High order nonlinear parabolic equations

171. Wolf yon Wahl, "Regularity of weak solutions to elliptic equations of arbitrary order," J. Different. Equat., 29, No. 2, 235-240 (1978). 172. K.-O. Widman, "Local bounds for solutions of higher order nonlinear elliptic partial differential equations," Math. Z., 12!., No. I, 81-95 (1971). 173. S. A. Williams, "A sharp sufficient condition for solution of a nonlinear elliptic boundary value problem," J. Different. Equat., 8, No. 3, 580-586 (1970).

HIGH ORDER NONLINEAR PARABOLIC EQUATIONS

UDC 517.956.45

Yu. A. Dubinskii

The basic results and methods of the theory of high order nonlinear parabolic equations are described. In the first chapter boundary problems for quasilinear parabolic equations having divergent form are considered. In the second chapter nonlinear parabolic equations of general form are considered~ Attention is mainly paid to methods of study of nonlinear parabolic problems~ In particular, the methods of monotonicity and compactness, the method of a priori estimates, the functional-analytic method, etc. are described.

PREFACE The present survey is devoted to one of the actively developing directions of contemporary nonlinear analysis, the theory of high order nonlinear parabolic equations. Just as in the corresponding elliptic theory, the theory of high order nonlinear parabolic problems counts nearly a quarter of a century in its development (M. I. Vishik's first paper on nonlinear parabolic equations of order 2m appeared in 1962.) However in this comparatively short period it has been enriched with such important results, and what is no less important, methods of study, that it has rightfully become one of the fundamental directions of contemporary research. The object of the present paper is the description of the basic achievements of this theory. The paper consists of two chapters. In the first chapter a survey of various results in the theory of nonlinear parabolic equations of divergent form is given. In the second chapter boundary problems for nonlinear parabolic equations of general form are described. Since a brief scientific and historical annotation prefaces each section (cf. also the comments in the "Literature" section) we shall not dwell on the content of the paper by chapters here in the preface but characterize it on the whole. First of all we note that in the choice of material we have chosen papers in which the methods of studying nonlinear boundary problems have been developed. Among such methods are the method of compactness, the method of monotonicity, the method of a priori estimates, the function-analytic method, etc. Further, as is known there are two general approaches to the theory of nonlinear parabolic problems. The first approach is based on the consideration of the initial-boundary value problem for a nonlinear parabolic equation as a Cauchy problem for the corresponding nonlinear differential-operator equation in a Banach space.

Translated from Itogi Nauki i Tekhniki, Seriya Sovremennye Problemy Matematiki, Noveishie Dostizheniya, Vol. 37, pp. 89-166, 1990.

0090-4104/91/5604-2557512.50

9 1991 Plenum Publishing Corporation

2557

The second approach is based on the proper study of the boundary problem as an object of the theory of nonlinear partial differential equations. At the present time both approaches have a sufficient amount of both factual material and methods of study in their arsenals and thus deserve completely independent accounts. However, in the literature, attention is paid mainly to the first approach. Here we have a number of detailed monographs and surveys among which we note the books of Browder [103], Lions [120], Brezis [94], Barbu [91], Gajevski, GrSger, and Zacharias [iii], Vainberg [Ii], Levitan and Zhikov [63], and Pankov [71], and also the surveys of Dubinskii [31], Krylov and Rozovskii [52], Krein and Khazan [48], published by the National Institute of Scientific and Engineering Information (VINITI) of the Academy of Sciences of the USSR in the series "Itogi Nauki i Tekhniki." The survey of Krein and Khazan cited has the most complete bibliography, containing more than 700 citations. Thus, nonlinear differential-operator equations are quite completely represented in the scientific and reference literature. As to the second approach, here, on the contrary, there are no general surveys as far as we know. In view of what has been said, in the present survey we do not deal with differentialoperator equations and we consider nonlinear parabolic problems only from the point of view of partial differential equations. Of course this choice is dictated not only by the hope of filling the lacuna cited in the literature. The fact is that by no means all the results of the theory of nonlinear parabolic problems can be included in the framework of the differential-operator approach, so that actually the theory of nonlinear parabolic partial differential equations is richer than a simple consequence of the theory of differential-operator equations in Banach spaces (the reader will see this particularly in the second chapter). Moreover, on this route one is able to consider the material recounted from single scientific and methodological points of view. CHAPTER 1 QUASILINEAR PARABOLIC EQUATIONS OF DIVERGENT FORM i.

Methods of Solution of Quasilinear Parabolic Problems

i.i. Method of Monotonicity. The method of monotonicity or method of monotone operators is an active contemporary method of study of nonlinear equations. We apply the nonstationary version of this method, described below, to a large class of initial-boundary value problems for quasilinear parabolic equations having divergent form [cf. (i)]: the first boundary problem, the problem with periodic conditions on x, the Neumann type problem, etc. To be specific we dwell, for example, on the case of the first boundary problem since the main facets of the method of monotonicity are identical in all cases. Let G c R n be a bounded domain with piecewise-smooth boundary F; Q = [0, T] • G be a cylinder in the space R ~+I of variables t~[0,T], x60 with lateral surface S=[0, 7]XF (T being an arbitrary fixed number). In the cylinder Q we consider the first boundary problem for the quasilinear parabolic equation of divergent form

Ou ~ (--1)mO~A~(t, x, DVu)=h(t, x), O_uOtq- L (t, x, D) (tt) =---~i-q-r~lr

u(O,x) =~p(x), x~a, D~uls=/~(s), sES, [co[~<m--1.

( 1) (2) (3)

Here u=u(t, x), ((t, x)6Q)is the function sought, D = ( D I ..... D,), Dj=O/Oxj(l-.<j-./I is an integer. The functions As(t, x, $7) are (generally) nonlinear functions }71~<~. Before formulating conditions under which the solvability of the problem (1)-(3) is established, we introduce the spaces needed. 2558

0m

Let W~(G) be the Sobolev space (p > 1 is a number, r n > 0 is an integer), and Wp(G) be the subspace of the space W~(G) consisting of functions u(x)EW~(O),which on F satisfy the conditions

D~ulr=O, I~ I < : m - - 1 . 0

We denote the dual space (W~(G))* by Wp~(G), p' = p/(p - ]).

u(t, x):lluIti.---

L.(0, T.

Further, let

Ilu(t, x)ll 2co) a t <

,

0

0 o

Definition I.i.

The space W is the closure with respect to the norm

of the smooth functions u(t, x) defined in the cylinder Q and satisfying the conditions

u(O, x)=O, D~ Definition 1.2.

Io~ l..<m--1.

If the function f(t, x) has finite norms

H film,p, l[Of/Otll-~,/,then

the

0

collection of functions of the form u(t, x) =- f(t, x) + z(t, x), wherez(t, k)GW, by definition forms the space W(f). Now we formulate conditions under which the problem (I)-(3) is solvable. Let us assume that the functions A=(I,x,~J, I~l~
For any g7 and almost all

(t,x)~Q

I~]
where K > 0 is a constant. For any functions g6Lp(O, T; W~(O)),

II [Strong Ellipticity of the Operator L(t, 0

@~Lp(O, T; IY/~(O)) such that u--m6Lp(0, T; l~z2(O)),one has

ReIL(t, x, D)(u)--L(t, x, D)(v), u - - w ] ---Re SI X

i A~(/' x, D~u)--A~(t, x, D ~ ) ]

X'-ff~(u--v)dxdt>aoHu.-.vl[~,p,

Q ]al<m

where a 0 > 0 i s a c o n s t a n t . III. The i n i t i a l and boundary f u n c t i o n s ~(x) and f ~ ( s ) admit e x t e n s i o n t o Q from t h e space W(f), i . e . , t h e r e e x i s t s a f u n c t i o n f ( t , x) h a v i n g f i n i t e norms I[fllm, p and Jlc)r/at[t--m,r, such t h a t f(O, x)=tp(x), DOf[s=fo(s), I ~ l ~ m - - 1 . THEOREM i.i.

If conditions I-III hold, then the map

~ + L(t,x, D) (u):W(y)-+ Lp.(O, T; W-~.m(O)) ot i s a homeomorphism. To prove the proper solvability of the problem (1)-(3) condition II is excessive and can be relaxed. Namely, let us assume that the following conditions hold: II'. one has

[Coercivity of the Operato r L(t, x~ D)(u).]

For any function u{[, x)@Lp(O, T; W2(O))

2559

Re [L (t, x, D)(u), ~] =----Reff X

where

c(~):R~-~R~ is

As(t, x, D~u)D'adxdt>c([[u[[m.~),

q l=l~m a continuous function

such that

c(g)/g

§ +~ a s r + +~.

II" [Monotonicity of the Operator L(t~ X~ D)(u).]

For any functions u(t, x ) , v ( t , 0 from the space Lp(0, T; W~(G)] such that u(t, X)--9(t, x)6Lp(0, T; ~7~(O~), one has

Re[L(t, x, O )(u)--L(t, x, D)(v~, u~v]>O. THEOREM 1.2. If conditions I and III hold as any right side h(t, x)ELp,(0, T; W;9(O)) there exists problem (1)-(3). If in addition the operator L(t, (4) equality to zero is only possible when u(t, x)

x)

(4)

well as conditions II' and ll", then for at least one solution u(t,x)fiW(~) of the x, D)(u) is strictly monotone, i.e., in ~ v(t, x), then the solution is unique.

The proof of Theorems I.i and 1.2 goes by the method of monotone operators, which has now become classical. Applied to the problem (I)-(3) this method consists of the following: I.

Investigation of the solvability of the finite-dimensional nonstationary approximations of the problem (1)-(3) (Galerkin-Hopf-Faedo approximations).

2.

Derivation in Lp(0, T; W~(G)) of uniform a priori estimates for the approximate solutions uN(t, x).

3.

Passage to the limit as N § ~.

It is important to stress (and this is of importance in the method of monotonicity), that condition II' lets one pass to the limit from the finite-dimensional approximations to the original problem (I)-(3) starting from only the simplest estimates of the approximate solutions in Lp(0, T; W~(G)) and consequently using only weak convergence of derivatives of order m. In other words, equations with monotone operators are remarkable in that they are weakly closed with respect to Galerkin approximations. Example.

We consider the equation

o~

Ot F X

(5)

a=m=(Im=uJP-~D~u)=h(t;x) ,

[al<m w h e r e a=~O, p > l Then f o r

are numbers. p~2

(5)

satisfies

all

the hypotheses

of Theorem 1.1 b e c a u s e

and as calculation shows,

Re [L (t, x, D)(u)-- L (t, x, D)( v), u-- 91 > a o [Iu -- v l[ ~,p (% > 0). Consequently, the operator of the left side of (5) defines a homeomorphism of the spaces W(f) and Lp,(0, T;w?m(o)). If 1 < p < 2 then the operator L(t, x, D)(u) does not satisfy condition II, however it is coercive, which is obvious, and strictly monotone, which follows from the inequality

RelL(t, x, D)Cu)--L(t, x, D)(v), u - - v ] > X a~

]D=u--D%[Pdxdt+2--fi:-7_,

D~u+

x

D=u_ Dav l2dxdt), where

Q+.=={(t,x)6Q:JD=u--D%I~
Q_,= ---Q\Q+,=.

Thus, when 1 < p < 2 all the hypotheses of Theorem 1.2 hold. 1.2. Equation with Semibounded Variation. We apply the method of monotone operators not only to problems satisfying a monotonicity condition but also to problems for which a weaker condition holds, the condition of lower semiboundedness of variation of the operator of the spatial part of the parabolic problem. This lets us study those classes of parabolic problems for which the lower terms are not monotonic but only subordinate in a certain sense to the principal part of the operator. Thus, in applications, the sphere of application of the methodology of the theory of monotone operators is essentially extended. We formulate a precise result.

2560

Let us assume that in the hypotheses of the preceding Theorem 1.2 instead of the monotonicity condition the following holds: Condition of Semiboundedness of Variation of the Operator

A(~, x, D)(zz)~ X (--I)I=ID=A=( t' l=l~<m x, DYu) : for any functions it(t,X)6 Lo(O, T; Wf(O)) and ~(t, ~)Egp(0, T; W~(O)) from a sufficiently 0 large ball fiN(#,x)]Im.o'-
(t,x, D) (u)--A (t,x, D)

Re[A where c(R,@)90 fixed.

(v), u--v]'~--c(R, llU--VIlm-h;),

is a continuous function such that c(R, ~p)/~ ~ 0 as $ § +0 and R and p are

THEOREM 1.3. Let the boundary problem (i)-(3) satisfy conditions I, If', III, and the condition of semiboundedness of variation of the operator A(t, x, D)(u). Then for any right side h([, X) 6Lv,(O, T; W~'m(O))it has at least one solution u(/, Jc)s 1.3. developed the class tives DSu which are

Method of Compactness. There are two modifications of the method of compactness for studying two classes of nonlinear parabolic equations of divergent structure: of "weakly" nonlinear equations, i.e., equations which are nonlinear in the derivato order m - i, and the class of "strongly" nonlinear equations, i.e., equations nonlinear in the derivatives to order m.

First we consider the case of weakly nonlinear parabolic problems. Let the functions A=8(t,x,~r), IdiOm, l~[<m, be defined in the cylinder Q=[0, T]XG, G~R ~ in the variables t and x, and everywhere in the variables Sy, where [7[ < m - i. Let us assume that these functions satisfy a Caratheodory condition. We study the first boundary problem O~ +(--I 0-7

)~

~

D a

(A~(t, x, DVu)D~u)=h(l, x),

(6)

Ir

u(O, x)=0, D~u[s=O, w h e r e S = [ 0 , T] • %G i s t h e l a t e r a l

surface

(7)

lot<m--l,

(8)

of the cylinder

O.

Let us assume that the following conditions hold: I" (Coercivit~,Cgndition.) values (7), (8), one has IRe

For any function

u(tx)6"Cl,m(Q) assumed

initial and boundary

~ [A~(t, x, DVu)D'Bu, D=ul>E(u)--K~ =----X [a(DVu) D~u, D=u]--l(, I~l=l~I=m [al=m

where the function a(~) (ITl--<m--1), satisfies a(~,)>ao(l+ K > 0 being a constant. II".

Z

I~,lq") ,

Here [ , ] is the integral

If the function

ao>O,

qv>O,

over the cylinder

u(t,x) :Q-+C I h a s f i n i t e e n e r g y i n t e g r a l A ~ (t; x, DTu) D~uELq(Q),

Q.

E(u),

then for all

e and B

w h e r e q > 1 i s a number. H e r e i f t h e s e t o f f u n c t i o n s u ( t , x) i s s u c h t h a t E(u)-<.K, w h e r e K > 0 i s a c o n s t a n t , t h e n t h e c o r r e s p o n d i n g s e t A ~ 8 ( t , x , DXu)D~u i s bounded i n L q ( Q ) . III".

x) has t h e form h ( t , x) - h z ( t , x) + h 2 ( t , x) + h 3 ( t , x ) , w h e r e k~(t,x)6L2(Q), k2(t,x)6L~(O,r;w~"r and ka(t,x)GLq,o(O,T;W-,(m-1)((~)) , w h e r e q0 = m i n q v + 2, q~ is the conjugate exponent. % u(t;

The f u n c t i o n

Definition 1.2. x) such that

i) E(u) <

h(t,

By a g e n e r a l i z e d

solution

of the problem (6)-(8)

is meant a function

~;

2) for any function tity

v(t,x)~Ci'm(Q), v(T,x)=---O,Dv~

[0o]

-- u, - ~ - +

~.~

lo!~.m--l, one has the integral iden-

[A~ (t, x, DVu) D~u, D=v] = ,

lal,l~l=m 2561

where i s t h e v a l u e of t h e f u n c t i o n a l on t h e f u n c t i o n (v, t , x ) .

h (t, x)ELe(Q) @Li(0, T; W~9m(G))OLqo (0, T; Wq-~(~z--1) (G))

The f o l l o w i n g theorem h o l d s . THEOREM 1 . 4 .

If conditions I"-III"

h o l d , t h e n t h e problem ( 6 ) - ( 8 ) has a t l e a s t one

generalized solution in the sense of Definition 1.2. The proof of the theorem goes by the method of Galerkin using the simplest a priori estimate

E (u) ~const. The presence of this estimate together with the estimate of the derivatives with respect to t of the family uN(t, x) of approximations in the space

L~ (Q)~L2 (0, T; W~ m(O))~Lg~(0, T; Wp Cm-~ (O)) guarantees the convergence uN(t, x) ~ u(t, x) (N + ~) almost everywhere together with the derivatives D~u~(t,x)-~DTu(t,x)to order l?[~.m--1. Condition II" and the famous theorems of de la Vallee Poussin and Vitali (cf. [68, p. 94]) let one make the passage to the limit as N § ~ in the integral identity defining a generalized solution. An essential role in the method of compactness (both for weakly nonlinear equations and for equations with strong nonlinearities which are considered below) is played by the following compactness theorem. Let E 0 be a seminormed* set (generally nonlinear), compactly imbedded in a Banach space E which in turn is imbedded in a third Banach space El. In other words, one has inclusions

EoCEnE1.

(9) M~Lq(O, T; Eo)(]C(O,T; El)

Compactness Theorem. Let the inclusions (9) hold. Then any set (q > i) which is bounded in Lq(0, T; E 0) and equicontinuous in C(0, T; E I) is compact in Lq(0, T; E) and C(0, T; El).

In the case of weakly nonlinear equations being studied the role of E 0 is played by the nonlinear space of functions with finite integral

Y, I a(D: )l D,,ulidx.

[al = m

(3

As E one u s e s t h e space W(m-O(O), and as E 1 t h e space q

E1 = L2 (O)| Example.

m (G) @WT,(~-~) (G).

We c o n s i d e r t h e e q u a t i o n

where a.,((~,) a r e c o n t i n u o u s f u n c t i o n s i n $,~ s a t i s f y i n g

ao I ~,'~ Iqv "-< av (~)-. 0, A 0 > 0, q~ > 0 are constants. All the hypotheses of Theorem 1.4 hold for this equation. Now we consider the method of compactness applied to the class of strongly nonlinear equations. This method was proposed by Vishik [14]. In the cylinder Q we study the boundary problem

Ou Ot + ~

(--l)l~ID~A~(t'x'DVu)=k(t'QO'

(i0) (]~l<m),

*The set Eo is called seminormed if with each element vEE0 there is associated a real number [v] such that: i) [v]~0, while [v] = 0 if and only if v = O; 2) for any

2562

X~C' one has [Xv] = ]A[[v].

u (0, x)-----, (x), x~O,

(ii)

D~ ls= f o (t, x), (t, x)~S, [~[~<m--1.

(12)

Let us assume that the functions A~(t, x, $) have polynomial growth in ~u at infinity, i.e.,

\ I'/l<:m where M 2>0, p ~ 2

are numbers.

Moreover, let us assume that the functions A~(t, x, ~ ) are differentiable in all variables while differentiation with respect to the variables t and x does not increase the growth in ~ at infinity and differentiation with respect to Ey decreases this growth by one. This means precisely that one has I

OA=

~

(t'dC' ~')I ~< J141 (i,,~m' ~v [P-~-[-1) '

IOA,........6_~_(t,X, ~.~)t ~
(13)

where Mi > 0 ( i = 1, 2, 3) are c o n s t a n t s . The following c o n d i t i o n s are fundamental c o n d i t i o n s a s s u r i n g the e x i s t e n c e for the approximate s o l u t i o n s uN(t, x) of the e s t i m a t e s of the d e r i v a t i v e s

~, in the Lebesgue m e t r i c s . Strong E l l i p t i c i t y Condition.

D"u~(t,x), / ~ l = m + l , For any values ~,, rk~, rl~ and

P"h lfll<;m

where

(t,x)~Q one has

fai=m

A=~(t,x,$O----OA=(t,x,~v)/c)~, ao>O, C > O are c o n s t a n t s .

Subordination Condition. The bilinear form A(~v;N=N's) is bounded above by the corresponding quadratic form A(~v;N=,N~). More precisely, one has

where M > 0 are constants. The analogous condition is also assumed to hold for the bilinear form obtained from the form A ~ ; N=, N~)by replacing the coefficients A=~(t,x,~) by their partial derivatives with respect to t and xi, i = l,...,n. Moreover, let us assume that the boundary and initial conditions (ii), (12) admit sufficiently smooth extension inside the cylinder Q satisfying the natural conditions of compatibility with the right side h(t, x). THEOREM 1.5. unique solution

Let the conditions listed above hold.

Then the problem (10)-(12) has a

Ou L u(t, x)~La(O, T; Wpm (G)), 3FE 2(Q,).

In addition the solution u(t, x) can be represented as the limit almost everywhere of Galerkin type approximations together with the derivatives D=u(t,x), l=l<~m. As is clear from what has been said the hypotheses of the method of compactness for strongly nonlinear equations are more restrictive than, for example, the hypotheses under which the method of monotonicity is valid. It is however essential that if in the method 2563

of monotone operators the strong convergence (I~I = m)

D~uN (t, x)--*D~u (t, x)

(N-+oo)

or convergence almost everywhere cannot generally be established, here on the contrary conditions (13)-(15) let one prove estimates guaranteeing not only weak convergence

D~'u~r(t, x)-'~D=u (t, x) for loci = m but also their convergence almost everywhere in Q. This fact follows from the estimates

dUN where (gGR~)and O' i s an a r b i t r a r y strictly i n t e r i o r subdomain of t h e c y l i n d e r O. One i s a b l e t o g e t t h e s e e s t i m a t e s t h a n k s t o t h e f a c t t h a t u n d e r t h e h y p o t h e s e s f o r m u lated the operator 0 Bu==-Mtt--xp(x)Au-- T{ ( ( T - - t) where M > 0 is constant, ~(x)EC~(G) the first boundary problem.

is a "cap" type function, is a separating operator of

Finally, the convergence almost everywhere

D=uN(t,x)-+D=u(t, x),

I~l~<m,

lets one pass to the limit in the moment equations defining the approximate solutions uN(t, x). 1.4. Parabolic Problems with Non Power Nonlinearities. The development of the method of monotone operators and also the method of compactness in the case of equations with rapidly or slowly growing nonlinearities is associated with a number of difficulties. These difficulties are caused first of all by the nonreflexivity of Orlicz-Sobolev spaces if their characteristic functions do not satisfy the familiar Ai-condition (cf. below). In the parabolic case the theory of monotone operators in the nonreflexive case was developed in Robert [131, 132], Donaldson [104], and afterwards Zini [139], etc. An essential role was played here by the earlier research of Brezis [94] and Gossez [113, 114] relating to the standard nonreflexive case. Before formulating the basic result on the solvability of a parabolic boundary problem in Orlicz-Sobolev spaces we introduce some notation and concepts concerning these spaces. Let Id(u):R~-+R$ be an N-function, i.e., a nondecreasing convex function satisfying the conditions

lim M(u)=O,

lim M(u)__~.+_~.

To each N-function M(u) corresponds an additional Jung (Young) N-function

/v/(u ~-- sup (vu v~0

- - M (~)).

L e t G ~ R n be a b o u n d e d d o m a i n , Q = [ 0 , T] x G be a c y l i n d e r Definition 13. By the Orlicz class ~ ( Q ) tions u(t, x) in Q for which

II M(l.(t,

(variables

t and x ) .

is meant the collection of measurable func-

ll)a,a <

(16)

Q

By the Orlicz space LM(Q) is meant the Banach space of functions u(t, x) satisfying (16) and h a v i n g f i n i t e norm (Luxembourg norm)

It is known (cf., e.g., Krasnosel'skii and Rutitskii [47]) that the spaces LM(Q) are reflexive if and only if M(u) satisfies the so-called A2-condition, i . e . , M ( 2 u ) ~ K M ( u ) , where K > 0 is a constant for all sufficiently large values of u.

2564

The closure of the set of bounded measurable functions in the norm H'II(M) will be denoted by EM(Q); this is a Banach space where EM(Q) c LM(Q). The coincidence of EM(Q) with LM(Q) holds if and only if M(u) satisfies the A2-condition. Further, by the Orlicz-Sobolev space of order m ~ O

is meant the space

W~LM(Q)--{u(t, x): Dau(t, x)ELM(Q), I~}~<m} with the norm 2

\1/~

Analogously,

WTEM(Q)---{u(t, x): D~u(r x~EM(Q), I~l~<m} w i t h t h e same norm

0

II~ll~,<~,

Further,

let

W~EM(Q) be t h e c l o s u r e

of ~(Q)

i n W~EM(Q) w i t h

0

respect to the norm H'IIm.(M) and W ~ L M ( Q ) be the closure of ~)(Q) in W~LM(Q) in the topology

O( g LM(Q), H E~(Q)), i.e.,

in the

Tml<m

\!alum

wk-e topology.

O

We identify the dual space

W~EM(Q) with

the space

W;~EM(Q)_-_i/E~'(Q)lf=~ D"f=(t,x), f=(t,x)Ge~(Q)}. Iml<m

Now we turn directly to the formulation of a nonlinear parabolic problem in the OrliczSobolev spaces. We choose a reflexive Banach space V such that (Q)cV c

L~(Q )cV' c ~" (Q),

0

and let us assume that the inclusion

(Q)cV' is also continuous. subspace of V'.

VcW~EM(Q)

is continuous.

Then the inclusion WxmLM •

We identify V with a subspace in W0~m L m(Q),

andW;~LM(Q)

with a

Let

W = {u6~D'(Q):u6V, o~EV'~ with the norm

L==-.~-:V.-+V'with ~u

and we introduce the operator

domain of definition

D(L) = { u 6 W : u ( O ) = 0 } . (We note that any function ator Lu is well-defined.)

u@W

has, for t = O, "trace"

ult=0~i2(O). Consequently, the oper-

Further, let J:V § V' be the dual operator. Let us assume that for any g > 0 the operator L + sJ is a pseudomonotone operator in the sense of Brezis (cf. Brezis [94], Lions [120], etc.) In practice, usually either LM(Q) c L2(Q) or L2(Q) c LM(Q).

In these cases the space V

0

can be chosen as follows: if LM(Q) c L2(Q) then V=W2s(Q), where s 9 1

is sufficiently large;

0

if L2(Q) c LM(Q) then V =L2(O,T; W21(Q)). In these cases the preceding assumption always holds since the dual Operator is a strongly elliptic linear operator (the gradient of the norm in the corresponding Hilbert space). These two cases also differ in theory which will be discussed below. 0 m

n~

Now we define an operator A:WxLM(Q).->W-;~ LB(Q). Namely, we consider a family of functions :Am(t,x, ~v.), ]~ [--
for any ~ the functions As(t , x, $7) satisfy Caratheodory conditions;

2565

II.

one can find functions a~(t,x)ELl(Q)

and constants ba > 0 such that

l~l<m III. one can find a function~(t,x)cL,(Q), ~(l,x)>O, such that for each a, l=I<~m, there exists a function g=(l, x ) @ E ~ ( Q ) a n d a constant C~ > 0 such that

X)]r

2~ [A,(t, x, r

~ C=Mt'I ~,I)--r

[a[<m

IV.

x);

[a[<m

for any SX and $ut [A,(t, X, ~v)--A~(t, x, ~.~)1(~..--~',0>0. I~[<m

Obviously conditions II-IV are generalizations of growth, coercivity, and monotonicity conditions to the nonpower case. From the chosen family of functions Ae(t, x, $7) we define the form

and operator A defined by this form

(A(u), v) -~

~x L~(o)•

o~

=a(u, v). Lm(Q) 0

The domain of definition of the operator A(u) is the set of functions u(t,x)~W~L~(Q) such 0 that the form a(u, v) is continuous (with respect to v) on W~EM(Q). The operator so defined is a pseudomonotone operator 0

A:w~ EM(Q)-~Wf~L~ (Q). Moreover, it is bounded on the subspace O

E =W~L~(Q)n L~(O, T; L~(O)). As already noted one should distinguish the cases / ~ ( Q ) c L 2 ( Q ) and/M(Q)~L2(Q). To be specific, let L~(Q)cL2(Q). of the operator

Then one requires in addition that the range of values

T,=L+eI+A:V-+V' should fill all of V', i.e., R(T s) = V'. Now we can formulate the basic result. THEOREM 1.6. Let all the conditions listed above hold and suppose in addition that for any fixed function /(t, x)16WTmEB(Q)the set U {T~I(/)} is bounded in L~(0, T:L2(G)). Then for any function /(t, x)EWI~E~(Q) there exists a function 0

u(t, x)~WfL~(Q)n L~(0, T; 4(a)) (which is unique in the case of strict monotonicity) such that

A(u)6Wx"~L~(Q), Ou ~Fe_W-ra x L ~t,Q ), while

Ou ~ + A(u)= f (t,X). The proof of the theorem is based on the ideas both of the method of monotonicity and the method of compactness. There is a detailed account [including the case LM(Q) D L2(Q) ] in Donaldson [104], Robert [131], and Zini [139]. To conclude we give examples of equations for which the theorem on unique solvability of the initial-boundary problem is valid: 2566

1. ~/-+(--1)

~

D'ID~u'exp(D%t)~l=f(t,x);

!"l=m

2. ~-t-~-~ ,, 1)= ~

D=iD=u'~n(l+tD=ul)l=f(t,x)

9

I-I=ra

.

Regularity of Generalized Solutions 2.1.

Partial Regularity of Morrey Typet

07~, + ~

We consider a real quasilinear parabolic system

(-- 1)'~' D~A~ (t, x, D~d) = O,

I~i<m

(17)

l~l<~m; ~ = 1 . . . . . N; where u = (ui,...,u N) is a vector-function. If (17) satisfies the strong ellipticity condition and the corresponding subordination conditions (cf. Sec. i, point 1.3), then the solution of the first boundary problem u(t, x) of this system has derivatives 0~ L i.e., the function u(t, x) itself and its indicated derivatives are generally generalized functions of the type of square summable functions. x),

The question arises can one assert that the solution u(t, x) and its derivatives D~u(t, [~[ = m are continuous functions in the cylinder Q. As examples show, this is not always

SO.

However, as established in Danilyuk and Skrypnik [23, 24], one can assert that the derivatives D~u(t, x), la] = m are continuous on an open set of full measure in Q, i.e., the solution u(t, x) has the property of "almost" regularity. We proceed to precise formulations. Let G be an arbitrary bounded domain in An. tions hold.

We shall assume that the following condi-

~k I. The functions A~(t,x,~) for(t,x)6Q-~[O,T]XO,~-~-{~v:I?l-.<m, k = 1 ..... N}, ~'~6~,~ are twice continuously differentiable in all arguments while all second derivatives satisfy a H61der condition in x and ~ with exponent p > 0 and in t with exponent ~/2m on each compact subset. II (Strong Ellipticity Condition.)

,v

One has

oA~ (t, x, D

N k

I

s

k,t=Z ["[=lPl=m where

(t, x)fQ, ~vER, t~ 1 "q~eR~,

(N

k=l 1131<m

)

while a o > O , p~>2.

III (Subordination Condition.

)

For the same values

IA~ (t,;, ~) I~< K =p-I(D, where K > 0 is a constant. Analogous inequalities also hold for

0 A~(t, x, ~), ~--FA~(t, x, ~), e ~ ( t , x , t ) , Ox 02 02 e= A~( ~ t ,x,~). Ot'Oi'b A~ (t, x, D, ~O~'Ox~A~ (t, x, t), We introduce the space of generalized solutions of the system (17). ~m denote by ~ , p (Q) the space of functions having finite norm

Namely, first we

2567

vraimax]]

u (t, x) llL.w)+ ~ [[D~tt {t, X) IIr.{Q).

O~t
(18)

lal--m

The closure of the functions which are smooth and finite near the lateral surface S of the cylinder Q in the norm (18) will be denoted by W2,. ~ (Q). Definition 2.1. By a generalized solution of (17) we mean a vector-function u(t, x) = (u1(t, x) ..... uN(t, x)) such that 0

u~(t, x)6W~ and for a n y function z)(t,

k----1 . . . . . N,

o Ov L 0,m (Q), ~-fE 2(Q), x)6W~,o

for

O
k = 1

,...

,N one has

f0 It=o

0

I~l~:m

The following theorem holds. THEOREM 2.1. Let conditions I-III hold and u(t, x) be a generalized solution of (17) having derivatives

O~6L2,1o~(Q),

k=l

N.

There exists a subset Z c Q of measure zero such that Q \ Z is open and in Q \ Z the functions uk(t, x), k = i .... ,N have continuous derivatives D~u(t, x) for [al = m. Thus, Theorem 2.1 shows that the famous result of Morrey on the partial regularity of generalized solutions of elliptic equations generalizes to parabolic systems. Marino and Maugeri [124] is also devoted to the problem of partial regularity of generalized solutions of systems of nonlinear parabolic problems (as is the preceding paper of Maugeri [125] for one equation). Let G ~ R n be a bounded domain with boundary of class C 2, Q = G x [-T, 0] be a cylinder with lateral surface S = 8G • [-T, 0]. In the cylinder Q we study the HSlder continuity of a generalized solution of the system of equations

Ti-+(--1)

~

D=(A~,~(t,x, D*u)DOu)--

laMf~l=m

:(,--1) ~ X

D:f=(t, x,D~tt) + ~.~

lal=m

(--1)I=tD:/=(t,

x, DVu),

(19)

IctI~m - I

Nl~<m--1, Ivl~<m.

where

H e r e u=(ut ..... u2r A=~(t,x,~.)are m a t r i c e s bounded f o r (t,x)EQ and a l l ~x"

of size N x N whichare

uniformly

continuous

and

In addition, let us assume that the following conditions hold. Condition I (parabolicity.). one has

For all (t,x)~Q and for any vectors gT and

(A~,(t,

x, ~ ) ~ , TI~)> v ~

lal,lfll=m

n=(Nl~<m--I, lal =~,

[[ "q~ II2 (v > 0),

1~I= m

where (', 9 ) i s t h e s c a l a r p r o d u c t and tl'll i s t h e E u c l i d e a n norm i n t h e f i n i t e - d i m e n s i o n a l space. Condition II. The vector-functions fa(t, x, ~ ) and fa(t, x, ST) satisfy a Caratheodory condition and the following conditions of growth of nonlinearity: a) for I~I = m

]lf~(t,x,~OH..
[l~fl ~

where

K > O , l~
2568

Ix])<~(n+2m)/(n+2lx[);

b) for

l~]~m--1 IIf , (t, x, ~) II--
iYl<:m

If~ h~

where

1~o(1~1, I'~[)~(n+gm--2l,~zl)/(n+2DI). M o r e o v e r , g=(t,x)fiLp f o r

[al = m, w h e r e p > n + 2m andg-~(t,x)s

I for I~i<m--I,

where

n + 4m--21al

n + 2m

7Jai--~ The basic result of [124] is THEOREM 2.2.

Let conditions I and II hold and the vector-function

u (t, x)~L2 (-- T, 0; W 7 (O)) A Leo ( - - T, 0; L2 (O)), be a generalized solution (in the sense of ~9') of (19). Then there exists a set Q0 c Q which is closed in Q such that for I~! = m - !

D~u(t,x)6 C~ (Q\Qo), v~ < 1 -

n+2----E-m; p

here mesn+2m_2(Q 0) = 0, where mesn+2m_ 2 denotes the (n + 2m - 2)-dimensional Hausdorff measure in relation to the parabolic metric 8(x, t; x', F)=max(]]x--x'l[,

]t--t'lw2m).

Further, under a somewhat altered condition II one establishes that u(t,x)~Lp,loe(Q). Now if u(t, x) is a generalized solution of the initial-boundary problem

u ( - - r , 0) = 0 (xEG), D'uls----O (l,coi~<m--1), t h e n u(t, x)6Lv(Q)

globally.

In the paper cited detailed proofs of the results formulated are given. 5.2. Some Special Weakly Nonlinear Problems. A number of papers of Wahl, Zhou, Yu-lin, Fu Hong-yuan, etc. are devoted to the question of the solvability of nonlinear parabolic problems in classes of summable functions of higher smoothness. Wahl [136] studies the equation

"~f't +

X

(--l)l~{D~(a=~(t,x)D~tt)----f (t,DVu), ["fl ~<m.

Let us assume that the following restrictions on the nonlinear function f(t, ~ )

1) If(g.~,.)t~<M

~

IVl
(20) hold:

ih, l*~ .+2,~t +1 ,

where M > O, m I > O, m 2 > 0 are constants. We consider a generalized solution u(t, x) of (20) satisfying the conditions of the first boundary problem

u(O,x)=~(x), xOO; D'uls=0, I~l<~m--1. THEOREM 2.3.

Let

n>2~--2, p > n + l .

Then if ~(x)6C~(G), then o

uCt, x)~L,(O, T; W ~ (a)) ~L, (0, T; Wg'(o)), o~ CLp (Q). Ot Zhou Y u - l i n

and Fu Hong-yuan [138] s t u d y t h e Cauchy p r o b l e m f o r t h e s y s t e m o f e q u a t i o n s

c)2mu

0~ (--1)~A(t) 0~, =f(~), (t>0, xeR'), Ot u(O, x) =~(x), 2569

where u = (ul..... u~), A(O-----(ai~(t))~• is a nondegenerate nonnegative definite matrix with continuous elements aij(t) for all i90. The nonlinear function f(u) is assumed to be smooth of class C 2re+h, k~>l, while 8f/3u is semibounded above. The following result is established: if ~(x)GW~m+k(l~1), then there exists a generalized solution of the Cauchy problem while the derivatives aZu/axl(l<,k--l) are continuous. In particular, for k>~2ra--I and the additional condition of smoothness of aij(t) the solution is classical. In the class of bounded functions a classical solution is unique. In the case of constant matrix the Cauchy problem is studied for periodic initial data. Zhou Yu-lin [137] studies the solvability of the first boundary problem for some classes of pseudoparabolic equations with one spatial variable

(_l)~yiT+

D ~'~~--B(t,~--

x,u, . . . . D~"-'u) D~tz + g (t, x, u . ... .

D~"-lu),

where u = (Ul..... uN), g = (gt..... gN), A (t, x)-- (Ai~(t,X))NxN is a symmetric positive definite matrix with continuous coefficients. The basic restrictions on the nonlinear summands are: I) the matrix B(t,x, u ..... D2n-tu) is semibounded above; 2) the function g_(t,x, ~0.... ,~2n-i) satisfies

(M > 0 being constant), $0, .... $ 2 n - 1 "

i.e., has no higher than linear growth in the variables

Under the restrictions cited the unique solvability of the first boundary problem in the classes W~,~ n is established, i.e., the classical solvability is established. Additional results are obtained for the equation 0u ( - - I ) n 7Ou{ + A ( t . x)D ~"OF ----f (u, Du,

. . . .

D~-Itt)

with special restrictions on the nonlinear function f(~0 ..... ~2n-l). 3.

Nonlinear Parabolic Equations of Infinite Order

3.1. Sobolev Spaces of Infinite Order. Nontriviality Criteria. bolic problems for an equation or system of infinite order

In considering para-

oo

0s

Ot + ~ (-I~!D~A:(t" x, DVtt)=h(t, x), t h e r e a r i s e n a t u r a l l y (as i n t h e spaces, but of infinite order.

For example,

elliptic

case too)

energy

t~ J~
(21)

of the type of Sobolev

if the elliptic part of (21) has the form

~(--l

)I~lO~(a~ [D~u ["-2D=u),

1el•0

then the corrsponding energy space is defined by the norm

IIu il~,.= If among the numbers a ~ O whether the space

'1 D ~u

(o>

p>l.

infinitely many are strictly positive,

then it is unclear a priori

w~176 p}={u(x): II ~(x)lloo., < ~o} is nontrivial,

i.e., contains at least one function u(x) which is not identically equal to

zero.

As it happens, the answer depends not only on the real parameters a=~>0 and p>~l, but also on in which domain O~l{'* the functions u(x) are defined and what boundary conditions they satisfy.

2570

We formulate criteria for nontriviality of the spaces W'{a~,p} in three cases (which in our view occur most frequently in the theory of differential equations): bounded domain G, complete Euclidean space R ~, and the torus T n. Bounded Domain.

Let

W~176

(x~C~(O),

a= ]D=u(x)lPdx< oo 9 1~1=0

Re c o n s i d e r

a sequence of nonnegative

0

numbers I

0

THEOREM 3 . 1 . a nonquasianalytic

The s p a c e W| ttadamard c l a s s

Complete Space R~.

(G) i s n o n t r i v i a l i f and o n l y i f C{MN} o f one r e a l v a r i a b l e .

Let

)~[=0

THEOREM 3.2.

t h e s e q u e n c e MN d e f i n e s

The space

W~{a=p}(R ")

Rn

is nontrivial if and only if the function co

z C", lal=O

is analytic in a neighborhood of zero. [We add that if ~(z) is an entire function, then the space Wco(a=,p} {R ~) is dense in Lp(R ~). ] Torus T n (Periodic Case).

In this case the space

W={a,, p}(Tn) =

a= [O=u(x)lPdx//~ oo \1~1=0

Tn

is nontrivial (and infinite-dimensional) if and only if the characteristic function oo

i~l=0

is an entire function. 3.2. Solvability of Parabolic Problems of Infinite Order~ Now we turn to precise formulations of the first boundary problem for (21), for example, with Dirichlet data of infinite order. In the cylinder Q=[0, problem

T]XG,

o-f+

G~R", with lateral surface S we consider the first boundary

(--1)~ID~A~(t,x,D~u)=h(t,x),

]Tt..
(21)

I-i=O

u(O,x)=O,

D'~uls=O, I~t
Just as before, let us assume that the functions As(t, x, ~X) s a t i s f y ditions.

(22) Caratheodory con-

In addition, let the following conditions hold: I.

For any (t, x)~Q, ~, ~=, where IyI<~.m, [=l=m, m = 0 ,

I,... one has

where K > 0 , a=~>0, p>l, b~j~O are constants while b 0 + b I +... < ~. II. For any (t,x)~Q, ~T

and ~=, Iml = m , one has the coercivity conditions la[=m

[al=m

2571

where 6>0, 6m~O are constants while 81+82+..,+Sm.3u...
The sequence a=~O and number p > i are such that the space W| trivial (cf. Theorem 3.1).

p}(O)

is non-

We introduce the spaces of solutions and of right sides of the problem (21), (22) needed. Namely, we set

z.(o, T; Ve~{a=, p}(O)) -- u(t, x]: Illu(t, x)lll~,~--

~

([Iiu(t,)x)ll~,.dt =I

{

<

,

here

D~

iol=0,

1.....

for almost all /6[0,T].

WT**{a=,p'} (G))

Further, we denote by L~, (0, T; by definition have the form

the dual space of ultradistributions which

co

k ( t , x ) = ~ ( - - 1 ) l a l a a D a / t a ( t , x), lal=o where ha(t,

(23)

x)fiLp,'(Q), while

aa IIha(t, x) IILp,(Q)< i~l=o

(24)

h(t, x)~Lt/(O, T; W-co {a=, p'}(O))

The v a l u e o f t h e f u n c t i o n a l W/~176

~.

on t h e f u n c t i o n

v Ct, x)6Lp(0, T;

i s defined by T

o0

0

T

0 a= 0

which is proper (well-defined) by (2.4).

Lo,(O, T;W-={aa, p'}(G)) a r e e q u a l by

Two functions h1(t , x) and h2(t , x) from t h e s p a c e definition if T

T

I
> dt = l < tt~, "o > dr.

0

0

0

for any function ~ (g, x)ELa ((3, T; lt~~ {a=, p}(G)). a Banach s p a c e i f as norm one s e t s

O b v i o u s l y t h e s p a c e Lo,(0, T;

W-~'{a=, p'}(G)) becomes

T

I!
Ill v 11'~,p

The following theorem holds. THEOREM 3.3. Let conditions I-III hold. Then for any right side lt(t,x)@Lp,(O,T;W-={a=, p'}(G)) there exists at least one function u(t, x) satisfying the following conditions: 1) tt(t,

x)ELp(O, T; W~ ~ {a a, p}(O)),

O,, (0, T; W-~ {a~, p'}(O)), -~{6Lp, 0

2) u ( 0 , x) = 0 i n t h e mean, n a m e l y i n t h e s e n s e t h a t one h a s

f o r any f u n c t i o n

v(x)EW~{ct.,P}(0)

linl < u (t, x), v (x) > = O,

t-~+O 0

3) f o r a n y f u n c t i o n T

0

v(t,x)s

T; ~

W={a=,p}(O)) one T

=0

The idea of the p r o o f i s t h e f o l l o w i n g . the partial sum o f t h e s e r i e s ( 2 1 ) p e r t u r b e d

2572

has the identity T

0

To b e g i n one c o n s i d e r s t h e b o u n d a r y p r o b l e m f o r by a " s m a l l summand" o f t h e f o r m

~.~ (--1)m+Ic~D2~u,

c~>O,

[a/=m+l

i.e., one considers the first boundary problem for the following equation of order 2m + 2:

c~D urn§ ~.~ (--1)mD~A~(t, x, DYu) ~-hm(& x), le*l=m+l

where hm(t, x) i s a p a r t i a l

(25)

]glUm

sum o f t h e s e r i e s

(23).

In view of t h e p r e s e n c e o f t h e p e r t u r b a t i o n , (25) i s a weakly n o n l i n e a r p a r a b o l i c equation. C o n s e q u e n t l y ( c f . Sec. 1, p o i n t 1 . 3 ) , t h e c o e r c i v i t y c o n d i t i o n I I g u a r a n t e e s t h e s o l v a b i l i t y of the first boundary problem for this equation. This same condition and the compactness lenmla (cf. Sec. 1 point 1.3) guarantees the possibility of passing to the limit as m § ~. We note that here the coefficients c a are chosen to tend to zero sufficiently rapidly. It suffices, for example, to choose them so that the intersection 0

0

W={c~,, 2}(O)r'lW={a~,, p}(O) is nontrivial which is always possible (in particular if p~2, one can set c a -= as). To conclude we note that Chan Dyk Van (cf. [88], Chap. 5, point 5.6) considered the first boundary problem for nonlinear parabolic equations of infinite order with degeneracy. A model example of an equation considered in this paper is the equation

O--f+

(--1)l~rDa(a~o~(x)lD~ulP-2D~u)=h(t,x),

I~!--0

where a ~ O

is a real sequence, p > 1 is a number, and the weight functions q~

q~

9~(X)=--Xl ...xn ~ X q~, where q~, I~[<~176are bounded. 4.

Nonlinear Stochastic Equations of Parabolic Type By a nonlinear evolutional stochastic equation is meant an equation l

~(t)= l 0

t

A(u(s))ds+ l B (u(s))dw(s),

(26)

0

where u ( t ) i s a random f i e l d , A and B a r e n o n l i n e a r o p e r a t o r s in a Banach s p a c e . e n t i a l form (26) has t h e form

In d i f f e r -

du-----A (u) dt-{-B (u) dw (t).

( 27 )

By w(t) we understand a Wiener process with values in a separable Hilbert space H. Let (~, F, P) be a complete probability space with expanding system of o-algebras {Ft} • (t6[0, ~, T < ~ ) , imbedded in F. We shall call a function co_q~atible with the o-algebra F t if it is Ft-measurable. By a solution of (26) [or (27)] we mean a function (random field) u(t) from the domain of definition of the operators A and B (independent of /~[0, ~), satisfying (26) [or (27)] and compatible with the same system of o-algebras with which w(t), A, and B are compatible. Such a solution is called strong. Nonlinear stochastic equations of parabolic type were studied from the beginning of the eighties in Bensoussan and Temam [92], Pardoux [129], Viot [134], etc. The ideas of the method of compactness were apparently first used in linear stochastic problems by Bensoussan [91]. In Bensoussan and Temam [92] the methods of compactness and monotonicity are applied to prove the solvability of a nonlinear stochastic problem. Here discretization of the equation and subsequent passage to the limit were used for the case of a nonlinear "drift" operator A(u). Moreover it was assumed that A(u) is a monotonic operator and B(u) = I. The subsequent development of the methods of monotonicity and compactness applied to stochastic evolution equations is found in Pardoux [129], Viot [134], and in Vishik and Komech

[16-18]. 2573

Pardoux and Viot consider a general evolutional stochastic equation with unbounded nonlinear "drift" and "diffusion" operators. Vishik and Komech study the Cauchy problem for quasilinear infinite-dimensional parabolic equations and the boundary problem for a stochastic Navier-Stokes system with white noise. Quite general results are obtained in Krylov and Rozovskii [52] in which in particular stochastic partial differential Ito equations of any finite order are studied. We formulate the results of this paper. The first boundary problem is considered for nonlinear stochastic equations of parabolic type

X ( -l)l~lD~A~(t'x'm; Dv~dt

du(t,x,~)~---

I~1<~

+B(t, x, ~; D ~ ) d w ( t , o ) + d z ( t , x , ~ ) , [?]..<m, u (0, x, ~) = ~ (x, ~), xEO, Dru[s-~f~(s), sES, l v l 4 m - , - - 1 ,

(28) (29) (30)

where G ~ R n is a bounded domain with regular boundary, S = [0, T] x F is the lateral surface of the cylinder Q = [0, T] • G. The functions A~(t, x, ~; gy) are real functions of real arguments, u(t, x, w) is a real function also. B is an operator-function with values in a Hilbert space E, w(t, ~) is a Wiener process with values in E. Let us assume that the following conditions hold.

s and (B(t, x, ~; ~y), I. For any ~TERI, [?[~ , and any e 6E the functio n s A ~ ( t , x, e) E are measurable in t, x, ~; for each rE[0, T] they a r e m e a s u r a b l e in (x, w) with respect to the product o-algebra on Ft; for each fixed t, x, ~ these functions are continuous in ~y. II. Let there exist a constant K > 0 and a nonnegative function f(t, x, w) having the same measurability properties as the functions A~(t, x, ~; 0) such that for all (t, x, m),

Wl<m

11B (t, x, o; ~v)1[~'-< K X

I ~v I~-~ + f (t, x, ~) + K I go IL

IWl~m

In addition, let the initial function u0(x, m) be measurable with respect to the Borel o-algebra on F 0 and for almost all oE~ be square summable in xEO. Further, suppose for almost all t~, T], o6~ the function f(t, x, w) is summable in x while T

I II/(t,x, ~ IlL,dr < ~ . 0 0

III.

For any f u n c t i o n u from t h e s t o c h a s t i c

S o b o l e v s p a c e W~ one has

( A(u),u > + l l B u q ~ ,u e~ +%[lul[~'- 0, K > 0 a r e c o n s t a n t s . IV.

(Lower S e m i b o u n d e d n e s s o f V a r i a t i o n . )

+ liB ( u , ) - - B 0

where K > 0 i s a c o n s t a n t ,

One h a s

(u~)ll2~ c~. r~ <~Kllu,-udl~,

0

~EW2, 89

In conditions III and IV

A (u)-- -- ~

(-- I)I=D=A= (t,x, m; O~@,

f~l<m

B (u)~B (t, x, ~; D~g), 0

< , > i s t h e d u a l i t y o f t h e s p a c e s W ; ~'W~'o' LatH'E) i s t h e s p a c e o f o p e r a t o r s r + E such that r are Hilbert-Schmidt o p e r a t o r s , where Q i s a s y m m e t r i c n o n n e g a t i v e k e r n e l o p e r a t o r i n H. The following theorem holds.

2574

THEOREM 4. i. Let conditions I-IV hold. Then the problem (28)-(30) has a unique strong solution (cf. the definition at the beginning of the section). This solution depends continuously on the initial data. One should note that in concrete situations the verification of conditions III and IV guaranteeing the strong parabolicity of the problem can be assured by the following sufficient conditions of algebraic character. Namely, let the functions A~(t, x, ~; $u be differentiable with respect to $7 everywhere except for a set which has a finite number of points of intersection with each line in the space of coordinates ~, IyI~<m, and the derivatives

A~(t, x, ~; ~T)-----OA~(t,x, o; ~)10~ be locally summable on each line in this space. Let us assume that there exist numbers s > 0 and N > 0 such that for all~T, N=, ~ (IYl, I~[, l~I~.m), and also t6~, T], xeG, ~ one has

I=l, I~l<m

I~l<m

I=[=m

I

~ ~<~N ~ O.

Then conditions III and IV hold. Example.

Let R a = E =R~

and (28) have the form

du----a(t,o)(--l) ~+~-~-~-\l~x~ (i ~ t

o~u )dt+b(t,

Ox m

o~u ~/2

where a(t, m) and b(t, m) are bounded measurable processes. Here the algebraic condition of strong parabolicity has the form

--2(p--1)a(t, co)+ P--~--4bi(t,~)..< --8, where ~ > 0 is a constant. 5.

Numerical Methods of Solution of High Order Nonlinear Parabolic Equations

As is known the standard explicit difference schemes for parabolic equations of order 2 m (m>/]) require quite strong restrictions on the smallness of the ratio T/h 2m, where T is the step of the net in time and h is the step of the net in the spatial variable, to guarantee stability. As a rule implicit difference schemes are free of this deficiency but their numerical realization leads to excessive "machine" expenditure since on each time layer one has to solve rather complicated difference systems. By economical difference schemes we mean schemes combining the virtues of ordinary implicit and explicit schemes: they should be absolutely stable and let one make the numerical passage from one temporal layer to another upon performance of a number of arithmetic actions of order of the number of nodes of the spatial net. Questions of the construction and study of economical difference schemes for nonlinear parabolic equations are considered in D'yakonov [34, 35]~ Namely, let Qc~-Rn be a domain composed of parallelepipeds. We denote the boundary of by F. In the cylinder Q = [0, T] • ~ we study the first boundary problem

~-+(--I)

~

D=(A=~(t,x,DVu)D~)=~(t,.x, DVu),

I=I,I~T=~

(31)

l~l<m. u(O,x)=~(x), D~uii~rl•

i~l<m--1, x,fir.

(32)

Basic assumptions: I.

The functions A~$(t, x, Sy) and also their derivatives with respect to $~ are continuous and uniformly bounded. Analogous requirements are imposed on the right side

h(t, x, ~u II. For any sufficiently smooth function v(t, x) the operator

L,(u)------(--1)= ~

O=(A~(t, x, DVv)D~u)

is a strongly elliptic operator. 2575

The economical difference schemes constructed for this problem are generated by familiar methods of variable directions and are based on the use of difference splitting operators on the upper layer. In the simplest form these schemes have the form

where uiq = u(qT, ilh,...,inh) is the value of the net function u at time q~ at the corresponding node and L h and A s are finite-difference approximations of the differential operators

L(u)=(--1) ~ ~ and (-l)mD~ m, respectively

D=(A=~(x,DYu)Dgu)

(s = I ..... n).

In addition, E is the identity operator and o > 0 is a parameter on whose choice the stability of the difference scheme depends. The presence in the difference scheme of a splitting operator lets us reduce the solution of the multidimensional systems which arise to the successive solution of n one-dimensional systems. The indicated schemes themselves can be considered as ordinary difference schemes for the problem (31), (32) with "viscous" perturbation. For example in the twodimensional case (n = 2) the perturbed equation has the form 2

where ~i > 0, g2 > 0 are constants. Under conditions ! and II the convergence of the difference scheme in the energy metry corresponding to the problem (31), (32) was studied. It was proved that if a generalized solution of the problem (31), (32) exists, then one has convergence in the metric of the net space of type L2(0, T; W~(~)). Moreover, under the additional condition of uniqueness of a sufficiently smooth solution of the original problem one gets an estimate of the rate of convergence of order h 2 + ~. W e note that analogous results are obtained on the Cauchy problem for (31) for the case of periodic conditions with respect to the spatial variables and also for the problems indicated in the case of systems of quasilinear equations of high order. Under other conditions difference schemes for quasilinear parabolic equations were studied in Lyashko [64-67], Lapin and Lyashko [62, 61]; Karchevskii, Lapin, and Lyashko [42]. We describe some of the results of these papers. We consider the first boundary problem

O_tu Ot + X ( -1)r~tD~A~(t'x'D~u)=f(t' x), I~fl~<m,

(33)

[=l~m

u(0, x ) - - ~ ( x ) ,

D~215

=0,

[~

(34)

The problem ( 3 3 ) , (34) i s c o n s i d e r e d in t h e c y l i n d e r Q = [0, T] • ~ whose base i s a s q u a r e , i . e . , f~={0<~.x,~
L(~)~ ~ (--1)'=ID~A~(t, x, D~u) lOq-<:ra o

defines a strongly elliptic nonlinear operator acting from wm(~) to w~m(~). More precisely, let us assume that the functions A~(t, x, $~) are continuous and have continuous derivatives with respect to g~ while for all (t,x)E[0,T ] X ~ and all $~

IA=(...)I--<M,(I+ ~ l~,J),IOA=(...)/O~I<M~, Iyl~m

where M i > 0, M 2 > 0 are constants.

2576

Condition

I.

One has

E

OA~ (~, x, ~ ) ~ o~ > co~

Ig],lBl<m

~,

I=l=m

where c o > 0 is a constant. Condition II. Let us assume that a solution u(x, t) of the problem (33), (34) (under condition I a generalized solution exists and is unique, cf. Secs. i, 2) is smooth and admits an extension of class C 2m+2 in x to some domain [0, T] • ~', where ~' m ~. Before formulating the basic results on the parabolic problem (33), (34) we consider difference schemes for the elliptic part L(u). At the base of these schemes lies the summator identity which is a discrete analog of the basic integral identity defining a generalized solution. Let

be one-dimensional

the Cartesian

net sets on ~, i = i ..... n, h = (N + i) -I.

product of one-dimensional n

as

/'1

%-~]Ico,, ;=]I~, ~=~\~o i =1

Let the operators

Nets on ~ can be defined

nets:

....

i=1

d i and dl z mean the simplest

finite-difference

relations

"forward"

and

"backward. " Then d"----d~'. .. d n% , d - = = d ~ -=I ,.. d n- = n, d ~ We associate with the multiindex +e = (a I ..... gn), where gi = 0 if ~i is even and z i = 1 if ~i is odd. Obviously the multi-index a = ~ - s has only even components. Further, - - e = ( - - e I..... --en)~ e+~-(el,-~e~ .... {--|~-~en), g_~---e+. The difference operators corresponding to these multi-indices are denoted

by d +~, d -~, d ~-, d'*. We define net spaces of the type of the Sobolev

spaces wm(~).

For this we introduce

the

sets " "

"'

~ 2 ~81

'

OJ+e~--H" 0)(I), (O_e----fi 0)(--0, I=I

and we consider

the difference

I=I

N

n

i=l

/=I

operators 0

0

0

O+"= d+~da, d-~ = d-~d a, O~+.~_d*+d% 0

0~- .~ d~-da, o

where da---d~ Let

-~

is the central-difference

relation

of order a, a ~ ( a I.... , an).

(u, v).=h"~u(x)~(~), [u,~]=h"~.(x)~(x~ (u, ~,)•

]~ .(x)v(x), (., (o .-I-8

.)~•

.(xlv(x), o)e.+,

~(o,,(x, OVu),O~)~=(O~(x, O+vu),O+.v)_~+ +(O=(x, O-~u), O-"v)+~+(O~(x, Ovu), Ov-vk +(o=(x, O~+u), O.+v)~_. 2577

Then

llu=If+,0=~88 ~(a=u, a=u)~is

0

the net analog of the norm in the space wm(~),

l=l=m

Iluli~=llull+,o+ (u,

u). i s the net analog of the norm in W~(Q),

Ilull_----sup!l~$~)l il§ i s t h e n e t a n a l o g o f t h e norm in w~m(~). Definition 5.1. The net function uh(x) is called a solution of the difference scheme of the Dirichlet problem (t fixed) for the equation

L(u)=f(x),

x~n,

if the following conditions hold:

l) llu~ll+,0<~, u~l~h=0; 2) for any function v h satisfying i), one has the identity

Ir

where

e

]=f-~-O(h2), b=(x, ~)=A=(x, ~v) at

points of the net.

Explicitly this scheme has the form

(35)

Lh(u)=y, where

L~(u)-~ ~., (--Dt='{a=b=(x, O-~u)+O-=b=(., O~u)+a=-~b~ a~.)+O=-b=(x, a~+u)} Icq~,m

THEOREM 5.1. The difference scheme (35) is uniquely solvable in the sense of Definition 5.1, and in addition the propriety inequality

tt u ~ - ~

t[+,o~< c~ 111Lhuh - L ~

IL.

holds. THEOREM 5.2.

The difference scheme (35) converges in the norm I1ull+with rate o(h2),

i.e.,

Ilu--uhll+~M.h2,

M>O.

Now we turn to the original parabolic problem (33), (34).

The net sets, norms, and

0

spaces of type W~(~) are introduced analogously to the stationary case. In correspondence with what was said above for the elliptic part L(u) the difference scheme for the problem (33), (34) can be given in the form of the differential-operator equation

B ~tnq - L h (u h)-- fn (t),

( 36 )

un(O)---'~h' un [~n---0'

(37)

which in turn leads, for example, to the explicit scheme

B

uh,qq-I t~ft,q

~-[- Ln(ttn'q)-~f h, uh,0--_~, uh.q [r----0.

Here [a=r(t,x)+O(h2-~z), ~ h = ~ (x) + O (h 2) a r e n e t a p p r o x i m a t i o n s o f t h e r i g h t s i d e f ( t ,

x) and

the initial function (p(x).. THEOREM 5.3. If the conditions on the problem (33), (34) formulated above hold, then the difference scheme (36), (37) converges with rate O(h 2 + T). 6.

Degeneration of a High Order Nonlinear Parabolic Equation and System

One distinguishes two forms of degeneration of nonlinear parabolic problems: problems with explicit (or fixed) degeneracy and problems with implicit degeneracy.

2578

The first type includes boundary problems for equations whose "characteristic" form vanishes on certain subsets independent of the possible solutions. An example is the following equation:

o~_a(t, x, u)~-~----'~ h(t, X), where

a(t,x,u)>~ao(t,x)>~O and

a(t, x) vanishes on fixed manifolds of lower dimension.

The second type includes boundary problems for equations whose "characteristic" form can degenerate at those points (t, x) at which the solution sought or some of its derivatives vanishes. Examples are the model equation of filtration of a gas in a porous medium @

0u

(u~ ~ ~= h (t,x) (~> 0)

or the familiar equation of flow of the boundary layer in the Prandtl-von Mises form (cf.~ e.g., Oleinik [70])

3F - - V ~

o-;r-2= k (t, x).

Obviously in both cases the degeneration of parabolicity is due to the possible vanishing of the solution u(t, x) sought. However in advance the set of values of the arguments (t, x), where u(t, x) = 0 is generally unknown. 6.1. Equations with Explicit Degeneration- Systems of quasilinear parabolic equations with explicit degeneration are studied by Kruzhkov [51]. In the rectangle Q = [0, T] • [-s

i] we consider the first boundary problem

(--1)'n ~-~ + ao (t, x, u . . . . . D~m-Iu) D ~ u = f ( t, x, u, . . . . D~-lu),

(38)

i----0,1 . . . . , m - - l ,

Dtu]x=•

(39)

u(o,x) =~(x). number), ao(t, x,~o

(4o)

Here u = (ul,...,UN) ( N ~ I is a natural ..... ~2m--l) is a square matrix of size N • N; the elements of this matrix are measurable in the variables (t,x)6Q and continuous in the variables ~=(~0 ..... ~2m-l)s 2ran (each component $i, i = 0, 1 ..... 2m - i is an N dimensional vector). We shall assume that the elements of the matrix a0(t, x, g) in any ball [~[~R are continuous in $ uniformly with respect to~(t,x)EQ and are bounded in modulus by a constant A(R) in domains of the form

e~={(t, x)eQ; I~l,--
f(t,x,~o..... ~2m-I)

oo, m..
is a vector-function f = (fl ..... fN) satisfying analogous

Basic Assumption. Let us assume that the matrix a0(t, x, ~0..... ~ - i ) lowing parabolicity condition for the system (N~RN):

satisfies the fol-

inf inf(~l, ao(t , x, ~o. . . . . ~2m_,)~])~-X(t, x)>O, ~6R2mNI~l=l ~-~(t,x)~Lq(Q),

q:>l.

In addition let the subordination condition of the right side vector ~]~N, (t, x)6Q and arbitrary ~ = (~0..... ~ - I ) one has

[ ( f ( t , x , g ) , ~ l ) J . . < K ( ~ , a o ( t , x , g ) n ) ~ t 2 ~;(r) 9

I~,l+ X t=1

where K > 0 is a constant, r~=~012+...+]~m_a] 2, ~(r)~>l satisfying the condition

f(t, x, ~0...... ~m-1):hold:

i~----m+l

]~l+/o(t,x)

for any

,

is a smooth, nondecreasing function

co

1

and t h e f u n c t i o n f 0 ( L x)fiL2(Q).

2579

THEOREM 6.1. Let the conditions listed above hold. Then the problem (38)-(40) has at least one solution u(t, x) belonging to the space ivf, 2m t ~ ], where q* = 2q/(q + i). r q*,q* ~ ~q. For the proof one first considers a nondegenerate system (38) with matrix of the form

ao(t, x, ~o. . . . . ~-,)+n-lE, n = i, 2,...; E = the identity matrix. The nondegenerate problem has a solution Un(t, x) I 2m (cf. Chap. 2, Sec. 2), where the norms of Un(t, x) in the space Wq'~,q,(Q) are uniformly bounded. The limit point of the sequence Un(t , x) is the solution of the problem (38)-(40) sought. 6.2. Implicit Degeneration of Filtration Type. In the cylinder Q =[0, T]XO, lateral surface S = [0, T] • 3G one studies the boundary problem 0_s

Ot +

X

OcR%

with

(--1)t~iD~(A"~ (t' x'D~u) D"u)--h(l'x')'

(41)

r=l,l~l<m

(f't D"u[s=0, loin<m--I, u(0, x) =q) (x), xO_G.

(42) (43)

Here, as usual, ~ = (~i ..... an) is the multi-index of differentiation (~, y, and ~ analogously). We shall assume that the functions A ~ ( t , x, $~) are measurable with respect to (t,x)EQ, and with respect to ~ are nonlinear continuous functions of polynomial growth. The latter means that these functions satisfy inequalities

IA=~(t,x, where /~(t, x, ~v)~0

~v)l~ X

l~(t, x, ~)15~1p~,

(44)

are functions which are continuous in ~y such that

l"=~(t, x, where M > 0 is a constant,

~v)< (Ivl<,n-1]~vl

O - . < p ~ < l are numbers.

Remark 6.1. In (44) the equality ~ = ~ - i means that the multi-indices (Pl ..... ~n) run through collections of the form (~i, .... ~i - i ..... ~n), i = I .... ,n. Now we formulate the basic coercitivity type condition which characterizes the degree of implicit degeneracy of the parabolic problem (41)-(43). Namely, suppose for any $~, ~ , and q~ one has

Re

~

A=~(t,x,~v)TI=-~ >ao

I~1, I~[<m

~.~

]~u[%ITI=]=--K,

(45)

I~l=m,~=~--I

where a 0 > 0, K > 0 are constants and p ~ 0

are numbers.

In weaker functional form (45) assumes the form

I~l,l~/~m Q

['l=m'P==--I Q

We n o t e t h a t a t t h o s e p o i n t s (t,x)~Q, a t which D~u(t, x) = 0 t h e p a r a b o l i c i t y

condition

has degenerate character. We introduce notation (for the set of solutions of the problem posed):

W={u(t, x):D, (I D~tzI;~t=D"u)~L~(Q), ~----=--I, I=l=m, where q * - ' + q - ~ = l ,

o,,

i = l . . . . . ~;-ff-

q=min(p~+2)/(p~+p~,+l). 6.1. The function u(t,x)~W

Definition is called a generalized solution of the problem (41)-(43) if the following conditions hold: i) the function u(t, x) satisfies (41) in the sense of generalized functions, i.e., for any smooth finite function v(t, x) in Q one has the identity

2580

[al ,i~l~rn

where

Q 2) t h e b o u n d a r y c o n d i t i o n s

(42) and i n i t i a l

Both conditions need explanation.

condition

(43) a r e s a t i s f i e d

" i n t h e mean. *~

The fact is that strictly speaking the function

u(l,x)6W cannot have derivatives D~u(t, x) for I~I = m which are square sum~able in Q [where DUu(t, x) = 0 the derivatives of order m may not even exist]. First of all we indicate the sense in which we understand the integrals

[A~ (t, x, Dru) D=u, D%]. For this we note that by (44) each function A ~ ( t , finite number of summands of the type of

(46)

x, ~y) can be represented as the sum of a

where

]m~f~(t,x,$v)[..<M(

Z

l ~ v ] + l ) p~'

\Ivl~<m-1 M > 0 being a c o n s t a n t and O ~ < p ~ < 1 . It follows from this that each integral (46) can be represented as a finite sum of in ~ tegrals of the type of

[rn~ (t, x, D Vu) ] D~u iar~D~tt, D~v], where the derivatives D~u obviously occur only in combination with factors makes sense if (by definition) we let

ID~u] p~ which now

p.

]D"tt I~'D~u= ID"u [p./2 p~+iDi(IDguj 2 2 D~u). We apply precisely this regularization as the values of the integrals Further, that

(46).

as already noted, from the fact that u(t,x)6W it does not generally follow for [al = m. However the "monotone" function

D=u~L2 ( Q)

cp~( D~u) ~ [D.u IvUIiD~uEW~(Q),

if u(I,x)~W, Consequently the functions %(D ~u) vanish on S in the sense of Li(Q). this in mind in saying that D~u[s=0, ]m]~rn--l,"in the mean~

We have

One has the following existence theorem for generalized solutions. THEOREM 6.2.

If the conditions formulated above hold then for any right side h(t,x)ELq,(0 , of the problem

'T;W~-,m(G)),q*=q/(q--1),there exists at least one generalized solution u(t,x)~W

(41)-(43).

The proof, of the theorem goes by the method of compactness analogous to the case of weakly nonlinear problems. The most essential point here is the application of the "nonlinear" version of the compactness theorem formulated in point 1.3 of Sec. i. 6.3. Implicit Degeneration of "Boundary Layer" TyEeo Now we turn to the case of implicit degeneration caused by the vanishing of derivatives DYu of order 2m - 2. Boundary problems for degenerate equations of the type indicated were studied by Soltanov in tlhe energy metric. In [84] he considers the first boundary problem for a quasilinear parabolic equation with one spatial variable ( (t,x) ~Q=[0, ~X[a, b])

ou

( - - 1 ) m ~ q-A([, .v, u . . . . . Dim-iu) Dimtt~h( [, x),

(47)

u(O, x)=O,

(48) (49)

DIg(t, a)=D]tt(t, b)~O, L e t us assume t h a t t h e n o n l i n e a r t h e o d o r y c o n d i t i o n and has t h e form

j = O , 1 . . . . . m--1.

f u n c t i o n A(t,x,~), ~= (~0, ~, . - . , ~=m-2), s a t i s f i e s

a Cara-

2581

A (t, x, ~)~A, (t, x, ~)I ~2~-21 ~, where 0 < a o ~ A , (t, x, ~)~.~M
tb

Eu(O= I [ D'~u(t' x) Izdx + I I [ D2~-2t~]~ID~=]Udxdt. a

0

a

In correspondence with this form we introduce the space of solutions. 0

,2m--2

~r~(q)-----{tt(t, x):q-+R'ltt(t,x)6Lv(O, T; Wp

Let

_

(a, b))N

9--I

0

f3 Loo(O, T; W~(a, b)); I D~-2ulT'D2"~-'ueL2q(O, T; W~q(a, b)); I D~m-utttoDU~-2tt~L~(O, T; W~(a, b)); [ D2~-~tt [~

0. u(0, .x)=0}, ~.~(o, T; W~(a, 0)); --~-~Cq(O), where p = p + 2, q = (0 + 2)/(p + i). 0

Definition 6.2. The function u(t,x)~(Q) is called a generalized solution of the problem (47)-(49) if it satisfies the equation in the sense of the space Lq(Q).'- Here the second summand is regularized as follows:

A ~, x, ~ .... D2~-2u) D ~

~ A1 (t, x, u . . . . . D2"~-2u)D ( I D~-2u [PD2~-lu)--

-- pAl ( t, x, ~ . . . .

D2m-2u) [ D2~-2u [~ D2~-2u ] D 2~-1u 12. o By virtue of the inclusions which hold for the functions u ~ t , x ) ~ ( Q ) , e a c h

(50) summand in (50)

belongs to Lq(Q) and consequently the regularization is proper. One has the following theorem. 0 z

THEOREM 6.3. solvable.

For any function h(t,x)CLq(0, T;W~(=,b)NWq(a,b))

the problem (47)-(49) is

The ~ goes by the method of compactness, but a new point here is the establishment and application of a nonlinear version of the compactness theorem (cf. Sec. i, point 1.3), for an adequate degeneracy of boundary layer type. We formulate this theorem (for a multidimensional domain G at once).

k

where X = | , 2; ~ l ,

~0

t,]=l

and B o i s a Banach space such t h a t

O b v i o u s l y SX,~, ~ i s a n o n l i n e a r

Let

set.

L~+~(G)~Bo.

I n S y , ~ , ~ we i n t r o d u c e

t h e seminorm

!

Further, we set

9'~,,,.~(Q)- .(t, x): tulsv,=,#* ' "=+~ "t < 0

where B l is a Banach space such that L=+e(Q)~Bl. One establishes that (in a certain sense) the inclusion

is compact if ai+~i~2+!~2, ~ 2 . Application of this theorem lets one pass to the limit in the compactness method.

2582

Analogous results are obtained in [82] for the multidimensional 0u O--f +

~

problem

(--l~'~'D~(Aa~(t, x, u . . . . . DVtt)D~tt)=fz(tx),

2<:[~ I ,ll~l
[col-..<m--l,

u(O,x)=O, D~

having degeneracy of boundary layer type. The case of conditions which are periodic in x and also the case of equations with nonpolynomial nonlinearities are also considered.. CHAPTER 2 NONLINEAR PARABOLIC PROBLEMS OF GENERAL FORM i.

General Boundary problems

i.I. A Priori Estimate and Solvability in the Large_~. Boundary problems for quasilinear systems which are parabolic in the sense of D o u g l i s - N i r e n b e r g - S o l o n n i k o v were studied by Pokhozhaev [72]. In this paper the following important fact for nonlinear problems of parabolic type is established: a sufficiently strong a priori estimate of all solutions implies the solvability of the problem in the large, i.e., for O,~t~T, where T > 0 is a given number. Thus the proof of the corresponding a priori estimate (without other additional constructions) guarantees the solvability of a parabolic problem. Let Q = [0, T] • G be a cylinder with base G ~ R n (with smooth boundary F) and lateral surface S = [0, T] • F. One considers a quasilinear system of equations of the form N

7=I l~1~=t7

Here i, k = 1 ..... N; tj = 2bt , where tj (j = 1 ..... n) are nonnegative

integers while

N

Z t =2o , .t=1

where b > 0, r > 0 are certain integers. Further, 9

are vectors with nonnegative

9 ..

integral components, I vlb=2b~'o+W+

O~/~

and by definition 9 9 9 +~,~

(analogously for [~k]b). The collections of multi-indices ~k are such that I~kI~g~--l, where k = 1 ..... N. The functions a~j(t,x,~=k),al{t,x,~=k) , and fi(t, x) are real functions while if t k = 0 then these functions are independent of the corresponding variables $~k. On the lateral surface S of the cylinder Q there are given boundary conditions N

(2) 7=1 1 3 % ~ l + t /

where ~ = 1 .... ,br; oj are negative integers; b~j(t, x) and ~ ( t , if o~ + tj < 0 then b~..(t, x) -- 0. x3 Moreover, for t = 0 let the initial conditions

o%ot~I ,=o=~ (j = 1 .... ,N) hold. tj = 0.

(3)

Here one does not impose initial conditions on the component uj(t, x) if

We introduce the spaces needed. N

i=o, 1,. .., t)--!

x) are real functions while

Let

0 t',t..

HI(Q)=HW~p'(Q) 1=I

be the space of solutions, 2583

N

]-]2(Q)~---]~Lp(Q)' be the space of right sides,

j=l

br

(s)= II

o

!-(

~- ~st - T~/i , - o ~-~1

(s)

f=1

be the space of boundary values. Norms are introduced in these spaces as the sums of the corresponding norms. Let us assume that the following conditions hold: I (Smoothness Condition.)

The functions a~j (t, x, ~=k) and a~(t,x,~ k) are defined and con-

tinuous for (t, x)6Q and all $~kcontinuous partial derivatives.

Moreover, let us assume that these functions have everywhere The coefficients b~j(t, x) belong to the class 1/

1

II

r,~-~l,-9-;,-~t-~-+s/e ~ 8 ~ 0 being arbitrary. t, x ~ol, II (Parabolicity Condition.) system

Let us assume that for any fixed function u(t,x)'~H1 the

N

~

mk

aTj (t, .x, m uk) DV~j= , j (t, x)

(] = 1 . . . . . N)

(4)

1=1 [ Vlb=t j

which i s l i n e a r [ w i t h respect to v ( t , Solonnikov (for s i = 0).

x)]

is parabolic

in the sense of D o u g l i s - N i r e n b e r g -

III (Complementability Condition. ) Suppose for any fixed vector-function u(t,x)~Hi(Q) the boundary conditions (2) relative to the linear system satisfy the familiar Lopatinskii condition (or complementability condition; in algebraic form applicable to the system considered these conditions are indicated, for example, in Ladyzhenskaya, Solonnikov, and Ural'tseva [60], Chap. VII, Sec. 9). IV (Existence of an A Priori Estimate.) problem (i)-(3) one has

Suppose for any solution u(t, x)EHz(Q) of the

IIu I! ,v cm < C ([l f IIH, CQ)+ it II where C(p) is a continuous function on [0, +~). Under the conditions formulated one has THEOREM i.i. if conditions 1-IV hold and p > n + 2b then for any vector-function f(t, x)~H2(Q) there exists at last one solution u(t,x)6Hl(Q)of the problem (1)-(3). If, in addition, one assumes that for any N

u(t,x)OHI(Q) and w(t,x)GHl(Q) the

system

1

!~lb~tj 0

which is linear with respect to v(t, x) is parabolic in the sense of D o u g l i s - N i r e n b e r g Solonnikov and the boundary conditions (2) satisfy a complementability condition [in v(t, x)] with respect to this system, then a solution u(t,x)~ff1{Q) of the original problem is unique. The proof is based on the application of the following operator theorem (Pokhozhaev, [73], Theorem 2). Let X and Y be Banach spaces while Y is uniformly convex. In addition, let A'X § Y be a nonlinear operator defined o n all of X and Frechet differentiable at each point x6X. THEOREM 1.2.

Let the following two conditions hold:

i) the domain of values A(X) is closed in Y; 2) for any

x~X

Ker(A'(x))*={0}, where A' (x) is the Frechet derivative.

Then A(X) = Y, i.e., for any y~Y the equation A(x) = y has at least one solution x@X. 1.2. Normal Solvability. Questions of normal solvability (in the sense of Fredholm) of general nonlinear parabolic boundary problems were considered by Babin. In [3] he establishes a nonlinear analog of the familiar "finiteness" theorems of Fredholm maps.

2584

Let D(S)u(t, t,x x), s = 0,...,lm mean the vector composed of all derivatives of the function u(t, x) of the form D~D~u such that lal + 2b$ < s, where b > 0 is a parabolic weig_h_t, i.e., ~(2m).

an integer which is a divisor of m.

We denote the dimension of the vector u~,x ~ by r.

Suppose in addition a(t, x, z), bj(t, x, z), j = i, .... m, where z ~ N , smooth functions of the variables (L x)OQT and z6R N.

are arbitrary

In the cylinder QT = [0, T] • G one considers the nonlinear parabolic problem

Lo(u)~a (t, x, O(lm)u)= f (r x),

(5)

B~ (~)-- Oi it, x, D ~ P ~ is =g] (t, x'),

( 6)

x'eOO, ]= 1, 2 . . . . . m

(mj~0

being integers).

We make correspond to the operators L0(u) and Bj(u) the polynomials

L~

Lx, z, ~, p)--

X

a ~ ( t , x , z)$~p~,

f~l+21~b=m

B~~(t, x, z, ~, p)----_-

2~

bm (t, x, z) ~p~,

l r [+ 2b[~=m ]

where ~ER n, zGR r, p E Q = { q @ C I : R e q ~ ; a=~ and bj~$ are the derivativs of the functions a(t, x, D(lm)u) andbj(t, x, D(mj)u) with respect to the variables D~D~u. Let us assume that the original problem is 2b-parabolic which means that the familiar conditions on the distribution of the roots p of the polynomial L~ x, z, $, p) and the complementability conditions (cf. point i.I) holdo 0 ~l,l

We shall seek a solution of the problem (5), (6) in the space Wl,2 (Qr), where the "zero" sign means that for t = 0 we consider null initial conditions. Further, let 0

0

fE 2,~ (Q~), gjew~,~ J(S), where c = l / 2 b , ~ = l - - 2 m , g j = l - - m l - - l / 2 ,

j = l . . . . . m.

In addition we shall assume that

where n = dimG.

Then from the familiar imbedding theorems for anisotropic Sobolev spaces it 0

follows that solutions tt(f,x)@l~z~,2 at.t{Qr) are classical solutions [we stress that no growth conditions with respect to the variable z are imposed on the functions a(t, x, z) and bj(t,

x, z)]. In order to formulate the result on the normal solvability of the problem (5), (6) precisely we introduce the Hilbert space (of right sides of the problem) 0

m

(QT)=WI,2 (QT)•

Wlff ](S). ]=1 0

We denote by oi, e 2 .... an orthonormal basis for the space

Wyffl,l

w2.2 (Qr) and let

B I,R,, RI, N) = {tt= t~+ v, y~L ~, II v II~, ~; R,, *~L -N, II ~ II~, ~R=}, where L N is the linear span of the first N vectors el, ...,eN; L -N is the orthogonal complement, R l > 0, R 2 > 0 are numbers; N = i, 2, .... One has the following theorem. THEOREM 1.3. Let the problem (5), (6) be parabolic. Then for any numbers R l > 0 and R > 0 and sufficiently large R 2 > 0 one can find a collection of functionals

F, (~, a) . . . . . ~ (n, a), defined on the set

{(,1, h):neR ~, t'1 (~
such that for any "right side" heel, IIhlIg~ N0(R I, R, R 2, T) and [~][~-~RI

(6)

has a solution

uOB(RI, R=,N)

F, (n, h) = 0 . . . . . FN(~, h) =0. This solution is unique and satisfies the conditions (u, e J = (rl, e~), ]=1 . . . . . N.

The p r o o f i s based on t h e t h e o r y of q u a s i l i n e a r e l l i p t i c a n o t h e r developed by t h e a u t h o r ( c f . [ 3 ] ) .

maps of one Banach space onto

1.3. P a r a b o l i c Problems w i t h L i n e a r P r i n c i p a l Part.; Weakly lems of general form in HSlder classes and also in HSlder classes successively by Vishnevskii [20, 21], Belonosov and Zelenyak [7], and Vishnevskii [6], Khachatryan [86], Solonnikov and Khachatryan

n o n l i n e a r p a r a b o l i c probwith weights were studied Belonosov [5], Belonosov [78], etc.

In these papers the solvability of the initial-boundary problems was studied as well as questions of stability of stationary solutions of quasilinear 2b-parabolic problems. We cite the basic result of the paper of Solonnikov and Khachatryan which is the most general result on the question of solvability in the indicated cycle of papers. Let u = (ul,... ,uN) be the vector-function sought. of the form

One considers a system of equations

Ou

Vi~u),

Ot + A ( t , x , D ) t t - ~ f (t, x, tt . . . . .

(7)

where the operator

~u~-~t +A(t, x, D)u is assumed to be a 2b-parabolic linear operator (matrix). Here 7qU is by definition the vector whose components are all derivatives D~(I<~.]~N) for I$I = q- The function f(t, x, u ..... 72bU) is a nonlinear function conditions on which will be formulated below. In the cylinder Q={0, satisfying the conditions

T]XG,

G c R n, with lateral surface S one seeks a solution u(t, x)

(8)

B(t ~ D)uls=O,

u(0, x ) = ~ (x),

(9)

where B = (Bij)Nx N is a matrix differential operator for which ordB~j<~.a~+ib~ where ~ m l . It is assumed that the operators B and ~ are related by the familiar complementability condition and the boundary of the domain G is of class C 2b+=, =>0. A solution of the problem (7)-(9) is sought in the weighted HSlder spaces C~li~,t(Q),the norm in which is defined by the following formulas:

2b=+lvl--s

l--s

2b~+Wl<st
~

o

t

o

where

Iv b=sup [ r

Q ~ = ( t / 2 , t ) X G,

X(~Q

[u]~o~ ,

, D i d ( t ,

=vl *'~" I

ot"

_

.

_

v or" 9h WI

XIx--yII'I-I+

~

sup IOv O=---2-u(t, x) --DV~0=------~-~ (*,x) I" It--*[

9s - - 2 0 < 2 0 ~ + W l < s x,t,'r,

c)t ~t

2b

Ox r

i f s i s n o n i n t e g r a l ( t h e seminorm [u]%! i s d e f i n e d a n a l o g o u s l y ) and t

2b=+wl=s

4-

2586

~

I

sup D v ~

s--it~<26~t+Wl<st ,~ ,x

Ot~

t,x

[

x)--D~

Ot =

Ot~t

I

(v,x) It--~l

Ivl

s

2~

~ ,

2O

if s is integral.

In addition it is assumed in the case of integral s that the derivatives

D~0~/0t~,2b~z-~-l?l=s, are continuous in Q up to the boundary. Now we give conditions on the nonlinear perturbation f(t, x, u .... ,V2bU). Let p(J) denote the vector with components p~, I~]=l, of the same dimension dj as the vector Vju. We letp=(p~0~ ..... p~b~) and assume that the vector f(t, x, p) satisfies the following conditions: a) f(t, x, p) is defined for all (t,x)6Q, pEll~~), where

ri~) = {pER=+d'+'§

p(~ I < r , . . . , [/~) I < r,

pU)ER~; j----k+l ..... 2b}, kE[O, 2b)is an integer; f(t, x, p) is twice continuously differentiable with respect to p~) and

/ (t, x, 0 ) = ~ b) for all inequalities

p~II~) the

#=f/Op~)Op~V)belong

derivatives

to

C=I~,= (Q), =6(0,1), and satisfy the

(=)

O=f (t, x, p)

W~'mOPT)"

O=f (t, X, if)

(L x, 0 . . . . . 0) = 0 ;

0 ..
if

1(or

apl~,apS~> [Q ..
161, [~'['-
if lPl-----k,l~[>k;

,o

where 2b

F(-:O=I+

~

I=I--/H-]

1

]pr

moreover, 0'f

|(=)

(t,x,p)Ij <el(r), if l~]>k,[~i>k,

0Pl

~2

JP l+a~ I> 2b +k; c) if I$I > k then for all p, qEllrc;), satisfying the condition p(J) = q(J) for j < h =

h(IPl)=2b+k--lPl,one has Of

(t, x, p)

w i t h any e x p o n e n t oE(0, 1] and c o n s t a n t

Of

0q~) (t, x, q)

(=)

< C~ (r) Jp - - q [%

C 2 i n d e p e n d e n t of p, q, ~.

x, 9)

In p a r t i c u l a r ,

for q = 0

C~(r) [po I~ IP -D,10

where

Po=(O ..... 0, p(~+l)..... p(2b)).

If [6] = 2b then C2(r) - 0 (f is a linear function of p ~ )

li31= 2 b ) . d) When k = 0 it is assumed in addition that b) holds with the function/~(p) for some m~(0, I) on the right side. The following theorem holds. THEOREM 1.4. Let a)-d) hold. Then one can find constants M(T) and 6(T) such that for any initial functions ~(x) = (~i (x)..... epN(x))~C~(O), s~[k,2b], satisfying the compatibility conditions N

Bqj (0, x, D) % 10o= 0, u q: ~q + 2b ~< s, ]=1

and such that ~p[Gts~f(T),the problem (7)-(9) has a unique solution

u(tx)6C~-=/2b.m~+=(Q), where

I ~ i$,t# <~+ =~< m (r)I ~ I~~. 2587

The proof of the theorem goes by the method of compressive maps on the basis of sharp H~ider estimates (of multiplicative character) of the nonlinear expression f(t, x, u .....

72bU). As examples of functions f(...) satisfying the hypotheses of the theorem one can take functions of the form

f(t,x,u

. . . . . V=vtt)=

~

O ~(

fe~(t,x,

~

= . . . . , Wu) O ~ u + / ~ (t, x, u . . . . . W u)), where f~fl and f ~ a r e smooth f u n c t i o n s

of their

variables.

A n o t h e r e x a m p l e : f(ff, Vu)----fo(u)exp(--lVu[~),

where f 0 ( u )

is a twice differentiable

func-

tion. 1.4. Asymptotic Stability of Quasilinear Parabolic Problems. As already noted in point 1.3, Vishnevskii, Belonosov, Zelenyak, Solonnikov, Khachatryan, etc. studied the question of stability of stationary solutions of quasilinear autonomous parabolic problems. In the lectures of Belonosov and Zelenyak [7] the asymptotic stability of the null solution of the following problem:

Ou A ( x , O ) t t q - f ( x , dt ----

tt,

""

.,Dmu),

(10)

Bq (x, D) tt lea = % (x, tt . . . . . Dmqtt) leo,

( 11 )

( q = l . . . . . mN), u(0, x) =uo(x), x~O.

(12)

Here u = (u I ..... u N) is a vector-function of the variables t~0 and x~O; A(x,D) is a linear differential matrix-operator subject to the condition of 2b-parabolicity in the sense of Petrovskii. The coefficients of the operators A(x, D) and Bq(x, D) (they are also matrixoperators) belong to the HSlder classes Ca(G) and Cim-'mq+=(O), respectively, 0 < ~ < i. Further, let the compatibility conditions which are the compatibility conditions of order zero in the linear case

Bqtto leo =

% (x, u o. . . . .

N _ ~ Bo(Auo+/(x, uo. . . . . 02"uo)loo_~ i f mq = 0 h o l d where { ' } j

Dimtto leo (q = 1 . . . . . m N),

O~q ( x ' u O { A u ~ O~.J

(x, "~. . . . . Dmu,)}j 10o,

d e n o t e s a component o f t h e c o r r e s p o n d i n g v e c t o r .

Besides what has been said let the function u(t, x) - 0 be a solution of the problem (10)-(12) [for u0(x) - 0]. Definition 1.2. We say that the zero solution of the problem (10)-(12) is asymptotically stable in the HSlder class C=m+~(O), if for any e > 0 one can find a number 6 > 0 such that for all u0(x) satisfying the inequality [u0[im+=,~6 and for which the compatibility conditions hold, the problem (10)-(12) has a unique solution U(t,x)~C~im+')/mn'=m+=(QT) for all T > 0 (QT = [0, T] x G) while

l u (t, x) 1~+.,~-~0 as t § +~. Let us assume that the following conditions hold: for any vl(t, x) and vi(t, x) such that

max{I v' 12a+~,Qr, I ~'212~+~,Qr} ~< ~ ~
l / (X, "ol, . . , D~"o I ) - / (x, ,o2. . . . . . O~'~ [q~r

"o', . . . . Dmqvx)--~q(x, v~ . . . . .

Dmqv~jl~_%+=,:Sr..
where the continuous function q(~) ~ 0 as ~ § +0. parabolic HSlder norm, i.e. r

sup

(t,x)OOr (t',x)6~ r 2588

I~,qr "< q (g)l vl _ "o~lz,-+a,Or;

( 13 ) ( q = l . . . . . . raN),

(1/4)

(In the last inequalities lo[~,0r is the

Iv(t.x)--v(u.x)] .~_ sup

I t - - t ' ttq~-m

(t,x)6~r (t,x')~0r

'!~(t.x)--v(t.x')[ ; the norms

I x - - x ' I~

1.12m+~,Qr;are defined analogously; S T is the lateral surface of the cylinder QT'] THEOREM 1.5. Let all the eigenvalues of the operator generated by the differential expression A(x, D) and the boundary operators Bq(x, D) lie in the left complex half-plane and let the formulated conditions (13) and (14) hold. Then the null solution of the problem (10)-(12) is asymptotically stable in the sense of Definition 1.2. At the base of the proof of this result lies an exponentially decreasing estimate of solutions with small initial norm. In its turn the derivation of this estimate is based on the familiar HSlder estimates of Solonnikov of solutions of linear parabolic problems. Lunardi [121] is devoted to the study of the asymptotic stability of the null solution of the first boundary problem for the nonlinear autonomous equation

O-K= ~.~ a=(x, D~u)D~u+b(x; DVtt), ["fl<2m--1, Ot

(15)

~[=2m

u(O,x)=uo(x), OOu!s=O, l,~l<m--1,

(16)

where S = [0, +~) x F is the lateral surface of an infinite cylinder. The fundamental assumption of the paper is that the linear elliptic operator

A(x, D)u-- ~.~ a.(x, O)D~u t~l=2rn

for null Dirichlet conditions is a positive operator. From the operator A one constructs a Banach space with norm of the type of the norm of the graph of an operator and establishes that if in this norm the initial function is sufficiently small, then the solution u(t, x) of the problem (15), (16) decreases exponentially as t § +~. Numerical characteristics of the exponential decreases depending on the spectral characteristics of the operator A and the subordinate operator b(x, DYu) are given. Durikovic [107, 108] are devoted to analogous questions. The asymptotic stability of stationary solutions is established in special HSlder classes. The basic machinery is estimates of Green's matrices. In conclusion we note the recent paper of Kirichenko [43] which is of an applied character. In this paper he considers a nonlinear parabolic problem which occurs in the theory of processes of laser heating of gaseous mixtures in which the effects of thermal conductivity, diffusion, and thermal diffusion are considered:

o---i+-~, -s

t>O, O<x
Ox

-'T Ox x=O-----"O-~ x-~' R

u(O,x)=~o(x), S uo(x)ctx=O. 0

The function f(u) has the form

where k~ > 0, g > 0 are constants. The basic result is the asymptotic stability of a solution of the problem posed. One notes that the results of the paper carry over to the case of an equation of the form

8u 8~n (8~u~. o~ +(--I)"+~-/~ \'6:~,--f(u)), n>1. 2.

Nonlinear Parabolic Equations with Two Independent Variables

2.1. Boundary Problem for Equations of Partially Divergent Form. The method[ of monotonicity considered in Chap. 1 is also suited for the study of certain classes of boundary problems having generally nondivergent form. The largest class of such problems relates to the case of one spatial variable.

2589

Let (k, ~, r, s) be an aggregate on collections of nonnegative integers where k + ~ = 2m ( m ~ l ! being an integer). We denote the maximum of the numbers % in these collections by ~0 the corresponding value of k by k 0 (k 0 = 2 m - ~0). Let us assume that ko.~<10and r~lo--l. Let r0 = (E0 - k0)/2For the given aggregate of collections we consider in the rectangle Q = [0, T] • [a, b] the boundary problem 0u

(--1) r. a--if+ ~

DkAk(t, x, u, .... OZu)=h(t, x),

(17)

u(O, x)---ep(X),

(18)

k+l~ 2m

Ulr = / o (~ ) . . . . . O ' ~ urr = :,._~ ( %

D='~

') . . . . . D'~

('0,

~r,

t 19)

where F is the lateral boundary of the rectangle O. We note that in two cases: r 0 = 0, r 0 = 2m the problem (17)-(19) is the first boundary problem. In the first case (17) has principal part of divergent form (L0 = k0 = m) and in the second case the principal part is essentially nonlinear, i.e., nonlinear in the derivative D2mu. In general the principal part of the operator has the form

Ok~

(t, x, tt . . . . . O2m-~~

This structure of the principal part also lets one study nonstandard problems with conditions (2.3). Definition 2.1. The function u(t, x) is called a generalized solution of the problem (17)-(19) if the following conditions hold:

1) uGLp(O, T; W~~

O)), Ou ~ e t ~, (0, T; Vg;,~o(a; 01);

2) for any function "o(t,

x~Lp(O, T;

0

W~~

b)) one

has the integral identity

3) t h e i n i t i a l and boundary c o n d i t i o n s ( 1 8 ) , (19) h o l d [we n o t e t h a t c o n d i t i o n 1) guarant e e s the existence of traces in the corresponding sense for t = 0 and on r]. THEOREM 2.1. If conditions of the type of I-III of Chap. 1 hold, then the problem (17)(19) has at least one generalized solution. The proof of this theorem goes by the method of monotonicity. We note only that to get a priori estimates needed, one takes for the separating operator for the problem (17)-(19) the operator D2r0u. Remark 2.1. The results of this point can obviously be carried over to certain multidimensional nonlinear parabolic equations if the symbol for differentiation D is formally replaced, for example, by an elliptic operator A whose cokernel is equal to zero. We consider, for example, in the cylinder Q=[0, T]XG, G c R n w i t h following problem:

lateral surface S, the

(--1) ~ 0 ~ + A (t, x, AINU)=h (t, x),

(20)

u(O,x) =0, O'uls=O, Iol~<m--1.

(21) (22)

If the function A(t, x, Sm) has polynomial growth in ~m of order p - 1 (p > I) and is monotonic, then for any right side h(t,x)ELr problem (20)-(22) has a unique solution u(t,

x~Wr 1 , 2 m

"

where

p !

= p/(p

--

1)

9

In particular the operator

-- 1..0~ ) ~/-+lA~ulm2A~u IF7 l'2ra/f]~

maps t h e space--u',p~-~J

2590

o n t o t h e s p a c e Lp,(Q) h o m e o m o r p h i c a l l y .

Interesting classes of divergent and partially divergent equations are studied in Kozhanov [44-46]. Namely, in the rectangle Q = [0, T] • [0, i] one considers the equation n~

Lu=-~-+ X ( - - I)~Dk%(u . . . . . Dkt0+~(u . . . . . Dktt)=h(t, x)

(23)

k=0

under initial-boundary conditions of two forms: u(0, x)=u0(x), Dku(t, 0)=Dhu(t, I)=0, 0~
(24) ( 25 )

THEOREM 2.2. Let the functions ~(~0 ..... ~k) be (k + i) times continuously differentiable on any compactum from R k+i, ~(~0..... ~m) be continuously differentiable on any compactum from Rm+l. If the following conditions hold:

0~(~00~ ..... ~ ) > ~'~ > 0 '

I~176 l%(

....

o~(~0o~ . . . . . ~) >0,

~)I-.<M, O ~ < l < k ,

0
O...
o. . . . .

o
1

0 0

0

max [ D~tt ]-.
w i t h c o n s t a n t s Ra, R~ d e p e n d i n g on t h e c o n s t a n t s t i o n s u0, h, and t h e number T.

from the hypotheses

The p r o o f o f Theorem 2 . 2 i s b a s e d on t h e r e g u l a r i z a t i o n

of the theorem,

the func-

o f ( 2 3 ) by t h e e q u a t i o n

m

L~ u - - Lu + 8 fit ~ ( -- 1)kD~% (u . . . . .

D~u)---- h (t, x),

obtaining a priori estimates, successive application of the method of extension with respect to a parameter, and passage to the limit as g * 0. The same theorem also holds for the problem (25). An analogous theorem is valid for the equations m--/

....

O~ "OF+ ~ ( - - 1)~-~Dm+l-k% (a, . . . . D~-l+kU) ~-- h (t, X) k=O

p r o v i d e d t h e c o r r e s p o n d i n g g r o w t h and p a r a b o l i c i t y c o n d i t i o n s on u 0 ( x ) and h ( t , x ) .

conditions

h o l d as w e l l as t h e a n a l o g o u s

The methodology for obtaining the estimates and existence theorems and applied analogously to prove Theorem 2.2 can also be applied for multidimensional equations of the following form:

L t t - - 3T + L ~,n'~( u . . . . .

D~r~-ltt, Limg) + ~ (tt . . . . .

D ~ t t ) = h ( t, x ),

where x E O c R a is a bounded domain with smooth boundary, D k denotes the collection of all derivatives of the form O~/Ox~,...Oxen, Lim is an elliptic operator of the form

[=l,l~l=m

L~m i s t h e f o r m a l l y a d j o i n t o p e r a t o r t o Lim. I f s m o o t h n e s s and g r o w t h c o n d i t i o n s on t h e f u n c t i o n s ~ and , h o l d as w e l l as c o n d i t i o n s on t h e f u n c t i o n s u 0 ( x ) and h ( t , x) a n a l o g o u s t o the conditions of Theorem 2.2, one can prove the existence of solutions of the problem 2591

0 J

(~-V)tl][O,r]• ,

~(0, X ) = ~ ( X ) , 0<j<2m--l, for which one has the estimates 2m

2m

t

k=O

~=0

0

I] L2~q~(tt . . . . .

D~'~-~tt, L2mtt)12d x <.M2.

G

Now we c o n s i d e r

in the

domain Q the

equation

L u ~ ~ + ( - - 1 )~Dm-~g) (u . . . . .

Dm+~) = k (t, x).

THEOREM 2.3. Let q~(~0. . . . . ~m+z) have continuous derivatives with respect to all variables to order m + s + 1 inclusive on any compactum from R~+~+I

O~ (~oC)~ra+t ..... ~m+l) > 0~0> 0 '

IOe#(~o, .~; ....~ra+l)I~.~ M ,

O..< j ..< m + l - - 1 for

a l l ( ~ 0 , . . . . ~m+t)GRm+t+l; i f

~l, ~a. . . . .

m + s is

an even number,

then

02k~( O, G, 0 . . . . . ~m+t-,, O) = 0

for

all

~m+t-1 a n d a l l k = O . . . . . ( m + I - - 2 ) / 2 ; t h e f u n c t i o n u 0 ( x ) b e l o n g t o C 2 ~ - ~ (0, l i , s a t i s f y DUquo(O)=D2quo(1)~--O, O ~ q ~ m + t - - l; t h e f u n c t i o n h ( t , x ) b e s u c h t h a t h , D 2 s a n d i f s i s a n e v e n n u m b e r , t h e n h, D2h . . . . . DZ-2h v a n i s h f o r x = O, x = 1;

the conditions belong to L2(Q) w h i l e i f s = 2s there exists a the initial and

o + 1 is an odd number, then h, D2h,...,D2s v a n i s h f o r x = O, x = 1. Then function u(t, x) satisfying the last equation a l m o s t e v e r y w h e r e i n Q, a s s u m i n g boundary conditions (25), and such that for it one has t

I

1

0 0

0

max I Dku ]-.
The proof of Theorem 2.3 is based on regularization of the original equation of the equation

L,u~Lu+a(--1)"~+*D'~+'e~(u . . . . .

D'~-'u) = h ( t , x).

2.2. Ouasilinear Equations and Systems of General Form. A large class of quasilinear equations and systems of general form with one spatial variable is studied in Kruzhkov [50, 51]. We describe part of the results of these papers. In the rectangle Q = [0, T] x [-s

s

we consider the parabolic system

ao(t, x, u,

(--1 ~0~' ' ~+

O~-lu)O~mu

(26)

. . . .

=f(t,

x, u . . . . . O~-~u), w h e r e u = ( u 1 . . . . . u N) (N i s a n a t u r a l n u m b e r ) , ao(t, x, ~o. . . . . ~m-1) i s a s q u a r e m a t r i x o f s i z e N • N w h o s e e l e m e n t s ao~(t, x, ~o. . . . , ~2~-1) a r e m e a s u r a b l e functions with respect t o (t, x)~Q and continuous

in the

variables

~=(~

.....

~2~-I)~R 2~N.

The right side is a vector-function satisfy analogous conditions, I<.]<.N. following

conditions

f = (fl ....

Let

the

I.

Strong Parabolicity Condition.

,fN)

whose components

~ ( t , x, ~ . . . . .

~-1)

hold:

For any values (t, x)6Q, ~ER 2 ~ and any vector ~ R ~

one has

ini

ini (% c~(t, x, ~ ) n ) = X > 0 .

(t,x,D In1=1

II. Condition of Subordination of the Vector f(t, x, ~). and any $ = ($0 ..... $2m-I)

2592

For any vector ~ R N , ( t ,

x) 6Q

m 2rn--I ] [ ( f ( t , x, ~), rl)l-. 0 is a constant,

r2=1~o12+...+1~2=-11 ~, ~b(r)~l

is a smooth nondecreasing function

s a t i s f y i n g the condition

1 [o(t,x)~L2(Q).

and the function

Under t h e h y p o t h e s e s

indicated

one h a s t h e f o l l o w i n g

u(t, x~W~:~m(Q) be D~lx=• i-~0, I . . . .

THEOREM 2.2.

Let the function

a priori

estimate

theorem.

a solution of (26) under the conditions ,m--l.

(27)

Let l

S [D~(O' x)lZdx+ --I

!71f~ x)l dxdt=M"

Then if conditions I and II hold, then

vralmaxl [D~u(t, x)12dx IS + (D~u, O~f~T

where t h e c o n s t a n t t i o n II.

(28)

ao(t, x . . . . . D ~ - l u ) ) D 2 ~ d x d t 4 C ,

Q

--I

C d e p e n d s o n l y on m, N, k, M, K, ~, T, and on t h e f u n c t i o n

~(r)

in c o n d i -

The estimate (28) is one of the basic elements of the proof of the following existence theorem. THEOREM 2.3. Let conditions I and II hold and in addition let the elements of the matrix ao(t,x,~o..... ~ - i ) be bounded in modulus by a constant A(R) in domains of the form

x)eQ;

i=o, 1..... m--l;

--

~

< ~ < + ~, ra~
1,..
In addition let the elements of the matrix a0(t, x, $) and the components of the vector f(t, x, g) in any ball I~I~R be continuous in $ uniformly with respect to (t, x)6Q. Further, let

~(x)6W=~(--l,l)) be

an initial function satisfying the compatibiity condi-

tions D~q~(Szt)=O,

i = 0 , 1. . . . .

m--1.

Then there exists a solution u(t, x)EW~I~m(Q) of (26) satisfying the boundary conditions (27) and the initial condition u(0, x)=~(x). Such a solution is unique if the elements of the matrix a0(L x, ~0,,:.,~2m-,) and the vector f(t, x, $0..... ~2m-l) satisfy a Lipschitz condition in domains of the form ~R" The proof of the theorem goes in two steps. First one considers quasilinear systems (26) with bounded nonlinearities, i.e., systems for which all elements of the matrix a0(t, x, $0,---,$2m-i) are bounded in all variables. The solvability of the boundary problem for such systems is established with the help of Schauder's principle and the sharp a priori estimates for linear parabolic systems. The general case is proved by "truncation" in g at infinity of the matrix a0(t, x, $0 ..... $2m-i) and subsequent passage to the limit. Here, as already noted, Theorem 2.2 on the a priori estimate plays an essential role. 3.

Functional-Analytic Solutions of Quasilinear Parabolic Equations

In a number of papers of M. I. Vishik and A. V. Fursikov, and afterwards Babin [3], Kuksin [54-58], etc. a functional-analytic approach is developed to the study of nonstationary quasilinear boundary problems. The results of these studies are summarized in Vishik and Fursikov [19] where there is an enormous bibliography and history of the question. The basic objects of applications of the theory of functional-analytic solutions are boundary problems for the nonstationary Navier-Stokes system and also for quasilinear 2593

Below we describe a number of results concerning parabolic equations. 3.1. Analyticity of the Inverse Operator9 In the cylinder Q = [0, T] x G, 0 < T < with lateral surface S one considers the quasilinear parabolic equation Ou

~(tt)----.--~- + Ag_m(X,O) t t + F ( x , D v u ) = f (t, x),

(29)

where Azm(X , D)u is a linear differential operator of order 2m-

A~m(x, D) ~---- X a=(X) Dc'u Ia I~<2rn with complex-valued coefficients a~(x)EC oo(0). Let us assume that for all Xt~G and all ~ 0

ReA~

X

a=(x)$ =>0,

(30)

l=i=2m i.e., the operator Azm(X , D)u is strongly elliptic in the closure of the domain9 Further,

let us assume that the nonlinear summand F(x, DYu) has the form co

F (x,

DVu)=~

Fk(x, D'~u),

k=2

where

Fk(x,D~u)=

X

O~Yi~2m

9k(x,?, ..... ~k)(Dv,O...(D~u),

l=l,...;k

while for each collection (~i ..... ~k) the complex-valued functions ~k(x, ~ ..... yk) are infinitely differentiable with respect to x in the closure of the domain G. Without loss of generality one can assume that the functions ~k(x, 71, ..., ?~) are symmetric functions with respect to the arguments Yz,...,~k. It is assumed that there exists an R > 0 such that for all ~ such that lal<~2m(r+l) for some integer r > 1/2 + n/4m the functions D~F(x, z) are analytic for [z I < R. On the lateral surface S the conditions of the first boundary problem 0Ju

Ovj 1s

O, 7=0,1,.

m--l,

(31)

are given and for t = 0 there is given an initial condition

u(O, x)~-~-Uo(X), x~O.

(32)

We introduce function spaces in which there is a functional-analytic dependence of the problem (29), (31), (32) on the right side f(t, x) and the initial function u0(x). Namely,

o ~ ( ,_j_T) ~r'2mr={12 (~,,x)~W;; 2mr(Q): ~/,oU~W2 1 (a),

let

j=O,1 .... , r--l},

(33)

where ~j, 0 are the trace operators

(~j,o~)(x) = o& (t. x) ] d#

t--0"

(34)

Let

~.,2,,, =

{u(t, x)~'

a.,.

9"OJu-lOv/ s--O' 0<j<m--l}.

0

The norms in ~r'2mr and~r.~mr

are defined as norms of Sobolev spaces.

0

Finally,

let

BoC~'+I'2m("+I) be

the ball of radius p > 0.

We consider the operator

(~ (u), ?o,o):Bo ~ a'~"'~' -F I'~r ~('+I/2) (G), 2594

(35)

where the operator ~(ui) is defined by (29) and the radius p > 0 is sufficiently small. The following theorem holds. THEOREM 3.1. Let r > 1/2 + n/4m, where we recall m = dimG, 2m is the order of the operator A2m(X, D). If the conditions on F(x, DXu) indicated above hold, then the operator ( ~ ( u ) , ~ is analytic for sufficiently small p > 0. If in addition (30) holds, then the oper0

ator (~(u), ~00) has an analytic inverse R(f, u 0) defined in a ball ~p,~r'2mr-~Z2m(r+I/2)(O) with sufficiently small radius Pz > 0. The proof of the theorem is based on the Use of the familiar theorem on an inverse operator for an analytic map. Namely, let: i) ~(u) be an analytic map of a neighborhood of zero of the Banach space E l into a neighborhood of zero of the Banach space E2; 2) the Frechet derivative d~(0) of the operator ~(u) at the point 0 effect an isomorphism of the spaces E~ and E 2. Then locally ~(u) has an inverse operator ~(v) which is an analytic operator in a neighborhood of the point ~(0)~E=. The operator ~(v) having these properties is unique. 3.2. Solvability Theorem. The functional-analytic approach lets one get a number of results on the solvability of boundary problems for parabolic quasilinear equations. In the cylinder Q = [0, T] • G we consider the quasilinear 2b-parabolic equation

s

~ ] a~(D~b)tt)D~D~tt(t, [~,~1=M

(zo)tt)__f (t,x), (36)

(t, x)GQ, where [~, v] = I v l / 2 b + p; m, b, M a r e n a t u r a l numbers, m < M; D( 2 b ) d e n o t e s t h e row o f a l l 11 v p a r t i a l d e r i v a t i v e s o f t h e form DtDxtt, [ix,~]~<m. The f u n c t i o n s a ~ v ( - ) and a ( ' ) a r e a n a l y t i c functions of all their arguments, a(0) = 0. Let us assume that the following conditions hold: I.

2b-Parabolicity Condition.

For any z the roots of the polynomial

~(z;r,p)=_ ~

[p,v]=M

a~(z)p~(i~) ~,

satisfy the inequality

Re p ~ - - 6 1 ~l 2~, where ~ > 0 is a constant. Further, for Eq. (36) one poses the following boundary and initial conditions:

Oju-----

gj~DtDxuls=~j, 1 < j <_Mb

(37)

Iv, v ]
(here S is the lateral surface of the cylinder Q which we shall assume sufficiently smooth), where rj > 0, 2brj are natural numbers; k

Dtu/t=o=~,(x),

O-.
(38)

II. Lopatinskii Condition. Let x'~OG be a vector tangent to 3G at the point x' and be the normal to 8G at the point x'. Then the equation

~(z; ~+yx, p ) = 0 with respect to y for [gl + lpl ~ 0 has exactly Mb roots ~t .... ' ~ b with positive imaginary part and the polynomials

~, eJ~P~(t(~ +iU))~ Y=1, .... rob,

[~,vl
linearly independent modulo the polynomial

p)) 8

for

R e p > - - 6 [ 2~,

IPt+l~l~0. 2595

We introduce the spaces in which we shall seek a solution of the problem (36)-(37). Mb

ga--

k=l

M--I

/

Let

I ",

j=0

Ik= M - - r ~ - - !/2bp,

where B~ ,~ are the Besov spaces (cf. [8, 69]). We define an operator N corresponding to the problem (36)-(38) and acting from the space WpM,2bM(Q) to the space ~ X L p ( Q ) :

N: u(t, x ) ~ ( K (u), L(u) ), where

K (u): u( t, x) ~ {G, (u) . . . . .

OMb (tt); tt [,=o . . . . .

Df-'tt It=o}.

The following theorem holds. THEOREM 3.2. Let conditions I and II hold and p > max((n + 2b)/2b(M - m ) , the following assertions hold: i) there exists a domain f ~ X L p ( Q )

I).

Then

such that the map

N : Wf'2oM (Q)--> f~ is an analytic diffeomorphism; 2) if ~ is a bounded domain and O~6f~ (]=I, 2....) while Oj-~O60f~ in the norm of ~Y~XLp(Q), then

]l N - t (07)It~vM,2b~t(O) -+ + ~ " P

In other words, upon approaching the boundary 8~ of the domain ~ analyticity is violated. Remark. Simple examples show that both the case f ~ = ~ X L p ( Q ) , and the case f ~ e ~ X L p ( Q ) are possible, Thus for the problem au

o-f = - A ~ + ~ ( t / ) = ~ ( t , x), ul~=o, u(o, x)=uo(x), i f , ( a ) = u 3, t h e n ~2=B~-21P(G)XLp(Q), and i f , ( u ) = - - u s, t h e n ~&B~-21P(G)XLp(Q) 9 4.

Parabolic Problems with Lowest Terms There are two approaches to the study of nonlinear parabolic problems with lowest terms.

The first is the description of bounded algebraic or functional character under which the lowest terms are subordinate, i.e., define either a map which is compact in relation to the map of the principal part or a map with small norm. For parabolic equations of first order in t this approach was already described in a number of cases (cf. Sec. 1 and Sec. 2 of Chap. i). For one nonlinear parabolic equation of second order in time, this approach is realized in Pokozhaev [74] to which point 1 of the present section is devoted. The second approach is that the lowest terms are considered independently and along with the principal part define those spaces in which the solution of the problem is sought. In this case as a rule a priori restrictions are not imposed on the growth of the nonlinearity of the lowestterms. This approach is applied in particular by Browder and Brezis in [97] which will also be described. 4.1. Nonlinear Parabolic Problem of Second Order in t. is considered in Pokhozhaev [74].

2596

The following parabolic problem

In the cylinder Q=[O, T]XG (GOR n being a bounded domain, T > 0 arbitrary) with lateral surface S one studies the boundary problem 0u

Uls=O, 74/s ou =0'

(40)

u It=o= ~o(X), ~ ]~=o= ~(x)'

(41)

where 8u/3v is the derivative with respect to the inner normal to S. One defines the class of functions a(t, x, u, Du, 3u/3t) for which the problem (39)-(41) is solvable in the space 2,4 W2nOQ). Here a(t, x, u, p, q) is a real function of its arguments, uGR I, P Elan, q @RI" Let us assume that the following conditions hold: I. The function a(t, x, u, p, q) satisfies a Caratheodory condition, i.e., for almost all values (t,x)~Q it is continuous in the collection of variables (u, p, q) and measurable in (/,x)eQ for all (u,p,q)6R1XRn)
inequalities:

a) if n > 4 then n

n

n+2"]

la(...) ]~
la(...) {~K[q(t, x) +exp lul~-*+ ]p l2+ l ql312]; c) if n = 3, then for ]u]~M

d) if n = 2, then for

la(...)J~
e) if n = i, then for lul~M, I p l ~ M

la(...) I<~K2(M)[q(t, x)+ I q Is]. Here

P [ = p ~ + . . . +p~,

q(t,x)fiL2(Q),

r > 0 is an arbitrary number, K > 0 is a constant, Ki(M), on M, where M > 0 is an arbitrary fixed number. Under the conditions

indicated,

i = I, 2 are constants depending

the operator

~4 A (a) =--a (t, x, u, Du, 7/-~ W ~:2(Q)~-4(Q) is completely continuous and consequently defines a subordinate operator in relation to the principal part ~.

2,4

Let us assume in addition that the function a(t, x, u, p, q) satisfies the following functional condition: III. Let us assume that for any function v(t, x) which is sufficiently smooth in Q, and satisfies the conditions

V]s----O,

ls=o,

Ov

Ov

=

one has t

OQ

o~, ~: ~, ,

t

0 G

f o r any t6[O, T]. 2597

Here C I > 0, C 2 > 0 are constants depending on ~0(x) and ~2(x); ~ is an arbitrary constant from the interval (a, ~), where

(s.. + s.)I/21 s:',

[Vwledx:

s**=sup

~ s7~/2, =

(Aw)2dx----1, ~[oe-----~)- oo = 0 9

G

Under t h e c o n d i t i o n s

indicated

one h a s t h e f o l l o w i n g

theorem.

THEOREM 4 . 1 . I f c o n d i t i o n s I - I I I h o l d , t h e n f o r any r i g h t s i d e f ( t , x ~ L 2 ( Q ) and f o r any functions ~o(x)EVr and ~I(x)CW/~(G), satisfying the compatibility condition (~0[0o--~O~o/Ov== ~llao=0), the problem (39)-(41) has at least one solution E(t,x )~W/24 ~:~(Q). Under certain additional restrictions on 3a/au, 3a/~p, and 8a/Sq the solution is unique. For the proof one uses the principle of extension with respect to a parameter (the LeraySchauder principle), and in addition an essential role is played here by the a priori estimates of possible solutions obtained under conditions II and III. 4.2. Equations with Lowest Terms of Arbitrary Growth. First of all we give a result of Brezis and Browder in whose proof one uses both the methodology of monotone operators and that of compactness. In [97] these authors consider the first boundary problem for a nonlinear parabolic equation of the form

0u O-t-+ A (t) (u) q- g (t, ~, u) = f (t, x)

(42)

without restrictions on the growth of the nonlinear summand g(t, x, u) with respect to u. Suppose for each fixed t~[0,T]

A(t)(u)~

~

(--1)I~ID~A~(t,x, DVa) ([?l<m) 0m

i s a s t a n d a r d m o n o t o n e o p e r a t o r a c t i n g from Wp(G) (p>~2) t o t h e d u a l s p a c e w~m(G) ( c f . p o i n t 1.1). F u r t h e r , as u s u a l , l e t g ( t , x, u) s a t i s f y a C a r a t h e o d o r y c o n d i t i o n , i . e . , be m e a s u r a b l e in(t,x)EQ and c o n t i n u o u s i n u. I n a d d i t i o n , on t h e n o n l i n e a r p e r t u r b a t i o n g ( t , x , u) we impose the following basic condition: Condition a). There exists a nondecreasing function h(r):Rt-~l~$, h (0)----O, - such that for all( t,x)GQ and all r,

rg(t, x, r)~O, [g(t, x, r)[~h(r), while

h(r)~C(Ig(t,

x, r) 1+ lrl ~+1),

where C > 0 is a constant. [Obviously these inequalities do not impose any a priori restrictions on the growth of the function g(t, x, u) in u. ] 0

If condition a) holds one seeks a solution it(t,Jc),ELp(0, T; I~/~(O)), u(0, x)=0, of (42). Condition b) (restriction on the domain G ~ n ) . Let G6' ={xEG:dist(x,~G)<6}, where 6 > 0. We shall say that the domain G satisfies condition b) if there exist constants C > 0 and 50 > 0 such that for all 5 < 50 and any ~(x)~'D(G) one has

I [~(x)IPdx
O6

THEOREM 4 . 2 . L e t c o n d i t i o n s a) and b) h o l d . Then f o r a n y r i g h t s i d e f ( t , x)6Lu,(O,T; exists at least one solution of the first boundary problem for (42) such that

W-~,ra(o)) there

0

tt(t, x)~Lp(O, T; W~(G)), ug(t, x, u)~Ll (Q). If

in addition

2598

the function

q(t,

x , u) i s m o n o t o n i c i n u, t h e n t h e s o l u t i o n

found is unique.

The basic difficulty in the proof of the theorem is the passage to the limit in the nonlinear summand g(t, x, u). This difficulty is overcome by the authors with the help of the following proposition which is a corollary of the general compactness theorem formulated in Sec. 1 of Chap. i. Proposition (compactness). Let G c R n be a bounded domain. (k = i, 2,...) be a sequence of functions such that:

2) . tk

In addition,

let uk(t, x)

x)+ zk(t, x),

where

Lp(O,T;Wp0)) and the sequence Zk(t, x) is weakly compact in LI(Q). compact in Lp(Q). Landes

[117] is development of Brezis and Browder

Then the sequence uk(t, x) is strongly [97].

In the cylinder Q = [0, T] x G one studies the solvability of the parabolic problem du

dt ~- ~ (--1)I~ID~A~(t' x, Dru)=h(t, x)

( 1 7 i'-<m),

(43)

l=l<m u(O~x)=U0(X),

D~

=0,

I~; "-
(44)

The solution is sought in the class of functions

ueL2(Q),

=m}

i.e., in a weaker sense than usual. In correspondence with this it is assumed that the nonlinear functions Am(t , x, D~u) are such that A~(t,x,D'u)~LI(Q), if u(t,x)E~, and the coercivity condition has the form

Ictl<m

whore

c(r)

+ +~ as

\l~l<:m-1

I~l = m

r § +~.

In addition a monotonicity condition is assumed either on all ~y or only on Sa for la[ = m.

Moreover, the growth of the functions Aa(t , x, $~) is connected by a two-sided inequality with a certain convex continuous function F (precisely these inequalities generalize condition a) of the paper of Brezis and Browder). Under the restrictions indicated one establishes the solvability of the problem (43), (44) in the class .~, i.e., with finite norm

lal=m

A peculiarity of the methodology of the paper is the nontrivial approach to overcoming the difficulty of passing to the limit in the Galerkin equations. The author singles out a certain convex linear combination of Galerkin approximations and directly establishes its convergence to a solution of the original problem. COMMENTARY A large part of the literature indicated below is used and is reflected in the basic text so we dwell on the books and surveys of general character and also on those papers which are either not discussed or are discussed insufficiently. As already noted in Chap. i, the methods of monotonicity and compactness have become classical by now. The concept of monotonicity of a map A:X + X* emerged on the border of the

2599

fifties and sixties independently in a number of papers of both Soviet and foreign mathematicians. As maps corresponding to nonlinear boundary problems monotone operators were intensively developed after the papers of Vishik [14, 15] and quickly became a full-fledged object of the general theory. The first basic results of the general theory of monotone operators were obtained by Minty [126-128] and Browder [99-103]. The subsequent development of the methods of monotonicity and compactness was obtained in the papers of Leray and Lions (cf. [120]), Brezis [93-96], Gossez [113, 114], Vainberg [ii, 12], Dubinskii [26, 27], Kachurovskii [13], and many other authors (cf. Literature Cited). Nonstationary versions of the theory of monotone were first proposed by Browder [99] and Lions [118]. The subsequent development of nonstationary methods of monotonicity and compactness were made in particular by Brezis [94], Ton [133], Barbu [gl], Donaldson [104], Robert [131], Zini [139], Bensoussan and Temam [92], Pardoux [129], Viot [134], Dubinskii [26], Zhikov [37], Krylov and Rozovskii [52], etc. Lions [120] which contains not only an enormous amount of factual material but also a large list of long range problems (a number of the problems posed are completely or partially solved at the present time) had essential influence on the development of nonlinear boundary problems and also on their broad popularization. Gajevski, Gr~ger, and Zacharias [iii] paid particular attention to the study of the convergence of projective processes for nonstationary problems of both the first and second orders in time. The theory of almost-periodic problems for nonlinear parabolic equations is described in Levitan and Zhikov [63]. Now we note the separate papers. Ton [133] suggested an abstract theorem on the solvability of an evolutional differential-operator equation containing initial-boundary problems for a Navier-Stokes system of equations or, for example, equations of the form

O K c)t

~

(_l)l~lO~(a~(t,x)O~u)+uiO~-,ul=O

as a p p l i c a t i o n s . A t o p o l o g i c a l approach to the study of boundary problems

Ou O-?--F(t, x, DVu)= f ( t , x) (]"~]~<2tn), (t, x)eQ, Oj(t,x, D~tt)==gt(t,x) (l~l~<m), j = l .... ,m, (t,x)6S, u(0, x)=0, x~G, i s proposed by Skrypnik [177].

The author developed a method of reducing the given problem

to an operator map A:X + X* for which the rotation ~(A, 3~) of the field A on the boundary of the domain ~ c X is defined if A(u) ~ 0 for U6afL Thus, to study the nonstationary boundary problem the topological methods developed in [76] are applicable. Goebel [112] considers the first boundary problem for the weakly nonlinear equation

-~i-+

2

O~(a=~( t, x)O~u)=g(u) "

lat ,ll31~<m

where the left side is a uniformly parabolic equation of order 2m, m > 2 with smooth coefficients in the cylinder Q = [0, T] x G. Under the conditions g(0)~>0, g'(s)<~.0,

IgC~(s) I<~C(l+lsto-~), v=O,1; soR ~, p = [r~rl- 2 (m-- 1)]/[n--2 (m-- 1)], n / > 6 m + 2 ,

the existence and uniqueness of a classical solution is established. Wahl [136] considers the first boundary problem for the equation

~-f q-Att-l- f (x, Drtt)=O, 2600

Ix142rn--1,

where A is a positive linear elliptic operator of order 2m and the nonlinear function f(x, Sy) is locally Lipschitz in ~0 and as I~I-+oo has order of growth I~TI2mml (!~I>0). The following conditional result is established: the existence of a specific a priori estimate of the difference of two solutions in terms of the difference of the initial and boundary data implies the existence of a classical solution for all t > 0. When m = ! sufficient conditions are given guaranteeing the a priori estimate needed. Kuchminskii [59] studies the problem of conjugation for the nonlinear equation 0a

--~-+ ~.~ ( - - 1)InLD~A~(t, x, D~'u)=h (t, x), [ T I < m , la!<m

where xfiO=Ol U02, t~(t, x)={tA (t,x), xfiOl; tt2(t,x), xE02; O
A~(t, x, ~v)={A(t, x, ~v, xfiG1; A2,~(t, x, ~v), xfiG~)}. The s u b d o m a i n s G 1 and G 2 a r e s u c h t h a t condition has the form

[D

a_[~_i t

OGI=L, OG2=FUL, OGINOG2=L, L N F = ~ .

The c o n j u g a t i o n

-->

Aa(t, x, DVu)cos(n, Xi)]L0,~)xL-----0

~+ti
for all ~ such that the m a - - ~ - - l ~ > 0 , l~ b e i n g t h e m u l t i - i n d e x o f d i f f e r e n t i a t i o n -->u l t i - i n d i c e s c o r r e s p o n d i n g t o 8 / 8 x i , n b e i n g t h e e x t e r i o r n o r m a l t o L, and t h e s u m m a t i o n b e i n g o v e r a l l a s u c h t h a t a~>~+li, t a l c < m , a r i s i n g i n t h e s u c c e s s i v e i n t e g r a t i o n by p a r t s w i t h r e s p e c t t o X1

~ 9 9 9 ,X n .

It is assumed that the functions A~(t, x, $7) satisfy the conditions of the theory of monotone operators. The basic result is the unique solvability of the first boundary problem with the indicated conjugacy conditions in the infinite cylinder Q = [0, +~) • G in the Sobolev classes

with weight exp (--% t), ?0~0. Kamynin [41] considers a sequence of quasilinear parabolic problems of high order with weakly convergent coefficients. Conditions are given under which a sequence of solutions converges to a solution of the limit problem. We note that the limiting equation is not generally the equation obtained by weak passage to the limit in the coefficients of the equation. Agaev [i] studies the solvability of the Cauchy problem for a differential-operator equation of the form

u" (t)+Kz (/<2(u)) =h(t), where K2(u ) is a nonlinear and K I a linear operator in Banach spaces. Adzhalova [2] and Khudaverdiev and Adzhalova [87] are devoted to the classical solvability of the mixed problem for the equation

0

L ~-

F i

Ou

Ou

O~u

where L is a second order uniformly elliptic operator. Shishkov [89] establishes estimates of the type of Sen-Venan estimates. Namely, in an unbounded domain Q lying in the layer /-/t~-{(t,x ) : 0 < t < T < oo, xER n} and having noncompact 0

boundary, one considers generalized solutions u(t,x)~L2(O, T; ~ zm2,to~(Q~))N ~z2(10,T; Qf]{t~-v} of the first boundary problem for the equation

L2,~o~(~t)),Q~--~

0u

-fir-+ ~.~ ( -- l )'='O= A= ( t, x, O~ u)=O ( [ 7 t < m ) , ]aI~<m

where the functions A~(t, x, ~ ) satisfy the polynomial nonlinearities (p = 2).

usual conditions of nonlinear problems with

Basic result: for any generalized solution one has the estimate (lower bound on the rate of growth at infinity of the energy integral) Q~

{(x,

):lx[<:'~}

2601

where C x > 0, C 2 > 0 depend on the parameters of the problem and the geometry of the domain Q. The results obtained are used for determining uniqueness classes of boundary problems. A. G. Zarubin studies the strong convergence of the Galerkin approximations of the first boundary problem

Ou Ot ' -1- L2m(x, D) t~+ g (t, x, O2"~-lu)= f (t, x),

u(O,x)=O,

~

TBl-.
O~(x)D~ttlo~=O, l~< ]~<m.

It is assumed that under the conditions given the linear part of the problem determines a self-adjoint, positive definite operator. Under natural monotonicity conditions and conditions on the growth of the nonlinear part g(t, x, Du 2m-l) and estimate of the rate of convergence of the galerkin approximations in the metric of W~; 2m(Q) is established. Bokalo [9, i0] are devoted to the question of uniqueness of a generalized solution of a boundary problem without initial conditions. For example, in the cylinder Q=RI>
Ou Ot

X (--1)l=tD"A=(t'x'DVu)-~h(t'x)'([ "fl'<m) i,,l<~m Dou ]R,xoa_~-0, fold<m--1.

all

Let us assume that for

tte ~.

(45)

(46)

t~R 1

< A=(t,x,D~u)--A=(t,x, DVv), D = ( u - v )

>>~.(O~(llu-~ll~,co~), 2

o

where

~(t)>/0, X(t)6L1,xoc(R1), I ~ ( t ) d t = + oo, --oo co

~ s ) > O , ~ ( 0 ) = 0 , j~' ~ as < oo. I 0

Then a solution u(t,x)@Lp.loc(l{l,W~(G)) of the problem (45), (46) is unique. LITERATURE CITED i. 2.

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