, 1.5])   #  t (τ ) = v(τ )    z (τ ) = f (τ, t(τ ), z(τ ), v(τ ))    t(a) = a, t(b) = b; z(a) = A, z(b) = B. where F (τ, t, z, v) = L (t, z, w(τ )) v, f (τ, t, z, v) = ϕ (t, z, w(τ )) v. Remark 2. Problem (Pτ ) is autonomous while (P ) and (Pτ [w(·)]) are not. The relation between problem (P ) and problem (Pτ [w(·)]) is discussed in the following two sections.

3

Relation Between the Solutions of the Problems

Let us begin to determine the relation between admissible pairs for problem (P ) and admissible quadruples for problem (Pτ ). Definition 3. The pair (x(·), u(·)) is said to be admissible for (P ) if all conditions in (2) are satisﬁed. Similarly, (t(·), z(·), v(·), w(·)) is said to be admissible for (Pτ ) if all conditions in (3) are satisﬁed. Lemma 4. Let (x(·), u(·)) be admissible for (P ). Then, for any function v(·) satisfying v(·) ∈ L∞ ([a, b]; [0.5, 1.5]) , b v(s) ds = b − a, t(τ ) = a +

τ a

(4) (5)

a

v(s) ds, z(τ ) = x (t(τ )) and w(τ ) = u (t(τ )), are such that (t(·), z(·), v(·), w(·))

is admissible for (Pτ ). Moreover, J [t(·), z(·), v(·), w(·)] = I [x(·), u(·)] .

(6)

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Proof. All conditions in (3) become satisﬁed: – Function t(·) is Lipschitzian: dt(·) dτ = v(·) ∈ L∞ ([a, b]; [0.5, 1.5]); – Function z(·) is absolutely continuous since it is a composition of the absolutely continuous function x(·) with the strictly monotonous Lipschitzian continuous function t(·): dt(τ ) = v(τ ) > 0; (7) dτ – Function w(·) is Lebesgue measurable, w(·) ∈ L1 , because u(·) is measurable and t(·) is a strictly monotonous absolutely continuous function; – Diﬀerentiating z(·) we obtain: z (τ ) =

dx (t(τ )) dt(τ ) dz(τ ) = . dτ dt dτ

In view of (2) and (7), one concludes from this last equality that z (τ ) = ϕ (t(τ ), x(t(τ )), u(t(τ ))) v(τ ) = ϕ (t(τ ), z(τ ), w(τ )) v(τ ); – From (5) and from the deﬁnition of t(τ ) we have t(a) = a and t(b) = b. It follows that z(a) = x (t(a)) = x(a) = A; z(b) = x (t(b)) = x(b) = B. It remains to prove equality (6). Since b L (t(τ ), z(τ ), w(τ )) v(τ ) dτ J [t(·), z(·), v(·), w(·)] = a

=

b

L (t(τ ), x(t(τ )), u(t(τ ))) v(τ ) dτ ,

(8)

a

from the change of variable t(τ ) = t,  dt = v(τ ) dτ  τ  =a⇔t=a τ = b ⇔ t = b,

(9)

it follows from (8) the pretended conclusion: b J [t(·), z(·), v(·), w(·)] = L (t, x(t), u(t)) dt = I [x(·), u(·)] . a

Lemma 5. Let (t(·), z(·), v(·), w(·)) be admissible for (Pτ ). Then the pair (x(·), u(·)) = (z (τ (·)) , w (τ (·))) , where τ (·) is the inverse function of t(·), is admissible for (P ). Moreover I [x(·), u(·)] = J [t(·), z(·), v(·), w(·)] .

(10)

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Proof. All conditions in (2) become satisﬁed: – Since deﬁned by the composition of the absolutely continuous function z(·) with the strictly monotonous absolutely continuous function τ (·), 1 dτ (t) = > 0, dt v (τ (t))

(11)

function x(·) is absolutely continuous; – We have that u(·) = w (τ (·)) ∈ L1 ([a, b]; Rr ), as it results from the composition of w(·) ∈ L1 ([a, b]; Rr ) with the continuous function τ : [a, b] −→ [a, b]; – Diﬀerentiating x(·) we obtain: dz (τ (t)) dτ (t) dx(t) = . dt dτ dt

x(t) ˙ =

From (3) and (11), one concludes from the last equality that x(t) ˙ =

ϕ (t(τ (t)), z(τ (t)), w(τ (t))) v (τ (t)) = ϕ (t, x(t), u(t)) ; v(τ (t))

– As far as τ (a) = a and τ (b) = b, it comes x(a) = z (τ (a)) = z(a) = A; x(b) = z (τ (b)) = z(b) = B. To ﬁnish we prove equality (10). By deﬁnition

b

I [x(·), u(·)] =

L (t, x(t), u(t)) dt a

=

(12)

b

L (t(τ (t)), z(τ (t)), w(τ (t))) dt. a

Doing the change of variable τ (t) = τ , it follows from (9) and (12) the pretended conclusion: b I [x(·), u(·)] = L (t(τ ), z(τ ), w(τ )) v(τ ) dτ = J [t(·), z(·), v(·), w(·)] . a

From Lemmas 4 and 5, the following two corollaries are obvious. Corollary 6. If (˜ x(·), u˜(·)) is a minimizer of problem (P ), then, for any function v˜(·) satisfying (4) and (5) (e.g. v˜(τ ) ≡ 1), the 4-tuple t˜(·), z˜(·), v˜(·), w(·) ˜ , τ deﬁned by t˜(τ ) = a + a v˜(s) ds, z˜(τ ) = x ˜ t˜(τ ) , w(τ ˜ )=u ˜ t˜(τ ) , is a minimizer to problem (Pτ ).

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Corollary 7. If t˜(·), z˜(·), v˜(·), ˜ is a minimizer of problem (Pτ ), then the w(·) pair (˜ x(·), u ˜(·)) deﬁned from t˜(·), z˜(·), v˜(·), w(·) ˜ as in Lemma 5 is a minimizer to problem (P ). Thus, let (˜ x(·), u ˜(·)) be a minimizer of problem (P ). From Corollary 6 we know how to construct a minimizer t˜(·), z˜(·), v˜(·), w(·) ˜ to problem (Pτ ). Obviously, as far as problem (P [ w(·)]) ˜ is the same as problem (P ˜ is ﬁxed, τ τ ) except that w(·) t˜(·), z˜(·), v˜(·) furnishes a minimizer to problem (Pτ [w(·)]). ˜ Choosing v˜(τ ) ≡ 1 we obtain. x(·), u ˜(·)) is a minimizer of problem (P ), then the triple Proposition 8. If (˜ t˜(τ ), z˜(τ ), v˜(τ ) = (τ, x ˜(τ ), 1) furnishes a minimizer to problem (Pτ [˜ u(·)]). It is also important to know how the extremals of the problems are related. This will be addressed in the next section.

4

Relation Between the Extremals of the Problems

At the core of optimal control theory is the celebrated Pontryagin maximum principle. The maximum principle is a ﬁrst order necessary optimality condition for the optimal control problems. It ﬁrst appear in the book [27]. Since then, several versions have been obtained by weakening the hypotheses. For example, in [27] it is assumed that functions L(·, ·, ·) and ϕ(·, ·, ·) are continuous, and have continuous derivatives with respect to the state variables x: L(t, ·, u), ϕ(t, ·, u) ∈ C 1 . Instead of the continuity assumption of L(·, ·, ·) and ϕ(·, ·, ·), a version only requiring that functions L(·, x, ·) and ϕ(·, x, ·) are Borel measurable can be found in book [4, Ch. 5]. There, in order to assure the applicability of the maximum principle, the following assumption is imposed: there exists an integrable function α(·) deﬁned on [a, b] such that the bound % % % ∂L % % % (13) % ∂x (t, x, u(t))% ≤ α(t) % % % ∂ϕi % % % (14) % ∂x (t, x, u(t))% ≤ α(t) (i = 1, . . . , n) holds for all (t, x) ∈ [a, b] × Rn . The existence and integrability of α(·), and the bound (13)–(14), are guaranteed under the hypotheses that L and ∂ϕ ϕ possess derivatives ∂L ∂x and ∂x which are continuous in (t, x, u), and u(·) is essentially bounded (these are the hypotheses found in [27]). Alternative hypotheses are the following growth conditions (see [8, Sec. 4.4 and p. 212]): % % % % % ∂L % % % % % ≤ c |L| + k , % ∂ϕi % ≤ c |ϕi | + k , (15) % ∂x % % ∂x % with constants c and k, c > 0. Those who are familiar with the Lipschitzian regularity conditions for the basic problem of the calculus of variations, will recognize (15) as a generalization of the classical Tonelli–Morrey Lipschitzian regularity condition (cf. e.g. [29]). From this fact, one can guess a link between the

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applicability conditions of the maximum principle and the Lipschitzian regularity conditions. The link between the applicability conditions of the classical Pontryagin maximum principle [27] and the Lipschitzian regularity conditions for optimal control problems with control-aﬃne dynamics, was established in [28]. Here, to deal with general nonlinear dynamics, we will need to apply the maximum principle under weaker hypotheses than those in [27]. This is due to the fact that when we ﬁx w(·) ∈ L1 ([a, b]; Rr ), functions F (τ, t, z, v) and f (τ, t, z, v) of problem (Pτ [w(·)]) are not continuous in τ but only measurable. Hypotheses (15) are suitable, as far as they can be directly veriﬁable for a given problem. Weaker hypotheses than (13) and (14) can also be considered. In this respect, important improvements are obtained from the use of nonsmooth analysis. For example, one can substitute (13) and (14) by the weaker conditions |L (t, x1 , u(t)) − L (t, x2 , u(t))| ≤ α(t) x1 − x2 |ϕi (t, x1 , u(t)) − ϕi (t, x2 , u(t))| ≤ α(t) x1 − x2 and formulate the maximum principle in a nonsmooth setting, in terms of generalized gradients (see [7,8]). Proving general versions of the maximum principle under weak hypotheses is still in progress and the interested reader is referred to the recent paper [30]. Definition 9. Let (x(·), u(·)) be admissible for (P ). We say that the quadruple n (x(·), u(·), ψ0 , ψ(·)), ψ0 ∈ R− 0 and ψ(·) ∈ W1, 1 ([a, b]; R ), is an extremal of (P ), if the following two conditions are satisﬁed for almost all t ∈ [a, b]: the adjoint system ∂H ˙ ψ(t) =− (t, x(t), u(t), ψ0 , ψ(t)) ; ∂x

(16)

the maximality condition H (t, x(t), u(t), ψ0 , ψ(t)) = sup H (t, x(t), u, ψ0 , ψ(t)) ;

(17)

u∈Rr

where the Hamiltonian equals H (t, x, u, ψ0 , ψ) = ψ0 L(t, x, u) + ψ · ϕ(t, x, u). Definition 10. Let (t(·), z(·), v(·), w(·)) be admissible for (Pτ ). The 7-tuple (t(·), z(·), v(·), w(·), p0 , pt (·), pz (·)) , n p 0 ∈ R− 0 , pt (·) ∈ W1, ∞ ([a, b]; R) and pz (·) ∈ W1, 1 ([a, b]; R ), is said to be an extremal of (Pτ ), if the following two conditions are satisﬁed for almost all τ ∈ [a, b]:

Regularity of Minimizers in Optimal Control

the adjoint system  ∂H   (t(τ ), z(τ ), v(τ ), w(τ ), p0 , pt (τ ), pz (τ )) , pt (τ ) = − ∂t  p (τ ) = − ∂H (t(τ ), z(τ ), v(τ ), w(τ ), p , p (τ ), p (τ )) ;  0 t z z ∂z

405

(18)

the maximality condition H (t(τ ), z(τ ), v(τ ), w(τ ), p0 , pt (τ ), pz (τ )) =

sup

H (t(τ ), z(τ ), v, w, p0 , pt (τ ), pz (τ )) ; (19)

v∈[0.5, 1.5] w∈Rr

where the Hamiltonian equals H (t, z, v, w, p0 , pt , pz ) = (p0 L (t, z, w) + pt + pz · ϕ (t, z, w)) v. Remark 11. Functions H and H, respectively the Hamiltonians in Deﬁnitions 9 and 10, are related by the following equality: H (t, z, v, w, p0 , pt , pz ) = (H (t, z, w, p0 , pz ) + pt ) v .

(20)

From it one concludes that ∂H ∂H = v, ∂t ∂t ∂H ∂H = v. ∂z ∂x

(21) (22)

Definition 12. An extremal is called normal if the cost multiplier (ψ0 in the Deﬁnition 9 and p0 in the Deﬁnition 10) is diﬀerent from zero and abnormal if it vanishes. Remark 13. As far as the Hamiltonian is homogeneous with respect to the Hamiltonian multipliers, for normal extremals one can always consider, by scaling, that the cost multiplier takes value −1. Remark 14. The (Pontryagin) maximum principle give conditions, as those discussed in the introduction of this section, under which to each minimizer of the problem there corresponds an extremal with Hamiltonian multipliers not vanishing simultaneously ((ψ0 , ψ) = 0 in the Deﬁnition 9 and (p0 , p) = 0 in the Deﬁnition 10). One can expect the set of extremals of problem (Pτ ) to be richer than the set of extremals of problem (P ). Nevertheless, there is a relationship between the extremals of the problems. Next lemma shows that to each extremal of problem (P ) there corresponds extremals of problem (Pτ ) lying on the zero level of the maximized Hamiltonian H.

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Lemma 15. Let (x(·), u(·), ψ0 , ψ(·)) be an extremal of (P ). Then, for any funcb tion v(·) ∈ L∞ ([a, b]; [0.5, 1.5]) satisfying a v(s) ds = b − a, the 7-tuple (t(·), z(·), v(·), w(·), p0 , pt (·), pz (·)) deﬁned by

τ

t(τ ) = a +

v(s) ds , a

z(τ ) = x(t(τ )) ,

w(τ ) = u(t(τ )) ,

p0 = ψ0 , pz (τ ) = ψ(t(τ )) , pt (τ ) = −H (t(τ ), x(t(τ )), u(t(τ )), ψ0 , ψ(t(τ ))) is an extremal of (Pτ ) with H (t(τ ), z(τ ), v(τ ), w(τ ), p0 , pt (τ ), pz (τ )) ≡ 0. Proof. From Lemma 4 we know that such 7-tuple is admissible for (Pτ ). The maximality condition (19) is trivially satisﬁed since we are in the singular case: from (20) the Hamiltonian H vanishes for pt = −H (t, z, w, p0 , pz ). It remains to ∂H prove the adjoint system (18). Since dH dt = ∂t along the extremals (see e.g. [27] or [4]) the derivative of pt (τ ) with respect to τ is given by dpt dH dH dt ∂H dt ∂H =− =− =− =− v. dτ dτ dt dτ ∂t dτ ∂t From relation (21) the ﬁrst of the equalities (18) is proved: pt = − ∂H ∂t . Similarly, d ψ(t) = − ∂H , it follows from (22) that as far as pz (τ ) = ψ (t(τ )) and from (16) dt ∂x dψ(t) dt ∂H ∂H pz = dt dτ = − ∂x v = − ∂z . It is also possible to construct an extremal of problem (P ) given an extremal of (Pτ ). Lemma 16. Let (t(·), z(·), v(·), w(·), p0 , pt (·), pz (·)) be an extremal of (Pτ ). Then (x(·), u(·), ψ0 , ψ(·)) = (z (τ (·)) , w (τ (·)) , p0 , pz (τ (·))) is an extremal of (P ) with τ (·) the inverse function of t(·). Proof. From Lemma 5 we know that the pair (x(·), u(·)) is admissible for (Pτ ). Direct calculations show that ∂H 1 dpz (τ ) dτ d =− . ψ˙ = pz (τ ) = dt dτ dt ∂z v From (22) the required adjoint system is obtained: ψ˙ = − ∂H ∂x . Maximality condition (19) implies that H (t(τ ), z(τ ), v(τ ), w(τ ), p0 , pt (τ ), pz (τ )) = sup H (t(τ ), z(τ ), v(τ ), w, p0 , pt (τ ), pz (τ )) w∈Rr

407

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for almost all τ ∈ [a, b]. Given the relation (20) one can write that H (t(τ ), z(τ ), w(τ ), p0 , pz (τ )) = sup H (t(τ ), z(τ ), w, p0 , pz (τ )) . w∈Rr

Putting τ = τ (t) we obtain the maximality condition (17). Lemmas 15 and 16 establish a correspondence between abnormal extremals of problems (P ) and (Pτ ). Corollary 17. If there are no abnormal extremals of problem (P ) then there are no abnormal extremals of problem (Pτ ). If there are no abnormal extremals of (Pτ ) then there are also no abnormal extremals of (P ). Definition 18. We call a control an abnormal extremal control if it corresponds to an abnormal extremal. Proposition 19. If (˜ x(·), u ˜(·)) is a minimizer of problem (P ) and u˜(·) is not an abnormal extremal control, then the minimizing control v˜ ≡ 1 of Proposition 8 is not an abnormal extremal control too. In the next section we show that the Lipschitzian regularity conditions we are looking for, assuring that all minimizing controls predicted by Tonelli’s existence theorem are indeed bounded, appear from the applicability conditions of the maxu(·)]). imum principle to problem (Pτ [˜

5

The General Regularity Result

Filippov [18] gave the ﬁrst general existence theorem for optimal control (the original paper, in russian, appear in 1959). There exist now an extensive literature on the existence of solutions to problems of optimal control. We refer the interested reader to the book [5] for signiﬁcant results, various formulations, and detailed discussions. Follows a set of conditions, of the type of Tonelli [31], that guarantee existence of minimizer for problem (P ). “Tonelli” existence theorem for (P). Problem (P ) has a minimizer (˜ x(·), u˜(·)) with u˜(·) ∈ L1 ([a, b]; Rr ), provided there exists at least one admissible pair, functions L(·, ·, ·) and ϕ(·, ·, ·) are continuous, and the following convexity and coercivity conditions hold: (convexity) Functions L(t, x, ·) and ϕ(t, x, ·) are convex for all (t, x); (coercivity) There exists a function θ : R+ 0 → R, bounded below, such that L(t, x, u) ≥ θ (ϕ(t, x, u))

θ(r) = +∞; r ϕ(t, x, u) = +∞ for all (t, x). lim

r→+∞

lim

+u+→+∞

for all (t, x, u);

(23) (24) (25)

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Remark 20. For the basic problem of the calculus of variations one has ϕ = u and the theorem above coincides with the classical Tonelli existence theorem. Analyzing the hypotheses of both necessary optimality conditions and existence theorem, one comes to the conclusion that the requirements of existence theory do not imply those of the maximum principle. Given a problem, it may happen that the necessary optimality conditions are valid while existence is not guarantee; or it may happen that the minimizers predicted by the existence theory fail to be extremals. Follows the main results of the paper. Theorem 21. Under the above hypothesis of coercivity, all control minimizers u ˜(·) of (P ) which are not abnormal extremal controls are essentially bounded on u(·)]) is [a, b] provided the applicability of the maximum principle to problem (Pτ [˜ assured. Remark 22. Convexity is not required in the regularity theorem. This is important since existence theorems without the convexity assumptions are a question of great interest (see e.g. [26] and the references therein). Applying the hypotheses (15) of the maximum principle to functions F and f of problem (Pτ [˜ u(·)]), the following result is trivially obtained. Theorem 23. Under the hypothesis of coercivity, the growth conditions: there exist constants c > 0 and k such that % % % ∂L % ≤ c |L| + k , % ∂L % ≤ c |L| + k , % ∂x % ∂t % % % % % ∂ϕ % % % % % ≤ c ϕ + k , % ∂ϕi % ≤ c |ϕi | + k (i = 1, . . . , n) ; % ∂t % % ∂x % imply that all minimizers u˜(·) of (P ), which are not abnormal extremal controls, are essentially bounded on [a, b]. A minimizer u˜(·) which is not essentially bounded may fail to satisfy the Pontryagin Maximum Principle. As far as essentially bounded minimizers are concerned, the Pontryagin Maximum Principle is valid. Corollary 24. Under the hypotheses of Theorem 23, all minimizers of (P ) are Pontryagin extremals. Proof. (Theorem 21) Let (˜ x(·), u˜(·)) be a minimizer of (P ). From Propositions 8 and 19 and by the assumptions of the theorem, we know that there exist absolutely continuous functions p˜t (·) and p˜z (·) such that for almost all points τ ∈ [a, b] v -−→ [−L (τ, x ˜(τ ), u ˜(τ )) + p˜t (τ ) + p˜z (τ ) · ϕ (τ, x˜(τ ), u ˜(τ ))] v is maximized at v = 1 on the interval [0.5, 1.5]. This implies that L (τ, x ˜(τ ), u˜(τ )) = p˜t (τ ) + p˜z (τ ) · ϕ (τ, x ˜(τ ), u ˜(τ )) .

(26)

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Let |p˜t (τ )| ≤ M and p˜z (τ ) ≤ M on [a, b]. Dividing both sides of inequality (26) by ϕ (τ, x ˜(τ ), u ˜(τ )) and using the coercivity hypothesis (23), one obtains θ (ϕ (τ, x ˜(τ ), u˜(τ ))) 1 + ϕ (τ, x ˜(τ ), u ˜(τ )) ≤M . ϕ (τ, x˜(τ ), u ˜(τ )) ϕ (τ, x ˜(τ ), u ˜(τ )) The coercivity condition (24)–(25) yields the essential boundedness of u ˜(·) on [a, b].

6

An Example

As far as Theorem 23 is able to cover optimal control problems with dynamics which is nonlinear both in the state and in the control variables, plenty of examples possessing minimizers according to the existence theory can be easily constructed for which our result is applicable while previously known Lipschitzian regularity conditions, such as those in [14] and [28], fail. Follows one such example with n = r = 2. Example 25.

0

1

2 u1 (t) + u22 (t) e2 (x1 (t)+x2 (t)) + 1 dt −→ min  (  x˙ (t) = u2 (t) + u2 (t) 1 1 2  x1 (t)+x2 (t) x˙2 (t) = u2 (t) e x1 (0) = 0, x1 (1) = 1, x2 (0) = 1, x2 (1) = 1.

Here we have: L(x1 , x2 , u1 , u2 ) = u21 + u22 e2 (x1 +x2 ) + 1 ; & ϕ u21 + u22 ϕ(x1 , x2 , u1 , u2 ) = 1 = . ϕ2 u2 ex1 +x2 The problem has a solution for x1 (·) , x2 (·) ∈ W1,1 ([0, 1]; R) and u1 (·), u2 (·) ∈ L1 ([0, 1]; R), as far as all conditions of Tonelli’s existence theorem are satisﬁed: – An admissible quadruple is (x1 (t), x2 (t), u1 (t), u2 (t)) = (t, 1, 1, 0). – Functions L(·, ·, ·, ·) and ϕ(·, ·, ·, ·) are continuous in R4 . – Function L(x1 , x2 , ·, ·) is strictly convex as far as the matrix 2(x +x ) ∂2L 2 e 1 2 +1 0 = 0 2 e2(x1 +x2 ) + 1 ∂u∂u is positive-deﬁnite for all (x1 , x2 ) ∈ R2 . The matrices   u22 u1 u2 − 3 ∂ 2 ϕ2 ∂ 2 ϕ1 00 (u21 +u22 )3   (u21 +u22 ) = = ; 2  u 00 ∂u∂u ∂u∂u − u21 u2 2 3 2 1 2 3 (u1 +u2 ) (u1 +u2 ) are non-negative and it follows that ϕ(x1 , x2 , ·, ·) is convex.

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– From the inequality L = u21 + u22 e2 (x1 +x2 ) + 1 ≥ u21 + u22 + u22 e2 (x1 +x2 ) , we have quadratic coercivity (θ(r) = r2 ); Smooth assumptions on data (1) are satisﬁed, since L(·, ·, ·, ·) and ϕ(·, ·, u1 , u2 ) are of class C ∞ . Theorem 23 allow us to conclude that all minimizing controls, which are not abnormal extremal controls, are bounded: ∂ϕ – The conditions on ∂L ∂t and ∂t are trivially satisﬁed as far as the problem is autonomous: L and ϕ do not depend explicitly on the time variable. ∂ϕ – The growth conditions on ∂L ∂x and ∂x are also satisﬁed:

∂L ∂L = = 2e2(x1 +x2 ) u21 + u22 ≤ 2L ; ∂x1 ∂x2 ∂ϕ1 ∂ϕ1 = = 0; ∂x1 ∂x2 ∂ϕ2 ∂ϕ2 = = ϕ2 . ∂x1 ∂x2

7

Final Remarks

In this paper we study properties of minimizing trajectories for general problems of optimal control in the cases where controls are unconstrained (like in the calculus of variations). We provide new conditions which guarantee Lipschitzian regularity of the minimizing trajectories for the Lagrange problem of optimal control in the general nonlinear case. These conditions solve the discrepancy between the optimality and existence results, assuring that minimizers predicted by the existence theory satisfy the optimality conditions. At the same time, undesirable phenomena, like the Lavrentiev one, are naturally precluded. We show that the conditions of Lipschitzian regularity are related with the applicability conditions of Pontryagin’s maximum principle. To deal with dynamics which are control-aﬃne, the classical Pontryagin maximum principle [27] is enough (see [28]). To treat the general case, a maximum principle under weak assumptions, like the one in [4], is necessary. Our approach is based on the relationship of the extremals of the Lagrange problem with the extremals of an auxiliary problem, and on the subsequent utilization of Pontryagin’s maximum principle to the later problem. The maximality condition of Pontryagin’s maximum principle together with the coercivity assumption of the existence theorem imply the Lipschitzian regularity of the corresponding minimizer of the original problem. This approach allows us to deal with more general class of problems of optimal control with nonlinear dynamics. It remains to clarify the interconnection between Lipschitzian regularity and abnormal extremality. For the problems of the calculus of variations studied in [13] and [15] no abnormal extremals exist. For the optimal control problems considered

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in [14] and [28], abnormal extremals are, like here, put aside. The question of how to establish Lipschitzian regularity for the abnormal minimizing trajectories seems to be a completely open question.

Acknowledgements This work is part of the author’s Ph.D. project which is carried out at the University of Aveiro, Portugal, under scientiﬁc supervision of A. V. Sarychev. The author is grateful to him for the inspiring conversations and for the many helpful comments on the present topic.

References 1. A. A. Agrach¨ev and R. V. Gamkrelidze, Symplectic geometry for optimal control, Nonlinear controllability and optimal control, Dekker, New York, 1990, pp. 263–277. MR 91f:49032 2. L. Ambrosio, O. Ascenzi, and G. Buttazzo, Lipschitz regularity for minimizers of integral functionals with highly discontinuous integrands, J. Math. Anal. Appl. 142 (1989), no. 2, 301–316. MR 91c:49060 3. J. M. Ball and V. J. Mizel, One-dimensional variational problems whose minimizers do not satisfy the Euler Lagrange equation, Arch. Rational Mech. Anal. 90 (1985), no. 4, 325–388. MR 86k:49002 4. L. D. Berkovitz, Optimal control theory, Springer-Verlag, New York, 1974. MR 51:8914 5. L. Cesari, Optimization—theory and applications, Springer-Verlag, New York, 1983. MR 85c:49001 6. C.-W. Cheng and V. J. Mizel, On the Lavrentiev phenomenon for optimal control problems with second-order dynamics, SIAM J. Control Optim. 34 (1996), no. 6, 2172–2179. MR 97g:49014 7. F. H. Clarke, The maximum principle under minimal hypotheses, SIAM J. Control Optimization 14 (1976), no. 6, 1078–1091. MR 54:3540 , Optimization and nonsmooth analysis, John Wiley & Sons Inc., New York, 8. 1983. MR 85m:49002 , Methods of dynamic and nonsmooth optimization, Society for Industrial and 9. Applied Mathematics (SIAM), Philadelphia, PA, 1989. MR 91j:49001 , An indirect method in the calculus of variations, Trans. Amer. Math. Soc. 10. 336 (1993), no. 2, 655–673. MR 93f:49002 , The calculus of variations, nonsmooth analysis and optimal control, 11. Development of mathematics 1950–2000, Birkhäuser, Basel, 2000, pp. 313–328. MR 1 796 845 12. F. H. Clarke and R. B. Vinter, On the conditions under which the Euler equation or the maximum principle hold, Appl. Math. Optim. 12 (1984), no. 1, 73–79. MR 85m:49051 , Regularity properties of solutions to the basic problem in the calculus of 13. variations, Trans. Amer. Math. Soc. 289 (1985), no. 1, 73–98. MR 86h:49020 , Regularity properties of optimal controls, SIAM J. Control Optim. 28 (1990), 14. no. 4, 980–997. MR 91b:49033

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Equadiﬀ 10, August 27–31, 2001 Prague, Czech Republic

Equadiﬀ 10 CD ROM Papers, pp. 413–435

Two-scale convergence with respect to measures in continuum mechanics Jiří Vala

Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, Technical University, Žižkova 17, 662 37 Brno, Czech Republic, Email: [email protected]

Abstract. The homogenization of variational formulations of nonlinear systems of partial diﬀerential equations with periodically oscillating coeﬃcients, needed in many problems of continuum mechanics with nonnegligible microstructural material properties, is studied, using the technique of the two-scale convergence with respect to Radon measures. (Unlike the classical approach, such technique can handle e. g. “domains with holes”, applied in problems of ﬂow of a liquid through a porous medium, without artiﬁcial geometrical assumptions.) The overview of basic lemmas (including corresponding proofs) is presented. The existence and convergence analysis for the variational formulation of a model elliptic problem demonstrates how the notion of the two-scale convergence is able to explain and simplify the complicated form of the macroscopic limit equation, thanks to the addition of a new microscopic hidden variable. MSC 2000. 35J60, 74Q15, 74M25 Keywords. Two-scale convergence, microstructural material properties, elliptic systems of PDEs

1

Remarks to the history of homogenization techniques

In most problems of continuum mechanics at least two length scales can be distinguished – a macroscopic one (usually in meters) and a microscopic one (typically in micrometers), which brings complications to all numerical calculations and simulations. Typically simple algorithms, based on the classical results from textbooks

Research supported by the Ministry of Education, Youth and Sports of the Czech Republic, Reg. No. CEZ : J22 : 261100007.

This is the ﬁnal form of the paper.

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of functional analysis and numerical methods, then do not reﬂect physical reality at a satisfactory level; in such sense [6] emphasizes that it is very important to distinguish between the veriﬁcation (whether the ideal mathematical problem is well-deﬁned, solvable, etc.) and the validation (whether and how some results similar to the computed ones can be observed both in the laboratories and in the nature) in all engineering applications. In this section we shall mention some practical approaches, their advantages and drawbacks, following their rough classiﬁcation from [39]. Most commercial software packages and technical standards prefer cheap computations and simple theoretical considerations; only in case when the results are expected to be too far from the realistic ones, some naive “averaging” procedure is applied to material characteristics. Unfortunately, there are many examples of strange and absurd numerical outputs, namely for physical processes in materials consisting of several phases with strongly diﬀerent mechanical quantities. Some of these diﬃculties can be overcome in a relatively simple way, demonstrated for various types of composites e. g. in [40], [35] and [37]: thanks to the assumed periodicity both of a material structure and of external loads, the processes of elastic deformation and high-temperature creep, based on the viscoelastic Maxwell model with one linear elastic component (from the well-known Hooke law) and one non-linear viscous component (e. g. in the power-law form by Norton), coupled with the diﬀusion and sliding along all phase interfaces, can be analyzed at a microscopic level directly to simulate the eﬀect of thin strengthening (nearly elastic) ceramic plates or ﬁbers, located in a matrix with low creep resistance at elevated temperature and aligned in the proposed direction of uniform tension. Nevertheless, the limits of this access are evident: if some loads are more complicated then the simple extension of results from a micro- to a macroscopic level is not available or requires non-realistic simpliﬁcations; attempts to cover both scales using standard FEM, BEM, FVM, RKPM, etc. numerical techniques then lead to unreasonably large, slow and expensive calculations. A natural way how to handle problems with a periodic microstructure, but without any a priori prescribed macroscopic symmetries, is to improve the “averaging” limit approach using better microstructural information. Such computational homogenization of periodic media has been designed in [5] yet. Consequently a large number of convergence techniques, studying and explaining the limit process from the microscopic (but ﬁnite) scale to zero one, has been developed in last two decades: at least the ideas of the asymptotic expansion (cf. [7], [33] and [43]), essentially adapted to the study of periodic problems, and the so-called G-convergence (cf. [34] and [14]), H-convergence (cf. [29]) and Γ-convergence (cf. [15] and [13]) should be mentioned. These ideas form the theoretical mathematical basis for the study of variational problems with a hidden microstructure, but some their unpleasant common properties cannot be ignored: they are formally complicated and user-unfriendly and the construction of adequate test functions for corresponding integral equations (namely for nonlinear systems) is often tricky, not transparent for physicists. A new approach occurred about ten years ago. All papers and books appreciate

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the pioneering role of [31], but their deﬁnitions, lemmas and theorems are usually taken (and slightly modiﬁed or generalized if necessary) from [2]. The main idea is, thanks to the addition of a microscopic hidden variable, to substitute the classical weak and strong convergence in Lebesgue and Sobolev spaces by the socalled two-scale (or multiple-scale) convergence incorporating certain compensated compactness phenomenon due to the particular (not very artiﬁcial) choice of test functions. This seems to be equivalent with the original idea of [4], based on certain transform of a spacial variable with respect to a hidden microstructural one, as discussed in [30]. In [10] the two-scale analysis has been applied to the homogenization of several linearized problems, as small-deformation elasticity, heat or wave equation. For linear or quasilinear problems, special families of homogenized FE decompositions supporting the two-scale convergence have been studied in [26] and [27] recently. The mathematical analysis of properties of the two-scale convergence (more detailed than in [2] where some expected result and important proofs are only sketched), extended to parabolic time-dependent problems, is presented in [22]. In many cases of practical interest the two-scale limit passage leads to some eﬀective equations for the original macroscopic problem, but in more complicated nonlinear problems such equations cannot be often written in a simple form, although the two-scale convergence may be guaranteed; this is e. g. the case of the “deck-of-card” model of creep ﬂow applied in [36], whose main idea of “unfractured” (reversible elastic) and “fractured” (irreversible plastic) deformation zones comes from [16]. All above mentioned methods are applicable to domains consisting of several material phases, but without any holes, cracks or perforations. This is evidently not satisﬁed in problems of ﬂow through porous media where important phenomena occur on the boundaries of pores (both of a macro- and a microscopic size), as demonstrated e. g. in [12] and [38]. In [3] the notion of the two-scale convergence has been generalized from classical non-perforated media from [2] to media with pores described by periodic surfaces; later such access has been used also for selected parabolic time-dependent problems from technical practice (as in [9] or [11] where, in addition, some special nonlinearities are taken into account). These studies introduce various special (and rather strong) assumptions on the shape of a domain under consideration; the proof technique must be then adapted to each case separately. This drawback can be removed by deﬁning the two-scale convergence more carefully with respect to measures (not only in classical Lebesgue spaces as in [2], [22] and their non-substantial modiﬁcations). The so-called scale convergence, introduced in [25], can be identiﬁed with the rearranged two-scale approach from [2], making use of the properties of Young measures, discussed in [32], with the close relation to the Γ-convergence. In the following sections of this paper we shall deal with a slightly diﬀerent generalization of this approach, compatible with [8], which is based on its redeﬁnition in Lebesgue spaces (and for gradients in Sobolev spaces) with respect to special periodic Radon measures, applying the tangential calculus, developed in [18]; the weak and strong convergence in such spaces has been characterized in [17]. This approach brings one non-negligible beneﬁt: a cor-

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responding microstructure can include both holes of complicated shapes and parts of lower dimensions without additional geometrical assumptions.

2

Deﬁnition and properties of the two-scale convergence

In this section we shall introduce the basic notations, present the deﬁnition of the two-scale convergence (with respect to measures) and make the overview of its useful properties, including corresponding proofs, although some of them are slightly modiﬁed versions of similar results from [2], [22] or [8]; the main reason for such form of publication is that some well-known lemmas from the classical theory of Lebegue and Sobolev spaces must be checked very carefully in the generalized spaces (and usually some non-standard assumptions are needed). Let us consider a n-dimensional domain Ω in the Euclidean space Rn (n ∈ {1, 2, 3}) with a boundary ∂Ω; we shall use Cartesian coordinates in Rn everywhere. The standard notation of function spaces (where ﬁxed real numbers p > 1 and q = p/(p − 1) occur) will be applied without comments and explanations (for more information see e. g. [32], p. 35) including such basic facts from functional analysis as the Hölder inequality (cf. [24], p. 65); the notation of spaces with respect to measures is taken from [18], the lower index # forces periodicity. Let Y be a unite cube in Rn with a boundary ∂Y. Let µ be some positive Y-periodic Radon measure in Rn and λ a Lebesgue measure in Rn . Let us choose an arbitrary positive ε. Let µε be a measure (“εscaling of µ”) deﬁned by the formula n ϕ(x) dµε (x) = ε ϕ(x) dµ(x/ε) ∀ ϕ ∈ C0 (Ω) Ω

Ω

such that for some positive constant ν |ψ(x, x/ε)|q dµε (x) ≤ ν sup |ψ(x, y)|q dλ(x) y∈Y

Ω

Ω

∀ ψ ∈ Lqλ (Ω, C# (Y))

(1)

holds independently of ε. In the following text all underlined symbols should be understood as sequences indexed with respect to selected positive ε or δ decreasing to zero and all overlined symbols as sequences indexed with respect to integer r increasing to ∞. For simplicity let us assume µ(Y) = 1 and µ(∂Y) = 0. This forces e. g. the convergence of µε to λ in sense |v(x)|p dµε (x) = |v(x)|p dλ(x) ∀ v ∈ Lpλ (Ω) ; (2) lim ε→0

Ω

Ω

more information about such convergence in the vague topology of measures (understood as in [23], p. 120) can be found in [8], p. 1200. Moreover in Lemma 12 one additional assumption on the connectedness of µ (cf. [8], p. 1210) will be needed: let there exist such positive constant c that the implication (the Poincaré-type inequality) p ϕ(y) dµ(y) = 0 ⇒ |ϕ(y)| dµ(y) ≤ c |∇µ ϕ(y)|p dµ(y) (3) Y

Y

Y

Two-scale convergence with respect to measures in continuum mechanics

417

ıp is valid for any ϕ ∈ Hµ# (Y). Now we are ready to introduce the boundedness, the two-scale convergence and the strong two-scale convergence of sequences from Lpµε (Ω) with respect to a measure µ:

Definition 1 (boundedness). Let uε be a sequence in Lpµε (Ω). We say that (uε , µε ) is bounded iﬀ there exists such positive constant a that |uε (x)|p dµε (x) ≤ a (4) Ω

for all elements of uε and any positive ε. Definition 2 (two-scale convergence). Let uε be a sequence in Lpµε (Ω). We say that it two-scale converges to some u0 ∈ Lpλ (Ω, Lpµ# (Y)) (brieﬂy uε \ \ u0 ) iﬀ

lim

ε→0

uε (x) ψ(x, x/ε) dµε (x) =

u0 (x, y) ψ(x, y) dµ(y) dλ(x)

Ω

Ω

(5)

Y

for every test function ψ ∈ Lqλ (Ω, C# (Y)). Definition 3 (strong two-scale convergence). Let uε be a sequence in Lpµε (Ω) that two-scale converges to some u0 ∈ Lpλ (Ω, Lpµ# (Y)) (in sense of Deﬁnition 2). We say that it strongly two-scale converges to u0 (brieﬂy uε → → u0 ) iﬀ lim |uε (x)|p dµε (x) = |u0 (x, y)|p dµ(y) dλ(x) . (6) ε→0

Ω

Ω

Y

Remark 4. Notice that (6) in Deﬁnition 3 can be rewritten in the weaker form lim sup |uε (x)|p dµε (x) ≤ |u0 (x, y)|p dµ(y) dλ(x) ε→0

Ω

Ω

Y

because uε \ \ u0 implies p lim inf |uε (x)| dµε (x) ≥ |u0 (x, y)|p dµ(y) dλ(x) ε→0

Ω

Ω

Y

automatically; for details see [8], p. 1202. Remark 5. Obviously for a stationary sequence uε where uε (x) := v(x) and a test function ψ(x, y) = |v(x)|p/q sgn v(x) (independent of y) (5) with u0 (x, y) = v(x) \ u0 degenerates to (2). More generally: observe that if u0 ∈ C(Ω, C# (Y)) then uε \ whenever uε (x) := u0 (x, x/ε) for all x ∈ Ω. If moreover u0 ∈ C(Ω, C# (Y)) then uε → → u0 ; for details see [8], p. 1203. The same is also true in case u0 (x, y) = u1 (x)u2 (y) for all x ∈ Ω and y ∈ Y where u1 ∈ Lpλ (Ω) and u2 ∈ C# (Y).

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Remark 6. It is easy to see that Deﬁnitions 1, 2 and 3 can be modiﬁed to cover the case that a sequence uε belongs to Lpµε (Ω)n and u0 to Lpλ (Ω, Lpµ# (Y)n ). Remarks 4, 5 and 8 and Lemmas 7, 9, 10 and 11 can be then reformulated without any diﬃculties. Another simple modiﬁcation of Deﬁnitions 1, 2 and 3 exchanges p and q mutually. (The ﬁrst type of generalization will be needed e. g. in Lemma 12, the second one in Lemma 10.) In the following text scalar products in Rj for j = n or j = n2 will be emphasized by · signs (unlike norms in the usual |.| notation). If ψ ∈ Lpµ# (Y) then the index in ψ will emphasize that ψ is considered as a constant (in 2-nd variable) extension of ψ from Ω onto Ω × Y. Standard symbols → and \ for the strong and weak convergence in various Banach spaces are used, too. In the rest of this section, assuming that uε is an arbitrary sequence in Lpµε (Ω) and v ε (if needed) an arbitrary sequence in Lqµε (Ω), we shall derive the most interesting and useful properties of two-scale convergent sequences: ∞ Lemma 7 (on test functions). C0∞ (Ω, C# (Y)) is dense in Lpλ (Ω, Lpµ# (Y)). Consequently, if (uε , µε ) is bounded and (5) from Deﬁnition 2 is true for any ψ ∈ ∞ C0∞ (Ω, C# (Y)) and certain u0 ∈ Lpλ (Ω, Lpµ# (Y)) then uε \ \ u0 . ∞ Proof. The density of C0∞ (Ω, C# (Y)) in Lpλ (Ω, Lpµ# (Y)) follows from [24], p. 73, ∞ and [8], p. 1204. We must only prove that if (5) holds for any ψ ∈ C0∞ (Ω, C# (Y)) q then it is true also for arbitrary ψ ∈ Lλ (Ω, C# (Y)). Let us consider uε (x) ψ(x, x/ε) dµε (x) = uε (x) (ψ(x, x/ε) − ψ r (x, x/ε)) dµε (x) Ω Ω Y r + uε (x) ψ (x, x/ε) dµε (x) − u0 (x, y) ψ r (x, y) dµ(y) dλ(x) Ω Ω Y + u0 (x, y) (ψ r (x, y) − ψ(x, y)) dµ(y) dλ(x) Ω Y + u0 (x, y)ψ(x, y) dµ(y) dλ(x) Ω

Y

r

r

∞ where ψ ⊂ C0∞ (Ω, C# (Y)) and ψ → ψ in Lpλ (Ω, Lpµ# (Y)) (using the density of ∞ C0∞ (Ω, C# (Y)) in Lpλ (Ω, Lpµ# (Y))). Since

|ψ(x, y) − ψ r (x, y)|q dµε (x) = 0 ,

lim sup

r→∞ y∈Y

Ω

the ﬁrst right-hand-side integral vanishes thanks to the assumed boundedness and (1) (with help of the Hölder inequality) for r → ∞. Similarly the convergence r ψ → ψ implies that the fourth integral can be removed, too. But Deﬁnition 3 guarantees that the second and third integrals together tend to zero for each integer r if ε → 0. Thus, the limit process r → ∞ and ε → 0 yields (5) with arbitrary ψ ∈ Lpλ (Ω, C# (Y)).

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419

Remark 8. Various other density results can be found in the cited references; e. g. in the proof of Theorem 15 we shall need the density of Lpλ (Ω, C# (Y)) both in ıp (Y)). Lpλ (Ω, Lpµ# (Y)) and in Lpλ (Ω, Hµ# Lemma 9 (on compactness). If (uε , µε ) is bounded then there exists such u0 ∈ Lpλ (Ω, Lpµ# (Y)) that, up to a subsequence, uε \ \ u0 . Proof. By (1) the choice of a measure µ guarantees the estimate ψε Lqµε (Ω) ≤ ν 1/q ψLqλ (Ω,C# (Y)) for each ψ ∈ Lqλ (Ω, C# (Y)) where ψε (x) := ψ(x, x/ε). For arbitrary positive ε let us introduce a linear operator Tε applied to arguments ψ ∈ Lqλ (Ω, C# (Y)), using the Riesz representation theorem (cf. [21], p. 33) [Tε , ψ] := uε (x)ψε (x) dµε (x) . Ω

Thus, due to the inequality (4) from Deﬁnition 1, we can estimate |[Tε , ψ]| ≤ uε Lpµε (Ω) ψε Lqµε (Ω) ≤ a1/p ν 1/q ψLqλ (Ω,C# (Y)) ; this guarantees that T ε is a bounded sequence in the space dual to Lqλ (Ω, C# (Y)) which can be identiﬁed with Lpλ (Ω, M# (Y)) where M# (Y) is the space of periodic Radon measures on Y (for details see [2], p. 1486, and [32], p. 40). In virtue of the Alaoglu theorem (see [21], p. 45) then a subsequence from T ε convergent in the weak ∗ topology can be extracted; this implies that such T0 ∈ Lpλ (Ω, M# (Y)) exists that, up to a subsequence, uε (x)ψε (x) dµε (x) = lim [Tε , ψ] = [T0 , ψ] lim ε→0 Ω ε→0 = u0 (x, y)ψ(x, y) dµ(y) dλ(x) Ω

Y

where the existence of some u0 ∈ Lpλ (Ω, C# (Y)) corresponding to T0 follows from the Riesz representation theorem again; but this is directly (5) from Deﬁnition 2. → u0 and v ε \ \ v0 (cf. Remark 6) for Lemma 10 (on function products). If uε → some u0 ∈ Lpλ (Ω, Lpµ# (Y)) and v0 ∈ Lqλ (Ω, Lqµ# (Y)) then

lim

ε→0

uε (x) vε (x) dµε (x) = Ω

u0 (x, y) v0 (x, y) dµ(y) dλ(x) . Ω

Y

Proof. Making use of the density guaranteed by Lemma 7, we shall apply for each positive ε the decomposition uε (x)vε (x) dµε (x) = (uε (x) − ϕδε (x))vε (x) dµε (x) + ϕδε (x)vε (x) dµε (x) Ω

Ω

Ω

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where ϕδ ⊂ C(Ω, C# (Y)), ϕδ → u0 in Lpλ (Ω, Lpµ# (Y)) and ϕδε (x) := ϕδ (x, x/ε) for all x ∈ Ω. Deﬁnition 2 with respect to Remark 5 gives lim ϕδε (x)vε (x) dµε (x) = lim ϕδ (x, y)v0 (x, y) dµε (y) dλ(x) δ,ε→0 Ω δ→0 Ω Y = u0 (x, y)v0 (x, y) dµ(y) dλ(x) ; Ω

Y

thus it is suﬃcient to prove (uε (x) − ϕδε (x))vε (x) dµε (x) = 0 , lim δ,ε→0

Ω

but this requirement, thanks to the boundedness of (v ε , µε ) by Deﬁnition 1 (cf. Remark 6), can be reduced (using the Hölder inequality) to the stronger one lim uε − ϕδε Lpµε (Ω) = 0 .

(7)

δ,ε→0

Two Clarkson inequalities uε − ϕδε pLpµ

+ uε + ϕδε pLpµ

uε − ϕδε qLpµ

+ uε + ϕδε qLpµ

ε (Ω)

ε (Ω)

≤ 2p−1 uε pLpµ (Ω) + ϕδε pLpµ (Ω) , ε ε q/p p p ≤ 2 uε Lpµ (Ω) + ϕδε Lpµ (Ω) (Ω)

ε (Ω)

ε

ε

ε

from [1], p. 37, are available. Using the Deﬁnition 3 and taking into account Remark 4, we obtain for p ≤ 2 from the ﬁrst one lim uε − ϕδε pLpµ (Ω) ε ≤ lim 2p−1 uε pLpµ (Ω) + ϕδε pLpµ (Ω) − uε + ϕδε pLpµ (Ω) ε ε ε δ,ε→0 ≤ lim 2p−1 u0 pLp (Ω,Lp (Y)) + ϕδ pLp (Ω,Lp (Y)) − u0 + ϕδ pLp (Ω,Lp (Y)) λ µ# λ µ# λ µ# δ→0 p p ≤ 2p−1 .2 u0 Lp (Ω,Lp (Y)) − 2 u0Lpλ (Ω,Lpµ# (Y)) = 0 δ,ε→0

λ

µ#

and for p ≥ 2 from the second one (respecting that q/p = q − 1) lim uε − ϕδε qLpµ (Ω) ε q/p p p q ≤ lim 2 uε Lpµ (Ω) + ϕδε Lpµ (Ω) − uε + ϕδε Lpµ (Ω) ε ε ε δ,ε→0 q/p p p q ≤ lim 2 u0 Lp (Ω,Lp (Y)) + ϕδ Lp (Ω,Lp (Y)) − u0 + ϕδ Lp (Ω,Lp (Y)) λ µ# λ µ# λ µ# δ→0 q ≤ 2.2q/p u0 qLp (Ω,Lp (Y)) − 2 u0 Lpλ (Ω,Lpµ# (Y)) = 0 . δ,ε→0

λ

µ#

Both these results together imply (7).

Two-scale convergence with respect to measures in continuum mechanics

421

Lemma 11 (on strong or weak convergence). Let there exist such u ∈ Lpλ (Ω) that lim |uε (x) − u(x)|p dµε (x) = 0 . (8) ε→0

Ω

Then uε → → u . Conversely: if uε \ \ u0 for some u0 ∈ Lpλ (Ω, Lpµ# (Y)) then lim uε (x)ϕ(x) dµε (x) = u "(x)ϕ(x) dλ(x) ∀ϕ ∈ Lqλ (Ω) ε→0

Ω

(9)

Ω

where u "(x) :=

u0 (x, y) dµ(y) . Y

Proof. By (2) u is included in all Lpµε (Ω) with a positive ε, hence (8) is welldeﬁned. To verify the ﬁrst assertion with uε \ \ u (instead of its stronger version → u ), by Deﬁnition 2 and Lemma 7 it is suﬃcient to prove with uε → uε (x)ψε (x) dµε (x) = u(x) ψ(x, y) dµ(y) dλ(x) lim ε→0

Ω

Ω

Y

for any ψ ∈ C(Ω, C# (Y)) and ψε (x) := ψ(x, x/ε). But Remark 5 shows that → ψ (cf. also Remark 6); thus the preceding equation can be rewritten as ψε → lim (uε (x) − u(x))ψε (x) dµε (x) = 0 ε→0

Ω

and the Hölder inequality gives the expected result. Moreover (using the estimate in the norm of Lpµε (Ω)) 1/p 1/p p |uε (x)| dµε (x) ≤ lim |uε (x) − u(x)| dµε (x)

p

lim

ε→0

ε→0

Ω

Ω

+ lim

ε→0

1/p |u(x)|p dµε (x) .

Ω

But the ﬁrst right-hand-side additive term is zero by (8) and dµε in the second one can be replaced by dλ using (2); this with respect to Deﬁnition 3 (with u0 (x, y) substituted by u(x) only) completes the proof of the ﬁrst assertion. The second assertion is a simple consequence of Deﬁnition 2 where ϕ ∈ Lqλ (Ω) is taken instead of ψ ∈ Lqλ (Ω, C# (Y)) (being constant in the second variable). In [2], pp. 1485 and 1488, only the special choice µε = µ = λ was studied; for this case Lemma 11 guarantees that uε → u

⇒

uε → → u

and

uε \ \ u0

⇒

uε \ u "

in classical sense of strong and weak convergence (→ and \ act in Lpλ (Ω)). In such case a lot of results from the standard theory of Sobolev spaces and functional

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analysis is available (e. g. Lemma 9 can be understood as a stronger version of the well-known Eberlein - Shmul’yan theorem – cf. [41], p. 201, and [20], p. 197), but this is not true in more complicated cases where namely an appropriate deﬁnition of gradients may be far from trivial. We shall introduce the gradients in the same ıp (Y) way as in [17], p. 4: for any ϕ ∈ Hµ# ∇µ ϕ(y) := Pµ (y)∇ϕ(y)

∀y ∈ Y

where Pµ ∈ Lpµ# (Y)n.n denotes the operator of orthogonal projection onto the local tangent space of µ (deﬁned in [17], p. 3, exactly). Moreover, following [8], p. 1206, we can make use of the operator divµ coming from the relation of the Green-Ostrogradskiˇı type q ∞ v divµ Φ dµ = − Φ · ∇µ v dµ ∀ v ∈ C# (Y) ∀ Φ ∈ Xµ# (Y) Y

Y

q (Y) (of vector ﬁelds “tangent to µ”) includes all elements of where the class Xµ# q p Lµ# (Y)n whose divergences (in distributional sense) belong to Lqµ# (Y); Xµ# (Y) ıp ∞ ∞ is deﬁned similarly. The space Hµ# (Y) is the completion of C0 (Ω, C# (Y)) in the norm ϕH ı p (Y) := ϕLpµ# (Y) + ∇µ ϕLpµ# (Y)n µ#

ıp for any ϕ ∈ then the space Lpλ (Ω, Hµ# (Y)) can be introduced, too. We intend to apply the projector Pµ especially in another context, taking a macrostructural variable x ∈ Ω (not only y ∈ Y) into consideration: instead of Pµ we shall have Pµε with a positive ε such that (similarly to Remark 5, cf. also → Pµ and Remark 6, for details see [8], p. 1203) P µε → ∞ C0∞ (Ω, C# (Y));

∇µε ϕ(x) := Pµε (x)∇ϕ(x)

∀x ∈ Ω

ıp for any ϕ ∈ Hµı pε (Ω); moreover if ϕ1 ∈ Lpλ (Ω, Hµ# (Y)n ) and ϕε is a sequence of ı p elements from Hµε (Ω) then

∇ϕε \ \ ϕ1

⇒

P µε ∇ϕε \ \ Pµ ϕ1

(10)

ıp (Y)) then the double dot in (this can be veriﬁed using Lemma 10). If ψ ∈ Lpλ (Ω, Hµ# ∇µ¨ ψ will indicate that the operator ∇µ is applied to the second (microstructural) variable only. (In the proof of Theorem 15 also the single or double dot as an index of ∇ will demonstrate that this operator is related to the ﬁrst or second variable only.) The following important generalization of Lemma 9 studies the behaviour of gradients of some sequences from the point of view of the two-scale convergence:

Lemma 12 (on gradients). Let there exist such positive constant c that (3) is ıp (Y). If (uε , µε ) and (∇µε uε , µε ) are bounded then some u ∈ valid for any ϕ ∈ Hµ# ıp (Y)) exist such that, up to a subsequence, uε \ \ u Hλı p (Ω) and u1 ∈ Lpλ (Ω, Hµ# and ∇µε uε \ \ ∇u + ∇µ¨ u1 (in sense of Remark 6).

Two-scale convergence with respect to measures in continuum mechanics

423

Proof. By Lemma 9 and Deﬁnition 2 (cf. also Remark 6) such u0 ∈ Lpλ (Ω, Lpµ# (Y)) and u1 ∈ Lpλ (Ω, Lpµ# (Y)n ) exist and from uε such subsequence can be extracted \ u0 and ∇uε \ \ u1 ; in other words: (5) must be true for all ψ ∈ that uε \ Lpλ (Ω, C# (Y)) and lim ∇uε · Ψ (x, x/ε) dµε (x) = u1 (x, y) · Ψ (x, y) dµ(y) dλ(x) (11) ε→0

Ω

Ω

Y

Lpλ (Ω, C# (Y)n ).

for all Ψ ∈ Especially for any x ∈ Ω and y ∈ Y let us choose q Ψ (x, y) = ϕ(x)Φ(y) where ϕ ∈ C0∞ (Ω) and Φ ∈ Xµ# (Y). Following [8], p. 1212, for every positive ε we are able to integrate by parts ε ∇uε ϕ(x) · Φ(x/ε) dµε (x) (12) Ω = −ε uε (x)∇ϕ(x) · Φ(x/ε) dµε (x) − uε (x)ϕ(x) divµ Φ(x/ε) dµε (x) Ω

Ω

and thanks to the boundedness of (uε , µε ) and (∇µε uε , µε ) to obtain 0 = lim uε (x)ϕ(x) divµ Φ(x/ε) dµε (x) ε→0 Ω = ϕ(x) u0 (x, y) divµ Φ(y) dµ(y) dλ(x) . Ω

(13)

Y

ıp (Y) forces The existence of a positive constant c from (3) independent of ϕ ∈ Hµ# that Φ(y) · Ψ (y) dµ(y) = 0 ∀ Φ ∈ W ∀ Ψ ∈ W⊥ (14) Y

with

q (Y) : divµ Φ = 0} , W := {Φ ∈ Xµ# p W⊥ := {Ψ ∈ Xµ# (Y) : Pµ Ψ = ∇µ v

and

ıp for some v ∈ Hµ# (Y)}

v(y) w(y) dµ(y) = 0 ∀ v ∈ V ∀ w ∈ V⊥

(15)

Y

with

p (Y)} , V := {v ∈ Lpµ# (Y) : v = divµ Φ for some Φ ∈ Xµ#

V⊥ := {w ∈ Lqµ# (Y) : w = a for some a ∈ R} ; the extensive veriﬁcation of these seemingly simple facts, generalizing the classical result (“orthogonals of divergence-free functions are exactly the gradients” if µε = µ = λ) from [2], p. 1492, with q = 2 (using the Fourier analysis), and from [22], p. 329, for a general q > 1 (based on the absolute continuity on line segments in standard Sobolev spaces by [44], p. 44), to periodic Sobolev spaces with general

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measures, can be found in [8], p. 1210 (in [42] the assumption q = 2 cannot be removed easily). Thus for a ﬁxed x ∈ Ω we know from (13) and (15) that u0 (x, y) is a constant on Y which gives a chance to set u(x) := u0 (x, y) independently of y ∈ Y. In particular let us consider Φ ∈ W only. Then (12) (divided by ε) degenerates to ∇uε (x)ϕ(x) · Φ(x/ε) dµε (x) = − uε (x)∇ϕ(x) · Φ(x/ε) dµε (x) , Ω

Ω

the limit passage (11) and (5) gives ϕ(x) u1 (x, y) · Φ(y) dµ(y) dλ(x) = − u(x)∇ϕ(x) · Φ(y) dµ(y) dλ(x) Ω Y Ω Y = ϕ(x)∇u(x) · Φ(y) dµ(y) dλ(x) Ω

C0∞ (Ω)

Y

and thanks to the density of in (the choice of ϕ in C0∞ (Ω) was arbitrary) (u1 (x, y) − ∇u(x)) · Φ(y) dµ(y) = 0 for λ - a. e. x ∈ Ω . Lpλ (Ω)

Y

¿From (14) we can deduce Pµ (y)u1 (x, y) − ∇u(x) = ∇µ¨ u1 (x, y) for λ - a. e. x ∈ Ω and for µ - a. e. y ∈ Y ıp with certain u1 ∈ Lpλ (Ω, Hµ# (Y)n ). Let us remember a sequence P µε from (10). Since ∇uε \ \ u1 (from (11)), ∇µε uε = Pµε ∇uε \ \ Pµ u1 = ∇u + ∇µ¨ u1 .

3

Analysis of a model elliptic problem

To make the notation as clear as possible, we shall introduce some special function classes. Let Carp (Ω, Y, Rm )k be a class of functions b : Ω × Y × Rm -→ Rk (m and k are positive integers) with the following properties: (a) b is Y-periodic. Moreover for each w ∈ Rm (later m = n or m = n(1 + n) together with k = n or k = n2 will be used), for λ - a. e. x ∈ Ω and for µ - a. e. y ∈ Y the growth condition |b(x, y, w)| ≤ βb (x, y) + γb |w|p−1 is satisﬁed with some positive γb and βb ∈ Lqλ (Ω, C# (Y)) (by (2) also βb ∈ Lqµε (Ω, C# (Y)) for any positive ε). (b) bϕ (x, y) := b(x, y, ϕ(x, y)) applied to all x ∈ Ω and y ∈ Y deﬁnes for any ϕ ∈ Lpλ (Ω, C# (Y)m ) a continuous mapping bϕ : Lpλ (Ω, C# (Y)m ) -→ Lqλ (Ω, C# (Y)k ). (In many cases the Nemytskiˇı mappings from [32], p. 36, are useful to verify these properties.) A class Carp (∂Ω, Rn )n can be deﬁned similarly with small changes: in particular m = k = n, Ω is replaced by ∂Ω, the modiﬁed growth condition from

Two-scale convergence with respect to measures in continuum mechanics

425

(a) is valid with some positive γg and βg ∈ Lpσ (∂Ω), σ is the Hausdorﬀ measure on ∂Ω, in (a) and (b) the second variable is missing, the requirement on periodicity in (a) disappears. Now we are ready to formulate our model variational problem: Problem 13. Find u ∈ Uε such that a(x, x/ε, u(x), ∇µε u(x)) · ∇µε v(x) dµε (x) Ω + f (x, x/ε, u(x)) · v(x) dµε (x) + g(x, u(x)) · v(x) dσ(x) = 0 Ω

(16)

∂Ω

for all v from certain subspace Uε of Hµı εp (Ω)n and their suitable extensions to Lpσ (∂Ω)n . To be able to discuss the solvability of this problem, let us suppose: (c) Some prescribed boundary conditions on ∂Ω (of the Dirichlet type) force the equivalence of norms vHµı p (Ω)n , ε

∇µε vLpµε (Ω)n.n ,

∇µε vLpµε (Ω)n.n + vLpσ (∂Ω)n

for any v ∈ Uε (which can be identiﬁed with the Friedrichs-type inequality). (d) The functions a, f and g (in classical mechanics: stress tensors determined by strain tensors and material characteristics from the constitutive law, volume loads and surface loads) belong to the following classes: a ∈ Carp (Ω, Y, Rn(1+n) )n.n ,

f ∈ Carp (Ω, Y, Rn )n ,

g ∈ Carp (∂Ω, Rn )n .

The estimates a(x, y, z, θ) · θ ≥ κ|θ|p − ϑa (x, y)(|z|p−s + |θ|p−s ) − ζa (x, y) , f (x, y, z) · z ≥ −ϑf (x, y)|z|p−s − ζf (x, y) , x)|z|p−s − ζg (" x) g(" x, z) · z ≥ −ϑg (" are true for λ - a. e. x ∈ Ω, for µ - a. e. y ∈ Y, for σ - a. e. x " ∈ ∂Ω and each z ∈ Rn and θ ∈ Rn.n (w = (z, θ) from (a) is considered) with some positive constant p/s κ, real s satisfying the inequality 1 < s < p, ϑa and ϑf from Lλ (Ω, C# (Y)), p/s ϑg from Lσ (∂Ω), ζa and ζf from Lλ (Ω, C# (Y)), ζg from Lσ (∂Ω) (by (2) also p/s ϑa and ϑf belong to Lµε (Ω, C# (Y)), ζa and ζf to Lµε (Ω, C# (Y)) for any positive ε). Moreover " · (θ − θ) " >0 (a(x, y, z, θ) − a(x, y, z, θ))

(17)

independently of the choice of θ" ∈ Rn.n other than θ (i. e. a is strictly monotone).

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In the rest of this section (and of the whole paper) we shall derive two results: the ﬁrst one that Problem 13 with a ﬁnite positive ε has already a solution (Theorem 14) and the second one that under certain physically realistic conditions the limit passage ε → 0 is possible (Theorem 15). These results generalize those from [2], p. 1503, naturally in several directions (probably the most visible is that the assumption “on non-perforation” µε = µ = λ was removed); the exact formulations and proofs follow:

Theorem 14 (existence result with a ﬁnite ε). Let (a), (b), (c) and (d) be satisﬁed. Then for any positive ε Problem 13 has at least one solution. Proof. Let Aε be a mapping of Uε into its dual space by the deﬁnition

Aε u, v = Ω +

a(x, x/ε, u(x), ∇µε u(x)) · ∇µε v(x) dµε (x) (18) f (x, x/ε, u(x)) · v(x) dµε (x) + g(x, u(x)) · v(x) dσ(x)

Ω

∂Ω

a coercive, b demicontinuous, c for every u, v ∈ Uε . By [19], p. 279, if Aε is d the estimate bounded and lim sup Aε v r − Aε v, v r − v ≤ 0

(19)

r→∞

together with v r \ v forces v r → v for any sequence v r ⊂ Uε and for a corresponding v ∈ Uε then Aε is also surjective; this implies that the integral equation

Aε u, v = 0

∀ v ∈ Uε

must have a solution u ∈ Uε . Thus the proof of the existence of at least one solution a , b of Problem 13 can be reduced to four steps consisting of the veriﬁcation of , c and : d a Following [24], p. 65, we shall use the inequality for any positive τ , η and ω η

τ

(τ ω)p−s ≤ p−s

s η p/s p − s s p−s p p (τ ω)p = τ −p(p−s)/s η p/s + τ ω + p τ p−s p p p

valid for any positive τ , η and ω; in the subsequent estimates several times special η and ω will be applied. Let us consider an arbitrary v ∈ Uε . ¿From

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Two-scale convergence with respect to measures in continuum mechanics

(d) we receive a(x, x/ε, v(x), ∇µε v(x)) · ∇µε v(x) dµε (x) ≥ κ∇µε vpLpµ

ε (Ω)

Ω

n.n

s p−s p p/s − τ −p(p−s)/s ϑa p/s − τ vpUε− ζa Lµε (Ω,C# (Y)) , Lµε (Ω,C# (Y)) p p f (x, x/ε, v(x)) · v(x) dµε (x) Ω

p−s p s p/s τ vpLpµ (Ω)− ζf Lµε (Ω,C# (Y)) , − ≥ − τ −p(p−s)/s ϑf p/s Lµε (Ω,C# (Y)) ε p p g(x, v(x)) · v(x) dσ(x) ∂Ω

p−s p s p/s τ vpLpσ (∂Ω)− ζg Lσ (∂Ω) − ≥ − τ −p(p−s)/s ϑg p/s Lσ (∂Ω) p p and consequently

Aε v, v ≥ κ∇µε vpLpµ (Ω)n.n ε p−s p τ vpUε + vpLpµ (Ω)n + vLσ (∂Ω)n − Sτ − ε p with certain real constant Sτ independent of v. Especially for τ small enough

Aε v, v ≥ κ1 vpUε − κ2

(20)

holds by (c) (due to the equivalence of norms) with some positive constants κ1 and κ2 independent of v; this implies the coerciveness of Aε (cf. [19], p. 266) evidently. b In Uε let us choose arbitrary u and v and any sequence ur → u. We have

Aε ur − Aε u, v = (a(x, x/ε, ur (x), ∇µε ur (x)) − a(x, x/ε, u(x), ∇µε u(x))) Ω

· ∇µε v(x) dµε (x)

(f (x, x/ε, ur (x)) − f (x, x/ε, u(x))) · v(x) dµε (x)

+ Ω

(g(x, ur (x)) − g(x, u(x))) · v(x) dσ(x) .

+ ∂Ω

Thus (b) (with help of the Hölder inequality) forces the demicontinuity of Aε (cf. [19], p. 270). c Let us consider u and v as in b again. ¿From (a) with a (where w has to understood as in (d)), f and g substituted to b we obtain |a(x, x/ε, u(x), ∇µε u(x))| ≤ βa (x, x/ε) + γa |u(x)|p−1 + |∇µε u(x)|p−1 , |f (x, x/ε, u(x))| ≤ βf (x, x/ε) + γf |u(x)|p−1 , x) + γg |u(" x)|p−1 |g(" x, u(" x))| ≤ βg ("

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for λ - a. e. x ∈ Ω and for σ - a. e. x " ∈ ∂Ω which directly from the deﬁnition (18) by the Hölder inequality yields

Aε u, v ≤ βa Lµε (Ω,C# (Y)) + γa up−1 vUε Uε vLpµε (Ω)n + βf Lµε (Ω,C# (Y)) + γf up−1 p Lµε (Ω)n vLpσ (∂Ω)n . + βg Lσ (∂Ω) + γg up−1 Lp σ (∂Ω) Making use of (c) we can thus see that independently of u and v such positive β and γ exist that vUε .

Aε u, v ≤ β + γup−1 Uε Let us introduce the unit ball Bε := {v ∈ Uε : vUε ≤ 1}. The norm of Aε u in the space dual to Uε then is sup Aε u, v ≤ β + γup−1 Uε

v∈Bε

which guarantees the boundedness of Aε (cf. [19], p. 266). d Unlike b in Uε let us choose an arbitrary v and any sequence v r \ v now. We have

Aε v r − Aε v, v r − v = (a(x, x/ε, v r (x), ∇µε v r (x)) − a(x, x/ε, v r (x), ∇µε v(x))) Ω

· (∇µε v r (x) − ∇µε v(x)) dµε (x)

+ Ω

(a(x, x/ε, v r (x), ∇µε v(x)) − a(x, x/ε, v(x), ∇µε v(x))) · (∇µε v r (x) − ∇µε v(x)) dµε (x)

(f (x, x/ε, v r (x)) − f (x, x/ε, v(x))) · v(x) dµε (x)

−

Ω

f (x, x/ε, v(x)) · (v r (x) − v(x)) dµε (x)

+

Ω

−

(g(x, v r (x)) − g(x, v(x))) · v(x) dσ(x) ∂Ω g(x, v(x)) · (v r (x) − v(x)) dσ(x) .

+ ∂Ω

By (c) v r → v in Lpµε (Ω)n ; thus the limit passage (based on the Hölder inequality) with respect to the continuity of a, f and g from (b) gives lim sup Aε v r − Aε v, v r − v r→∞ = lim sup (a(x, x/ε, v r (x), ∇µε v r (x)) − a(x, x/ε, v r (x), ∇µε v(x))) r→∞

Ω

· (∇µε v r (x) − ∇µε v(x)) dµε (x) .

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429

Let us suppose v r → / v and believe that the inequality (19) is fulﬁlled. Then a we deduce using (17) from (d) (the only part of (d) that was not needed in ) that lim sup Aε v r − Aε v, v r − v > 0 r→∞

which is in contrary with (19) immediately. Theorem 14 informs us that a set Uε0 of solutions of Problem 13 cannot be empty. Let U be a set of all two-scale limits v of such sequences v ε with elements from Uε that vε \ \ v and ∇µε vε \ \ ∇v + ∇µ¨ v1 . Let U be a subset of U (deﬁned in the same way) where moreover ∇µε vε → → v is required. For the study of limit behaviour of (16) with ε → 0 (to avoid divergence sequences of solutions from Uε0 ), using the simpliﬁed notation a(x, y, v(x)) := a(x, y, v(x), u 0 (x, y)) , u 0 (x, y) := ∇u(x) + ∇µ φr (x, y) + δφ(x, y) ,

(21)

u ε (x) := u 0 (x, x/ε) for each x ∈ Ω and y ∈ Y with any positive δ, φr ∈ Lpλ (Ω, C# (Y)n ) (r is a positive integer), φ ∈ Lpλ (Ω, C# (Y)n.n ) and v = u or v = uε , we shall slightly regulate the choice of “loads” f and g and “strain-stress relations” a with respect to µε : (e) If uε and v ε are some sequences of elements from Uε with two-scale limits u and v and and Φε is some sequence of elements from Lpλ (Ω, C# (Y)n.n ) with a two-scale limit Φ0 where u, v ∈ Hλı p (Ω)n and Φ0 ∈ Lpλ (Ω, Lpµ# (Y)n.n ) then a(x, x/ε, uε (x)) · Φε (x) dµε (x) lim ε→0 Ω = a(x, y, u(x)) · Φ0 (x, y) dµ(y) dλ(x) , Ω Y lim f (x, x/ε, uε (x)) · vε (x) dµε (x) ε→0 Ω = f (x, y, u(x)) dµ(y) · v(x) dλ(x) , Ω Y lim g(x, uε (x)) · vε (x) dσ(x) = g(x, u(x)) · v(x) dσ(x) . ε→0

∂Ω

∂Ω

The property (e) looks rather diﬃcult to be veriﬁed; thus (for illustration) we shall demonstrate how it could be simpliﬁed (using suﬃcient conditions) in special cases: If a, f and g are independent of the third variable explicitly then the ﬁrst two relations can be seen as simple consequences of Deﬁnition 2, only the third one needs vε → v in Lpσ (∂Ω)n which (except pure Dirichlet problems, favourable for matematicians, but rare in practice) may not be trivial (the properties of extensions of vε onto ∂Ω have to be studied). If (for general a, f and g again) µε = µ = λ then Lemma 11 results that uε has a weak limit in U = U ε (which

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is a subspace of Hλı p (Ω)n here) immediately, hence (if the Sobolev imbedding theorem holds), up to a subsequence, uε → u in U and (if the trace theorem holds) uε → u in Lpσ (∂Ω)n . (This is even true for a large class of domains Ω without any respect to a, f and g; the geometrical properties of such class are studied in [28], pp. 58 and 219, in great details, including perverse conﬁgurations uncovered by standard theorems.) Then (e) can be checked using continuity arguments from (b) only; Lemma 10 can be helpful, too. Also if we only know that (for any reason) uε is bounded in Hλı p (Ω)n then the Eberlein - Shmul’yan theorem implies, up to a subsequence, uε \ u in Hλı p (Ω)n and the same approach can be applied. In more complicated cases simple criteria are not known; nevertheless, we shall try to formulate a general convergence result: Theorem 15 (existence result with ε → 0). Let the assumptions of Theorem 14 and (e) be fulﬁlled. Then the limit process ε → 0 converts the integral equation (16) from Problem 13 into its limit form a(x, y, u(x), ∇u(x) + ∇µ¨ u1 (x, y)) dµ(y) · ∇v(x) dλ(x) (22) Ω Y + f (x, y, u(x)) dµ(y) · v(x) dλ(x) + g(x, u(x)) · v(x) dσ(x) = 0 Ω

Y

∂Ω

ıp for all v ∈ U which has at least one solution u ∈ U , u1 ∈ Lpλ (Ω, Hµ# (Y)n ).

Proof. For the sake of brevity let us introduce the notation αε (x) := a(x, x/ε, uε (x), ∇µε uε (x)) for every x ∈ Ω. (Since by Theorem 14 Uε0 = ∅, the choice of some uε ∈ Uε0 is c in the proof of Theorem possible.) Using (a) we obtain in the same way as in 14 p−1 q |αε (x)|q ≤ βa (x, x/ε) + γa |uε (x)|p−1 + |∇µε uε (x)| ≤ 2q−1 βaq (x, x/ε) + 22(q−1) γaq (|uε (x)|p + |∇µε uε (x)|p ) for λ - a. e. x ∈ Ω and consequently αε qLqµ (Ω)n.n ≤ 2q−1 βa qLµ ε

ε

q−1 γa uε pUε (Ω,C# (Y)) + 2

.

(23)

c The convergence of µε (cf. (2)) enables us to ﬁnd κ1 and κ2 in (20) from independently of both v and ε; in particular (for v = uε ∈ Uε0 ) (20) can be written in the form 0 = Aε uε , uε ≥ κ1 uε pUε − κ2 showing that both (uε , µε ) and (∇µε uε , µε ) are bounded (in sense of Deﬁnition 1 and Remark 6). Thanks to (23) the same is true for (αε , µε ). According to \ u , αε \ \ α0 and Lemma 9 and Lemma 12, up to subsequences, then uε \

Two-scale convergence with respect to measures in continuum mechanics

431

∇µε uε \ \ ∇u + ∇µ¨ u1 for some u ∈ Hλı p (Ω)n , α0 ∈ Lpλ (Ω, Lpµ# (Y)n.n ) and u1 ∈ ıp (Y)n ). Unfortunately no reasonable relation between α0 and u with u1 Lpλ (Ω, Hµ# is available now. To investigate it, let us start with the integration of (17) from (d) with y = x/ε, z = uε (x), θ = ∇µε uε (x) and θ" = u ε (x) over Ω. We receive (a(x, x/ε, uε (x), ∇µε uε (x)) − a(x, x/ε, uε (x), u ε (x))) Ω

· (∇µε uε (x) − u ε (x)) dµε (x) ≥ 0 which with help of (16) gets the form

−

f (x, x/ε, uε (x)) · uε (x) dµ −

g(x, uε (x)) · uε (x) dσ(x)

Ω

∂Ω

Ω

a(x, x/ε, uε (x)) · ∇µε uε (x) dµε (x) a(x, x/ε, uε (x)) · u ε (x) dµε (x) − αε (x) · u ε (x) dµε (x) ≥ 0 .

− + Ω

Ω

But the initial three left-hand-side integrals are exactly those from (e), only vε = uε and also Φε = ∇µε uε in the third one had to be set. Since u ε → →u 0 (by Remark uε for the fourth one; for the last one 5), the same can be repeated with Φε = − αε \ \ α0 has been derived yet. In this way we obtain −

f (x, y, u(x)) · u(x) dµ(y) dλ(x) −

Ω Y −

g(x, u(x)) · u(x) dσ(x) ∂Ω

a(x, y, u(x)) · (∇u(x) + ∇µ¨ u1 (x, y)) dµ(y) dλ(x) Ω Y ( a(x, y, u(x)) − α0 (x, y)) · u 0 (x, y) dµ(y) dλ(x) ≥ 0

+ Ω

Y

and in another order (using δ, φ and φr ) −

f (x, y, u(x)) · u(x) dµ(y) dλ(x) −

Ω Y −

g(x, u(x)) · u(x) dσ(x) (24) ∂Ω

α0 (x, y) dµ(y) · ∇u(x) dλ(x) Ω Y

−

α0 (x, y) · ∇µ φr (x, y) dµ(y) dλ(x) Ω Y

a(x, y, u(x)) · ∇µ¨ (φr (x, y) − u1 (x, y)) dµ(y) dλ(x) +δ ( a(x, y, u(x)) − α0 (x, y)) · φ(x, y) dµ(y) dλ(x) ≥ 0 .

+

Ω

Y

Ω

Y

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J. Vala

But by Deﬁnition 2 and (16) (with respect to (e) again) we have f (x, y, u(x)) · u(x) dµ(y) dλ(x) + g(x, u(x)) · u(x) dσ(x) Ω Y ∂Ω + α0 (x, y) dµ(y) · ∇u(x) dλ(x) Ω Y = lim f (x, x/ε, uε (x)) · u(x) dµε (x) + lim g(x, uε (x)) · u(x) dσ(x) ε→0 Ω ε→0 ∂Ω + lim αε (x) · ∇u(x) dµε (x) = 0 ε→0

Ω

and the ﬁrst, second and third left-hand-side integrals in (24) vanish. For the fourth one we have α0 (x, y) · ∇µ φr (x, y) dµ(y) dλ(x) Ω Y = lim αε (x) · ∇µε φr (x, x/ε) dµε (x) , ε→0

Ω

but also ∇µε φr (x, x/ε) = Pµε (x)∇.. φr (x, x/ε) = ε (Pµε (x)∇φr (x, x/ε) − Pµε (x)∇. φr (x, x/ε)) and (due to the boundedness of αε ) the last limit is zero, too. In particular it r ıp is always possible to choose φ → u1 in Lpλ (Ω, Hµ# (Y)n ) (as in Remark 8); the limit passage r → ∞ then removes the ﬁfth integral. Finally, divided by δ, (24) degenerates to lim (a(x, y, u(x), ∇u(x) + ∇µ¨ φr (x, y) + δφ(x, y)) − α0 (x, y)) r→∞

Ω

Y

· φ(x, y) dµ(y) dλ(x) ≥ 0 which, in particular for δ = 1/r, can be (thanks to the continuity of a from (b)) rewritten as ς(x, y) · φ(x, y) dµ(y) dλ(x) ≥ 0 (25) Ω

Y

where ς(x, y) := a(x, y, u(x), ∇u(x) + ∇µ¨ φ(x, y)) − α0 (x, y) . Let us consider (consulting Remark 8 again) a sequence ς r of elements from Lpλ (Ω, C# (Y)n.n ) with the strong limit ς in Lpλ (Ω, Lpµ# (Y)n.n ) . It remains to prove that the norm of each element of this sequence in Lpλ (Ω, Lpµ# (Y)n.n ) is zero. Indeed, if this is true then the norm of ς must be zero, too, and α0 (x, y) = a(x, y, u(x), ∇u(x) + ∇µ¨ u1 (x, y))

Two-scale convergence with respect to measures in continuum mechanics

433

for λ - a. e. x ∈ Ω and for µ - a. e. y ∈ Y which yields (22). Applying the famous Minty trick (cf. [20], p. 261) φ(x, y) = U|ς r (x, y)|p−1 sgn ς r (x, y) with U ∈ {−1, 1} to (25), we obtain U |ς r (x, y)|p dµ(y) dλ(x) ≥ 0 Ω

Y

which (independently of r) forces ς r to be the zero point of Lpλ (Ω, Lpµ# (Y)n ). Now it is easy to ﬁnish this proof: ¿From the just veriﬁed two-scale convergence of αε we have lim a(x, x/ε, uε (x), ∇µε uε (x)) · ∇µε v(x) dµε (x) ε→0 Ω = a(x, y, u(x), ∇u(x) + ∇µ¨ u1 (x, y)) dµ(y) · ∇v(x) dλ(x) Ω

Y

which, making use of Lemma 10, demonstrates how the ﬁrst left-hand-side additive term from (16) tends to the corresponding one from (22) if ε → 0. The same for the second and third terms follows from the second and third equations of (e) (with vε unchanged).

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Two-scale convergence with respect to measures in continuum mechanics

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35. Svoboda J., Vala J., Micromodelling of Creep in Composites with Perfect Matrix / Particle Interfaces, Kovové materiály (Metallic Materials, Slovak Rep.) 36, No 2, 1998, 109-126. 36. Vala J., On One Approach to Numerical Modelling of Creep Flow in Crystalline Materials, in Proc. Prague Mathematical Conference (Acad. Sci. Czech Rep.), 1996, 325-330. 37. Vala J., On One Mathematical Model of Creep in Superalloys, Appl. Math., 43, No 5, 1998, 351-380. 38. Vala J., On Mathematical Modelling of Heat and Moisture Transfer through a Porous Medium (in Czech), in Proc. Programy a algoritmy numerické matematiky 10 (Acad. Sci. Czech Rep., L. Libverda), 2000, 188-194. 39. Vala J., Two-Scale Limits in Some Nonlinear Problems of Engineering Mechanics, in Modelling 2001 Abstracts (Univ. West Bohemia, Pilsen), 2001, 65-66, and Proc., to appear. 40. Vala J., Kozák V., Svoboda J., Čadek J., Modelling Discontinuous Metal Matrix Composite Under Creep Condition: Eﬀect of Diﬀusional Matter Transport and Interface Sliding, Scripta Metall. Mater. 30, No 9, 1994, 1201-1206. 41. Yosida K., Functional Analysis (in Russian), Mir, Moscow, 1967. 42. Zhikov V. V., Connectedness and Homogenization – Examples of Fractal Conductivity, Mat. Sbornik 187, No 8, 1996, 3-40. 43. Zhikov V. V., Kozlov S. M., Oleˇınik O. A., Homogenization of Diﬀerential Operators and Integral Functionals, Springer, Berlin, 1996. 44. Ziemer W., Weakly Diﬀerentiable Functions, Springer, Berlin, 1989.

Equadiﬀ 10, August 27–31, 2001 Prague, Czech Republic

Equadiﬀ 10 CD ROM Papers, pp. 437–441

Irregular boundary value problems for ordinary diﬀerential equations Yakov Yakubov School of Mathematical Sciences, Tel-Aviv University, 69978 Tel-Aviv, Israel Email: [email protected]

Abstract. Birkhoﬀ-irregular boundary value problems for quadratic ordinary diﬀerential pencils of the second order have been considered. The spectral parameter may appear in a boundary condition, the equation contains an abstract linear operator while the boundary conditions contain internal points of an interval and a linear functional. Isomorphism and coerciveness with a defect are proved for such problems. Two-fold completeness of root functions of corresponding spectral problems is also established. As an application of the obtained results, an initial boundary value problem for second order parabolic equations is considered, and the well-posedness and completeness of the elementary solutions are proved. These and some other results have been published in [1] . MSC 2000. 34L10,34B05,35K15 Keywords. Birkhoﬀ-irregular boundary value problems, isomorphism, completeness, well-posedness

1

Isomorphism and Two-fold Completeness

Consider a principally (because of the addition of B, T , and xji ) boundary value problem for ordinary diﬀerential equations L(λ)u := λ2 u(x) + λ [a1 u (x) + b1 u(x)] + [a2 u (x) + b2 u (x)] + + Bu|x = f (x), x ∈ (0, 1), (1)

Research supported in part by the Israel Ministry of Absorption

This is an overview article.

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 N0    L (λ)u := λ α u(0) + β u(1) + η u(x ) +  1 11 11 0i 0i    i=1     + [α10 u (0) + β10 u (1) + γ10 u(0) + δ10 u(1)] +    N1 N2  + η u (x ) + η2i u(x2i ) + T u = f1 ,  1i 1i    i=1 i=1     N3     L2 u := [α20 u(0) + β20 u(1)] + η3i u(x3i ) = f2 , 

(2)

i=1

where ak , bk , ανk , βνk , γ10 , δ10 , ηji , fν are complex numbers, f (x) is a given function; xji ∈ (0, 1); B is a linear operator in Lq (0, 1) and T is a linear functional in Lq (0, 1), a real q ∈ (1, ∞). We assume that a2 = 0 and denote by 1

ω1 :=

−a1 + (a21 − 4a2 ) 2 , 2a2

1

ω2 :=

−a1 − (a21 − 4a2 ) 2 2a2

the roots of the equation a2 ω 2 + a1 ω + 1 = 0, 1

1

where z 2 := |z| 2 ei Further,

arg z 2

, −π < arg z ≤ π. We also assume that arg ω1 = arg ω2 . ω := min{arg ω1 , arg ω2 + π}, ω := max{arg ω1 , arg ω2 + π},

and values arg ωj are chosen up to a multiple of 2π, so that ω − ω < π. Introduce now the following notations: α11 + α10 ω1 β11 + β10 ω2 , θ0 (ω1 , ω2 ) := α20 β20 −α10 b1 +b2 ω11 + γ10 β10 b1 +b2 ω21 + δ10 2 2 (a1 −4a2 ) 2 (a1 −4a2 ) 2 θ1 (ω1 , ω2 ) := . α20 β20 Definition 1. Problem (1)—(2) is called regular (regular with defect 1) with respect to the numbers ω1 , ω2 if: (1) a2 = 0, arg ω1 = arg ω2 ; (2) xji ∈ (0, 1), for some real q ∈ (1, ∞) the operator B from Wq1 (0, 1) into Lq (0, 1) is compact and the functional T is continuous in Lq (0, 1); (3) θ0 (ω1 , ω2 ) = 0 θ0 (ω1 , ω2 ) = 0, θ1 (ω1 , ω2 ) = 0 .

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Theorem 2. Let problem (1)—(2) be regular with defect 1 with respect to the numbers ω1 , ω2 . Then for any ε > 0 there exists Rε > 0 such that for all complex numbers λ that satisfy |λ| > Rε and lying inside the angle 3π π − ω + ε < arg λ < − ω − ε, 2 2 the operator L(λ) : u → L(λ)u := (L(λ)u, L1 (λ)u, L2 u) ˙ C2 is an isomorphism and for these λ the following from Wq2 (0, 1) onto Lq (0, 1) + estimate holds for a solution u(x) of problem (1)—(2) 2

|λ|

1−k

uWqk (0,1)

2 ν− 1q ≤ C(ε) f Lq (0,1) + |λ| |fν | . ν=1

k=0

Theorem 3. Let |αν0 |+|βν0 | = 0, ν = 1, 2 and let homogeneous problem (1)—(2) be regular with defect 1 with respect to the numbers ω1 , ω2 and regular, or regular with defect 1 with respect to the numbers ω2 , ω1 , for q = 2. Then the spectrum of homogeneous problem (1)—(2) is discrete and a system of its root functions (eigenfunctions and associated functions) is two-fold complete in the space H := {v | v := (v1 , v2 ) ∈ W21 (0, 1) ⊕ L2 (0, 1), L2 v1 = 0}.

2

Well-posedness and Completeness of Elementary Solutions

Consider, in [0, T ] × [0, 1], the following initial boundary value problem ut (t, x) + [auxx(t, x) + bux (t, x)] + Bu(t, ·)|x = f (t, x),  L1 u := [α10 ux (t, 0) + β10 ux (t, 1)] + [γ10 u(t, 0) + δ10 u(t, 1)] +     N1 N2     + η1i ux (t, x1i ) + η2i u(t, x2i ) + Qu(t, ·) = 0, i=1 i=1    N3     η3i u(t, x3i ) = 0,  L2 u := [α20 u(t, 0) + β20 u(t, 1)] +

(3)

(4)

i=1

u(0, x) = u0 (x).

(5)

Let E, E1 , and E2 be Banach spaces. Introduce two Banach spaces Cµ ((0, T ], E) := {f | f ∈ C((0, T ], E), f = sup tµ f (t) < ∞}, µ ≥ 0, t∈(0,T ]

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Cµγ ((0, T ], E) := {f | f ∈ C((0, T ], E), f = sup tµ f (t) + t∈(0,T ]

+

sup

f (t + h) − f (t)h−γ tµ < ∞}, µ ≥ 0, γ ∈ (0, 1],

0
and a linear space (in the case E1 ⊂ E2 ) C 1 ((0, T ], E1 , E2 ) := {f | f ∈ C((0, T ], E1 ) ∩ C 1 ((0, T ], E2 )}, where C((0, T ], E) and C 1 ((0, T ], E) are spaces of continuous and continuously diﬀerentiable, respectively, vector-functions from (0, T ] into E. Theorem 4. Let the following conditions be satisﬁed: (1) a = 0, | arg a| > π2 , α10 β20 + α20 β10 = 0, γ10 β20 − δ10 α20 − ab α10 β20 = 0; (2) the operator B from Wq1 (0, 1) into Lq (0, 1) is compact; (3) the functional Q is continuous in Lq (0, 1). (4) f ∈ Cµγ ((0, T ], Lq (0, 1)) for some γ ∈ ( 12 , 1] and µ ∈ [0, 12 ); (5) u0 ∈ Wq2 ((0, 1), Lν u = 0, ν = 1, 2). Then problem (3)—(5) has a unique solution u ∈ C([0, T ], Lq (0, 1)) ∩ C 1 ((0, T ], Wq2 (0, 1), Lq (0, 1)) and for the solution the following estimates hold u(t, ·)Lq (0,1) ≤ C(u0 Wq2 (0,1) + f Cµ ((0,t],Lq (0,1)) ), and u(t, ·)Wq2 (0,1) + u (t, ·)Lq (0,1) ≤ Ct−1 (u0 Wq2 (0,1) + f Cµγ ((0,t],Lq (0,1)) ), for t ∈ (0, T ]. Consider now a spectral problem coresponding to the homogeneus (3), (4): λu(x) + [au (x) + bu (x)] + Bu|x = 0, x ∈ (0, 1),

 L1 u := [α10 u (0) + β10 u (1)] + [γ10 u(0) + δ10 u(1)] +     N1 N2     + η1i u (x1i ) + η2i u(x2i ) + Qu = 0, i=1 i=1    N3     η3i u(x3i ) = 0,  L2 u := [α20 u(0) + β20 u(1)] +

(6)

(7)

i=1

A function of the form kj t tkj −1 λj t uj (t, x) = e uj0 (x) + uj1 (x) + · · · + ujkj (x) kj ! (kj − 1)!

(8)

becomes an elementary solution of the homogeneus (3), (4) if and only if a system of the functions uj0 (x), uj1 (x), . . . , ujkj (x) is a chain of root functions of problem (6)—(7), corresponding to the eigenvalue λj .

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Theorem 5. Let conditions of the previous theorem be satisﬁed with q = 2. Then problem (3)—(5) (with f (t, x) ≡ 0) has a unique solution u ∈ C([0, T ], L2(0, 1)) ∩ C 1 ((0, T ], W22 (0, 1), L2 (0, 1)) and there exist numbers cjn such that lim

sup u(t, ·) −

n→∞ t∈(0,T ]

lim

sup t(ut (t, ·) −

n→∞ t∈(0,T ]

n

n

cjn uj (t, ·)L2 (0,1) = 0,

j=1

cjn ujt (t, ·)L2 (0,1) +

j=1

+ u(t, ·) −

n

cjn uj (t, ·)W22 (0,1) ) = 0,

j=1

where u(t, x) is a solution to problem (3)—(5) (with f (t, x) ≡ 0) and uj (t, x) is an elementary solution (8) of homogeneus problem (3)—(4).

References 1. Yakubov Ya., Irregular Boundary Value Problems for Ordinary Diﬀerential Equations, Analysis, 18 1998, 359–402.

Equadiﬀ 10, August 27–31, 2001 Prague, Czech Republic

Equadiﬀ 10 CD ROM Papers, pp. 443–460

Singular Solutions of the Briot-Bouquet Type Partial Diﬀerential Equations Hiroshi Yamazawa Department of Language and Culture, Caritas College, 2-29-1 Azamino, Aoba-ku, Yokohama, Japan, Email: [email protected]

Abstract. In 1990, Gérard-Tahara [2] introduced the Briot-Bouquet type partial diﬀerential equation t∂t u = F (t, x, u, ∂x u), and they determined the structure of singular solutions provided that the characteristic exponent ρ(x) satisﬁes ρ(0) ∈ {1, 2, . . . }. In this paper the author determines the structure of singular solutions in the case ρ(0) ∈ {1, 2, . . . }. MSC 2000. 35A20, 35C10 Keywords. Singular solutions, Characterirtic exponent

1

Introduction

In this paper, we will study the following type of nonlinear singular ﬁrst order partial diﬀerential equations: t∂t u = F (t, x, u, ∂x u)

(1)

∂ ∂ , ∂i = ∂t ∂xi for i = 1, . . . , n, and F (t, x, u, v) with v = (v1 , . . . , vn ) is a function deﬁned in a polydisk 8 centered at the origin of Ct × Cnx × Cu × Cnv . Let us denote 80 = 8 ∩ {t = 0, u = 0, v = 0}. The assumptions are as follows: where (t, x) = (t, x1 , . . . , xn ) ∈ Ct × Cnx , ∂x u = (∂1 u, . . . , ∂n u), ∂t =

(A1) F (t, x, u, v) is holomorphic in 8, (A2) F (0, x, 0, 0) = 0 in 80 , ∂F (0, x, 0, 0) = 0 in 80 for i = 1, . . . , n. (A3) ∂vi

This is the ﬁnal form of the paper.

(2)

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Definition 1. ([2], [3]) If the equation (1) satisﬁes (A1), (A2) and (A3) we say that the equation (1) is of Briot-Bouquet type with respect to t. Definition 2. ([2], [3]) Let us deﬁne ρ(x) =

∂F (0, x, 0, 0), ∂u

(3)

then the holomorphic function ρ(x) is called the characteristic exponent of the equation (1). Let us denote by 1. R(C\{0}) the universal covering space of C\{0}, 2. Sθ = {t ∈ R(C\{0}); | arg t| < θ}, 3. S (U(s)) = {t ∈ R(C\{0}); 0 < |t| < U(arg t)} for some positive-valued function U(s) deﬁned and continuous on R, 4. DR = {x ∈ Cn ; |xi | < R for i = 1, . . . , n}, 5. C{x} the ring of germs of holomorphic functions at the origin of Cn . "+ of all functions u(t, x) satisfying the following Definition 3. We deﬁne the set O conditions; 1. u(t, x) is holomorphic in S (U(s)) × DR for some U(s) and R > 0, 2. there is an a > 0 such that for any θ > 0 and any compact subset K of DR max |u(t, x)| = O (|t|a ) x∈K

as

t → 0 in

Sθ .

(4)

We know some results on the equation (1) of Briot-Bouquet type with respect to t. We concern the following result. Gérard R. and Tahara H. studied in [2] the structure of holomorphic and singular solutions of the equation (1) and proved the following result; Theorem 4 (Gérard R. and Tahara H.). If the equation (1) is Briot-Bouquet type and ρ(0) ∈ N∗ = {1, 2, 3, . . . } then we have; (1) (Holomorphic solutions) The equation (1) has a unique solution u0 (t, x) holomorphic near the origin of C × Cn satisfying u0 (0, x) ≡ 0. "+ -solutions of (1). (2) (Singular solutions) Denote by S+ the set of all O {u0 (t, x)} when Reρ(0) ≤ 0, (5) S+ = {u0 (t, x)} ∪ {U (ϕ); 0 = ϕ(x) ∈ C{x}} when Reρ(0) > 0, "+ -solution of (1) having an expansion of the following form: where U (ϕ) is an O U (ϕ) = ui (x)ti + ϕi,j,k (x)ti+jρ(x) (log t)k , ϕ0,1,0 (x) = ϕ(x). (6) i≥1

i+2j≥k+2,j≥1

445

Singular Solutions of the Briot-Bouquet Type

In the case ρ(0) ∈ N∗ , Yamane [7] showed that the equation (1) has a holomolphic solution in a region {(t, x) ∈ C × Cn ; |x| < c|t|d 7 1} for some c > 0 and d > 0, but the solution is not in S+ . The purpose of this paper is to determine S+ in the case ρ(0) ∈ N∗ . The following main result of this paper is; Theorem 5. If the equation (1) is Briot-Bouquet type and if ρ(0) = N ∈ N∗ and ρ(x) ≡ ρ(0), then (7) S+ = {U (ϕ); ϕ(x) ∈ C{x}} , "+ -solution of (1) having an expansion of the following form: where U (ϕ) is an O U (ϕ) = u01 (x)t + ue00 (x)φN (t, x) + uβi (x)ti ΦβN i+|β|≥2,|β|<∞,

+

0 w0,1,0 (x)tρ(x)

+

|β|∗ ≤i+|β|−2

i+j+|β|≥2,

k≤i+|β|0 +|β|1

|β|<∞,j≥1,

+2(j−1)

β wi,j,k (x)ti+jρ(x) {log t}k ΦβN ,

|β|∗ ≤i+j+|β|−2 0 (x) = ϕ(x) is arbitrary holomorphic function and the other where u0N (x) ≡ 0, w0,1,0 β β 0 coeﬃcients ui (x), wi,j,k (x) are holomorphic functions determined by w0,1,0 (x) and deﬁned in a common disk, and

l = (l1 , . . . , ln ) ∈ Nn , |l| = l1 + · · · + ln , β = (βl ∈ N; l ∈ Nn ), |β| = βl , |β|p = βl for p ≥ 0, |β|∗ = (|l| − 1)βl , |l|≥0

ΦβN

|l|=p

|l|≥2

< ∂ l φN βl tρ(x) − tN x . = , ∂xl = ∂1l1 · · · ∂nln , φN (t, x) = l! ρ(x) − N |l|≥0

The following lemma will play an important role in the proof of Theorem 5. At ﬁrst, we deﬁne some notations. We denote for l ∈ Nn , el = (βk ; k ∈ Nn ) with βl = 1 and βk = 0 for k = l and for p ∈ N, e(p) = (i1 , . . . , in ) with ip = 1 and iq = 0 for q = p, and denote that l1 < l0 is deﬁned by |l1 | < |l0 | and li1 ≤ li0 for i = 1, . . . , n. Lemma 6. Let ρ(x), φN and ΦβN be in Theorem 5. Then we have; β−e +e 1. ∂p ΦβN = |l|≥0 βl (lp + 1)ΦN l l+e(p) for i = 1, . . . , n, 2. t∂t φN = ρ(x)φN + tN , 0 1 ∂ l −l ρ(x) β−e +e 0 3. t∂t ΦβN = |β|ρ(x)ΦβN + β0 tN Φβ−e + |l0 |≥1 l1
446

H. Yamazawa

2. By t∂t φN = (ρ(x)tρ(x) − N tN )/(ρ(x) − N ), we have the result 2. 3. By 2, we have l l β β −1 ∂x φN l ∂x φN l ∂xl (ρ(x)φN + tN ) . t∂t = βl l! l! l!

(8)

Therefore we have l β ∂x φN l t∂t = l!   β0 ρ(x)φβN0 + β0 tN φβN0 −1 if l = 0 l βl −1 l βl 1 1 = ∂xl−l ρ(x) ∂xl φN ∂x φN ∂x φN  βl φ(x) + 0≤l1 0. l! l! Hence we have the desired result. Q.E.D.

2

Construction of formal solutions in the case ρ(0) = 1

By [2] (Gérard-Tahara), if the equation (1) is of Briot-Bouquet type with respect to t, then it is enough to consider the following equation: Lu = t∂t u − ρ(x)u = a(x)t + G2 (x)(t, u, ∂x u)

(9)

where ρ(x) and a(x) are holomorphic functions in a neighborhood of the origin, and the function G2 (x)(t, X0 , X1 , . . . , Xn ) is a holomorphic function in a neighborhood of the origin in Cnx × Ct × CX0 × CX1 × · · · × CXn with the following expansion: G2 (x)(t, X0 , X1 , . . . , Xn ) = ap,α (x)tp {X0 }α0 {X1 }α1 · · · {Xn }αn (10) p+|α|≥2

and we may assume that the coeﬃcients {ap,α (x)}p+|α|≥2 are holomorphic functions on DR for a suﬃciently small R > 0. We put Ap,α (R) := maxx∈DR |ap,α (x)| for p + |α| ≥ 2. Then for 0 < r < R Ap,α (R) tp X0α0 X1α1 × · · · × Xnαn (11) (R − r)p+|α|−2 p+|α|≥2

is convergent in a neighborhood of the origin. In this section, we assume ρ(0) = 1 and ρ(x) ≡ 1 and we will construct formal solutions of the equation (9). Proposition 7. If ρ(0) = 1 and ρ(x) ≡ 1, the equation (9) has a family of formal solutions of the form: uβi (x)ti Φβ1 (12) u = ue00 (x)φ1 + m≥2 i+|β|=m |β|∗ ≤m−2 0 (x)tρ(x) + + w0,1,0

m≥2

i+j+|β|=m

k≤i+|β|0 +|β|1

j≥1,|β|∗ ≤m−2

+2(j−1)

β wi,j,k (x)ti+jρ(x) {log t}k Φβ1

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Singular Solutions of the Briot-Bouquet Type

0 where w0,1,0 (x) is an arbitrary holomorphic function and the other coeﬃcients β β 0 ui (x), wi,j,k (x) are holomorphic functions determined by w0,1,0 (x) and deﬁned in a common disk.

Remark 8. By the relation |β|∗ ≤ m − 2 in summations of the above formal solution, we have βl = 0 for any l ∈ Nn with |l| ≥ m. We deﬁne the following two sets Um and Wm for m ≥ 1 to prove Proposition 7. Definition 9. We denote by Um the set of all functions um of the following forms: u1 = u01 (x)t + ue00 (x)φ1 , uβi (x)ti Φβ1 for m ≥ 2, um =

(13)

i+|β|=m |β|∗ ≤m−2

and denote by Wm the set of all functions wm of the following forms: 0 (x)tρ(x) , w1 = w0,1,0 wm = i+j+|β|=m

β wi,j,k (x)ti+jρ(x) {log t}k Φβ1 for m ≥ 2

k≤i+|β|0 +|β|1

j≥1,|β|∗ ≤m−2

+2(j−1)

β where uβi (x), wi,j,k (x) ∈ C{x}.

We can rewrite the formal solution (12) as follows: u=

(um + wm ) where um ∈ Um , wm ∈ Wm .

(14)

m≥1

Let us show important relations of um and wm for m ≥ 2. By Lemma 6, we have ∂p um =

m−1 β−e +e ∂p uβi (x)ti Φβ1 + (lp + 1)βl uβi (x)ti Φ1 l l+e(p) , |l|=0

i+|β|=m |β|∗ ≤m−2

∂p wm =

β ∂p wi,j,k (x)ti+jρ(x) {log t}k Φβ1

i+j+|β|=m

k≤i+|β|0 +|β|1

j≥1,|β|∗ ≤m−2

(15)

+2(j−1) β +j∂p ρ(x)wi,j,k (x)ti+jρ(x) {log t}k+1 Φβ1

+

m−1

β−el +el+e(p)

β (lp + 1)βl wi,j,k (x)ti+jρ(x) {log t}k Φ1

|l|=0

448

H. Yamazawa

for p = 1, . . . , n, and we have 0 {i + (|β| − 1)ρ(x)}uβi (x)ti Φβ1 + β0 uβi (x)ti+1 Φβ−e Lum = 1

(16)

i+|β|=m |β|∗ ≤m−2

+

0

1

{i + (j + |β| − 1)ρ(x)}

i+j+|β|=m

k≤i+|β|0 +|β|1

j≥1,|β|∗ ≤m−2

∂ l −l ρ(x) β i β−el0 +el1 , u βl 0 x 0 (x)t Φ 1 i (l − l1 )! 0

|l0 |=1 l1

Lwm =

m−1

+2(j−1) β ×wi,j,k (x)ti+jρ(x) {log t}k Φβ1

β β 0 + kwi,j,k (x)ti+jρ(x) {log t}k−1 Φβ1 + β0 wi,j,k (x)ti+jρ(x)+1 {log t}k Φβ−e 1

+

m−1

|l0 |=1 l1
∂xl −l ρ(x) β β−e +e wi,j,k (x)ti+jρ(x) {log t}k Φ1 l0 l1 . 0 1 (l − l )! 0

βl 0

1

We show two lemma. Lemma 10. If um ∈ Um and wm ∈ Wm , then Lum ∈ Um and Lwm ∈ Wm . Proof. We prove Lum ∈ Um . We will see all powers of each terms in (16). For the second term in (16), we have i+1+|β −e0 | = i+|β| = m and [β −e0 ] = [β] ≤ m−2. For the third term, we have i+|β−el0 +el1 | = i+|β| = m and [β−el0 +el1 ] = [β] (if |l0 | = 1), = [β] − (|l0 | − 1) (if |l0 | > 1 and |l1 | ≤ 1), = [β] − |l0 | + |l1 | (if |l0 | > 1 and |l1 | > 1). Further by l1 < l0 , we have [β − el0 + el1 ] ≤ [β] ≤ m − 2. Hence we have Lum ∈ Um . We can prove Lwm ∈ Wm as Lum ∈ Um , and we omit the details. Q.E.D. Lemma 11. If um ∈ Um and wm ∈ Wm , then the following relations hold by the relation (15) for i, j = 1, . . . , n 1. a(x)Um ⊂ Um and a(x)Wm ⊂ Wm for any holomorphic function a(x), 2. tUm , φ1 Um ⊂ Um+1 and tρ(x) Um , tWm , tρ(x) Wm , φ1 Wm ⊂ Wm+1 , 3. um × un , ∂i um × ∂j un , ∂i um × un ∈ Um+n , 4. wm × wn , ∂i wm × ∂j wn , ∂i wm × wn , ∈ Wm+n , 5. um × wn , ∂i um × wn , um × ∂j wn , ∂i um × ∂j wn ∈ Wm+n . Proof. This is veriﬁed by the relations (15) and (16) but tedious calculations. We may omit the details. Q.E.D. Let us show that um and wm are determined inductively on m ≥ 1. By substituting (um + wm ) into (9), we have m≥1

(1 − ρ(x))u01 (x) + ue00 (x) = a(x),

(17)

449

Singular Solutions of the Briot-Bouquet Type

for m ≥ 2 Lum =

α0 <

ap,α (x)tp

αj n < <

um0,h0

∂j umj,hj ,

(18)

j=1 hj =1

h0 =1

p+|α|≥2 p+|mn |=m

Lwm =

ap,α (x)tp

α0 <

(um0,h0 + wm0,h0 )

∂j (umj,hj + wmj,hj )

j=1 hj =1

h0 =1

p+|α|≥2

αj n < <

p+|mn |=m

−

ap,α (x)t

p

p+|α|≥2

α0 <

um0 ,h0

αj n < <

∂j umj,hj ,

(19)

j=1 hj =1

h0 =1

p+|mn |=m

n where |mn | = i=0 mi (αi ) and mi (αi ) = mi,1 + · · · + mi,αi for i = 0, 1, . . . , n. 0 (x) = ϕ(x), and We take any holomorphic function ϕ(x) ∈ C{x} and put w0,1,0 e0 0 by (17), we put u1 (x) ≡ 0 and u0 (x) = a(x). For m ≥ 2, let us show that um and wm are determined by induction. By Lemma 11, the right side of (18) belongs to Um and the right side of (19) belongs to Wm . Further by mj,hj ≥ 1, we have mj,hj < m for hj = 1, . . . , αj and j = 0, . . . , n. Then for m ≥ 2, we compare with the coeﬃcients of ti Φβ1 and ti+jρ(x) {log t}k Φβ1 respectively for (18) and (19), then put {i + (|β| − 1)ρ(x)}uβi (x) 0 + (β0 + 1)uβ+e i−1 (x) +

(20)

m−1

0

(βl0 + 1)

|l0 |=1 0≤l1
=

fiβ ({ap,α }2≤p+|α|≤m ,

1

∂xl −l ρ(x) β+el0 −el1 u (x) (l0 − l1 )! i

{uβi (x)}i +|β |<m )

and β β (x) + (k + 1)wi,j,k+1 (x) {i + (j + |β| − 1)ρ(x)}wi,j,k β+e0 + (β0 + 1)wi−1,j,k (x) +

m−1

|l0 |=1 0≤l1
0

(βl0 + 1)

1

∂xl −l ρ(x) β+el0 −el1 w (x) (l0 − l1 )! i,j,k

(21)

β = gi,j,k ({ap,α }2≤p+|α|≤m , {uβi (x)}i +|β |<m , {wiβ ,j ,k (x)}i +j +|β |<m ).

We deﬁne an order for the multi indices (i, β) and (i, j, k, β) to show that uβi (x) β and wi,j,k (x) are determined by (20) and (21). Definition 12. The relation (i , β ) < (i, β) is deﬁned by the following orders; 1. i + |β | < i + |β|. 2. If i + |β | = i + |β|, then i < i.

450

H. Yamazawa

3. If i + |β | = i + |β| and i = i, then |β |0 < |β|0 . 4. If i + |β | = i + |β|, i = i, |β |0 = |β|0 , . . . , |β |l = |β|l , then |β |l+1 < |β|l+1 . The relation (i , j , k , β ) < (i, j, k, β) is deﬁned by the following orders; 1. i + j + |β | < i + j + |β|. 2. If i + j + |β | = i + j + |β|, then i < i. 3. If i + j + |β | = i + j + |β| and i = i, then j < j. 4. If i + j + |β | = i + j + |β|, i = i and j = j, then |β |0 < |β|0 . 5. If i + j + |β | = i + j + |β|, i = i, j = j, |β |0 = |β|0 , . . . , |β |l = |β|l , then |β |l+1 < |β|l+1 . 6. If (i , j , β ) = (i, j, β), then k > k. For m ≥ 2, we have i + (|β| − 1)ρ(x) = 0 and i + (j + |β| − 1)ρ(x) = 0 by ρ(0) = 1. β (x) are determined in the order of Therefore all the coeﬃcients uβi (x) and wi,j,k Deﬁnition 12. Hence we obtain Proposition 7. Q.E.D.

3

Convergence of the formal solutions in the case ρ(0) = 1

"+ . In this section, we show that the formal solution (12) converges in O Proposition 13. Let γ satisfy 0 < γ < 1 and let λ be suﬃciently large. Then for any suﬃciently small r > 0 we have the following result; For any θ > 0 there is an U > 0 such that the formal solution (12) converges in the following region: {(t, x) ∈ Ct × Cnx ; |η(t, λ)t| < U, |η(t, λ)2 tρ(x) | < U, |η(t, λ)tγ | < U, t ∈ Sθ and x ∈ Dr } , where η(t, λ) = max {|(log t)/λ| , 1} . β β In this section, we put wi,0,0 (x) := uβi (x) and wi,0,k (x) ≡ 0 for k ≥ 1 in the formal solution (12). Then the formal solution (12) is as follows: e0 0 u = w0,0,0 (x)φ1 + w0,1,0 (x)tρ(x) β + wi,j,k (x)ti+jρ(x) {log t}k Φβ1 .

(22)

m≥2 i+j+|β|=m k≤i+|β|0 +|β|1 |β|∗ ≤m−2

+2(j−1)

Let us deﬁne the following set Vm for (22). Definition 14. We denote by Vm the set of all the functions vm of the following forms: e0 0 v1 = w0,0,0 (x)φ1 + w0,1,0 (x)tρ(x) , β vm = wi,j,k (x)ti+jρ(x) {log t}k Φβ1 i+j+|β|=m k≤i+|β|0 +|β|1 |β|∗ ≤m−2

+2(j−1)

(23) for m ≥ 2.

451

Singular Solutions of the Briot-Bouquet Type

We deﬁne the following estimate for the function vm . Definition 15. For the function (23), we deﬁne ||v1 ||r,c,λ = ||v1 ||r,c :=

||vm ||r,c,λ :=

e0 ||w0,0,0 ||r 0 + ||w0,1,0 ||r , c β ||wi,j,k ||r λk

i+j+|β|=m k≤i+|β|0 +β1 |β|∗ ≤m−2

(24) for

c<β>

m≥2

+2(j−1)

for c > 0 and λ > 0, where β β ||wi,j,k ||r = max |wi,j,k (x)| and < β >= x∈Dr

(|l| + 1)βl .

(25)

|l|≥0

We will make use of Lemma 16. For a holomorphic function f (x) on DR , we have ||∂xα f ||R0 ≤

α! ||f ||R (R − R0 )|α|

for

0 < R0 < R.

(26)

Proof. By Cauchy’s integral formula, we have the desired result, and we omit the details. Q.E.D Lemma 17. If a holomorphic function f (x) on DR satisﬁes ||f ||R0 ≤

C (R − r)p

for

0
(27)

then we have ||∂i f ||R0 ≤

Ce(p + 1) (R − r)p+1

for

0 < r < R,

i = 1, . . . , n.

(28)

For the proof, see Hörmander ([5]lemma 5.1.3) Let us show the following estimate for the function Lvm . Lemma 18. Let 0 < R0 < R. Then there exists a positive constant σ such that for m ≥ 2, if vm ∈ Vm we have ||Lvm ||r,c,λ ≥

σ m||vm ||r,c,λ 2

for

0 < r ≤ R0

for suﬃciently small c > 0 and suﬃciently large λ > 0.

(29)

452

H. Yamazawa

Proof. Let us give an estimate the second, the third and the fourth term in the right side of the second relation in (16) respectively. For the second term, since k ≤ i + |β|0 + |β|1 + 2(j − 1) ≤ 2m by i + j + |β| = m we have

T2 :=

k

β ||wi,j,k+1 ||r λk−1

c<β>

i+j+|β|=m k≤i+|β|0 +|β|1 |β|∗ ≤m−2

≤

2m ||vm ||r,c,λ . λ

(30)

+2(j−1)

For the fourth term, we have

T4 :=

m−1

i+j+|β|=m k≤i+|β|0 +|β|1 |l0 |=1 l1
+2(j−1)

≤

m−1

0

0

c|l

0

|−|l1 |

βl 0

i+j+|β|=m k≤i+|β|0 +|β|1 |l0 |=1 l1
1

β ||r λk ||∂xl −l ρwi,j,k βl 0 (l0 − l1 )! c<β−el0 +el1 >

(31)

1

β k ||∂xl −l ρ||R0 ||wi,j,k ||r λ . (l0 − l1 )! c<β>

+2(j−1)

By Lemma 16, we have l1
l0 −l1 ρ||R0 |l0 |−|l1 | ||∂x c (l0 − l1 )!

|l0 |−|l1 | c ||ρ||R ≤ R − R0 l1
(32)

for suﬃciently small c > 0. Therefore by (31) and (32), we have T4 ≤ κ(c)

i+j+|β|=m k≤i+|β|0 +|β|1 |β|∗ ≤m−2

m−1

βl 0

|l0 |=1

β ||wi,j,k ||r λk

c<β>

+2(j−1)

cn ( R−R0 )n ||ρ||R . where κ(c) := R−R 0 R−R0 −c For the third term, we have

T3 : =

β0

β ||wi,j,k ||r λk

i+j+|β|=m k≤i+|β|0 +|β|1 |β|∗ ≤m−2

=

+2(j−1)

i+j+|β|=m k≤i+|β|0 +|β|1 |β|∗ ≤m−2

c<β−e0 >

+2(j−1)

cβ0

β ||wi,j,k ||r λk

c<β>

.

(33)

453

Singular Solutions of the Briot-Bouquet Type

m−1 Therefore, since cβ0 + κ(c) |l0 |=1 βl0 ≤ σ3 m by the conditions κ(0) = 0 and i + j + |β| = m ≥ 2 for suﬃciently small c > 0 and some σ > 0 we have 2m σ T2 + T3 + T4 ≤ + m ||vm ||r,c,λ . (34) λ 3 Further we have |i + (j + |β| − 1)ρ(x)| ≥ σm by the condition ρ(0) = 1 and i + j + |β| = m ≥ 2. Therefore we have 2m σ ||Lvm ||r,cλ ≥ σm − − m ||vm ||r,c,λ . (35) λ 3 Hence for suﬃciently small c > 0 and suﬃciently large λ > 0, we obtain the desired result. Q.E.D. Let us estimate the function ∂i vm . Definition 19. For the function vm ∈ Vm we deﬁne β Dp vm := ∂p wi,j,k (x)ti+jρ(x) {log t}k Φβ1

(36)

i+j+|β|=m k≤i+|β|0 +|β|1 |β|∗ ≤m−2

+2(j−1)

for p = 1, . . . , n. Lemma 20. If vm ∈ Vm , then for i = 1, . . . , n, we have 3m − 2 ||vm ||r,c,λ ||∂i vm ||r,c,λ ≤ ||Di vm ||r,c,λ +c0 λm||vm ||r,c,λ + c Proof. We have

(lp + 1)βl ≤

|l|≥0

m−1

for

(|l| + 1)βl = 2|β| + [β] ≤ 3m − 2.

0 < r ≤ R0 . (37)

(38)

|l|=0

We put c0 = max {||∂i ρ||R0 }, and by the relations (15), (38) and j ≤ m we obi=1,...,n

tain the desired estimate. Q.E.D. Therefore by the relations (18), (19) and Lemma 18, 20, we have the following lemma. Lemma 21. If u = vm is a formal solution of the equation (9) constructing m≥1

in Section 2, we have the following inequality for vm (m ≥ 2): ||Lvm ||r,c,λ α0 < ≤ ||ap,α ||r ||vm0,h0 ||r,c,λ p+|α|≥2

h0 =1

p+|mn |=m

×

αi n < <

{||Di vmi,hi ||r,c,λ + c0 λmi,hi ||vmi,hi ||r,c,λ +

i=1 hi =1

3mi,hi − 2 ||vmi,hi ||r,c,λ }. c

454

H. Yamazawa

Let us deﬁne a majorant equation to show that the formal solution (22) converges. We take A1 so that e0 ||w0,0,0 ||R 0 + ||w0,1,0 ||R ≤ A1 , c e0 ||∂i w0,0,0 ||R 0 + ||∂i w0,1,0 ||R ≤ A1 c

for i = 1, . . . , n. Then we consider the following equation: σ σ Y = A1 t1 2 2 1 + R−r

(39)

p+|α|≥2

αi n < Ap,α (R) 3 p α0 Y t Y λY + . eY + c 1 0 c (R − r)p+|α|−2 i=1

The equation (39) has a unique holomorphic solution Y = Y (t1 ) with Y (0) = 0 at (Y, t1 ) = (0, 0) by implicit function theorem. By an easy calculation, the solution Y = Y (t1 ) has the following form: Y =

Ym t1 m with Ym =

m≥1

Cm (R − r)m−1

(40)

where Y1 = C1 = A1 and Cm ≥ 0 for m ≥ 1. Then we have; Lemma 22. For m ≥ 1, we have m||vm ||r,c,λ ≤ Ym

for

0 < r ≤ R0

(41)

||Di vm ||r,c,λ ≤ eYm

for

0 < r ≤ R0 ,

(42)

for i = 1, . . . , n. Proof. By A1 = Y1 and the deﬁnition of A1 , (41) and (42) hold for m = 1. By induction on m,let us show that (41) and (42) hold for m ≥ 2. By substituting the solution Y = m≥1 Ym t1 m into the equation (39), we have the following relation:

σ 1 Ym = 2 R−r

p+|α|≥2

α0 < Ap,α (R) Ym0,h0 (R − r)p+|α|−2 h0 =1

p+|mn |=m

×

n < αi < i=1 hi =1

eYmi,hi + c0 λYmi,hi

3 + Ymi,hi c

(43)

455

Singular Solutions of the Briot-Bouquet Type

for m ≥ 2. Therefore if we assume that (41) and (42) hold for mi,hi < m, by (43), Lemma 18 and Lemma 21 we obtain σ σ m||vm ||r,c,λ ≤ (R − r) Ym . 2 2

(44)

m||vm ||r,c,λ ≤ (R − r)Ym ≤ Ym .

(45)

Therefore we have The relation (45) is rewrited as follows:

m

β ||wi,j,k ||r λk

c<β>

i+j+|β|=m k≤i+|β|0 +|β|1 |β|∗ ≤m−2

≤

Cm . (R − r)m−2

(46)

+2(j−1)

By (46) and Lemma 17, we have m||Di vm ||r,c,λ ≤

(m − 1)eCm (R − r)m−1

(47)

for i = 1, . . . , n and 0 < r < R < 1. Therefore we have ||Di vm ||r,c,λ ≤

eCm = eYm . (R − r)m−1

(48)

Hence (41) and (42) hold for m ≥ 2. Q.E.D. Let us show that the formal solution (22) converges by using (41) in Lemma 22. We put (22) as follows: 0 (x)tρ(x) u = ue00 (x)φ1 + w0,1,0

+

m≥2 i+j+|β|=m k≤i+|β|0 +|β|1 |β|∗ ≤m−2

β wi,j,k (x)λk

c<β>

ti+jρ(x)

log t λ

k Ψ1β ,

+2(j−1)

where Ψ1β =

β < ∂ l φ1 l . c|l|+1 x l!

(49)

|l|≥0

Firstly let us estimate (49). For ||φ1 ||R , we have the following lemma. Lemma 23. For any γ with 0 < γ < 1, there is an R > 0 such that ||φ1 ||R = O (|t|γ ) as t → 0 in Sθ holds for any θ > 0.

(50)

456

H. Yamazawa

Proof. We put φ1 = tγ

tρ0 (x)+α − tα ρ0 (x)

(51)

with α + γ = 1 and ρ0 (x) = ρ(x) − 1. Then we can take R > 0 with ||ρ0 ||R < α

(52)

by ρ0 (0) = 0. Therefore we have ρ (x)+α t 0 − tα α−||ρ0 ||R →0 ≤ | log t||t| ρ0 (x) R

as t → 0 in

Sθ

(53)

for and any θ > 0. Hence we have the desired result. Q.E.D. By Lemma 23, there exists a positive constant c1 such that ||φ1 ||R ≤ c1 |t|γ

in Sθ .

(54)

By Lemma 16 and (54), for |l| ≥ 0 we have ||∂xl φ1 ||R0 ≤

l! l!c1 ||φ1 ||R ≤ |t|γ (R − R0 )|l| (R − R0 )|l|

for

0 < R0 < R.

(55)

Therefore, we have ||Ψ1β ||R0

< c|l|+1 ≤ |l|≥0

βl c1 γ |t| ≤ (R − R0 )|l|

c R − R0

<β>

(c1 (R − R0 )|t|γ )|β| (56)

for 0 < R0 < R in Sθ .

k log t Let us estimate t Ψ1β . λ t c , 1 , c = max , 1 and c3 = c1 (R − R0 ). We put η(t, λ) = max log 2 λ R−R0 Since we have < β >≤ 2|β| + |β|∗ ≤ i + j + 3|β| (57) i+jρ(x)

and k ≤ i + |β|0 + |β|1 + 2(j − 1) ≤ i + |β| + 2j,

(58)

we obtain k i+jρ(x) log t Ψ1β ≤ t λ r

≤ {|c2 η(t, λ)t|}

i

j

|β| ||c2 η(t, λ)2 tρ(x) ||r |(c2 )3 c3 η(t, λ)tγ |

in Sθ . For any suﬃciently small U > 0, there exists a suﬃciently small |t| in Sθ such that |c2 η(t, λ)t| < U, ||c2 η(t, λ)2 tρ(x) ||r < U, |(c2 )3 c3 η(t, λ)tγ | < U,

(59)

457

Singular Solutions of the Briot-Bouquet Type

i+jρ(x) log t t Ψ1β ≤ Um . λ

and we obtain

(60)

r

Then by Lemma 22, we have ||u||r ≤

Ym Um

(61)

m≥1

for suﬃciently small |t| in Sθ . Hence the formal solution (22) converges for x ∈ Dr and suﬃciently small |t| in Sθ . Q.E.D.

4

Completion of the proof of Theorem 5 in the case ρ(0) = 1

In this section, let us complete the proof of Theorem 5 in the case ρ(0) = 1. We know the following theorem. "+ (i = 1, 2) are solutions of (9), we have; Theorem 24. If ui (t, x) ∈O "+ . 1. For any a < ρ(0) = 1, we have t−a (u1 − u2 ) ∈O −b " "+ . 2. If t (u1 − u2 ) ∈O+ for some b ≥ ρ(0) = 1, we have u1 (t, x) = u2 (t, x) in O For the proof, see Gérard and Tahara ([2] Theorem 3). By the discussions in sections 2, 3 and 4, we already know the following results; (C1) If ρ(0) = 1 and ρ(x) ≡ 1, for any ϕ(x) ∈ C{x}, the equation (1) has a "+ -solution U (ϕ)(t, x) having an expansion of the form unique O e0 0 U (ϕ) = w0,0,0 (x)φ1 + w0,1,0 (x)tρ(x) + uβi (x)ti Φβ1 (62) m≥2 i+|β|=m

+

m≥2

|β|∗ ≤m−2

i+j+|β|=m

k≤i+|β|0 +|β|1

j≥1,|β|∗ ≤m−2

β wi,j,k (x)ti+jρ(x) {log t}k Φβ1

+2(j−1)

β 0 with w0,1,0 (x) = ϕ(x), where all the coeﬃcients uβi (x), wi,j,k (x) are holomorphic n in a common disk centered at the origin of Cx . If we take ϕ(x) = 0, then the solution u0 (t, x) has the expansion u0 (t, x) = U (0) = ue00 (x)φ1 + uβi (x)ti Φβ1 . (63) m≥2 i+|β|=m |β|∗ ≤m−2

"+ of the equation (1) (C2) If ρ(0) = 1 and ρ(x) ≡ 1, and if a solution u(t, x) ∈ O is expressed in the form "+ , (64) t−1 u(t, x) − ue00 (x)φ1 (t, x) − ϕ(x)tρ(x) ∈ O

458

H. Yamazawa

then the coeﬃcient ue00 (x) is uniquely determined by the equation (1), and they are independent of ϕ(x). If ρ(0) = 1 and ρ(x) ≡ 1, by (C1) we have S+ ⊃ {U (ϕ); ϕ(x) ∈ C{x}} .

(65)

Hence it is suﬃcient to prove the following proposition to complete the proof of the main theorem. Proposition 25. Assume (A1), (A2) and (A3). Let u0 (t, x) and U (ϕ)(t, x) be as above. If ρ(0) = 1 and ρ(x) ≡ 1, and if u(t, x) ∈ S+ , then we can ﬁnd a "+ . ϕ(x) ∈ C{x} such that u(t, x) ≡ U (ϕ)(t, x) holds in O The proof of this proposition is almost the same as that of Proposition 2 in Gérard and Tahara [1]; so we may omit the details. Q.E.D. By (65) and Proposition 25 we obtain the main theorem 5 in the case ρ(0) = 1 and ρ(x) ≡ 1. Q.E.D.

5

Proof of Theorem 5 in the case ρ(0) = N

In Section 2, 3, and 4, we have proved Theorem 5 in the case ρ(0) = 1. In this section, we will prove Theorem 5 in the case ρ(0) = N ≥ 2 and ρ(x) ≡ N . We put u(t, x) =

N −1

ui (x)ti + tN −1 w(t, x),

(66)

i=1

"+ . where ui (x) ∈ C{x} (1 ≤ i ≤ N − 1) and w(t, x) ∈ O Then by an easy calculation we see Lemma 26. If the function (66) is a solution of the equation (9), the functions u1 (x), . . . , uN −1 (x) are uniquely determined and w(t, x) satisﬁes an equation of the following form: (t∂t − ρ(x) + N − 1)w = ta(t, x) + tA0 (t, x)w + t

n

Ai (t, x)∂i w

(67)

i=1

+

|α|≥2

where a(t, x) =

t(N −1)(|α|−1) Aα (t, x)wα0

n <

(∂i w)αi ,

i=1

1 (G2 (x)(t, w0 , ∂x w0 ) + ta(x) − (t∂t − ρ(x))w0 ) tN

(68)

459

Singular Solutions of the Briot-Bouquet Type

with w0 =

N −1 i=1

ui (x)ti and

1 ∂G2 (x)(t, w0 , ∂x w0 ), i = 0, 1, . . . , n, t ∂Xi 1 ∂ |α| G2 Aα (t, x) = (x)(t, w0 , ∂x w0 ), |α| ≥ 2. α! ∂X α Ai (t, x) =

Since the equation (67) satisﬁes the conditions (A1), (A2), (A3) and the characteristic exponents ρN (x) = ρ(x) − N + 1 satisﬁes ρN (0) = 1, we can apply the results in sections 2, 3 and 4. Further, by the form of all the nonlinear parts of the equation (67), we see that the formal solution constructed in Section 2 has the following form: N

N,0 0 (x)φN,1 + w0,1,0 (x)tρ (x) w = uN,e 0 i + uN i (x)t + i≥2

+

m≥2

i+j+|β|=m

N,β wi,j,k (x)ti+(N −1)(j+|β|−1)+jρ

N

(x)

{log t}k ΦβN,1

k≤i+|β|0 +|β|1

j≥1,|β|∗ ≤m−2

where ΦβN,1 =

(69)

i+|β|=m |β|∗ ≤m−2,|β|≥1

m≥2

uN,β (x)ti+(N −1)(|β|−1) ΦβN,1 i

+2(j−1)

N < ∂ l φN,1 βl tρ (x) − t x . Therefore we have and φN,1 = N l! ρ (x) − 1

|l|≥0

u=

N −1

N,0 0 ui (x)ti + uN,e (x)φN + w0,1,0 (x)tρ(x) 0

i=1

+

i+N −1 uN + i (x)t

i≥2

+

m≥2

m≥2

i+|β|=m

uN,β (x)ti ΦβN i

(70)

|β|∗ ≤m−2,|β|≥1

i+j+|β|=m

k≤i+|β|0 +|β|1

j≥1,|β|∗ ≤m−2

N,β wi,j,k (x)ti+jρ(x) {log t}k ΦβN .

+2(j−1)

We put uN i (x) -→ ui+N −1 (x) N,β (x) wi,j,k

-→

β wi,j,k (x)

for i ≥ 2,

uN,β (x) -→ uβi (x) i

for |β| ≥ 1,

for any (i, j, k, β),

and we have u0N (x) ≡ 0 by the form of the solution (69) and the above relations. Hence this completes the proof of Theorem 5. Q.E.D.

460

H. Yamazawa

References 1. 2.

3. 4. 5. 6. 7.

Briot, Ch. and Bouquet, J. Cl., Recherches sur les propriétés des fonctions déﬁnies par des équations diﬀérentielles, J. Ecole Polytech., 21(1856), 133–197. Gérard, R. and Tahara, H., Holomorphic and Singular Solutions of Nonlinear Singular First Order Partial Diﬀerential Equations, Publ. RIMS, Kyoto Univ., 26(1990), 979–1000. Gérard, R. and Tahara, H., Singular Nonlinear Partial Diﬀerential Equations, Vieweg, 1996. Hill, E., Ordinary diﬀerential equations in the complex domain, John Wiley and Sons, 1976. Hörmander, L., Linear partial diﬀerential operators, Springer, 1963. Kimura T., Ordinary diﬀerential equations, Iwanami Shoten, 1977 (in Japanese). Yamane, H., Nonlinear Singular First Order Partial Diﬀerential Equations Whose Characteristic Exponent Takes a Positive Integral Value, Publ. RIMS, Kyoto Univ., 33(1997), 801–811.

Author Index

Adamec, Ladislav, 9 Apushkinskaya, Darya E., 13 Ayrjan, Edik A., 17

Marín, Julio, 275 Marini, Mauro, 279, 289 Maryška, Jiří, 297 Matucci, Serena, 279, 289 Melnik, Valery S., 57 Mikyška, Jiří, 25 Minhós, Feliz Manuel, 183

Baštinec, Jaromír, 21 Beneš, Michal, 25, 83 Bonheure, Denis, 37 Bouchala, Jiří, 45

Nazarov, Alexander I., 13, 323

Cabada, Alberto, 53 Caraballo, Tomás, 57, 67 Cavani, Mario, 77, 275 Chalupecký, Vladimír, 83 Chrastinová, Veronika, 103 Cid, José Angel, 53 Ciocci, Maria-Cristina, 109

Ortegón Gallego, Francisco, 169 Papageorgiou, Nikolaos S., 327 Pisani, Lorenzo, 119 Pokorný, Imrich, 17 Pospíšil, Zdeněk, 339 Pouso, Rodrigo L., 53 Puzynin, Igor V., 17

d’Avenia, Pietro, 119 Diblík, Josef, 21, 129, 133, 135, 139 Doležel, Ivo, 143 Ďurikovič, Vladimír, 159 Ďurikovičová, Monika, 159

Rebiai, Salah-Eddine, 349 Řehák, Pavel, 279, 289 Růžičková, Miroslava, 135

Fabry, Christian, 37

Sanchez, L., 203 Sanz, Javier, 357 Schilder, Frank, 363 Segeth, Karel, 379 Severýn, Otto, 297 Siegmund, Stefan, 221 Smets, Didier, 37 Smyrlis, George, 327 Šolín, Pavel, 143, 379 Švec, Marko, 391 Svoboda, Zdeněk, 139

González Montesinos, María Teresa, 169 Grobbelaar-Van Dalsen, Marié, 177 Grossinho, Maria do Rosário, 183 Guzmán-Gómez, Marisela, 191 Habets, P., 203 Hayryan, Shura, 17 Hricišáková, Daniela, 217 Hu, Chin-Kun, 17 Khusainov, Denis Ja., 133 Koksch, Norbert, 221

Tarallo, M., 203 Terracini, S., 203 Torres, Delﬁm F. Marado, 397

Lacková, Dáša, 267 Langa, José Antonio, 57, 67

Ulrych, Bohuš, 143

Mamedova, Violeta G., 133

Vala, Jiří, 413 461

462 Valero, José, 57, 67 Vogt, Werner, 363 Vohralík, Martin, 297 Yakubov, Yakov, 437 Yamazawa, Hiroshi, 443

H. Yamazawa