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0 and f ~ const, then V(t) strengthens positivity if and only if U(t) strengthens positivity. 1.29. Comparison of Semigroups. Let A, B be complete infinitesimal operators of semigroups U(t), V(t) in E. We shall compare the solutions of the equations du/dt = Au, dv/dt = Bv with common initial condition x. Simon [622] showed that if A, B are self-adjoint operators which are semibounded above in Li(s) = E, where S is a space with measure and U(t)E+cE+, then the relation IV (t)x I..
0). The f o l l o w i n g a s s e r t i o n was proved and used f o r t h e s t u d y of the dependence of a solution of the equation u t = A ~ ( u ) conditions by Khazan [154]. 0 let the function F~+IP(x, r, ~)--tor not increase for any x, ~. We set Bu = F(x, u, 7u) on D(B)={I~C'(~): -- 0g/0v~y (g) on F}, where ~ 6 ~ [cf. (54)]. Then the operator B - c0I is completely dissipative and. dispersive in E = C(~-~. If in addition, ]F(x, u,~)]..
z~D(B)~Jz]~D(A t/2)'and for z~D(B), O-.<x6D(A 'I~) one has the
(A~:=x, A*/~ I z l) > Re (x
sgn
z, Bz).
H e s s , S c h r a d e r , and U h l e n b r o c k [ 3 9 2 ] i n d e p e n d e n t l y p r o v e d t h e a n a l o g o u s f a c t f o r n o t n e c e s s a r i l y self-adjoint contracting semigroups: for U(t)E+cE+ , (43) is equivalent with the generalized Kate inequality (A'x, [z[)~>Re ( x s g n z, Bz)
for z~D(B), O<.~D(A*). "In [392] there is given a generalization of the result to the case when B acts on the complex Hilbert space F, and A on the real Hilbert space E with cone E+,
2185
while there is given a map z + [zJ from F to E+ having certain natural properties; thiszgeneralization is applied in quantum field theory. If both semigroups U(t), V(t) in E = L (S) preserve positivity, then (43) is equivalent with V (t)x ~ < U (1)x Vx6E+,
(44)
and for self-adjoint A, B which are semibounded above, with
(Ax, z)>(x, Bz) @~<xED (A), 0 ~ z 6 D (B)) (Bratelli etc. [226]). The latter condition is preserved under replacement of A, B by f(A), f(B), where f:R+ + R+ satisfies the conditions of point 1.27. If (44) holds [i.e., U(t) -V(t) preserves positivity], U(t) strengthens positivity and A ~ B, thenU(t) -- V(t) strengthens positivity (Kishimoto and Robinson [421]). A number of examples are considered in [226, 421]. Another approach to the comparison of semigroups is connected with estimates of Jlu(t)x -V(t)xJJ. Thus, Weilenmann [671, 672] showed that if the densely defined operators A, B in the Banach space E are such that
fIR(l, A)II, fiR(l, B)JI~JW/III for
JarglJ-.<~q-~,
where 0 < ~ < r ,
so that U(t), V(t) are bounded analytic semigroups, then for i = 0 and i = I, . ~ ( IE'-s~ +M') I J U ,t ~ ,~- - V,t~ ~,I,~
c~(t)Si(A,B);
where 60(A, B) is the sup of the distance from the point (x, Ax)6E X E to the graph of B over all xs with JJxJJ=+JJAxJJ2=l , and 6,(A, B ) = sup {60~ A , %B) :0
Nonlinear First-Order Evolution Equations 2.1.
Introduction.
We consider the Cauchy problem
du(t)/dtoAu(t) (t~O), u(O)=x,
(1)
where A is a nonlinear and possibly multivalued operator in the Banach space* E, which assigns to each zED(A) a nonempty set A z c E . The study of equations with multivalued operators encounters difficulties both in the theory (for example, the nonlinear analog of the Hille--Yoshida theorem has not been able to be formulated in the realm of multivalued operators), and the applications (to variational inequalities). To facilitate reading we shall denote by Az not only the image of the point z~D(A) , but also an arbitrary (not fixed, but "general") element of the image; it is always clear from the context what we have in mind: image or element. If the Cauchy problem (1) for any X ~ O c E has in some class of solutions (in general generalized) a unique solution u:R+ § D, then setting U(t)x = u(t), we get a semigroup of nonlinear operators U(t):D § D(t ~ 0), i.e., U(t + s) = U(t)U(s) and U(0) = I on D. If the generalized solutions are continuous, i.e., belong to C(R+; E), then the semigroup U(t) is strongly continuous on D for t ~ 0; in what follows we consider only semigroups. If the solution depends continuously on the initial condition, then the operators U(t) are continuous. At the present time there is no somewhat complete theory which establishes a connection between semigroups of arbitrary continuous operators and the corresponding Cauchy problems. A number of results on the continuity of the map (t, x) § U(t)x and the smoothness of the maps x § U(t)x, (t, x) § U(t, x) were found by Ball, Chernoff and Marsden, Dorroh and Graff [184, 273, 276, 277, 343]; these results are partially recounted in Marsden and McCracken [95]. If the operators U(t) are Lipschitz-continuous, then, as in the linear case, there exist M ~ ] , o~R, such that the Lipschitz constant lJU(t)lJup~<Me~( [95]; however, in contrast with the linear case, it is impossible in general, to make M = I by an equivalent renorming of E. If M = I, i.e., *In this section we consider only real spaces.
2186
IIv ( 0 x - u
(t)yll<eo'
IIx'yll (x, yED),
then the semigroup U(t) is called a semigroup of type ~, and for ~ = 0, contracting; semigroups of type ~ > 0 are said to be quasicontracting. To contracting semigroups there correspond [inn sense made more precise in point (2.10)] dissipative operators A (the definition is given in the Introduction to the survey, cf. also point 2.3), to semigroups of type ~, operators of class
Diss(E,~) = {A :the operatorA--~l in E .lsd~sipattve}
(2)
(such operators are sometimes called quasidissipative, or u-dissipative, but the first term is also used in completely different meanings [149, 427, 559], and the second can be confused with the term "m-dissipative"). At the present time there are sufficiently deep and rich results and applications of the theory of the Cauchy problem with operators of the class Diss(E, ~) and nonlinear semigroups of type ~. The foundations of this theory were laid in 1967 by the papers of Komura, Kate, Browder (cf. the references in Dubinskii [36]) and Oharu [536]; in 1969-1972 fundamental results were obtained by Crandall and Pazy, Dorroh, Brezis, Brezis and Pazy, Crandall and Liggett, Crandall, Miyadera, Oharu, Martin, Benilan, Webb, Barbu; the following authors participated in the further development of the theory and its applications in addition to those named (in chronological order with respect to this theme): Calvert, Konishi, Chambers, Picard, Goldstein, Attouch, Damlamian, M. I. Khazan, Aizawa, Fafermos, Slemrod, Kurtz, Kenmochi, Watanabe, Kobayashi, Plant, Evans, Baillon, Bruck, Reich, Pierre, Le, Pavel, Burch, Haraux, V. V. Kutuzov, Pazy, Nevanlinna, Vrabie, Schechter, Okochi, Lapidus, etc. (cf. [67, 69-74, 106, 124, 142-154, 163-164, 168-173, 175-178, 180-182, 188, 189, 191, 194-198, 200, 201, 213-220, 225, 227-241, 243-247, 248-251, 253-258, 264, 265, 285292, 295, 296, 299, 302, 322-325, 339, 342, 348-350, 357, 358, 368, 383, 385, 386, 394, 395, 397-399, 401, 404, 410-417, 425-433, 436-448, 455, 463, 464, 469, 497-503, 509-516, 518-520, 533, 536, 537, 541, 547-549, 551-552, 558-560, 565-584, 587-597, 605, 614-616, 627, 642, 645, 646, 651, 652, 660-664, 666-668, 670, 674, 676-678, 685-687, 690], points 2.2-2.17, 2.22-2.24). A wider and more important class for applications than quasicontracting semigroups is made up of semigroups satisfying for some functional p:E + [0, +~] (for example, norms in E or something stronger) the stability condition p(U(t)x)..<~(t, r) Dr p(x)~
IIU (t) x - - U (0 y II"-<e~
IIx - y IIj Dr p (U(t)x), p(V(t)V ) < r.
(4)
To such semigroups t h e r e c o r r e s p o n d i n t h e Cauchy problem (1) o p e r a t o r s of t h e c l a s s there exists an' ~:R+--+R+, inch that Al{x,,p(x)
Dlss (E, p (.)-loc)={A:
(5)
i
(to guarantee the stability (3) an additional condition on A is necessary). A number of resuits on the solvability of the Cauchy problem with operators A6Diss(E; ~) carry over to this case (Chambers and Oharu, Konishi, Oharu, Kutuzov, Khazan [71-73, 106, 151, 153, 265, 445], cf. point 2.16). A considerable part of the theory of which we spoke above generalizes to the nonautonomous case, i.e., to equations of the form
du (t) / dt~ A (t) u (t) ( h e r e t h e semigroups a r e r e p l a c e d by e v o l u t i o n s y s t e m s ) , c f . p o i n t 2 . 1 7 . there is constructed the theory of the Cauchy problem for the equation
I n Khazan [153]
au (t) / at~ a (t, u (t)) u (t) under the assumption that for fixed z the operators u § A(t, z)u belong to the class Diss(E, p(.)-loc) [the corresponding function ~ can depend on p(z) and lllAzlll] and satisfy the conditions which guarantee the solvability of the Cauchy problem for the equation with "frozen coefficient" du(t)/dt~A(t, z)u(~) (cf. point 2.18 for more details). There are other results on nonlinear evolution equations with operators not necessarily belonging to the class (5) in the papers of Khazan, Altman, da Prate and Grisvard, lanelli, Koi and Watanabe, Otani, Vrabie, Galaktionov [28, 142, 150, 151, 167, 318, 319, 397-399, 432, 550, 660-663] (local and global existence and nonexistence, uniqueness and approximation of solutions, cf. points 2.19-2.21), Babin and Vishik, Ladyzhenskaya, Ball [6-8, 79-80, 186] (asymptotic behavior of solutions, cf. point 2.25). 2187
Nonlinear evolution equations which are unsolvable with respect to du/dt were studied by Barbu, Gr~ger and Necas, Suquet [199, 200, 375, 641], cf. point 2.26 (there are a number of papers on this theme carried out in the realms of the theory of monotone operators A:E + E*). Since for ~ < O Diss(E, ~)cDiss(E~ 0) in all formulas where ]~[ occurs, one can, for < 0, replace [~I hy zero. 2.2.
Classes of Solutions of the Cauchy Problem.
We consider the Cauchy problem
du (t) IdtE A u (t) -}- f (t) (0 ..< t -.< T), u(~=-x,
(6) (7)
where /eLi(O, T;E), xED(A) . A function u:[0, T] + E, which satisfies (7), is called: a) a Cl-solution of the problem (6), (7), if uECI([0, T]; E), u(t)6D(A) and (6) holds; b) a wC lsolution, if u is a Cl-solution of the problem (6), (7) in the locally convex space Ew, i.e., in E provided with the weak topology; c) a strong solution, if u6C(]0, T]; E) du/dl~Ll~=(O, T; E) and (6) holds a.e. on (0, T) [in particular, a(t)@D(A) ]; d) a weak solution, if u(t) = limun(t) uniformly on [0, T], where u n are strong solutions of Cauchy problems with initial conditions x n + x in E and free terms fn § f in LI(0, T; E). We note that if u is a wC:-solution, then it is Lipschitz continuous (u~Lip~[0, T]; E)) and is a strong solution (since a weakly continuous function is bounded and strongly measurable). If A is linear and generates a semigroup of class Co, then, as is evident from the results of Paragraph 1.24, the definition given above of a weak solution is equivalent with the definition of Ball [185] (cf. Paragraph 1.1), and also the formula for variation of parameters. The "more generalized" solutions introduced below-- limit-differences (LD-solutions) and integrals -- are weak (this follows from the corresponding uniqueness theorems and the results of Paragraph 1.7), but in the nonlinear case this is not so in general (in a nonreflexive E). If E is dual to the Banach space G, then we denote E, provided with the weak* topology, by Ew, , and by a w*Cl-solution of the problem (6), (7) we shall mean its C1-solu tion in Ew, (it may not be a strong solution, if E is nonreflexive). We shall say that the Cauchy problem (6), (7) admits implicit difference approximation, if there exist a sequence of partitions ~.={O=g.,o< .. 9 < I.,mtn)-1
tim (1~". I+ Ilu.,0 -- xl{ + Ill. - - / l i d -- O, where l~l=max{ts--t~_,}
for
~----{ts},II lh us--us-, ts--ts.,
is the norm in L'~,T;E)f.(t)=f.,k
/sEA~k ( k = l . . . . .
(8)
(t.,s-,
m; rrt.-.-~-m(n))
(9)
for each n [in (9) the index n has been omitted from u, t, and f]. The sequence of functions Un:[0 , T] + E, defined by the equations un(t) = Un, k for tn,k- 1 < t ~ tn,k, 0 < k ~ m(n), is called an admissible approximating sequence. The function u 6 C[0, T]; E) is called an LDsolution of the Cauchy problem (6), (7), if u(t) = limun(t) uniformly on [0, T], where u n is an admissible approximating sequence [from which it follows that u(t) ~, D--~) for 0 ~ t < T]. All the definitions formulated carry over in an obvious way to an equation with variable operator A(t) [in (9) it is necessary to replace A by A(tk)]. If there exists a strong solution of the Cauchy problem, then it is an LD-solution [without any assumptions about the operators A(t)!], while for some sequence Pn one can take Un, k = u(tn, k) (the proof is implicitly contained in Evans [348]); from this it follows easily that a weak solution, if it exists, is an LD-solution. The class of LD-solutions was introduced and investigated [for A~Diss(E, m), f = 0] by Kobayashi [426, 427], but the concept itself was actually contained in Benilan [215], to whom the following definition is also due: A function u6C(,[0, T]; E), which satisfies (7), is called an integral solution of type m of the problem (6), (7), if for all z6D(A), 0 ~ s ~ t ~ T one has t
e-re' II u ( t ) - - z !12< e - ' ~ II u (s) -- z ll 2 + 2 ~ e - 2 ~ < f (.) + A z , u (T)-- z ) +a., $
where the functionals (
,
) ~:EXE-+R
are defined as follows:
< v, x > ~ = • 1 7 7
s
(I0)
e~o
[here llm e~O
2188
can be replaced by Inf ; in a Hilbert space, obviously,
details, cf. Paragraph 2.3]. Since < , >• is upper semicontinuous on E x E (cf. [289]), in the definition of an integral solution one can replace A by A. Benilan [214, 215] showed that if the Cauchy problem (6), (7) with A6DIss(E, ~) has an LD-solution, then it is the unique integral solution of type m. All definitions of the present paragraph can be carried over to the problem (I) as follows: the corresponding conditions must hold V T > O . 2.3 Classes of Operators Diss (E, m). with the fact that A6 Diss (E, m~:
Each of conditions (11)-(13) below is equivalent
the operator
(I--LA)-hR(I--LA)-+D(A)
(11)
for L>0, k ~ < l
is single-valued and Lipschitz with constant (I -- ~ ) - I ;
IIx--v--~,(Ax--Av)II>(I--X~)IIx--Yll (x, y6.D(A), < Ax--Ag, x--V>
~<~IIx--vll ~
L>0);
(x, gGD(A))
(12) (13)
( c f . [200, 289, 294]). The f u n c t i o n a l s < , >+ are defined i n (10) and have the f o l l o w i n g properties [200, 289, 603]: a)--<--z, x)+---~-(z, x)_-.< (z, x ) j ; b) l
inf over e > 0); d) (~z, ~ x ) • x ) + for ~, ~6R, ~ > ~ 0 ; e) ( z + ~ x , x > + = (z, x ) • =llxll2 for =ER ; f ) ( y , x > + ( z , x > _ ~ <
( z,
x ) ,=rnax{z(s)x(s):sGS, Ix(s)l=llxll~},
and
i n t h e s p a c e L~(S)
( z, X > .----llm ess sup {z (s) x (s):J x ( s ) ] > IIx II . - 8 } , el0
and
x>_ i s o b t a i n e d b y r e p l a c i n g
sup b y i n f ;
in Ll(S)
! z(s)sgn(x(s))ds+- I 'z(s)[ds}; x(
~0
x(
~0
in LP(s) for I < p < ~
> +-=Iz(s)x(s)lx(s)lP-ids/llxllP-2
(x--/=O)
( i n t h e l a s t t h r e e c a s e s S i s an a r b i t r a r y space with measure). All these formulas generali z e t o s p a c e s o f f u n c t i o n s w i t h v a l u e s i n a Banach s p a c e ( o f . [ 2 1 5 ] ) ; i t i s n e c e s s a r y t o r e p l a c e z ( s ) x ( s ) b y < z ( s ) , x ( s ) > + o r < z ( s ) , x ( s ) > _ d e p e n d i n g on w h i c h o f t h e f u n c t i o n a l s
III f Ill For an operator so t h a t D:(.4~ for x~D (A) ]
ln~ {11 z II:z6F},
F0= {z~f ill z I!
=
Ill f III}.
A i n E we d e f i n e i t s r e s t r i c t i o n A ~ t o b e e q u a l t o A~ = (Ax)0 f o r (Ax)0 ~ ~ , if A is single-valued, A ~ = A. F u r t h e r , we s e t [ a s s u m i n g III Axlil= + oo
The s e t I3(A) i s
D(A)cD(A)cD(A) on E and IAxl
=
=
I A x l = lnf llm suPlli ax= Ill,
(14)
D (A)={xO.E:I Axl < + ~},
(15)
called the generalized domain of definition o f t h e o p e r a t o r A. O b v i o u s l y and IAxl ~< lllAxlll; m o r e o v e r , t h e f u n c t i o n a l x+lAxl i s l o w e r s e m i c o n t i n u o u s IAxl (of. [ 4 1 i ] ) . It is clear that if E is reflexive and A is closed in 2189
E x E w (here and below, when it is convenient, we identify an operator with its graph), then D(A) = D(A); this is true also for an m-dissipative operator in a reflexive E [411], but false in general for an m-disslpative operator in an arbitrary E. In the latter case the value of the set D(A) is the following: firstly, an LD-solution of the Cauchy problem (1) [its existence and uniqueness for x 6 D(A) follow from the m-dissipativeness of A, cf. Paragraph 2.6) is Lipschitz if and only if x~O(A) , while the Lipschitz constant is equal to IAxl; secondly, x~O(A)~u(t)~D(A), so that in contrast with D(A), the set D(A) is invariant, and the condition u(~)~O(A) characterizes the "smoothness of a solution with respect to the spatial varlable"for "smooth"initialdata. These results are contained in Paragraph 2.7 in more general form: for A~ Diss (E, ~), satisfying the condition
llm InIL-~d(R(I--XA), x ) = 0
VxED (A).
The latter condition automatically holds if for each x~D(A)" solvable for small ~ > 0, i.e.,
(16)
the equation z -- ~Az = x is
U N R(I--LA)"~b(A).
(17)
~>00<~<~
In particular, i f AEDiss(E, o) and R ( I -- XA) = E f o r some X > 0 w i t h ~m < l , t h e n R ( I -- XA) = E f o r a l l s u c h X, i . e . , A -- ~I i s m - d i s s i p a t i v e [ 2 0 0 ] , and (17) h o l d s . From t h e r e s u l t s o f P i e r r e [ 5 7 5 ] , K o b a y a s h i [ 4 2 7 ] , and B e n i l a n [ 2 1 4 , 215] i t f o l l o w s t h a t f o r A~Dlss(E, o) u n d e r the condition (16),
b (A) = 15 (A) = D (A) = D (A~
I Ax I = I ~ x I = Ill ~ x Ill,
where A is the extension of the operator A to E**, defined as follows:
A----- A U {(x, z):xED (A),
A U {(x, z)}EDlss (E**, o)},
and under condition (17), in addition
I Ax I = lira k-' II(1 - LA)-' x - - x II,
(18)
hi0
so t h a t 1)(A)={xED(A):II(I--LA)-Ix--xII=O(~ f o r X + 0} ( t h e o r i g i n a l d e f i n i t i o n s o f IAxI and D(A) w e r e g i v e n by C r a n d a l l [287] i n p r e c i s e l y t h i s f o r m , and h e a l s o i n t r o d u c e d t h e o p e r a t o r A). We note that if the operator A is m-dissipative, then it does not have dissipative extensions to E, i.e., it is maximal dissipative; in Hilbert space the converse is also true (Minty's theorem, cf. Barbu [200]). A convenient technical tool in a number of questions is the approximation of an operator AEDiss(E,m) by Lipschitz-continuous operators AE = e-X((l -eA) -I --I), defined for E > 0, e m < I on R ( I - eA) = D(Ae); here A~EDIss{E;o(I--s~)-J~; ]IAelILip-.< S~l(1 - +(1--SO)-~); AexEA(I--eA)-~x; [IA~x n -.< (1-- so)-~ I A x (cf. [294]). I f x~D (A) ND(A,) (0 < c < u ) , t h e n a s s $ 0 (i'BA)-~ x - ~ x and OIIA~xlI-~IAxl . I f am < 1 and R(l--LA)~convD x (A) = C ( < L < = ) , thenR(I--kA~)-~C ~<X,s<=) and ( I - - k A ~ ) - I x - ~ ( l - - k A ) - l x as s$O, x ~ C [294]. I f t h e norm i n E* i s F r e c h e t d i f f e r e n t i a b l e (for example, if E is uniformly convex) and R ( I - - L A ) ~ D ( A ) (0<~<~), t h e n D(A) = D(A) = D(A ~ and AEx § A~ a s s ~ 0 , x~D(A) ( c f . [514]). The operator A e is called the Yoshida regularization for A. The operator A -- ~I is called completely dissipative (and the operator--A+ ml Browder accretive), if (13) holds with < , >- replaced by < , >+. The role of this concept is explained by the following: if A is completely dissipative, and B is dissipative, then A + B [defined on O ( A ) N O ( B ) ] is dissipative (this follows from property f) of the functionals < , >• If E* is strictly convex, then the functionals < , >+ and < , >_ coincide so that any dissipative operator is completely dissipative; in general this is not so even for m-dissipative operators [215]; however, if A is the infinitesimal operator of a semigroup of type m (cf. Paragraph 2.10), then A-- ml is completely dissipative [200], in particular, a linear m-dissipative or nonlinear everywhere defined continuous dissipative operator is completely dissipative. 2.4. Functional KA(t, x). In investigating the character of continuity in t of an LDsolution of the Cauchy problem (I) with x~D(A) , and also in estimating the rate of convergence of the difference method, it is convenient to use the interpolation functional
tCa (t, x)---- mf ~ II~ - - z II+ t 11A z II:zeD (A)}O b v i o u s l y VxED(A) KA(t, x)$O as t r 0. The r a t e o f d e c r e a s e o f K A ( t , x) as t + 0 c h a r a c t e r i z e s the "smoothness of x with respect to A." It is easy to verify that
2190
KA (t, X} = 0 ( t) ( t Jro).~xeb (A)a.d I f AeDlss (E, ~)
and
that
KA (t, X} ..< t I Ax I.
R (I--~.A)~D (A) (0 < ~ < ~ ) , t h e n VxeD(A~ (1 - t I~ I)11( ! - t A)-'x-- x II~< Ka (t, x) < 3 I1(/-- t A)-' ~-- x II
(0~.t[~[.
Some Notation for ~ L t ( 0 , ~ E ) .
We set
ot **(s)----sup {i II/(t + x ) - f
(t)lldt:lxl--<s},
assuming f(t) = 0 outside [0, T]. Then p~l)(~)i0 as e + 0, and i f f : [ 0 , T] + E h a s bounded variation: f~BV([O, T]; E), then pll)(a)--
Existence and Uniqueness of LD-Solutions for an Equation with Constant Operator Rate of Convergence of the Difference Method. For A~Diss(E,~) the following assertions are valid.
A~Diss(E, to);
I~ The existence of an LD-solution of the Cauchy problem (6), (7) is equivalent with the fact that this problem admits implicit difference approximation [from which, in particular, it follows that x~D(A) and f~Lii0, T; E) ]. 2 ~ . If an LD-solution of the problem (6), (7) exists, then it is the unique integral solution of type ~ (and all the more the unique LD-solution) of the problem (6), (7). 3~ An LD-solution of the Cauchy problem with operator A is also an LD-solution of the Cauchy problem with operator A for the same f, x. 4 ~. The set of all pairs (x, f), for which there exists an LD-solution of the problem (6), (7), is closed in E x LI(0, T; E). 5~ The map (x, f) § u, where u is an LD-solution of the problem (6), (7), is Lipschitzcontinuous from E x LI(0, T; E) to C([O, T]; E). 6~
The rate of convergence of the difference method can be estimated as follows:
IIu~ i t ) - u (011 < M (r) {K~ (~'/~, x) + pI.') (~'/~) +a}, a = a (x, f ) = l l u0--x I1+11/~--/I1,,
(19)
where ~={tk} is a partition o f [0, T], I ~ l = x , u~(t)=u~, f~(/)-----f~ f o r tk_~
If
feL'(O, T; E),
F~E,
then for the existence of an LD-solution of the problem (6), (7) for all
such that
f(t)eF
a.e., and all
l l m Inf ~-'d
x#0
I f F = E, t h e n (20) 8 ~.
If
implies
that A-
xeD(A) , it is sufficient that
(R (I--kA), x + ky)-~-0 Vx(D (A), yeF.
(20)
~I i s m - d i s s i p a t i v e .
the condition Vr>0
R(I--kA)D{xED (A)+).F:llxll~
w h i c h i s s t r o n g e r t h a n ( 2 0 ) , h o l d s , and x , f a r e as i n 7 ~ t h e n an L D - s o l u t i o n ( 6 ) , (7) e x t s t s a n d can be r e p r e s e n t e d i n t h e f o r m u(t)-----llmgX(t), w h e r e
(21) of the problem
z~0
:,x (t)_~x (t--x)eAttx (t)+ fT (t)(t> 0), u" (t)m-x (t<0),
(22) 2191
D(A)gx--+x
in ~ , ff(t)=f~6F for (k--l)z
u (t) = H (I - ds (A + f (s)))-'x
(23)
o
for
xED(A) , and i f A i s c l o s e d ,
then also
for
xED(A).
9 ~ . For u:[0, T] § E the following conditions are equivalent: a) u is an LD-solution of (6); b) for any (or some) s6(0, T) the restrictions of u to [0, s] and [s, T] are LD-solutions of (6) on these segments~ c) for any (or some) s > 0 v(t) = u(t -- s) is an LD-solution of dv(t)/dt6Av(t)+[(t--s) (s<~t<~T+s).
Assertions 2 ~ and 5 ~ w e r e p r o v e d b y B e n i l a n [ 2 1 5 ] ( c f . a l s o [214] and l o w ) ; t h e u n i q u e n e s s o f an L D - s o l u t i o n t o g e t h e r w i t h a s s e r t i o n s 1 ~ and 5 ~ f e r e n t m e t h o d i n C r a n d a l l and E v a n s [288] and ( i n a m o r e g e n e r a l s i t u a t i o n ) and the uniqueness of an integral solution [under a condition of type (21) and (r) = const] in Barbu [200]. For f = 0, F = {0} assertions I ~ 3 ~ , 40, proved by Kobayashi [427]; here (19) has the form
P a r a g r a p h 2.9 b e i s p r o v e d by a d i f in Khazan [153], with F = c o n v F 6 o, 7 ~ , 90 were
II U~ (t) - - u (t)I[ ~ e41~'~ll{K.~ ((tz) 1/2, x) + A}.
(24)
In the general case 6 ~ was proved by Khazan [153], 7 ~ follows from the special case F = {0} by the results of Benilan [215], 3 ~ and 9 ~ follow directly from the definition of an LD-solution, and 4 ~ from this definition and assertion I ~ . If (21) holds and ~, is an arbitrary sequence of partitions of [0, T] with I~nJ-~T,-+O (for example, ~n={kr,}). 21~I~',
U,,,~=(I--T. hA)-l (tt.,~-,+~.kf.~). which
is possible by (21), and by the a priori estimate
T.,~=t.,~-- t.,k-,, (cf. [427] or [153])
il u,,,~ _ x ll < e21otr { li u.,o - - x ll + K a (T,
x)+llf,,lh},
(25)
we c o n s t r u c t an i m p l i c i t difference approximation of the problem (6), (7). I f /EBV([0, T ] ; E ) o r fEC ([O, T]; E), ~ = { t , } is a partition o f [ 0 , T] and f ~ ( t ) = f k ~ f ( [ t , _ , t , ] ) f o r tE(t,_,t,) , then for I~]=T-+O Hf~--fll,-+O with rate indicated in assertion 8 ~ . S i n c e a n y fED(O, T ; E ) i s t h e l i m i t i n LX(0, T; E) o f a s e q u e n c e o f s t e p f u n c t i o n s (which therefore belong to BV([0, T]; E)) with values from any set which is dense in f(0, T) (cf. [215] or [348]), one has that VfELI(0, T;E) with values in F there exist step functions f~ (~ > 0) with the properties indicated in 8 ~ If F is convex, an explicit formula for fT, which guarantees the estimate II]~--/lh=O(p}"(zln)), is given in [153]. Thus, assertion 8 ~ follows from I ~ and 6 ~ [for (23) it is necessary to consider in addition that (I -- TA)-I(u + Ty) = (I -- ~(A + y))-lu]. We note that the existence of the limit
limuT(t)-----u(t) x40
of solutions
of the problems
(22)
with /~(t)=0, x~=xED(A) under the condition I@(I--%A)~D(A) ( 0 < k < = ) is proved in the classical paper of Crandall and Liggett [289], and if /@C(10, T~;E), /~----/(kx), R(I--%A)----E" (0 < ~ < e) in the paper of Crandall and Pazy [294]. The basic difficulty in applying the theorems formulated is the verification of (20) or (21), i.e., the proof of the solvability of the stationary equation. Here theorems on perturbations of m-dissipative operators (cf. Paragraph 2.15) and also the approximation of the operator A by operators for which (21) is easily verified, become useful. There are examples of the application of these methods to specific problems in [145, 147, 163, 164, 200, 215, 218, 232, 234, 244, 257, 286, 350, 383, 434, 436, 440-444, 542, 642], etc. Sometimes one can manage without verifying (20) or (21) by applying the theorem of Khazan [151] cited below in Paragraph 2.19. 2.7. Tests for LD-Solutions to Be Lipschitz, H~Ider, and Absolutely Continuous, Invar~ance of the sets 0(A), D(A)~ D=(A), and Also Some Estimates~ In this paragraph we assume that A,~Diss(E,~) and u is an LD-solution of the Cauchy problem (6), (7). We use some notation from Paragraphs 2.3, 2.4. We recall that Aci. D(A)cD(A)cb(A)~D(A) , and if E is
2192
reflexive and A~Diss(E,~) is maximal (for example, A -- ml is m-dissipative), then A = A. The following assertions, the first of which was proved by Benilan [214, 215], are valid. I~.
fEBV
If
a) x E O ( A ) ;
([0, T]; E) , then the following conditions are equivalent:
b)
uELIDOO,T];E) ;
u(T)ED(A).
c)u(t)~D(A)(O
Here
~mh~'~[u(t-Fa)-u(r hJ, O
(26)
IIu(t +h)-u(O the function t~+[llAu(t)~f(t-P0)[l[ except possibly a countable set. 2 ~ . If fEBV([0, T];E), equivalent :
a)
x6f)(Ai;
b)uO_Lip([O,
H e r e i n (26) and ( 2 7 ) , n[Au(t) Assertion
(27)
is right continuous on [0, T) and continuous at all points
f(t)QF Tl;f);
II~
a.e. and (20) holds,
then the following conditions are
u(t)eb(A) (O..
2 ~ f o l l o w s f r o m 1 ~ and t h e f a c t
that
(O~
under condition
(16),
as P i e r r e
[575]
showed, IIIA~III----IAxlVxEE. For f(t) = 0, under (17) assertions I ~ and 2 ~ were proved originally by Crandall [287]. The following assertion follows from 2 ~ . 3 ~ . If under the assumptions of assertion 2 ~ , D(A) = D(A) (for example, if E is reflexive or dual to a separable Banach space, and A is sequentially closed in E x Ew*, or E is reflexive and A -- ml is m-dissipative), then xOD(A)~u(t)~_D(A) (0~
Cf. P a r a g r a p h 2.8 f o r o t h e r c o n d i t i o n s w h i c h g u a r a n t e e t h e i n v a r i a n c e e r a l D(A) i s n o t i n v a r i a n t , as t h e f o l l o w i n g r e s u l t o f Webb [667] shows. 4 ~.
xED(A)
o f D(A).
In gen-
T h e r e e x i s t s an m - d i s s i p a t i v e o p e r a t o r A i n E = C ( [ 0 , ~ ) ) , such t h a t f o r c e r t a i n f o r an L D - s o l u t i o n o f t h e p r o b l e m ( 6 ) , (7) w i t h f ( t ) = 0, we h a v e : tt(t)~DiA) (O
T). Here A = A0 + B, where A0 is linear and m-dissipative, E is continuous and dissipative.
D(A) = D(A0)
is dense in E, B:E §
5 ~ . Let the Cauchy problem (6), (7) admit implicit difference approximation [cf. (8), (9)], for which sup{]Aun.ol-q-2ilfnllBv}=C<~ (for this, it is necessary, and if f(t)O.F and n
(21) holds,
then also sufficient,
that
xEs
fEBV([O,T];
E)). Then for this approximation
(28)
Ill Au,,,k III~< ~i~/[u.,~ --u.,~_, II+ II/.,. I1-
u(t)6_s Let
(O..
a.e.
and (20) h o l d .
Then t h e
following conditions are equivalent:
xO_D=(A):{xO_E:Ka(t,x)=O(t=) " a s t + 0}; e) uO_C~ ([0, T]; E); f ) tt(t)O_D=(A) (O~
d)
Here
9
[f]==
h~.o
rim sup
h-=p~'J(h).
The
as0
e) is valid without (20). Assertion 6 ~ for f = 0 and /~ ([-- L A ) ~ D (A) implication d) (0 < k < ~0) was proved by Plant [582] and Westphal [676]; the general case follows from the inequalities (valid for 0 ~< t <~ t + h ~< T) h
Ka (h,
x) ~ 9 e l ~ l h / s u p
[0<x~h
Iiu(x)--x]lq-- S l] f
(T) d r + d (R ([ -- k A), x)},
(29)
0
t+h
(3O)
IIu i t + h ) - u (011 ~< eo'+lol'Ka (h, x) + etol~t+')0}'~ (h),
(31)
2193
which are derived from the inequality [215, 582]: VzED(A), O-.<s
e-~' Ilu ( 0 - z II< e-~s II u ( s ) - z II ~ ~ e - ~ [/(x) + a z , u (3)-- zl a.~
(32)
$
which is equivalent with the definition of an integral solution, as follows: (29) is obtained by Plant's method [582, 584] (it is necessary to take s = O, t = h, to consider that [v, w] ~< h-l (II~-~ hv II-II ~ID and to apply some triangle inequalities, in particular, llx--z ]I~
b] § E is differentiable a.e. on
(ct, b) tt'~L'(al b; I~), and tt(t)--tt(a)-~-]tt'(s)ds . Any reflexive O
space has this property (cf., e.g., Barbu [200]), and also the space Z1 and certain Lorentz and Besov spaces. The following assertions are valid [in I~
~ A 6Diss(E,(0) ].
I~ If A is closed, f(t)6F a.e. and (21) holds, then for a function u:[0, T] -> E the following conditions are equivalent: a) u is a strong solution of the Cauchy problem (6), (7); b) u is an LD-solution of the problem (6), (7) and du/dt614oci(O,T; E) Under these conditions, du(t)/dt6(Au(1)~-f(t!)~ a.e. on (0, T). If one also assumes that E has the Radon--Nikodym property (for example, is reflexive), then conditions a), b) are equivalent with the following: c) u is an LD-solution of the problem (6), (7), and u is absolutely continuous on any [a, b] ~ (0, T ) ;
d) u i s an L D - s o l u t i o n o f t h e p r o b l e m ( 6 ) , g r a b l e on any [a, b ] c ( 0 , T).
(7),
and t h e f u n c t i o n
t + IIIAu(t)III
is inte-
The e q u i v a l e n c e o f c o n d i t i o n s a) and b) f o r f = O, F = {0} g o e s b a c k t o B r e z i s and P a z y [ 2 4 6 ] , C r a n d a l l and L i g g e t t [ 2 8 9 ] , and M i y a d e r a [ 5 1 1 ] . I n g e n e r a l t h e i m p l i c a t i o n b) =~ a) [even without the assumption of (21)] was proved by Pavel [559], somewhat generalizing a resuit of Evans [348]; the rest of assertion I ~ [without condition d)] is contained in Benilan [214]; it is obvious that a) and b) together imply d), and from assertion 7 ~ of Paragraph 2.7 it follows that d) => c). 2~ If A is closed, f~BV([O,T]; E), f(t)6F a.e., (21) holds and x%D(A) , and E has the Radon--Nikodym property (for example, is reflexive), then there exists a unique strong solution of the Cauchy problem (6), (7). If the norm in E* is Frechet differentiable (this condition is stronger than reflexivity and it holds if E is uniformly convex, in particular, for E = LP, I < p < =), then d+u(t)/dt = (Au(t) + f(t + 0)) ~ (0 ~< t < T), and if the norm in E is also Frechet differentiable, then d+u(t)/dt is right continuous on [0, T), and du(t)/dt exists and is continuous everywhere except possibly for a countable set of points, i.e., u is an almost" Cl-solution. The first part of this assertion follows directly from assertions 2.8.1 ~ , 2.7.1 ~ , and 2.6.2 ~ and is contained in Benilan [214]; the second follows from the results of [214] and Miyadera [514], and the third generalizes a result of Dorroh [342] for f = 0 and m-dissipative A and is proved analogously (from the differentiability~of the norm in E it follows by virtue of the results of [215, 514], that ~Diss(E,~) and (A) ~ = A ~ where A is the closure of A in E x Ew; together with 2.7.1 ~ this gives the weak right continuity of d+u(t)/dt on [0, T) and the weak continuity outside a countable set; from the
2194
differentiability of the norm in E* it follows [514] that the weak convergence of x n to x in E implies the strong convergence, if llxnll+ Ilxll, and this gives the strong continuity). We recall that for foBV([0, T]; E) ~(t~-0). exists on [0, T) and is equal to f(t) outside a countable set. 3 ~ . If E is dual to a separable Banach space [for example, E = L ~ = (LI) *] or reflexive, A is single-valued and sequentially closed in EXEw, ~ ~BV([O,T]~ E), xOD(A), ~(1)O~ a.e. and (21) holds, then an LD-solution u of the problem (6), (7) (it exists and is unique by virtue of 2.6.8 ~ and 2.6.2 ~) is an almost w*CX-solution, and if f is weak* continuous, a w*Cl-solution (in a reflexive E the weak* topology coincides with the weak, so that u is a wC1-solution and hence a strong solution). Instead of (21) it is sufficient to assume that the problem (6), (7) admits an implicit difference approximation with [Aun,0[nu[[fn[[sv~.c. Assertion 3 ~ is a special case of results of Khazan [142, 144, 151] and follows from 2.7.5 ~ and the following assertion 4 ~ , which is also contained in [144, 151]. 4 ~ . Let E be imbedded in a separable locally convex space E0 and A be the sequential closure of A in E • E0. If A is single-valued and the Cauchy problem (6), (7) admits implicit difference approximation, for which the set {Au,.~:hEN, ~,= I..... m ~ ) ~ is relatively sequentially compact in E0, and u is an LD-solution of the problem (6), (7), then u(L)6D(A)% (0 < t ~ T), the function t + Au(t) is continuous from [0, T] to E0, the function t + u(t) -t
~/(s) ds
belongs to CI([0, T]; E0), and du(t)/dt = ~u(t) + f(t) at all Lebesgue points of the
0
function f:[0, T] + E0 (hence, a.e., and if addition, ]IAun,kll ~ c, then
/6C([O,T];
E~,
then everywhere on [0, T].
If in
(33)
In this and the following assertions, A is an arbitrary operator [not necessarily of the class Diss (E, m)], and E can be a locally convex space. It is important to note that in applications to partial differential equations, in order to guarantee that the closedness type conditions hold (and sometimes the dissipativeness ones also), which figure in 3~ ~ one should often take not the maximal operator A, generated in the corresponding space by the differential expression and boundary conditions, but its restriction to some set D (possibly depending on x and f), where un,hEO (n~N. k=l .....m(n)), so that u is an LD-solution of the Cauchy problem with operator AID. There are examples in the papers of Kutuzov, Khazan, Aizawa, Burch [69-72, 142, 144-146, 148, 163, 164, 250] (cf., in particular, the examples in Paragraph 2.13) and others (cf. Paragraph 2.16). The set D can be defined by monotonicity or convexity conditions on the functions occurring in it or by other inequalities. In order that implicit difference approximation of the Cauchy problem (6), (7) satisfy the condition u.,k~D, it suffices that u,,o6D and for small X > 0 (I--MI) 7' (O--b~F)cD. (34) If D={u~E:p(u)~.r) , where p:E + [U, +=o] is some functional, then an LD-solution of the Cauchy prob-iem with operator AID is a p-bounded LD-solution of the Cauchy problem (6), (7), i.e., has difference approximation with p(Un,k) ~ const. Sufficient conditions for the existence and uniqueness of p-bounded LD-solutions are given in Paragraph 2.16. Any LD-solution can be considered as p-bounded for p(u) E 0. Considering that by virtue of the definition of an LD-solution the set {Un, k} is relatively sequentially compact in E, we derive the following assertion directly from 4~ 5~ If A is single-valued [possibly Cauchy problem (6), (7), and
ACDiss(E,~) ], u is a p-bounded LD-solution of the
(D(A)~z.-~zBE, p (z.)-.
(35)
then u is a strong solution satisfying (33). If in addition f:[0, T] + E is continuous (weakly continuous), then u is a C1-solution (wC1-solution) of the Cauchy problem (6), (7). Under the hypotheses of assertion 5 ~ , as follows from 2.6.2 ~ and 2.8.1 ~ , u is the unique
2195
strong solution, if A6DIss(E, ~) , and the unique strong solution with p(u(t)) --< const, if A6DIss (E, p(.)-10c). Since (35) obviously holds (in the strong version) for continuous A with closed D(A), in this case an LD-solution of the problem (6), (7) with continuous f(t) is automatically a Cl-solution [we take p(u) - 0]. However this does not exhaust the applications of assertion 5 ~ Thus, in studying the problem ttr-Fa(t,x , u) ttx+b(t,'x , u)=0(0-.
Let As
be single-valued,
the Banach space (E,, [] ]],) be imbedded in E,
D (A)cE~ , and let there exist a function oI:R+-+R+, such that Vr>~0 the restriction of A to the set Yr ={zED(A):IIzII+IIAzJI'-
Further, let
(D (A)Dz. -+ z in E,, [lAz. If,--
(36)
let (21) hold, /(1)6F, / E B V ([O,T]; E,) N C([O, T]; E). xED (A), AxEEI . Then an LD-solution u of the Cauchy problem (6), (7) which exists by 2.6.7 ~ is its unique Cl-solution in E. Moreover,
uELip (10, T]; EO. Assertion 6 ~ proved by Khazan [148] (for f = 0), was the first test for the existence of C1-solutions for nonlinear evolution equations. With its help, in [148] the unique classical solvability of the problem c#(ttt)=ttxx(t>~O,O~x..
][u(t)--z~(t)[l~e~' 2~ If instead of A6DIss(E;~) dispersive, i.e.,
2196
[Ix--y[[+o
e-~*ll f ( s ) _ g ( s ) l [ d s
.
(37)
one assumes that E is a Banach structure and A -- ml is
fl ( ( I - ~ A )-, x - -
( I - - ~ A )-,y) § II ~
(I -- ~)-~ II (x-- Y)§
for ~>0, X ~ < I , x, yER(I--LA) , then instead of (37),
II (~ (t) - ~ (t))+II < e~'
e-g" II ( / ( s ) -
II ( x - yFll +
g (s)Fl; as .
(38)
0
from which, in particular, it follows that for x ~ y, f(s) ~ g(s) one will have u(t) ~ v(t). It is obvious that in E = LP(S) (I < p ~ ~) or E = C(S) dispersivity implies dissipativity; in general a dispersive operator becomes dissipative after equivalent renorming of E (for example, llxlll = [Ix+t1 + Jlx-ll), so the results of Paragraphs 2.6, 2.8 (and considering the renorming 2.7 also) remain valid for equations with dispersive operators (cf. Konishi [436], Picard [574] for f = O, Benilan [215] for f ~ 0). The condition
( A x - - Ay, (x--y) +) ~<~ll(x-y)+ll2
(x, yEO(A))
is sufficient for the dispersivity of A -- ~I in an arbitrary Banach structure [436]. Nonlinear evolution equations with dispersive operators were also studied by Calvert [253-258] and Sato [605]. We note that if A is dispersive, then--A is called T-accretive. Now let u be an LD-solution of the Cauchy problem (6), (7), and v be an LD-solution of the Cauchy problem
dv(t)/dt6B*(t)+g(t)
~..
~)----Y,
(39)
( Ax--By, x - - y ) _<~llx-yll2q-811x-yll+6
(40)
where A, B are arbitrary operators in E, while
for all x@D(A), y6D(B), where p, c, 6 > O, then
II u ( 0 - ~
(t)II ..< e,,
IIx - y
I1+.[ I I / ( s ) - g ( s )
(41)
0
(Khazan [ 1 5 3 ] ) .
Here D(A) and D(B) can be d i f f e r e n t ,
f o r example, A and B can be d i f f e r e n t i a l
operators with different highest terms and different boundary conditions [154]. For A = B6D%~(E, ~) one can take ~ = ~+, c ~ ~ = 0 and (41) becomes Benilan's estimate (37). In [153J the case is actually considered when A, B depend on t, and ~. 8EL I (0, T 1 . We note that if the solutions considered are absolutely continuous, weakly differentiable, and satisfy the equations a.e. on (0, T), then (37), (41) can be proved rather simply with the help of property h) of the functionals < , >• from Paragraph 2.3, while the dependence of the operators on t does not introduce any additional difficulties, and m, p, e, 6 can depend on the solutions themselves, i.e., (40), for example, need only be verified on solutions (cf. [142] for an estimate of the type of (37)); an analogous approach is also possible to (38) (cf. [605]). For arbitrary LD-solutions, proofs of comparison theorems (in particular, uniqueness) are technically quite complicated; the need for them is explained hy the fact that in nonreflexive E the set of x for which there exists a weakly differentiable solution of the Cauchy problem (I) with m-dissipative A can be empty (Crandall and Liggett [289]; cf. also 2.7.4 ~ above). To verify the condition A6Diss(E, ~) for (37) or condition (40) for (41) one can restrict the operators to any set containing the approximations Un,k, Vn,k, in particular, to a set of the form {u:p(u) s const}, if u, v are p-bounded LD-solutions (in this connection, cf. Paragraph 2.8, 2.16). 3 ~ . Comparison theorems of another type, which cannot be proved even for strong solutions without the use of difference approximation, were obtained in Khazan [153, 154], Benilan and Diaz [219]. Let AEDiss(E, ~), R ( [ - - L A ) ~ D ( A )
(k>O, ko
S (x, A, T)={(l--~A)-kx:X >O, 2 ~ < t ,
(42)
O-.
(43)
Wenote t h a t f o r z~S (x, A, T) o n e w i l l have l]z--xlJ..<e~l~lrKa (T ' x) and IIIAzlll<e2t~ (28)] ; i f p : E + [0, q-~]is a f u n c t i o n a l s u c h t h a t Vzs (A) w i t h p(z)>ro>p(x), ~ > 0
p (z --kAz) ~ (1 --k~0) p (z) --~/M
Axl
[ e l . (25) and (44)
[ f o r example, p((l--kA)-~u)-.
on the function ~
and the boundary 2197
4 ~ . Let (42) hold and the same condition for the operator B, X~)(A), yf~(B) there exist 6,870, rE[0, I) such that for zES (x, A, T), ~ S (y, B, T), ~ > 0 , 2~r < I,
, and let
I (I--~A)-'z--(I -- ~B)-'~ II--<(1 --X~)-' (11 z - ~ II+ 6X= +s~)
(45)
[by virtue of the preceding remark, 6, ~ can depend on llzll, llwll, lllAzlll, lllBwlll (if x~O(A), gED(B)), and under condition (44) for A and B also on p(z), p(w) (if p(x), p(y) are finite)]. Then for LD-solutions of the Cauchy problems (6), (7) with f = 0 and (39) with g = 0 (which exist by 2.6.8 ~) one has [with r = I/(3 -- 2~)]
IIu (t) - ~ (t)11 ~< e4,.,, {11x - y II+ c (t'-=8)r + 12~ I,-=t6 + t~}. (46) if either xED(A), y~D(B) [here c r 2(IAxl + IByl) + I], or 6 = 0 (here one can replace e 41~It by e~t).
If A -- ~I is m-dissipative, then (45) [with ~6S(y, B, T), z@~) ] follows from
II (I - k A ) - ' w - - (I - - k B ) - ' w II ~< (1 -- ~r [with
wES(y, B, T ) ] .
In
this
case,
along
with
(46),
Jlu(t)--v(Ol!<eo'llx-yli+e'~~
(6~ =-F sk)
x~D(A),!~_D (B)
for
(47) one has
+ 10 (t=t2-=)P-~)'+ 12~ P-=t6 + 181
(48)
w i t h r = 1 / ( 3 -- 2 ~ ) ; f o r a = 0 o n e c a n r e p l a c e 10 b y 2, a n d f o r ~ = 1 / 2 , b y 3. We n o t e t h a t i f to=%=-A4-----0, R ( A ) 9 0 , ~ (B)~0 , a n d 6 a n d e i n ( 4 5 ) d e p e n d o n l y on Ilzll, Ilwll, IIIAzlll, IllBwlll, p(z), p(w), then (~5) holds for z~S(x, A, -~-~),r B, -F ~ ) , so that (46) is valid for all t ~ 0; the analogous remark is true for (47), (48) also. A simple sufficient condition for (45) (with = = 0) is given in the following assertion.
5~ Let A, B6DIss(E, ~), R(I--~B)~D (B)(~>0, %~ < I) (for A such a condition is not necessary), x~D(A), y~D(B ) , and let there exist a single-valued operator Q:D(B) ~ D(A) (generally nonlinear) such that V ~ 8 (g, B, T) 2[[Q~-~][<6,
[] A Q w - - B ~ ] I - . < s .
Then if u is an LD-solution of the Cauchy problem tion of the Cauchy problem (39) with g = 0, then
IIu ( 0 -
~ (t)ll ~< e~'
IIx
(6),
(7) w i t h
(49) f = 0, and v is an LD-solu-
- y II + e~ ~ ' {(31B~ I + 1 + s) t~ls6x/s-l- 2 [~ I t~ +
t. + 61.
(5 o)
Moreover, (45) holds with a = 0 for all ~S(y,B, T), z@R(I--~A) so that if y~D(A), x~D(A) and R(I--%A)~D(A) (~>0, k ~ < l ) then in (50) one can throw out the summand ~, replacing 3[Byl + I + e by 2([Ax[ + [By[) + I. The first part of assertion 5 ~ is proved in [153] [the case B = B(t), A = A(t) is also considered there], and the second in [154]. The conditions A BEDiss(E, ~) can be relaxed to A, B6Dlss{E,p(.)--loc),, if u, v are p-bounded LD-solutions and Q carries p-bounded sets into p-bounded ones (the last condition is omitted in [153]). The idea of using the nonlinear operator Q:D(B) § D(A) for passing from a difference approximation of the problem (39) to a difference approximation of the problem (6), (7) and getting a comparison theorem on the basis of this appeared simultaneously and independently in Khazan [151, 153, 154], Benilan and Diaz [219]. The following assertion is proved in [219]. 6 ~. Let A, B be a dispersive operators in the Banach structure E, u be an LD-solution of the problem (6), (7)/6L~(0, T; f), x6D(A) , v be an LD-solution of the problem (39) with g----0, yED_(B) , where D• • Let V L > 0 R(I--XB)~D_(B) [this guarantees the existence of an LD-solution of (39)]. Let there exist a continuous operator Q:D-[~ + E such that V ~ D ( B ) Q*6D(A), A Q ~ B ~ . Then if I -- Q preserves order in E, then t
IICu( t ) - Q~ (t))+ll < II(x- QWII + I II/ (~)+ II 0
in particular
(x~
I preserves order in E, then t
II(Qv ( t ) - u (t))* II--< il ( Q y - x) + II+ ~ II/ (~) II a~. If in the hypothesis of this theorem one replaces D_(B) by D+(B) (i.e., roughly speaking, instead of By < 0 one assumes that By >10), then in the conclusion the two estimates given are interchanged. Now we proceed to the comparison of an LD-solution of the problem (6), (7) (for simplicity we assume f = 0) with the solution v T approximating it, obtained by discretization of the
2198
time according to the implicit scheme and replacing A by some approximation B of it, i.e., v~(t) = Vk((k -- 1)T < t ~ kx)
(Vk--~k_,)/rGBvk+g~ ( k = l
. . . . . m; m ~ > r ) .
7~ Let (42) hold with A replaced by B, ~06D (/3) , let there exist an operator Q:D(B) + D(A) such that for ~ S (~0, B, T) [cf. (43) and the text after this formula], let (49) hold (so that d and e characterize the error of approximation). Then (cf. [151])
II u ( t ) - =* (Oil < e,I-t,iI I x -
~0 If+ c (t,),/, + ~-,~ + ~) t + 6 + 11g Ih},
w h e r e c = m l n { I Axl, IB*01+s}, Ilglh=*EIIg~ll ; i f A s a t i s f i e s (16) and v~D (A), t h e n t h e c o e f f i c i e n t o f IIx -- v011 can be r e p l a c e d by e mt due t o ( 3 7 ) . On t h e o t h e r h a n d , i t f o l l o w s f r o m (24) f o r IIv(t) -- v Z ( t ) l l and (50) t h a t
IIu ( 0 - ~ (t) I1--<e~' IIx -
v0 II+ e~l' {[ Bvo I (tx)'/2+ (31 aVo J+ 1 + s) t=/~6 '/a + 21 ~ I t~ + te q-8 + IIg Ih}.
I n [151] t h e c a s e i s a l s o c o n s i d e r e d operator does not necessarily belong f o r p(w) > r0 ~ p ( v 0 ) w i t h A r e p l a c e d that for II Q ~ - -
when (42) d o e s n o t h o l d f o r B, i . e . , t h e a p p r o x i m a t i n g t o t h e c l a s s D i s s (E, ~ ) . I f ~(I--xB)~D(B), (44) h o l d s b y B, and t h e r e e x i s t s an o p e r a t o r Q:D(B) + D(A) such ~
II+ II AQ~ --B ~ II--< ~ (1 + I] B ~ IlL
then
II u (0 - ~ C0 II< e4t'~ {11x - ~0 II+1 Ax I (tx) ~/~+ cl (x-28, r) (1 + x -~) 8 + IIg Ih}, where r ~, T) = ~ + T) {I + ([[B y 0 [[+2~T) eOi~l+4fl)r},~ = T-26. 2.10. Nonlinear Semigroups and Various sems on the Generation of Semigroups of Type A on E is called a pseudogenerating operator D(A) + D(A) (of type ~), such that Vx6D(A) the Cauchy problem (I); here one says that A (0 < ~ < e) and
U (t) x =
Types of Complete Infinitesimal Operators. Theo~ and Order Preserving Semigroups. An operator (of type ~), if there exists a semigroup U(t): the function u(t) = U(t)x is an LD-solution of generates U(t). If in addition R(I--IA)DD(A)
lim (I --xA)-lt/rlx= lira (I --~t A)-"x x ~0
(51)
n~
[i.e., the solution of the problem (I) is a limit of solutions of the problem (22) with x x = x, fT = 0], then A is called an exponential generator of the semigroup (the convergence in (51) is meant to be uniform on any [0, T]). The infinitesimal (generating) operator of the semigroup U(t):D + D (t > 0), where D c E , is defined by the equation A0x=lim(U(h)x--x)/~ on the set D(A0) of all x6D, for which the limit exists. Analogously one defines the weak infinitesimal operator A,~Ao. The following assertions are valid. I~ For the semigroup U(t) + D (t > 0) of type m the weak infinitesimal operator A w E Diss (E, m); moreover, A w -- ml (and hence also A0 -- el) is completely dissipative (Barbu [200]). For all xED the limit lim[lU(k)x--xllih.-~-L(x)s +~] exists, while L(U(t)x) h~O
emtL(x). The set D o = { x E D : L ( x ) < + ~ } coincides with the set of all x, for which U(.)x is Lipschitz on any [0, T], and U(t)DocD o, while llU(t+h)x--U(t)xll~
2199
5 ~ . There exists an m-dissipative operator A on E = C[O, I], such that there is no weak infinitesimal operator of the contracting semigroup generated by it [cf. (51)]. There exists a contracting semigroup in E = R 2 (with norm [](a, b)ll=max{la [, Ibl}) , having two different mdissipative exponential generating operators (Crandall and Liggett [289]). 6 ~ . If U(t) is a semigroup of type m in a reflexive E, then D(A0) = D0, where Do is defined in I = (Barbu [200]). If in addition, U(t) is generated by some AEDiss(E,~),, then D(A0) = D(A) and v x E D 0 = O (A) u(t) = U(t)xis a string solution of the Cauchy problem (i) ; if A -mI is m-dissipative, then AocAO (cf. 1 ~ and 2.8.1~ 7~ conv D c
If the norm in E is uniformly Frechet differentiable (on the unit sphere), D = E and U(t):D § D (t I 0) is a semigroup of type m, then for it there exists an ex-
ponential
generating
infinitesimal
operator
operator
AEDiss(E, 0~) with
AocAO
with D(A0)
(l--;kA)-Ix=llm(I---~(U (h)-- I))-'x
and an
hi0 = D(A) = D (Baillon [178]).
8~ If (42) holds, A is closed, the norm in E* is Frechet differentiable and U(t) is the semigroup generated by A [cf. (51)], then D(A0) = D(A ~ = D(A) = D(A) and A0 = A ~ (Miyadera [514]). 9~ Let the norm in E be uniformly Gato differentiable, D~E be a nonexpanding retract of E (in Hilbert E any closed convex D ~ E has this property), and U(t):D + D be a contracting semigroup. Then there exists a uniquely determined m-dissipative operator A with D(A) = D, generating U(t) [cf. (51)]. If in addition the norm in E* is Frechet differentiable, then there-exists a one-to-one correspondence between the contracting semigroups U(t), acting on nonexpanding retracts D ~ E, and m-dissipative operators A on E, where for U(t) and A, which correspond to one another we have: D(U(t)) = D = D(A), (51) holds and the infinitesimal operator A0 of the semigroup U(t) coincides with A~ D(A0) = D(A ~ = D(A) and A0 = A ~ These results ("nonlinear Hille--Yoshida theorem") are due to Reich [590, 595], and in the Hilbert case to Crandall and Pazy [292]. 10 ~ If E is Hilbert, D = D ~ E and U(t):D -> D is a contracting semigroup, then there exists a contracting semigroup V(t):C § C, where C = convD, which coincides with U(t) on D (Komura [434]; cf. [245] for a simpler proof). 11 ~ . In E = C(R+) there exists a contracting semigroup U(t):E + E with densely defined infinitesimal operator A0 [which is m-dissipative and generates U(t)], which does not leave D(A0) invariant (Webb [667]). 12 ~ . Let E be a Banach space, D = D c the following conditions are equivalent: a) A is the infinitesimal
E, A:D § E be single-valued
Then
operator of some semigroup U(t):D -> D (t ~> 0) of type m;
b) A EDiss (E, c0) and lim %,-ld(x-l-,l.Ax, D ) = 0 c) A E D i s s ( E , ~ )
and continuous.
VxED;
and (16) h o l d s ;
d) A i s a p s e u d o g e n e r a t i n g
operator
of t y p e ~.
Under t h e s e c o n d i t i o n s , u ( t ) = U ( t ) x i s a C Z - s o l u t i o n of t h e Cauchy p r o b l e m (1) VxED (the e q u i v a l e n c e o f a) and b) i s a t h e o r e m o f M a r t i n [ 4 9 9 ] , t h e i m p l i c a t i o n b) 3 c) ~ d) was p r o v e d by K o b a y a s h i [ 4 2 7 ] , and f r o m 2 . 8 . 5 ~ i t f o l l o w s t h a t d) i m p l i e s a ) ) . 13 ~ . Let E be a Banach structure, the operator A -- mI be dissipative in E (cf. Paragraph 2.9) and R(I--%A)~D(A)(0<~<~). Then A is the exponential generating operator of a semigroup U(t):D(A) § D(A) (t i> 0) such that
II (U(Ox-U(t)@• II < eo' II ( x - v ) • II, so t h a t exist
U(t)
preserves
order
in E (Konishi [436],
lira II(U (h)x--x)+ll/h..
and i(AU(Ox)•
h~0
IAxl i n (:14) w i t h _+Az N E + v ~ 9 }
[i.e.,
IIIAxnl]l r e p l a c e d I(Ax)-I
= 0],
by IIl(Axn)-+lll).
Picard
[574]).
(52) F o r any x6.D(A)
there
o'' I(Ax)+I [ h e r e ] (Ax)+l a r e d e f i n e d l i k e I f x6D+(A) , w h e r e D•
t h e n U ( t ) x /> U ( s ) x f o r t t> s ; now i f x6_D_(Ai , i . e . ,
I(Ax)+I = 0, then U(t)x ~< U(s)x for t i> s (Benilan [215], Benilan and Diaz [219]). If A0 is t h e infinitesimal operator of a semigroup U(t) satisfying (52), then A0 -- ml is completely dispersive [436, 574, 605] (cf. 2.12 for the definition).
2200
ones)
Additional information on semigroups of nonlinear operators (of type ~ and more general can be found in Paragraphs 2.1, 2.6-2.9, 2.14, 2.16, 2.19, 2.24, 2.25. 2.11.
Some Classes of m-Dissipative Operators in Hilbert and Banach Spaces~ are densely and continuously imbedded Banach spaces, while E is Hilbert, flexive, A:V § V* is semicontinuous, monotone, and coercive [the latter means that ][U[]E2)/[]U][V--~-'] -OO a s []U[]v,--+-[-oo) ], then the restriction of --A to the set {xOV:AxEE} dissipative operator on E (cf., e.g., [200]). Many examples can be found in Lions Dubinskii [36].
VcECV*
2~. with
Let E be Hilbert,
I~ If V is re(
$:E-+(--oo, +oo] be a convex lower semicontinuous functional Its subdifferential O~ is the operator defined by
D(~)={u6E:~(u)< ~ - o o ~ L O .
onD(O~)={uEE:O~(u)--/=O}. H e r e t h e o p e r a t o r A = - - O ~ i s m - d i s s i p a t i v e i n E, D ( A ) c D ~ ) , and D ( A ) = D ( ~ ) . I f ~ is Haraux d i f f e r e n t i a b l e a t t h e p o i n t u, t h e n O~(u)={~'(u)} . I f A i s a linear self-adjoint dissipative operator with D(A) = E, then A = - - O ~ , where ~(u)----l[A,/~u][21 2 for u ~ D ( A I/2) [here and below in analogous cases we indicate the value of ~ only for u ED(~)]. If K ~ E is closed and convex and ~-----]K is the indicator function of K (equal to 0 on K and to += outside K), then D(~I K) = K and dfK(U)={W~E:(~, u - - ~ ) > 0 vvEK} , so that the variational inequality (auldtq-Alt(t)--f(t), u(t)--v)~<0, vvEV, f@[O, T] , where A is a singlevalued operator on E, can be written in the form du(t)/dt-~-A~(t)-}-O~(~t(t))~/(1). All these results are contained in Barbu [200], Brezis [234], Haraux [386]. In these books and in Brezis [232] there are many examples of nonlinear operators of the form 0~ (a number of examples are given below in Paragraph 2.13). The importance of this class of operators is explained by the fact that the equation du(t)/dlq-O~(u(t))~(t) has a number of properties analogous to the properties of the equation du(t)/dt = Au(t) + f(t), where A is the complete infinitesimal operator of an analytic semigroup, and which are lacking for Eq. (6) with arbitrary m-dissipative A (cf. Paragraph 2.14); considerably later than in the general case such aspects of the theory as the solvability of nonautonomous equations, the asymptotic behavior of solutions, the theory of perturbations, progressed for equations containing subdifferentials (cf. Paragraphs 2.17, 2.24, 2.15, 2.20). 3~ If E is a Banach space, A:E § E is single-valued, continuous, A is m-dissipative and completely dissipative (Martin [497]).
and dissipative,
then
4 ~ . If E is a Banach space, B :DcE:-+E satisfies a Lipschitz condition with constant ~, then B~Diss(E~ l) ; moreover, B -- ~I is completely dissipative, and V u 0 R B--~l~Diss(E, I--~) in particular, if ~ = I, i.e., B is a nonexpanding operator, then u(B -- I) is dissipative for any ~ ~ 0. If in addition B:D + D and D is a closed convex cone, then R(]--%B)--~D (~ > 0, ~ < I), so that B is an exponential generating and infinitesimal operator of a semigroup of type ~ on D; in particular, if D = E, then B -- ~I is m-dissipative. If D is a closed convex set and B:D § D, then R(I--s for ~ > 0 , % ( / - - I ) < I , so that u the operator a(B -- I) is an exponential generating and infinitesimal operator of a semigroup of type =(~ -I) (for ~ = I a contracting semigroup) on D. These results are, for example, in Barbu [200]; cf. also Miyadera and Oharu [516]. 5 ~.
If the operator B in E is single-valued, p:E + [0, +~] is some functional and for p(x) ~ r, p(y) ~ r, then B6Diss(E, p(.)-loc) (this follows from 4~
[[Bx--By[[~l(r)[[x--y[[
2.12. Tests for Operators in LP Spaces and C Spaces to Dissipative and Dispersive: In this paragraph we shall denote by LP, LP(~), where ~ is a space with nonnegative o-finite measure, and by C we shall denote C(S), where S is compact, or C0(S), where S is locally compact [functions from C0(S) tend to zero "at infinity"]. The standard norm in Lp ( l ~ p < ~ ) is denoted by I[ ]Ip, and that in L ~ and in C, by [[ [[~. In applications, usually ~cR", S-----~,, so in this paragraph and those following, points of ~ or S are denoted by x, the measure in by dx; u ~ v are arbitrary elements (functions) from D(A), where A is an operator on L p or C. wherever the domain of integration is not indicated, it is equal to ~; we set ]~] =
.[dx.
The operator A is dispersive
if and only if--~'(~--~;
--A~+A~)
, where
~(~)=[I~+I],
~,(=;~)=llm~(=nus~)--~(~))/8 [215, 436, 574, 604, 605]; under the stronger condition, ~'(=-e&0 v; Au -- Av) ~ 0, A is called completely dispersive; tests for dissipativeness have the same form with ~(=)-----]I=II (cf. Paragraph 2.3). The tests I ~ , 3 ~ 4 ~ below follow from the formula for ~'(=; ~) given by Sato [603-605] (for ~(i)=ll=]l, cf. Paragraph 2.3). 2201
I ~ . In the space C the dissipativeness (complete dissipativeness) of the operator A means that (Au -- Av)(x0) ~ 0 at some (any) point x0, at which (u -- v)(x0) = flu -- vlI~, and dispersivity m e a n s that (Au -- Av)(xo) ~ 0 at some point x0, such that (u -- v)(x0) = ll(u -v)+ll~. Kutuzov and Laptev [74] showed that if the differential operator A in C(~-~, where ~ c R " , is dissipative and D(A) admits multiplication by smooth functions with compact carrier, then (Au -- Av)(x0) ~ 0 at any point x ~ , where u -- v has at least a local maximum, and hence, the order of the operator A does not exceed 2. From this and from the necessary conditions for a maximum, one gets easily algebraic tests for dissipativeness [or for belonging to the class Diss(E, p(.)-loc) with p(u) = llull~ and p(u) = llull~ § llVul1~)] of the operator Au = F(x, u, DSu) (Isl ~ 2), if the functions from D(A) vanish on the boundary of ~ (other boundary conditions require special consideration; cf., in particular, examples 2.13.3 ~ , 6 ~ ,
16~176 2~.
Let A~ be a family of completely dissipative
operators
in C.
We set
(Au)(x)-~-
sup (A=u) (x), (Bu) (x)= inf (A=u) (x) for u6 N D (A~)~ . Then A a n d B are completely dissipative; this follows easily from I~ 3 ~.
Let
dissipative, then
cf. Evans
I < p < ~.
[350].
The operator A on LP is dissipative,
which means that
if and only if it is completely
~(A=--Av)(x)(=--v)(x)I(u--v)(x)lP-2dx..
if it is dispersive
I (Au-- Av) (x) ((=--~) (x))p-'dx..
If an operator
is dissipative
(completely dissipative)
on L I, then
I (Au--Av) (x) sgn (u--v) (x)dx(-~) .II(Au- Av) (x)ldx ..
U=~
and if i t is dispersive (completely dispersive), then
I (Au-- A~)!(x)dx(~) S(Au-- A~)i(T)~x)dx..
Let
1 ~ p ~ ~ and t h e o p e r a t o r
A i n LP b e s u c h t h a t
I(Au--Av)(x)~((u--v)(x))dx~
any smooth ~ : R - + R w i t h ~)-----0, ~ ' ( r ) > 0 and c o m p a c t s u p p o r t o f ? ' . Then A i s d i s p e r and hence dissipative; if in addition R(I -- A) is dense in LP, then A is m-dissipative, the closure of A-n(L I X L l) in L I satisfies the conditions of assertion 6 ~ (Benilan [215], also Attouch and Damlamian [173], Brezis [232]). 8~
Let the operator
is closed with respect are equivalent:
A on L 1 b e s u c h t h a t
to taking
IiAu)(x)dx=O Vu~D(A)
t h e maximum o f two f u n c t i o n s .
and Vk > 0
Then t h e f o l l o w i n g
R(I--kA) conditions
a) A i s d i s s i p a t i v e ; b) A i s d i s p e r s i v e ; c)
f..
for
f, g6R(I--~A).
C o n d i t i o n s a ) , b ) , c) a r e a l s o e q u i v a l e n t f o r a n o p e r a t o r and Vr@R ArgO ( C r a n d a l l and T a r t a r [ 3 0 2 ] ) .
2202
A on L ~ i f D(A) c o n t a i n s
constants
9~ Let ~(x, .):OxcR-+R not decrease for a.a. (all) x@~., and the operator A on E = L = (E = C) be dissipative or dispersive. Then the operator 8A, defined by (~A)u{x)=~(x, Au(x)) on 19{~A)={u6O(A):Att(x)EO"x, (~A)u6E} also has this property; for E = C completely dissipativity and complete dispersivity are also preserved (Ha [383], Konishi [439]). 10 ~ . Let 8(x, ") be a maximal monotone graph in II, which depends measurably on xEQ (cf. 2.13.1 ~ , 2.13.2 ~ below), while
Vr~R ~vEL 0~ [J (x, ~ (x))gr a.e~ m ~.
(53)
L e t A be an m - d i s s i p a t i v e operator in E=L ~ D(~A)~O. We c o n s i d e r t h e c o n d i t i o n s : a) ( I - - A) - 1 i s c o m p a c t ; b) Ifil < = and A i s c l o s e d i n E x Ew, ; c) r + 8 ( x , " ) - Z ( r ) i s a c o n t i n u o u s f u n c t i o n f r o m R t o L=; d) 8 ( x , ") i s s i n g l e - v a l u e d f o r a . a . x , and t h e c o r r e s p o n d i n g operator 8 on L ~176(cf. 2.13.2 ~) is continuous on sets of the form {tt:I[~(tt)[[=~
11 ~ . Suppose given for a.a. xE[~ a maximal monotone graph 8(x, ") in R, 8 is the corresponding operator in L I (cf. 2.13.1 ~ , 2.13.2~ A is some operator on L I, and the operator A8 on L I is defined as follows: ~z~.(A[~)tt~=~6~(tt)fID(A):~EA~]. If A is completely dissipative, then A8 is dissipative, and for single-valued 8(x, "), completely dissipative. If A is dissipative, 8(x, ") and 8(x, .)-z are single-valued, then A8 is dissipative. These assertions are valid for dispersivity; they follow from 4 ~ and (for 8 independent of x) are contained in [200, 215, 438, 442]. Suppose further [fl[ < =, 8 is independent of x, ~(r0)~O, A is m-dissipative, and satisfies the conditions of Assertion 7 ~ . We consider the following conditions: a) 8-I is single-valued; b) the operator A-Z:R(A) + L z is single-valued, continuous, and u ,~<~(lllA~tlll~); c) the set {u:luII~q-lllAulll~
.
I~ A maximal monotone graph means a (possibly multivalued) operator 8 in R, which is monotone [i.e., Yr, sED(~) ([J(r)--~(s))(r" s)> O] and has no monotone extensions (so that -- 8 is m-dissipative in R). Examples: a) ~:(a_,a+)-+R, is a continuous nondecreasing function, while if a+ ~ -+=, then 8(r) + += as r + a+; b) 8(r) = sgnr for r ~ O, 8(0) = [--I, I]; c) D(8) = {0}; 6(0) = R; d) D(8) = [0, +=), 8(0) = (--=, 0], 8(r) = 0 for r > 0. Any maximal monotone graph is a subdifferential: G-(r) is equal to: minS(r)
for
8 = ~j , w h e r e j(r)=~_(s)ds,
r61ntD{~)~-(a_,ct+); --oo
roED (~), and
the function
for r ~< a_; += for r I> cz+.
2~ Let E be one of the spaces L~(Q)(l~
is a maximal monotone graph,
~(0)90},
(54)
while the function x + G-(x, r) is measurable for each r~R [if E = C(fl), instead of this one needs the continuity of (I + 8(x, -))-Z(r) in x]. For tt~ we set ~(tt)={tv'~i+R:~(x)E~{x, u(x)) a.e.}. This defines an operator 8 in E with~D~;E)={=EE:~(=)!f~E~=O} 9 Here 8 is mdispersive and (if E # L ~ or 8 is single-valued) completely dispersive, D ~ ; E)={=@E:=(x)~D(~) a.e.}. If (for simplicity) B(x, .) = 8 is independent of x and 8 = 3j in R, then in L = ~ = @ ~ ,
2203
where Cf.
qJ(tt)-~Ij(tt(x))dx
[200,
234,
for
](tt)QL l . The c o n d i t i o n
~(x, 0)30
can be replaced
by D(~;E)=#O.
383].
3 ~. Let ~E~ [cf. (54)]. We d e n o t e b y 47 t h e L a p l a c e o p e r a t o r on fl u n d e r t h e b o u n d a r y c o n d i t i o n --tgtt/OvEy(tt) a . e . on F [ t h e D i r i c h l e t boundary condition is obtained for D(y) = {0}]. More precisely, for E=LP(2~
case D (Av; L')={us
(n):AuE
p p < - or y is locally Lipschitz (c depends on n, ~, p, and y). I ~< p < 2 basically follow from E = L ~ the operator by is closed 4 ~. O(Av;
Let ~, 7 6 ~
[cf.
LOND(~;L~)=D(Ap)
and n < For p >2.12.7 ~ in E x
2p < -, then D(Av; Lp) cW"P(f~) and llull~,~
(54)] and I < p ~< =o. The operator Ap = A in LP(~2), defined on (cf. 3 ~ and 2 ~ by Au = A u - B(u) is m-dispersive and for p <
also completely dissipative.~ Moreover,--A~=0~, ~j = 8 and ~h = y in R (L~ [469]).
where ~(~)=I(21ZTuI2- ~](u))dx-~-Ik(u)dI', r
5 ~ . Let l..
(([ - ~A)-'~) ~< ~ (~)
in E = on F, so A7 by A dispersive is m-dissi-
(5 5 )
(for E = Ll(fl) cf. Benilan [215], and for the Dirichlet boundary condition, i.e., D(y) = {0}, also Konishi [447]; for E = LI(R n) cf. Benilan and Crandall (Ref. Zh. Mat., 1982, IB645)). In connection with this example, cf. also [154, 219, 299, 349, 350, 444, 570]; the equation u t = AB(u) describes processes of nonlinear thermal conductivity, diffusion of gases, filtration in a porous medium. In the one-dimensional case Konishi [442] also considered the case of periodic boundary conditions; here (as also for the Dirichlet problem in the multidimensional case [447]) A admits extension to a single-valued dissipative operator in E = ( C 2 ~ [ - - ~ , w])* [respectively, E = C0(~)*], closed in E x Ew,. If 8 is an arbitrary maximal monotone graph in R with ~(0)~0, then the operator Au = AS(u) is m-dissipative and satisfies (55) in E = LI(R n) [Benilan and Crandall, Ref. Zh. Mat., 1982, IB645, for n --<-2 it is necessary that %For p = I complete dissipativity
2204
is only for single-valued
8.
061ntD (8)), and also in E = LI(~), if the boundary condition has the form ~(u)IF Barbu [200], and also 2.12.11~ here if D(8) ~ R, then instead of ~ = E one D(A)={uEE:u(x)ED(~) a.e.} and the semigroup generated by A acts on this set. considered the operator Au = (inu)xx in E = LI(R) which occurs in gas kinetics, that A is dissipative and R(I--%A)~D(A)=E+ for h > O. 8~
= 0 (of. will have Kurtz [455] and showed
Let 8 be a maximal monotone graph in R, R(8) = R and the operator A in E = W-I,2(~)
be defined by Au = bB(u) on D(A)={uEEND(~;L'):~(u)N~V"2(~2)~=| (i.e., the Laplacian is considered under the Dirichlet boundary condition). Then A is m-dissipative a n d - - A = O ~ (cf. 2.11.2~
where
~(tt)=lj(tt(x))dx
tional that 8 = ~j in R (cf. Barbu [200]). 9~
I~
for
ttEEN Lt, j(u)EL',
if D(8) = R, then'D(A)
We consider the differential
]:R-~(--=o, q-col
= E (Brezis
is such a func-
[229, 234]; cf. also
operator
Au=
~
(~01(U))xt--=div Q(u),
(56)
w h e r e ~hECI(R), ~;~(0)=0 . I n t h e s p a c e E = Lz(R n) = L 1 t h e r e e x i s t s a s i n g l e - v a l u e d d i s p e r s i v e o p e r a t o r A such t h a t f o r n E D ( A ) (56) h o l d s , CoI ( R ~ ) c D ( A ) C L 1IlL ~~ D ( A ) = L l, V L > 0 R ( I - LA)~LIN L*~ and (55) h o l d s w i t h p(u)=l!u~llq, l-.
tinuously
differentiable
i n a n e i g h b o r b o o d o f some p o i n t
= lai(x, U)Uxi [the If a i are con-
o~at,
( x 0 , r0) and f o r some i0--~--(x0, r0)v~0
[i.e., the operator is not linear], t h e n V~ER A E D i s s ( L 2 , ~ ) ; m o r e o v e r , AEDiss(L2,11"ll~-loc). This fact is also valid for systems [i.e., f o r tt(x)ERm)], i f i n one o f t h e m a t r i c e s a i t h e r e is a nonlinearity on t h e d i a g o n a l . On t h e o t h e r h a n d , i f L2-=-L2(~), Pit-r, .D(A)c{uEW1"P(~): ulc,=f(x)}, p > n , p>2, and f o r xEFII~t; ttED(A) Zv~(x) a,(?c, u(x))-.<0, t h e n AEDiss(L 2, II.ll~,.=-loc), and if eiai(x, r) is nondecreasing in r, where ~i = +1, then the restriction of A to the set {u:eiUxi ~< 0} is a dissipative operator in L 2. These results also carry over to systems, if the matrices ai are symmetric and O~z~;1O/~=O~z~lOrs (Khazan [146]). 11~ Let d;(x, r), y(x, r):[O, ] ] X R - + R for a.a. x, y be nondecreasing in r and
be measurable
in x for each r and continuous
Au ( x ) = - - {~; (x, u (x))x+ V (x, u Cx))}
in r (57)
on D(A)={~LI:@(~)=@(.,~(.))EW~'~(0, l),?(~)6L~,9(=)ix=0==}, where aER is given. Then: a) the restriction of A to the set {=EC[0, l]:@(u) ix=t=b} is a dispersive operator in L I, and if ~(I, r) is nondecreasing in r, then AIC[0,I] is dispersive in L ~ (this can be verified directly with the help of 2.12.4~ b) if ~(x, r) is nondecreasing in r for a.a. x and satisfies (53) with 8 = ~ [for ~ independent of x, (53) means that R(~) = R], and y(x, r) -- 0, then the operator A is m-dispersive and completely dispersive in L 1 [for ~(x, r) -= r the dispersivity follows from a), the equation R(I --A) = L ~ is obvious, and the complete dispersivity follows from 2.i0.13~ from this, by 2.12.11 = the complete dispersivity in general follows; finally, if /EL l and ~EW~'~(O, I) is a solution of the problem ~ q - @ ( X , ~ x ) = ~ q X
I/(;)d;, ~(0)----0, cf. example
13 ~ below,
then u = V x is a solution of the equation u-- Au = f,
0
so that A is m-dispersive
and satisfies
(55) with p(u) = ll~(u) ll=.
12 ~ . We consider the operator (57), where ~, y: [0, I ] X R - + R is continuous and nondecreasing in r for r > p, on the domain of definition f)(A)={uEC[O, I] : u ( 0 ) = ~ , u i s n o n d e c r e a s i n g ,
2205
~(u) 6Wi'| l)}. Let one of the following conditions hold: a) ~(x, r) = ~(r) is convex, y(x, r) = y(r), y(~) = 0; b) ~ is convex in r, continuously differentiable in x, strictly increasing in r, ~x(0, p) + y(0, ~)--<0, ~x(X, r) + y(x, r) is nonincreasing in x and nondecreasing in r. Then the operator A is dispersive in L x, L =, and L ~, and is an exponential generating operator, infinitesimal (in L =) and weak* infinitesimal (in L =) , which is order-preserving and contracting in L I, L 2, L = of a s emigroup U(Q:D-+(t~O), where D = { u ~ C [ 0 , l] :u(0)=St, u is nondecreasing}, while U(I)D(A)c D(A). In case a) moreover {/--ZA)-'(Wt'| l:= and (55) holds with p(u)=llu=II= , so that p(U(t)Uo)<--p(uo) (Khazan [142, 144, 145], except for the dispersivity in L x which follows from ii ~ 13 ~ . Let ~(x, r) : [0, ]] X R - + R be continuous and nondecreasing in r for a.a. x, measurable in x for all r, and satisfy (53) (with 8 = ~). We define an operator A in E = L=(0, I) or E -- C[0, I] [in the latter case we assume in addition that ~ is strictly increasing in r and ~(x, .)-Z(r) is continuous in the collection of variables] by (Au)(x) = --~(x, Ux) on D(A) = {u:u,'ux, AuEEI u(0)=0} . Then A is m-dispersive, and for E = C[O, I] also completely dissipative (for ~(x, r) --- r this is obvious and the general case follows from this with the help of a theorem of Ha [383], cf. 2.12.10~ We consider the Banach space El={u6~TH(0, l) : u(0) = 0} with norm IlulJ~,=l]uxl]l. Then, as is evident from 11 ~ , the restriction of A to the set {u:Au6E1} is a dissipative operator in El, so that, according to a theorem of Khazan [148] (cf. 2.8.6 ~), the semigroup generated by A gives a C1-solution in E for the problem ut +
r u=)=0 ( t ~ 0 , 0 ~ x ~ I ) , u(t, 0 ) = 0 , u(0, x)=uo(x) , i f uoOD(A; E), Auo6El In p a r t i c u l a r , i f ~ ( x , . ) - Z ( r ) i s c o n t i n u o u s , one can t a k e E = C[0, 1], and f o r u0OCl[0, l]. u0(0)-z0, ~ ( . , u=)OE, t h e i n i t i a l - b o u n d a r y problem i n d i c a t e d above f o r t h e Hamilton--Jacobi e q u a t i o n has a u n i q u e solution
u(3Ct(R+X[O, 1]).
14 ~ . Let ~:R § R be continuous, ~(0) = O. Then in the space E = L=(R) there exists a single-valued dispersive operator A such that V l > O R(1--ZA)=WL=(R)~D(A),VuED(A), Au = --~(Ux) EWt,=, D(A)=R(I--)~A)==BUC(R)-={uEL=(R):t~ is uniformly continuous}, and (55) holds with p(u)=ilUxll=; thus, A generates a semigroup U(t) on the set BUC(R), while U(t)(W'.=)C ~7,.=; if u0E]~v'.=(R) and (Uo)x@BUC(R)-~L'(R), then the function u(t, x) = (U(t)u0)(x) belongs to W ' . ~ ( R + x R ) and is a solution of the Cauchy problem for the equation ut + ~(Ux) = 0 (Aizawa [ 163]). 15 ~ Let ~:R n + R be a convex function of class C z, ~(0) = 0. In E = L=(R n) there exists a dispersive operator A, such that VttED(A)uE~/I.~(R~) , and Au = -~(7u), while R(I --
~.A)~D(A) ~Wa,=(R n) and D(A) = BUC(Rn) . u(x--h))/Ihl~
For px (u)=l] V=I[=
(55) h o l d s (Aizawa [ 1 6 4 ] , Burch [250];
and p=(u) = sup (u(x+h)--2u(x)-l-"
x.n~en
cf. also [642]).
16 ~ . L e t ~:R-+R be c o n t i n u o u s and s t r i c t l y i n c r e a s i n g , ~ ( 0 ) = 0 and ~ = ~ - ~ , so t h a t [ c f . (54)] and D ~ ) = R ( ~ ) 9 -We c o n s i d e r i n E = C[0, 1] t h e o p e r a t o r Au = ~(Uxx) w i t h D(A) = {uEC~[0, 1]: UxxED (P), ,,(u, (-. 1)'+~ux, uxx)lx=,=O, iE{0, 1}}, where ,,:Ra-+R, 4,(0, 0, 0 ) = 0 , ( r > s, r ~ > s , , r = > s = ) ~ ( r , r~, r=)>~(s, sz, s~). Examples of a d m i s s i b l e b o u n d a r y c o n d i t i o n s [ f o r x ~- iE{0, 1}): a) (--1)~UxEy~(u), where y~E~; b) (--1)~UxxEy~(Ux) , where y~E~ ; c) Uxx=f~((--1)~+=Ux, u) , where f i : R 2 § R i s n o n i n c r e a s i n g in b o t h a r g u m e n t s , f i ( 0 , 0) = 0. Then t h e o p e r a t o r A i s m - d i s p e r s i v e and c o m p l e t e l y d i s p e r s i v e . Under an a d d i t i o n a l c o n d i t i o n on ~i [which h o l d s in example b) f o r any y~fi~0~ , in example a) i f Yi:R + R i s l o c a l l y L i p s c h i t z , i n example c) i f f i ( r , rz) a r e l o c a l l y L i p s c h i t z i n r , c o n t i n u o u s l y d i f f e r e n t i a b l e i n r~, and 3 f i / ~ r z < 0] t h e ~E~
llull=+llAull=-.
restriction of the operator A to the set {g:AgE~#~.='(O, I), is a dissipative operator in El = Wz,=(0, I) with normHgl[e,=rnax{c[l#ll=, ll#xll=} , where c depends on r [in example b) c = I]. Consequently, the semigroup generated by A, gives a classical solution of the problem with the boundary conditions indicated above for the equation ep(ttt)~-ttxx. , if the initial condition uoED(A) and A#0EE~ (Khazan [148]; the result is also valid for ~, depending (continuously) on x). 17 ~ .
Let
~
, cf. (54), R(B) = R, and the operator A in E =L~(O, I)" (l..
be
defined by Au = (8(Ux))x on-D(A;E)={l~EfP't'~(O,l):uxED(~),~(Ux)NWLP(0,1)~=~} 9 Then A is mdispersive, and for p < = also completely dissipative (for p = I, for complete dissipativity one also needs 8 to be single-valued).
In E = L 2, moreover,
--A=O~P,
where
~(u)=S](Ux)dx
for ~tEW ''t, ](Ux)@L. ~, ~i=O] in R (L~ [469]); a generalization to 8 depending on x and to the Neuman problem is also possible (Attouch and Damlamian [173]). If 8 and 8-z are single-valued,
2206,
one can also consider nonlinear boundary conditions- if Au = (B(ux))x on D (A; E)={uEC' [0, I]: x~-iE{0, I}} , where 'Yi~0~ , then A is m-dispersive in E = I], and for E ~ L ~ also completely dispersive [the equality R(I -- A) = E follows from a result of M. I. Khazan, recounted in example 160: if ~C2[0, I]
ttxED(~), Au~E. (--l)tUx~y~(u) for LP(O, I) (l~p..
Jr
is a solution of the problem
~)--~(Vxx)=yf(~)d~ (O~<x~
x=i~{0, I} , then
0
u = v x is a solution of the equation u -- Au = f; the dispersivity follows from 2.12.7 ~ , the complete dispersivity in C from 2.12.1 ~ , and in L I from 2.14.3 ~ and 2.10.13~ For E = L ~, A is closed in E x Ew,. In the case of Neuman boundary conditions (Ux = 0 for x~{0, I}) or Dirichlet ones (u = 0 for xE{O,l}) the restriction of A to the set {u:AuEEl} , where El = WI,I(0, I) [respectively, E,=~~ I)) ] is a dissipative operator in El (this follows from example 7 ~ ), so that the semigroup generated by A gives a Cl-solution in E of the corresponding boundary problem for the equation ut = (B(Ux))x, if u0fiD(A, E), Au0~E l and B is singlevalued (for E = C[0, I] one gets a classical solution, cf. 2.8.6~ a strong solution exists
Vtto~D(A;L2i
even for multivalued B, cf. 2.14.1~
18 ~ Let ~{x, .)~-~ depend measurably on xEf] (cf. 2 ~) and satisfy (53), YE~. We define an operator A in E = L~176 by Au = B(x, Au) on D(A)={ttEI){Av;L~176 [i.e., we consider the boundary condition--Ou/dv~y(u) , cf. example 3~ Then A has m-dispersive restriction (which coincides with A if $ is single-valued), while the corresponding semigroup gives a w*Cl-solution in L ~ of the above-indicated boundary problem for the equation ut~~(x, Au) for any initial condition ttoED(A) (Benilan and Ha [220], Ha [383]; the case of the Dirichlet problem for a strictly increasing single-valued 8:11 § It which is independent of x, was considered earlier by Konishi [447]). If in addition R(y) = II, then the analogous result is valid for the operator Bu = 8(x, (~(Ux))X)inL~176 I) under Dirichlet or Neuman conditions [383], and if in addition y and y- are single-valued, then also under the boundary conditions -- (1)lUx~yt (U) for x=i~{O, I}, where y t ~ [this follows from example 17 ~ , 2.12.10 ~ and 2.14.4~ the precise description of D(B) is evident from 17~ In the case of single-valued 8(x, ") and y the restriction of the operator B under Dirichlet (Neuman) boundary conditions to the set {u:BuEEl}, where E,~---~'l'1(O,I) [respectively, EI=W~'I(O, I))] is a dissipative operator in El with norm IItz[[E.=iltt][ooq-ilttxHl[for y(r) - r this can be verified with the help of 2.12.7 ~ , and the general case follows from this with the help of 2.12.11~ so that the theorem on the existence of Cl-solutions from 2.8.6~ is applicable. 19 ~ .
Let
8, Y, YI, Y2E~
the operator A in E = LI(O,
[cf. (54)], R(y) = R, y and y-i be single-valued. I), defined byAtt=~xy(~-~(tt))=(y((~(tt))x))x
on
We consider
D(A)={ttEL*~
IntD(~), ~ ( u ) E C ' [ 0 , 1], AuEL~(O, 1), (--1)~(~(u))x~yl~(u)(x=i~{O, 1})}. T h e n f r o m 17 ~ a n d 2 . 1 2 . 1 1 ~ i t follows that A is dispersive, lP,(I--)~A)~D(A) f o r ~ > 0 a n d (55) h o l d s withp('tt)=I[tt• I f f o r i = 0 o r i = 1, D ( y ~ ) ~ / ~ ( ~ ) , t h e n a s s e r t i o n 2 . 8 . 4 ~ w i t h E0 = ( C [ 0 , 1 ] ) * * i s a p p l i c a b l e to the restriction of A to the set {u:IIuiloo--
We note other applications to equations containing operators of the form Au -- g(Yix.))xi with monotone Yi (Attouch and Damlamian [173], Vrabie [663]), to variational inequalitie~ (Barbu [200], Brezis [232-234], Tanabe [645]), to control theory (Barbu [197, 198, 200], Barbu and da Prato [201], Slemrod [627]), to infinite-dimensional equations of Hamilton-Jacobi--Bellman type (Benilan and Catte [218]) to second-order hyperbolic equations (Barbu [200], Brezis [230, 232, 234], Damlamian [325]), to the theory of elastoplasticity (Kuksin [67]). Many papers are devoted to applications to functional-differential" equations in Banach spaces, but we are not concerned with this theme. 2.14. Additional Results on the Existence of Strong Solutions for Certain Special Classes of Equations with m-Dissipative Operators. I~ Let E be Hilbert and A = - - ~ (cf. "2.11.2 ~ and examples 2.13.2~ ~ , 2.13.8 ~ , 2.13.17~ Then the Cauchy problem (6), (7) has a unique solution f o r a n y x~D (A), f~L~(O, T; E) w h i l e tl]/'-t-dtt/dtll~
du/dteL~(O, T; E)
and 2~(u(/))+~ll A ~
= 9 For x ~ D ( A ) . f ~ B V ([O, rl; e)
0
T) tt(l)6D(A), d+uldt=(Au(t)q-f(tdrO))~ t lld+u/dlll--
we h a v e : Vt~{0,
particular,
group U(t),
and
for
the
semi-
IIAoU(OxlI..
These results are due to Brezis
[229, 234].
Suppose further A = - - O ~ n U B
.
Then a strong
solution of the problem (6), (7) exists and is unique for x~D(~) ~D(B), f61V"I~,T;E) if B is m-dissipative and I n t D ( B ) N . D ( 0 ~ ) q = ~ (Brezis [234]), and for xED(~), f~L2(0, T;E) if B E D i s s ( E , ~ ) , D(B) is convex and contains D(A),[JBx][
be densely imbedded Hilbert spaces, B:V § V* be semicontinuous, (Bx--By, x--y) ~[[x--yl[~, ~>0. Then the Cauchy problem (6), (7), w h e r e A i s the restriction of B to {x:BxGE} , has a strong solution for any xEE , if f, trdf/ dtEL2(O, T; V*) for some r ) I, and then trdu/dl~L~(O, T;E) (Barbu [196, 200]). 3 ~ . If the operator A in E = L1(fl) satisfies the conditions of assertion 2.12.6 ~ , then the Cauchy problem (I) has a strong solution for any x6D(A) ; moreover, the infinitesimal operator A0 of the semigroup U(t), generated by A, is defined on D(A0) = D(A), and Vx6D(A) Aox6Ax, U,#~)xeD(A) and AoU(t)x is right continuous on [0, +=) (Le [469]; cf. examples 2 . 1 3 . 3 ~ 5 ~ , 17~ 4~ Let the operator A, which is m-dissipative in E = L~(fl), where lfll < =, be the restriction of some operator of the form - - 0 ~ , acting on L2(~). Then the Cauchy problem for the equation du/dt6~Au , where the operator BA is described in 2.12.10 ~ , has a w*CZ-solution in L=(~) for any initial condition from D(BA), if one of the conditions a), d), given in 2.12.10 ~ holds [condition b) holds due to 2.12.5~ This result and its applications were found by Benilan and Ha [220, 383]; cf. 2.13.18 ~ . 2.15. Perturbation of m-Dissipative Opera~ors. In this paragraph A is an m-dissipative operator, B is a dissipative operator in the Banach space E. Below we give conditions 1 ~ ~ , each of which guarantees that the operator A + B is m-dissipative. I~ B:E § E is continuous (for linear A -- Webb [295] for an intelligible proof). 2~
B:D(A) § E is continuous
and completely
[667],
in general -- Barbu [195];
dissipative
(Kobayashi
[427], Pierre
cf. [576]).
3~ One of the operators A, B is completely dissipative, D(B) ~ D(A), B is singlevalued, ][Bx]]~.<~(I]x ][,;]IAx[]l) , where ~ is nondecreasing in both arguments and Vr ~ 0 lirasup ~ (r,
a)/a
the operator B(I -- %A) -I is compact for X > 0 (Khazan [143]; the result and proof remain valid if B = BI + B2, B I ( I - XA) -I is compact,[lB=x--B2ylI~b(r) IIx--yllq-al]IAx--AylI] for !IxII, IlyIl~r, where a < I). E is Hilbert, A and B are single-valued, A 0 = B O = O , D(B)~O(A), i]Bx--ByH
y]i-}-aIIAx--Ay]I ,
5 ~ . E* is uniformly convex, D(B) ~ D(A), B is m-dissipative, and Vx~D find a < I and b, r > 0 such that IIIBxlll~alllAxlll+b for x~D(A), llx--xoll~
(A)
6~ E* is uniformly convex, D (A)~ D (B)=/=O,( Ax, Bex ), > 0 for any B e is the Yoshida regularization, cf. Paragraph 2.3 (Barbu [200]).
(A), s > 0 , where
xED
one can (Crandall and
7 ~ . E* is uniformly convex, B is linear and m-dissipative, D(B) ~ D(A) and
D (A -k- B) = D
(A) fl D (B).
c f . 2 . 1 2 . 1 0 ~ , 2 . 1 2 . 1 1 ~ , 2 . 1 4 . 1 ~ and [ 1 9 1 , 194, 200, 2 2 8 , 2 3 4 - 2 3 6 , 2 4 0 , 241, 2 4 4 , 2 5 6 , 257, 3 5 8 , 4 2 8 , 4 3 7 , 668] f o r o t h e r t h e o r e m s on p e r t u r b a t i o n s . We n o t e t h a t i n a s s e r t i o n s 2 ~ and 3 ~ one c a n w a i v e t h e c o m p l e t e d i s s i p a t i v e n e s s and e v e n t h e d i s s i p a t i v e n e s s o f B: o n l y the dissipativeness o f A + B i n 2 ~ and t h e d i s s i p a t i v e n e s s of A+sB (1/2~e~l) in 3 ~ are necessary. 2.16. E q u a t i o n w i t h C o n s t a n t O p e r a t o r o f C l a s s D i s s (E, p ( . ) - l o c ) . I f A E D i s s ( E , p(.)-loc) [ c f . ( 5 ) ] and t h e Cauchy p r o b l e m ( 6 ) , (7) a d m i t s i m p l i c i t d i f f e r e n c e approximation (cf. Parag r a p h 2 . 2 ) w i t h ~n.~Zr~-{z:p(z)~r} , w h e r e r < =, t h e n o n e c a n r e p l a c e A b y AIzr~Dlss(E, ~(r)),
2208
so that a p-bounded LD-solution of the problem (6), (7) exists and is unique, and the results of Paragraphs 2.6-2.8 are applicable. If the condition indicated holds for the problem (6), (7) with f = 0 for any xED(A) ND(p), T > 0 [here D(p)={z:p(z)< ~ ] , then the Cauchy problem (I) has a unique LD-solution V x E D = U (D (A) NZr) , and the map x + u(t) defines a semigroup r>0
of operators U(t):D + D, satisfying conditions (3), (4) (Chambers and Oharu [265], Konishi [445] for p(x) = llxll, Oharu [106] in general; for p(x) = [[X]lEl, where E t c E , this was noted independently by Kutuzov [71, 73]; in [265, 446, 71, 73] the concept of LD-solution is missing and only the approximation (9) with fn.k = 0 is considered). If p(x) ~ [[xll, then in general D is not closed and the semigroup U(t~ does not extend to D; however in this case too, we can call AID(p ) its pseudogenerating operator. The results of 2.10.1~ ~ on semigroups of type m generalize to semigroups with conditions (3), (4). For example, for the infinitesimal operator A0 of such a semigroup, one will have AO[zrEDiss(E, o(r)) , if ~(t, r)$r as t + 0. If AEDIss(E, p(.)-loc) satisfies (21) [with llxll replaced by p(x), F = {0}] and (44), then AID(p ) is the pseudogenerating operator of a semigroup for which (3) and (4) hold, since (44) guarantees the a priori estimate of p(un, k) for the difference approximation (9) with fn,k = 0, and (21) guarantees the existence of such an approximation. 2 ~ . Konishi [444, 445] applied the results cited above to prove the existence of nonnegative solutions of initial boundary problems for systems of the form u t = Au + Euv, v t = --euv with ~ = ml (E = L = x L =, p is the norm in E) and more general systems of the type of (Ui)t = A~l (U~)-- UiVx+ ( - - 1)tUt~ 2, (~l)t = (-- ~lUl~l -- ua_lvi (i ~- I, 2), w h e r e B~ :R + R a r e c o n t i n u o u s and strictly i n c r e a s i n g [E = L 1 x L z x L x L =, p i s t h e norm i n ( L = ) ~ ] . Analysis of these exa m p l e s shows that it is reasonable to generalize condition (44) as follows: p(u)=max{p~(u): 0 < k 6 m}, where p0 satisfies (44), and for I ~ k ~ m and pi(z) 6 r ('0 6 i 6 k -- I) one has Pk ( ( I - XA)-lz) -.< (1 - - h ~ k (r))-l(Pk (z) + X~q, (r)). 3 ~ . Now we c o n s i d e r t h e i n h o m o g e n e o u s e q u a t i o n . L e t f(t)EF a . e . , fELl(0, T; E) , and the functional p satisfy the condition p(x+y)..
[with ]]xll replaced by p(x)] and (44), and ~q(f(t))dt
is finite, then there exists a unique
0
p-bounded LD-solution of the problem (6), (7); if in addition p is lower semicontinuous, then
/
'
}
p (u(0)-.<e ~176p(x)--k ~q(f(~))e-*'~
(Khazan [ 1 5 3 1 ) .
0
2.17. Equation with Variable sider the Cauchy problem
Operator
A(t)EDtss(E,m)
or A(0EDtss(E, p(.)-loc).t ~
We c o n -
du (t)/dtEA (t)u (t) (s..< t ..< T), u ( s ) = x , where 0 ~< s ~< T.
(58)
Let the following condition hold:
(A.I) There exists a functional p:E § [0, +=], such that if s,
tel0, T],
x@D(A(s)),
. yED (A (t)), p (x), p (y) -.
( A (s) x - - A (t) y, x - - y ) _-.< r (r)ll x - VII~+ , (r, "a)l h ( s ) - tt (t)l I['x- V II, w h e r e h~L x (0, T; H), (H, [ 9 [)-+
Banach s p a c e .
C o n d i t i o n ( A . 1 ) [ i t f o l l o w s f r o m i t t h a t A(t)EDiss(E, p ( . ) - l o c ) ] i s e q u i v a l e n t ( c f . [ 1 5 4 ] ) w i t h (45) w i t h 6 = 0 , 8 = ~ ( r , a)l h (s) - - h (t)[ f o r A = A ( s ) , B = A ( t ) , z=x--XA(s)x, ~=y--~.A(t)y.. It o b v i o u s l y h o l d s i f A(t)EDiss (E, p ( . ) - l o c ) , D ( A ( t ) ) = D a n d V x E D [[A(t)x--A(s)xl[..
{z:p(z)~
c > 0 o n e c a n f i n d T = x ( r , c) > 0 s u c h t h a t f o r O..<s-.
here
I?,(I--XA(t))~D(A(s))N
,(r, [[[A(s)z][[..
2209
(A.4) Either (A.4.1) ~(r, a) in condition (A.I) is independent of a, and the initial
c o n d i t i o n xO.O~= U l) (A (s)) N {z:p(z) ~< r}, o r (A. 4 . 2 ) r>0
Ul~ (A(s)J{z,p(z)<,})
r>O
in condition
(A. 1) hO.BV ([0, T]; I-l), x ~ b s =
+~
and
Idal(a+2,(ro,
a)>varn[s, r l,
where a o = ]
A(s){z,p(,)
and ro is d e t e r -
ae
mined by r, to correspond to x by (44). Then the Cauchy problem (58) has a unique p-bounded LD-solution, while under (A.4.1) and under (A.4.2) u([)6~, (s~t~T), u6Lip([s, T]; E) and there exists an implicit difference approximation with p(u,,h)-}-IllA (t~,k)U,,hIII ~const (Khazan [ 153] ; earlier analogous results were found by: Crandall and Pazy [294] for continuous h, ~(r) = const, p(x)=l[xI[, @(r, a) =@(r) (l+a) in case (A.4.2); Evans [348] for m-dissipative A(t) with constant (as in [294] also) ff)(A(t)), p(x)=IlxIl, @(r, a) =~(r)(l+a) ; Oharu [106] for continuous h and (A.4.1); Pavel [559] for p(x) = llxll and (A.4.1)). In [153] it is shown that under conditions (A.I)-(A.4) for p-bounded LD-solutions, analogs of the comparison theorems (37) and (41) are valid [in (41) ~ and ~ are replaced by the L1-norms of the corresponding functions], from which, in particular, it follows that for a solution of the problem (58) one has
u(t)ODt (s~t~T),
Iiu(t+h)--V,(h)u(t)ll=o(h)
a.e. on (0,
T)
(59)
where Ut(h):Dt--+Dt(h>~O ) is the semigroup generated by the operator A(t)ID(p) (cf. Paragraph 2.16); under the restrictions indicated above, the estimate (59) is also found in [348] and [559]. From (59) and 2.10.2 ~ , one derives by the method used for the equation with constant operator in [246, 289, 511] [using (A.3)], that if Vt, r the operator A(Ol(z,p(z)
s):Os-+/)t ( 0 < s ~< I ~ T), if
Sdal(C~+@(r,
(~))=oo V r > O ;
now if the latter condition does not
1
hold, we get a local evolution system, since h can, without loss of generality, be assumed to be right continuous and (A.4.2) for any xEDs will hold for small T -- s. If the function h is t
continuous or has bounded variation, then
U(t, S)X=~I(I:d~A(T))-IX
[cf. (23)].
$
2 ~ . If E is Hilbert and --A(t)---~O~t are subdifferentials of convex functionals opt:E-+ (--~, +~] (cf. 2.11.2 ~ and examples 2.13.2~ ~ , 8 ~ , 17~ the results recounted above can be improved. This case was studied in [170, 172, 231, 234, 413, 502, 571, 664, 685, 690]; the most general theorem is due to Yotsutani [690], who showed that the problem dtt(t)/dt+a~pt(tt(t)) ~
f(1) (O~
has a unique strong solution for any XED(~o), fEL2(0, T; E), if there =fi[O, 1] and o p e r a t o r s Qt,:D(q~t)--+DFp~) (t, sE[0, T]), s u c h t h a t f o r H x J l < r IIQ,.~--xl[ < Jg (s)-- g (.t) I (~, (x) + M (r))? and ~s (Qt, x) ~<~t (x) + J'h (s) -- h (t) I (~t (x) + M (r)), where gEW'.P (0,. T) and h~.BV[O, T] can d e p e n d on r , w h i l e p = 2, i f 0 ~< a ~< 1 / 2 ; p = 1/(1 -- ~ ) , i f 1/2 < ~ < 1; p = ~, if ~ = I. The regularity of the solution is the same as in the case of ~ not depending on t (cf. 2.14. I~ Applications to variational inequalities and the Navier--Stokes equation in noncylindrical domains were given by Brezis [233, 234], Kenmochi [414, 415], Otani and Yamada [551]. We note in addition Moro [518, 519], in which ~Pf~-I~c(t) are the indicator functions of convex sets K[t)cE (of. 2.11.2~ here the restrictions on the dependence of ~p~ on t are weaker.
exist
3 ~ . Martin's theorem from 2.10.12 ~ generalizes to the nonautonomous case: if /)cE is closed and A(t)x:[0, T] x D + E is continuous and for each t, A = A(t) satisfies 2.10.12 ~ b), then Vx~D there exists a unique C~-solution of the problem (58) (Kenmochi and Takahashi [417]); there is further generalization in Martin [500] and Schechter'[615]: Carathdodory 2210
condition instead of continuity of A(t)x, o=o(t)~L t (0, T) ; local closedness instead of closedness of D (in the latter case the solution exists locally). 2.18. Equation du(t)/dtEA(t, [u] (Q) u (t). under the Condition That A(t, w) Are Pseudogenerating Operators. Here we expound results of Khazan [149, 153]. Suppose for any t~[O, r~ T], w ~ c 0 {~:[0, S]-+E} there is given an operator A(t, w) in E. For brevity we assume that D(A(t, w)) = D t is independent of w and P[/~[JWr,t, where Wr,t={~:[0, t]-+E:~(s)6O*, p(~v(s))..
II A ( t . . , )
O-.
x - - A (t. ~.) x I1--<e Cr, a) la ( t , ) - h (t=) J+ . , (r, a)II ~ C t , ) - * (t~)II + ~ Cr, a) II * , - * ~
one has
II ~=~0 ,,;~,.
( B . 3 ) E i t h e r ( B . 3 . 1 ) i n (A. 1) and ( B . 2 ) t h e f u n c t i o n s ~, ~, ~1, ~2 a r e i n d e p e n d e n t o f a , and i f h~C(i0, 7"]; ]-/)OBV([0, T];I'I) , t h e n ~l = 0; o r ( B . 3 . 2 ) h6BV([O, T]; ]-]). Then t h e Cauchy p r o b l e m
du(t)/dt~A(t,[u](t))u(t) where [u](t)
is the restriction
To > 0 f o r any
of u to [0, t],
xEWNDo, D 0 ' = U D~
C0
u(0)=~,
has a unique p-bounded LD-solution under condition
(B.3.1)
r>O
f o r some
and a u n i q u e L i p s c h i t z
f)(A(O,X)IT,:o~z>~})~
p-bounded LD-solution for some To > 0 for any xE[.~'Nm0, where D0-----0 r>0
DoND
(p), under (B.3.2), if h is continuous at zero; in the latter case this solution is strong, if E is reflexive. Under (B.3.1), To depends on r, where x~.Do~{z:p(zi..
{z:p(z)..
c a s e one can t a k e To = T, i f
g drl(ro~o(r)+lH(r))----oo ,
and i n t h e
1
second,
if
in addition
~0(r, a)
is independent of a and
~dal(a+a~A(r, a)hu~;( r, a ) ) = oo V r > 0 1
(there are no restrictions on ~2), here the continuity of h at zero is not needed. The determination of an LD-solution in this case is modified as follows: in (9), Au k is replaced by A(tk_l, [U~](l~_t))=~, where =~(1)==~_, on [tk-l, tk) , i.e., the scheme is no longer purely implicit; an LD-solution is dalled Lipschitz if it can be obtained from a difference approximation with bounded difference quotients. The differentiability of an LD-solution in a nonreflexive E can be studied with the help of assertions of type 2.8.3o-2.8.6 ~ (cf. 2.19.1 ~ , 4~ 2.19. Existence Theorems Based on Approximation of Operators and (or) Topological Properties of the Space. Here we recount results of [142, 144, 151]. I~ We consider the Cauchy problem (58) in a locally convex (to include the case of weak approximation) space El, imbedded in E0. Let the operators A(t) and their sequential closures A(t) in El x E0, be single-valued, and An(t) be a sequence of operators in El, approximating A(t) in the following sense: for a.a. t~[0, ~] one has a)
[O,T]~)tn-+t,D(An(t.))gxn-+x
in El,
Here n § ~, possibly on some subsequence. problem
(~,~ -- u., ._,) / (t..~"
An(tn)Xn'-+Z
in
Eo)~(x6.D(fi~(t)),~4(t)x=z).
We approximate the problem (58) by the difference t..~-0 = A.,,z~.. + f~,,.
=..o = x~,
where ~q~n-----{In,~} is a sequence of partitions of [s, T] with I..aa.I-->-O, An,~--=An(Sn,~), Zn,~=OlZn,a+ (]-- 0) =.,k--i, l.,~_,-.<S.,a-.<~.,~,O-.
d) fn § 0 in Ll(s, T; El), x n + x in El. Then for some subsequence n + ~ in Ex there exists the uniform l i m i t l i m u n ( t ) = u(t) on Is, T], where u~Lip([s, T]; Et) flCl([s, T]--Z, E0)~ where E is a set of measure 0, on which condition a) fails, and du(O/dt=~(t)u(t) (s~t~T, t~) (cf. [142, 144] for An(t) = A(t), Zn, k = Un,k, 0 = I, fn,k = 0 [151] in general; the inhomogeneous equation with f~Lt(0, Ti El), is also considered there, cf. 2.8.4~ 2*. Let the Banach space E be reflexive or dual to a separable Banach space, and let El = Eo = Ew*. The conditions b), c) of assertion I ~ follow from just the one boundedness of the set {An,kZn,k}. From this, for example, there follows the following theorem [142]. If the operators A(t) in E are single-valued, the set uR(A(t)) is bounded, condition a) holds with An(t) = A(t), El = E0 = Ew*, and for some sequence T + 0 for Tn ~ t ~ T one has R(I -~.A (t))~D (A (t'--~.)), t h e n t h e Cauchy p r o b l e m (58) h a s a s o l u t i o n f o r a n y s6[0, T), xED(A (s)). I n t h e h y p o t h e s e s o f t h i s t h e o r e m D ( A ( t ) ) can d e p e n d on t , c f . t h e e x a m p l e i n [ ] 4 2 ] . 3 ~ . U n d e r t h e same c o n d i t i o n on E as i n 2 ~ , i f t h e o p e r a t o r s A ( t ) m a t i o n by o p e r a t o r s A n ( t ) , w h i c h , u n i f o r m l y w i t h r e s p e c t t o n , s a t i s f y g r a p h 2 . 1 7 . 1 ~ t h e n t h e p r o b l e m (58) h a s a w * C l - s o l u t i o n ; t h e a n a l o g o u s the situation of Paragraph 2.18 [t51]. We f o r m u l a t e t h e r e s u l t f o r an operator. Let condition a) of I ~ hold with A(t) = A, An(t) = An, El = Diss (E, m) and for some sequence Tn = Xn(r) + 0
a d m i t weak* a p p r o x i t h e c o n d i t i o n s of P a r a assertion is valid in equation with constant E0 = Ew*, while An 6
R ( I - - r.An)~D (A.) 0 {z:llzll + ItA.zll--< r}
(60)
for each r > 0. If for a given x one can find a sequence x n with D(A,)gx,-+x in Ew*, IIAn • Xnll ~ c, then there exists a w*Cl-solution of the problem (I). Here, possibly, AqD|ss(E,'~). This theorem is convenient in that if D(A n) = D n are closed linear subspaces in E and An: Dn § Dn are continuous, then (60) holds automatically by 2.11.3 ~ In [151] the case is also considered when the approximating operators do not belong to Diss (E, m), and this condition holds asymptotically. 4~ Let E be arbitrary, AEDiss (E,p(.)-loc) , and let there exist a sequence of singlevalued operators A,6Diss(E,p(.)-loc), which, uniformly with respect to n, satisfy (44), and also a sequence of operators Qn:D(An) § D(A) such that for {~II§247 one has IIQ.x-xll<6.(r)-+O, I]IAQnx--AnxIII~<8.(r)-+O and R ( I - - x . A n ) g x f o r some s e q u e n c e Tn = ~ n ( r ) § 0. If in addition Vx~D (A) one can find a sequence x n with D (A.))gxn-+x, p(x.)+llA.x.ll..
Operator,
Perturbed by a Nondissipative
Operator.
du (t) I dt6A u (t) + (B~(t) ~..< t ~
We
(61 )
where A is m - d i s s i p a t i v e , B:D ( B ) c C ( [ 0 , T]; E)-~LX(0, T; E). The f u n c t i o n u i s c a l l e d an LDsolution of this problem if ufD(B) and u is an LD-solution of the problem (6), (7) with f(t) = (Bu)(t); a strong solution is defined analogously. The map f + u for the problem (6), (7) for any xED(A) is defined on LI(0, T; E) (cf. Paragraph 2.6); Benilan's comparison theorem [cf. (37)] and the theorem of Vrabie [660] and Baras [188] on the compactness of the map f § u from Lq(0, T; E) (q > ]) to C([0, T]; E) under the condition that the operators U(t) (t > 0) of the semigroup generated by A are compact, allow us to use fixed point theorems to prov 9 the solvability of the problem (61). The following results are obtained in this way. ~ i 1~
I f D(B) = C ( [ 0 ,
r];
D(A)) and
.~l](Bu - 0
is the restriction of u to [0, s], =ELI~,T) xED(A) (Crandall and Nohel [291]).
B~)(s)Hds -.
, then (61) has a unique LD-solution
for any
2 ~ . Let operators U(t) (cf. above) be compact for (t > 0) [sufficient condition: E is Hilbert, A satisfies the assumptions of 2.14.1 ~ or 2.14.2 ~ and (I -- A) -I is compact]. If B:C([0, T]; D) § L~(0, T; E) is continuous, where D e E is open, then problem (61) has at least one local LD-solution for any xED(A)OD (Vrabie [660]). The analogous result in the
2212
case of continuous and bounded B:[0, T) x D + E, where D c E is locally closed, is obtained in [661, 662]. A typical application is the problem with nonlinear boundary condition (cf. 2.13.3 ~) for the equation u t = Au + f(t, x, u) with continuous f. In [663] the case of Hilbert E and A satisfying the assumptions of 2.14.2 ~ is considered for continuous B:LP(0, t; V) -> Lq(0, t; E), p /> 2, I/p + I/q = I, 0 < t ~< T, which permitted the consideration of the nonlinearity f(t, x, u, Vu) in the example mentioned. 3~ Koi and Watanabe [432] proved the local strong solvability of the problem du(t)/ dt +O~(U(t))--O1p(u(t))~)f(t),u(O)=x in Hilbert E for any x60(9), /6_L2(0,7"; E) under the condition that D (~)cD(#) and that any bounded set scm(~) on which the functional ~ - - $ is bounded, is relatively compact in E, and (3~0)0 is bounded on S. In the presence of an a priori estimate ~(u
+ E be such
that
p(z)..
and
IlBz--Bvll-.<,(r, a)liz-vl[
for
p(z),
p(v)
[][Az[[[,[[[Avlll..
uniformly convex, then a) is also equivalent with the condition (AOx, X--PDX> +~<~IIX--POXll2 in case of convex D (Martin [501]). Cf. also 2.10.12 ~ . If E is Hilbert, A is m-dissipative, CcE is closed and convex, and PD-7~CcC, then condition a) for D = C n D ( A ) is equivalent with the fact that
([-iLA)-ICcC
[245].
Cf. also [578, 579, 652].
2.23. Periodic Solutions. Let E be Hilbert and A = - - 0 ~ (cf. 2.11.2~ For the existence of a T-periodic solution of (6) with /G/2(O, T;E) it is necessary that the mean value of f on (0, T) belong to the closure of R(A), and it is sufficient that it belong to the interior of R(A) (Haraux [385, 386]). If /6L~oc(R.;E) is T-periodic and the equation du(t)/ dt +c)~P(u(t))s (t>0) has at least one T-periodic solution v, then for any solution u of this equation one can find a T-periodic solution w of it such that u(t) -- w(t) -> 0 weakly as t -~ (if the sets of the form {x:IIxll-{-q~(x)-.
Behavior of Solutions at Infinity.
Let A be a dissipative operator in the Banach
space E, R(I--~,A)DD(A) (0<~<=), and U(t) be the semigroup generated by it on D(A) (cf. 2.10.4~ and F = A-10 be the set of its fixed points (possibly empty). By m(x) and mw(X) we denote, respectively, the set of limit points, and weak limit points of U(t)x as t + =. I ~ . If m(x) x @, then U(t):m(x) § m(x) is a semigroup of isometries, admitting extension to a strongly continuous group on re(x), and ,,~(x) coincides with the closure of the orbit {U (t)a:t >/0} of any point a ~ ( x ) . If F ~ @, then Vz6F the set re(x) lies on the sphere with center z and radius r-.
2213
of affine operators (in Hilbert E, isometrics) on the closed linear span of m(x), which coincides with U(t) on cony m(x) = Cx. If Cx is weakly compact (for example, if F ~ @ and E is t
reflexive), then C x N F = / = O . If (Dafermos and Slemrod [324]).
CxNF=/=O,
then
CxNF~{a}.,
where a = l i m t - I ~ U (~) za~ t~ 0
vz~C.~
2 ~ . Let E be Hilbert and A be m-dissipative. If the orbit of the point x is relatively compact, then there exists a single-valued almost-periodic function v:R § ~(x), which is a weak [and for x~D(A) a strong] solution of the equation du(t)/dt~Au(t) on R, such that IIu(t)--v(t)II-~0 as t + = (Dafermos [323]; cf. also Haraux [386]). For any x~D(A) we have:
~(x)cD(A);
if F = ~, then
VxED(A)
for
zE~(x) ND(A)
one will have
z>
0 [386].
t
If F ~ ~, then VI
there exists the weak limit
~-|imt-i~U(x)xd~=Px~F t~
(ergodic theorem
of
0
Baillon and Brezis [181]); conversely, if Px exists for at least one x, then F = ~ and P: D(A) + F is a nonexpanding retraction (Hirano and Takahashi [395]). Generalizations and refinements of the theorem of Baillon and Brezis are given in [238, 239, 395, 568]. For x6 D(A-----)the following conditions are equivalent: a) w-|imUit)x exists; b) F = @ and ~),(X)CF ; c) F ~ ~ and V h > 0 U(t+h)x--U(t)x-~O weakly as t + ~ (Pazy [565], Sch~nberg [616]; cf. [182, 394] for a partial generalization to Banach E [565, 569, 580] for other results on weak convergence). Under condition a)~-limU(t)x=limPU(t)x [565]. For the existence of lira(t)x as t §
it is necessary that
O~=~(x)cF
, and sufficient that ~(x)nF=/=~
[565].
3 ~ . Let E be Hilbert, A-----O~ and F ~ @, i.e., ~ achieves its minimum. Then Vx6D(A) there exists a ~.limU(t)x~F; if ~ is an even functional or the sets E,={x:!Ixll+~(x)~r } are relatively compact V r > 0 , then the convergence is strong (Bruck [248]). Okochi [549] gave the generalization: if ~ is symmetric with respect to the closed affine subspace I / ~ E , and the projections of the sets Er on V are relatively compact, then one has strong convergence of all trajectories; for arbitrary ~ this is not so (Baillon [177]). 4 ~ . Let E and E* be uniformly convex, A be m-dissipative, F ~ @ and P:E § F be defined by [Ix -- PxI[ = d(x, F). In order that
Vx6D(A) it
suffices a) b) c)
that
any of t h e f o l l o w i n g
conditions
Slim
U (t)xEF
(62)
hold:
.~ofiD(A)--F; (.Ax, x--Px)+>~.(r).ilx--Pxll ~ f o r Ilxll+HlAxlll
A) - x
is
c o m p a c t and
Px>+ > 0 f o r
> 0, o > 0;
Under condition b), the convergence in (62) occurs exponentially, if ~ = I, in finite time, if ~ < I, with rate O(t-r), r = I/2(~ -- I), if ~ > I. In the Hilbert case these results and their applications are in Pazy [567], the generalization to Banach E was made by Nevanlinna and Reich [533]. Cf. [182, 373, 431, 566, 569, 646] for other theorems on strong convergence. 5~
If the norm in E* is Frechet differentiable, then
Vx~D
(A) llm U(t)x/t-~-v
,
where
v is an element of least norm in R(A) (Miyadera [514], Reich [592, 593]). Some of the results cited can be generalized to the equation du(O/dteAu(t)+[(t) (t~O) [200; 234, 401, 515, 567]. There is another type of theorem connected with the asymptotic behavior of nonlinear semigroups in Alikakos and Rostamian [165, 166], Brezis and Browder [237], Crandall and Pierre [299], Diaz [339], Kartsatos and Taro [404], Turinici [652], Webb [670], Wexler [678]. In Pazy [570] there is constructed an analog of the Lyapunov theory for investigating the asymptotic behavior of nonlinear contracting semigroups in an arbitrary Banach space; the results obtained are illustrated by the example of the semigroup generated by an equation of type u t = A(lulm-lu) in LX(~) (cf. with [299]). here.
2214
In the same paper there are also many other results, which it is impossible to elucidate This is also true of Ball [186], where Lyapunov's method is applied to not necessarily
contracting nonlinear semigroups and evolution systems. tions.
Cf. [258] also on Lyapunov func-
2.25. Attractors. Let U(t):E § E be a semigroup (not necessarily contracting) of nonlinear operators in the Banach space E. The set ~ is called a maximal attractor [95] of the semigroup U(t), if U ( t ) ~ = ~ and Vr>~ llm d(U it)x, ~ ) = 0 uniformly with respect to llxll ~ r. Babin and Vishik [6-8] showed that if the operators U(t) are compact for t > 0 and there exist c:R§ c06R+, satisfying the conditions: llU(t)xlf
exceed [2n In ([/V 2n/(I- b~)l+ l)/In(2/(I-l-62))I9 2.26. Equations, Unsolved with Respect to the Derivative. Arai (RZhMat, 1979, 12B907) proved the global strong solvability of the problem Bu'(t)+Au(t)~f(t) (O~
7.
8. 9. 10. 11. 12.
13. 14.
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