Differential eq in Banach space

315. W.-S. Young, "Weighted norm inequalities for multipliers," in: Harmonic Anal. in Euclidean Spaces. Symp. Pure Math. Am. Math. Soc., Williamstown, Mass., 1978, Part I, Providence, R.I. (1979), pp. 133-139. 316. W.-S. Young, "Weighted norm inequalities for the Hardy--Littlewood maximal functions," Proc. Am. Math. Soc., 85, No. I, 24-26 (1982). 317. M. Zinsmeister, "Courbes de Jordan verifiant une condition corde-arc," Ann. Inst. Fourier, 3_/2,No. 2, 13-21 (1982). 318. A. Zygmund, "On certain len~as of Marcinkiewicz and Carleson," J. Appr. Theory, ~, No. 3, 249-257 (1969).

DIFFERENTIAL EQUATIONS IN A BANACH SPACE S. G. Krein and M. I. Khazan

UDC 517.986.7;517.983.5

A systematic survey of the theory of linear evolution equations in Banach spaces, reviewed in the period 1968-1982 in Ref. Zh. Matematika, is presented.

INTRODUCTION The present survey reflects the basic results (and typical applications) of the theory of linear and nonlinear evolution equations in Banach spaces, contained in papers reviewed in Ref. Zh. Matematika in 1968-1982 (for completeness some earlier results not elucidated in monographs and surveys and also some later results known to the authors are included in the account). By an evolution equation we mean a differential equation with respect to a function u(t) with values in a Banach space E, where the domains of definition and the sets of values of the operators which appear in the equation lie in E; the real variable t plays the role of time. We exclude from consideration ordinary differential equations, i.e., equations with continuous operators, defined on the whole space or on a set with nonempty interior. Also we do not recount results which relate to equations in locally convex spaces, if they give nothing new in the Banach case. There are two basic approaches to evolution equations. The first is connected with the theory of semigroups of operators (linear and nonlinear), and the second with the theory of monotone operators and variational inequalities (here, in contrast with the semigroup approach 9 one considers equations with operators, acting from a Banach space to its dual). In the present survey we do not touch on results obtained in the framework of the second approach. We refer the reader to the monographs of Gaevskii, Gr~ger, and Zaharias, Glovinski, Lions and Tremoller, Duveau and Lions, Lions [27, 29, 42, 88], which have been published in Russian, and also to the survey of Dubinskii [36]. Of later papers in this direction we note the cycle of papers by Belov, Savvateev, and Yanenko [16-22, 121], Pankov [107, 108], Campanato (cf., e.g., [259]), Naumann [524-526]. As to the semigroup theory of evolution equations, after the book of Krein [63] (whose continuation is the linear part of the present survey), there was no systematic and sufficiently complete account in Russian, although the number of papers both on the theory itself and on its applications to partial differential equations, functional-differential and integrodifferential equations, mathematical physics, mathematical biology, controlled systems, etc. grows continuously. A brief but successful introduction to the theory and applications of linear semigroups is in the book of Richtmayer [119]; there are more details in the books of Yoshida, Kato, Midzokhata [55, 58, 97]. A beautiful and modern account of many aspects of the theory of linear and semilinear evolutions is in the monograph of Reade and Simon [115-118], but results obtained after 1978 are not reflected in it and such themes as semigroups with singularities at zero, semigroup-distributions, behavior of solutions as t + | equations of elliptic type, equations of higher order, nonlinear semigroups and evolution equations, etc. are not touched on at all. Nonlinear semigroups were considered in the Translated from Itogi N a u k i i Tekhniki, Seriya Matematicheskii Analiz, Vol. 21,, pp. 130-264, 1983.

2154

0090-4104/85/3003-2154509.50

@ 1985 Plenum Publishing Corporation

survey of Dubinskii [36], where results obtained up to 1970 are recounted. Finally, we recommend to the reader Chaps. 8A and 9 of the book of Marsden and McCracken [95], where there is a rather modern account (although most of the results have been strengthened at the present time) of the theory of linear nonautonomous evolution equations, and also semilinear equations of the first order and tests for smoothness of nonlinear semigroups. When necessary we shall refer to the books mentioned above. L Our survey is divided into two parts. In Part I we consider linear evolution equations of the first order with constant operator, and also first-order nonlinear evolution equations. In Part II, which will be published in one of the following issues of this series, we shall consider: linear nonautonomous, semilinear and quasilinear first-order equations; cosine functions of operators and hyperbolic second-order evolution equations (linear, semilinear, and quasilinear); linear and nonlinear second-order elliptic equations; higher-order equations. Both in Part I and Part II we study basically the following questions: existence, uniqueness, and regularity of solutions; their dependence on initial conditions, right sides, parameters and operators which appear in the equations; behavior as t § ~; some methods of approximation of solutions (theorems of Trottem-Kato type, multiplicative formulas, etc., and also implicit difference approximation, which in the nonlinear case lies at the base of a considerable part of the theory). We note that in recent decades many books and surveys devoted to these questions have been published abroad: the monographs of Barbu, Belleni-Morante, Brezis, Carrol and Showalter, Chernoff, da Prato, Davies, Haraux, Martin, Tanabe, Zaidman [200, 210, 234, 263, 272, 316, 334, 386, 500, 644, 712], the survey papers of Evans, Goldstein, Kisynski, Lions, Pazy, Sova, Trotter [350, 370, 422, 472, 569, 631, 637, 650]. In the bibliography we include all papers of Soviet authors on the questions cited which were reviewed in Ref. Zh. Mat. in 1968-1982 (except for deposited papers or abstracts of reports, whose contents are reflected in other publications), and also the most important papers of foreign authors, reviewed in 1978-1982. Moreover we give bibliographical citations on certain questions of the theory of evolution equations, on which we shall not dwell in more detail. The papers of Aliev, Zhikov, Zhikov and Levitan, Pankov, Bart and Goldberg, Biroli, Dafermos, Haraux, Nakao, Zaidman, etc. are devoted to the existence of almost-periodic solutions and solutions bounded on the real axis [2, 45, 46, 87, 107, 108, 204, 222, 322, 379, 385, 386, 523, 692, 695, 698, 699, 701, 704, 706, 708, 709]. Questions of admissibility and dichotomy were studied by Zhikov and Levitan, Krein and Savchenko, Megan, Popescu, and Preda [46, 66, 506, 585, 586], questions of reducibility by Guichet [376]. On the existence and stability of invariant central manifolds and bifurcation of periodic solutions, cf. the books of Marsden and McCracken [95, Chaps. 2, 8, 9], Carr [262], Hale [384], the papers of Crandall and Rabinowitz, Holmes and Marsden, Itoh, Marsden, de Mottoni and Schiaffino [300, 301, 396, 402, 496, 521]. The papers of Niremberg (cf. the supplement to [100]), Ovsyannikov, Baouendi and Goulaouic, Nishida [101-103, 197, 534, 553] are devoted to the abstract form of the Cauchy-Kowalevska theorem in the realms of Banach spaces. Improper and conditionally proper problems for evolution equations were studied by Abdulkerimov, Atanbaev, Bakaev, Bakaev and Tarasov, Bakushinskii, Karasik, and Lavrent'ev, Lavrent'ev and Amirov, Leites and Lions, Mel'nikova [I, 5, 9, 10, 11-14, 57, 75-78, 86, 96]. We shall not touch on the spectral theory of differential equations in Banach spaces and scattering theory. Due to lack of room we shall not deal with such aspects of the theory of evolution equations as Galerkin's method, difference methods (except for those of which we spoke above), etc. For the same reason we do not consider functional-differential (in particular, with deflected argument and integrodifferential) equations in Banach spaces. Throughout the entire survey we use the following notation: R and C are the sets of real and complex numbers; R+ = [0, ~); I is the identity operator; D(A), R(A), A are the domain of definition, set of values, and closure of the operator A; AID is the restriction of the operator A to D(A)ND ; E* is the dual space to E; is th~ value of the functional g~E* on the element x~E ; (u, v) is the scalar product in Hilbert space. The s e t ~ D ~ D ( ~ ) is called the core of the operator A, if A is the closure of its restriction to D (in Russian this is called "the essential domain of definition"). The operator A in the Banach space E is called dissipative, if for all % > 0 the operator (I -- %A) -z [defined on R(I -- %A)] is single-valued and satisfies a Lipschitz condition with constant I, and m-dissipative, if in 2155

addition, R(I -- %A) = E for all (equivalently, for some) % > 0 (the operator --A in these cases is called accretive, respectively, m-accretive). For a linear A, to be m-dissipative means the existence for % > 0 of the resolvent R(%, A) = (%1 -- A) -z and the estimate ]JR(%, A) II

1Ix. We note that (by virtue of traditions established in the literature), in Sec. I and Sec. 2 the term "strong solution" is used in somewhat different senses (cf. Paragraphs 1.1 and 2.2). The numbering of formulas is separate in each section, and here the number of the section is indicated before the number of the formula only in references to formulas from another section. In Sec. 2, in view of the large number of inner references, in a number of paragraphs separate assertions are numbered by figures I ~ , 2 ~ , etc. Here, for example, assertion 2 ~ of Paragraph 2.7 is mentioned inside this paragraph as "assertion 2 ~ or simply "2 ~ , and outside of Paragraph 2.7 as "2.7.2o. '' Paragraphs 1.1-1.6, 1.8-1.18, 1.21-1.27 were written by Krein, the introduction and Paragraphs 1.19, 1.28, 1.29, 2.1-2.26 by Khazan; Paragraphs 1.7 and 1.20 were written together.

1.

First-Order Linear Differential Equations with Constant Operator

Coefficient and Semigroups of Operators 1.1.

Cauchy Problem.

We consider the equation du -~=

(1)

Au,

where u(t) is the unknown function with values in the Banach space E, A is a linear operator in E, the derivative du/dt is understood as the limit in the norm of E of the difference ratio Au/At as At § 0. By a (strong) solution of (I) on the interval [0, T) (0 < T ~ ~) is meant a continuously differentiable function u(t) with values in D(A), satisfying the equation for t6[0, T). By a weakened solution of (I) is meant a function u(t), continuous on [0, T], continuously differentiable on (0, T), assuming values in D(A) for t~(0,T) and satisfying the equation on this interval. By a weak solution is meant a function u(t) which is continuous on [0, T) and has the property that for each linear functional g 6 O ( A * ) c E * the function is absolutely continuous and almost everywhere satisfies the equation d

a7 < g, tt (t) > -- ( A % u (t) >. By a weakest solution is meant a function which is continuous on [0, T) and satisfies the i n t e g r a l identity T

I
t=o

0

for any continuously differentiable function g(t), which is finite on (0, T), with values in O ( A * i c E * and such that A~g(t) is continuous on [0, T). By the (weakened, weak, weakest) Cauchy problem is meant the problem of finding a (weakened, weak, weakest) solution of (I), satisfying the initial condition

(2)

u(O) =x~E.

1.2. Laplace Transform and Resolvent. If one formally applies the Laplace transform to (i) assuming that upon integration by parts the integrated terms vanish at infinity, then for the Laplace transform image of the solution

(~)=~e-~'u (t) dt one gets

the

equation

( k I -- A ) v ( k )

= u(0)

= x.

If

the

resolvent

(3) R ( X , A) = ( k I -- A) - x

of

the

operator is defined for a sufficiently wide set of k, then one can try to reconstruct the solution of the Cauchy problem from the function v(%) = (%1 -- A)-Zx (for the case when the solutions are defined only on a finite interval, Lyubich [92] developed a theory of local Laplace transforms). Basic results on the existence and properties of solutions of the Cauchy problem have been obtained along these lines. However an example was given in [63] of a system of partial differential equations for which the corresponding operator A has no resolvent 2156

for any k~C , but the Cauchy problem is sufficiently well posed. The desire was expressed for a "resolvent-free" theory of the Cauchy problem. Independently, the first essential step in this direction was made by da Prato [303], who introduced the n-th order resolvent (cf. below). In what follows, various estimates of the norm of the resolvent figure, and to shorten the account here in writing fIR(k, A)II we shall already assume that % is a regular point of the operator A (the same thing relates to the resolvent of order n). I .3. Operator A. In what follows we shall always assume that A is a linear operator with domain of definti0n D(A) and range of values R(A). If such an operator is densely defined and has at least one regular point, then it will be called regular. By 0(A) we denote the set of all regular points of A. For a closed operator A the domain of definition D(A n) of the operator A n becomes a Banach space D n with respect to the norm

llx II~=]~ IIA'xll. We shall speak of the smoothness another of the spaces D n.

of an element x, depending on its belonging

We can consider

the set

D~=

~ D ( A n)

to one or

as a Frechet space with

respect to the countable system of norms Ilxlln (n = I, 2,...). We denote the restriction of A to D~ by A~. The operator A~ acts continuously on the space D~. We denote the closure of D~ in the norm of the space D n by Dn. If the operator A is regular, then D n = Dn and the closure in E of the operator (A~) n coincides with A n (cf. [655]). 1.4. Uniqueness of the Solution. The basic result is due to Yu. I. Lyubich Suppose for sufficiently large positive ~ and llm

(cf. [63]).

InllR(k, A)II = h < T .

Then a weakened solution of the Cauchy problem is unique on the segment [0, T -- h] and can ramify for t > T -- h. If h = u, then the solution is unique on the whole half-line. In the sense of the behavior of R(%, A) as % § ~, this assertion is sharp. Some generalizations of this result were obtained in [162, 113]. Cf. [3, 625, 693, 694] for another type of uniqueness theore m . 1.5. Dense Solvability. By this term we mean for a set of initial data which is dense in E. The (cf. [63]), and was sufficiently completed by Beals function on [0, ~], with ~(s) § ~ as s + ~, s-l~(s)

i s-~ (s) as
0,,x~E) .

A semigroup belongs

to the class Co, if | I m U { t ) x = ~ -

for any X~E.

2157

A semigroup is said to be equicontinuous, if for any continuous seminorm p(x) on E one can find a continuous seminorm q(x) on E such that p(U(t)x) < q(x) for all t ~ 0 and x6E. If the space is barreled (in particular, a Frechet space), then a semigroup of class Co is locally equicontinuous [433]. By the infinitesimal operator A0 of the semigroup U(t) is meant the operator defined by

Aox=lim U(h)x--x h~O

h

on those x where the limit exists. If A0 admits the closure A, then A is called the generating operator of the semigroup (in the foreign literature the term "complete infinitesimal operator" is used). By the image of the semigroup U(t) is meant the linear set

Eo=

UU (t)E. t>O

By the kernel of the semigroup, Ker U, is meant the collection of all x for which U(t)x = 0 for all t > 0. The domain of definition of the operator A0 is dense in E0. In what follows, strongly continuous semigroups in the Banach space E are called simply semigroups. For a semigroup in a Banach space E the limit

O~o= lim t-' In IIu (011 < oo t .-.~oo

exists and we call it the type of the semigroup. If sects the closure of the image E0 in zero only, then of the semigroup exists [47]. If K e r U = 0, then the algebraic inverse ( X I - A) -l, i.e., from ( X I - A)x =

the kernel of the semigroup K e r U interthe complete infinitesimal operator A operator X I - A for ReX > ~0 has an 0 it follows that x = 0.

In order that the semigroup U(t) in a Banach space belong to the class Co, it suffices that weakly limU(t)x=x for any xOE [55]. For a semigroup of class C0 the operator A0 is t-~0

closed, D~ is dense in E and is the core of the operator A -- A0. If A0 is the infinitesimal operator of a semigroup in a Banach space E, then for x6D(Ao) the function u(t) = U(t)x satisfies, for t > 0, the differential equation u' = A0u; if there exists a complete infinitesimal operator A = A0, then also for xED(A) the function u(t) = U(t)x satisfies (I) for t > D. This establishes a connection between the theory of the Cauchy problem for the differential equation (I) and the theory of semigroups. I .7. Uniformly Proper Cauchy Problem. A Cauchy problem is said to be proper on the set F and the interval [0, T) (0 < T ~< ~), if for each xEF there exists a unique solution u(t) of the problem and for any sequence of solutions un(t), it follows from Un(0) + 0 that Un(t)+ 0 for all t6[0, T). The Cauchy problem is said to be uniformly proper, if in the preceding conditions the convergence is uniform on each segment [0, Ti]c0, T) If F = D(A), then one does not mention F in the definitions. Analogously one defines the propriety and uniform propriety of the weakened, weak, and weakest Cauchy problem on the set F and interval [0, T). Frequently the propriety follows from the unique solvability. If the Cauchy problem (I), (2) has a unique solution on [0, ~) for any x~D(A), then the problem is uniformly proper on [0, =) (Phillips [573]). If the Cauchy problem is uniformly proper on the interval [0, T), then it is uniformly proper on [0, =) and its solution is given by the formula u(t) = U(t)x, where U(t) is a semigroup of class C0. Conversely, for the complete infinitesimal operator A of a semigroup of class Co the Cauchy problem is uniformly proper (cf. [63]). For an operator A with D(A) = E the following conditions are equivalent: I) A is the

complete

infinitesimal operator of the semigroup U(t) of class Co;

2) there exist (06R and M > 0 such that

34 for ~,>a), rt~---l, 2, "" .; []R" (~,, A)I[< ()~--co). 3) the operator A-- ~I is m-dissipative in (E, Ill.Ill) for some c00R equivalent with the original norm in E;

(6) and some norm Ill'ill,

4) there exists a functional [., .] :E x E + C (semiscalar product), which is linear in the first argument, such that ~[xa[[~[x,x] O, x6Z).

The complete infinitesimal generator A~ of this semigroup will be the restriction of the infinitesimal operator A0 to all elements x of D(A0), for which x~Z and Aox6E (cf. [517]). This operator A~ is called the Feller generating operator of the semigroup U(t). For ~ > ~ for any xEE

there is defined the Laplace transform

( 1 O)

J (~) x = ~ e-~'U (0 xdt 0

[the integral converges with respect to the norm N(x) and with respect to llxll]. It is shown in [356] that N (x) =

Further,

for

a n y e > 0 and" xEE

sup

n>O,~>.

II( x - ~)"J" ~) x II.

one can find

a KE > 0 a n d %o = % o ( e , x)

II~-~)":"~)xll-.x0

:a.d

such that

n>(x-~)8.

It follows from the strong continuity of a semigroup that the image E0 of the semigroup is imbedded in E (E0r-E/ and if E0 is dense in E, then Z is also dense in E. Thus, there is associated to each semigroup a uniformly proper Cauchy problem d#

-dF=A|

#~)=x~D (A|

on some space Z, normally imbedded (lixll ~ N(x)) in the space E. Wg note that for elements x of D(Ao)NZ [and if A0 = A exists, then also for elements of D(A)NE ] there exists a weakened solution of the homogeneous Cauchy problem. The question arises of how to determine for 2159

an operator A whether it is a Feller generator of some semigroup and how to describe its set of continuity. Let F be a dense linear subset of E and A be a linear operator acting on F. sume that

Let us as-

I) there exists an ~ R , such that for all ~ > m and x6F the equation (~I --A)y = x is uniquely solvable in F [we denote the solution by y = R(%)x]; 2) the space F is Banach with respect to the norm N (x)-----.~"~>

II(~- ~)" R" (~)x If;

( 11 )

3) D(A) is dense in F in the norm N(.); 4) for each e > 0 and n >i (% -- re)e,

x'EF there exist K e > 0 and l0 = 10(e, x) such that for % > l0 and

I{(a-- m). R" (X) x [I 0 one be able to find an a b and Cb, such that

ReL>a~--bllm~l).

IIRe(L A)[l~ m. Another sufficient condition: if for some ~ > ~0 (m0 is the type of the semigroup)

ll--mln[xlllR

(~+ix,

A)l[=to t0 and x~E. For to = 0 this result was found earlier by Hille [393]. An example of Phillips [572] shows that the property of differentiability for t > 0 may be violated under perturbation of the complete infinitesimal operator A by a bounded one. If the sufficient condition (16) holds, then the differentiability is preserved under perturbation by a bounded operator [561]. Sometimes one can judge the smoothness of a semigroup from its behavior at zero: if for each C > 0 there exists a ~C such that

iil_U(till 0 there exists an M~ such that Izl=llU(z)ll<M,

~r z~Zg_~, I z l < 1 ,

w h e r e ~ i s a f i x e d number f r o m [ 0 , 1 ) . A closed operator A is the complete infinitesimal o p e r a t o r o f s u c h a s e m i g r o u p o n l y i f D(A) = E and f o r e a c h e > 0, s u c h t h a t

II~(X, A)II--<M, IXl=-' ~r xe~0+~_ ,.

(18)

It is noted in [683] that from an estimate of type (17) there can follow an estimate of type (18) in certain intermediate interpolation spaces between E and DI. This allowed the author to consider the question of solvability of the Cauchy problem in the corresponding interpolation spaces. Okazawa (preprint) investigated the semigroup U(z), generated by a regular operator, for which instead of (17), for some integer k and 0 6 (0, ~/2] one has

[I ~ (%, A) X ]1-. 0) for any xED(A~). It turns out that U(z) is holomorphic in the sector larg zl < 0, while dU/ dz = AU is a bounded operator for Izl > 0. Further, for x~D(Ak)? U(z)x-+x, when Izl + 0 and largzl ~ 0 -- E one has

IIU(z)xll..<M~e~Re'llxll~r

Iart zl-.. 1.

Da P r a t o [303] (22)

Following [303], the boun~ed operator Sn(X, A) is called a resolvent of order n of the operator A in the domain G ~ C , if it depends holomorphically on %EO , one has

Sn (~,,A) A x = AS~ (X, A)x(xeO (A)), S~ (L, A) (Xl-- A)n§ = x (xED (a~')) and from the fact that Sn(X, A)x = 0 for all XOO it follows t h ~ x = 0. Obviously the resolvent of order 0 coincides with the ordinary resolvent. If D(A n+i) = E, then the operator A can have only a unique resolvent of order n for which there is a maximal domain of holomorphy On(A), called the resolvent set of order n of the operator A. In [303], for integral =, and in [48] for any, it is shown that if (22) holds for the semigroup U(t), and K e r U = 0, E = E, then the complete infinitesimal operator A of the semigroup has resolvent of order n for n > ~ - I and ReX > m, while

2165

S,, (k, A)= 1 i t"e-X'U (t)dt 0

and

d~S.'(~,,

11

A)

d~,k

..
~

and m = k(n + I) (k = I, 2,...) M

r (m--m)

II(Zl- A)-mxll'-< (~-i), (x_~),.-= IIx II for some M > O. 3) D(A) is dense in E and D(A n+2)

is the core of A.

4) For some__Iz > ~ the operator (IiI -- A) n+l admits a closure (here it is also assumed that K e r U = 0, E0 = E). Earlier, Sobolevskii [135] found a sufficient condition in similar terms for an operator to generate a semigroup of growth ~. We note that for a semigroup with (22) the domain D ( A ~ ] +*) is containedin t h e set of continuity of the semigroup, and consequently, such semigroups belong to the class (o[e]+ I) (cf. [517, 513] for more details). Zafievskii [48] considered the more general situation when the functions teiiu(t)xll are summable on [0, I] for each x~E (or the function iiu(t) ll is summable on [0, I]). Here healso found a description of generating operators in terms of estimates of the resolvent of order n, analogous to those which were given for summable semigroups (i.e., for e = 0) (cf. Paragraph 1.14). For the semigroups indicated, conditions are found for their differentiability and analyticity in a sector [49]. The more general case when the functions t=llU(t)xll or teU(t) belong to the space LP, was investigated by Wild [681]. Okazawa [546] considered semigroups U(t) of growth n, which in addition have the property that U(t)x + x as t § 0 for any x~D(A") [semigroups of class (Dn)]. For such semigroups he gave a complete description of their complete infinitesimal operators. 1.16. Infinitely Smooth Initial Data. Ushijima [655] made a detailed investigation of semigroups generated by closed operators A in the space D=, assuming that D= = E. Here he gave up the assumption of regularity of the operator. He introduced the concept of resolvent of the n-th order as follows: the point I belongs to the resolvent set Pn(A), if the operator (II --A~) n is invertible on D= and the inverse operator (II -- A~) -n is defined on all of D= and is bounded in the norm of E. Its closure with respect to continuity is called the resolvent of order n and is denoted by Rn(l , A). If there exist a and b > 0, such that li(ll -A=)-*xIi~ b l n + l h n ~ I and x E D . , then the operator A~ generates in D= a semigroup U~(t) of class Co. This semigroup will be exponential, i.e., for some the semigroup e-mtU= is equicontinuous only if one of the following conditions holds: I) there exist m and k, such that

ll(kl--A=)-"xlI~ m, n = I, 2,...,

x 6D=;

[I(~I--A~']x[[-.o', xED=; [[(k/--A.)-lxl[-.o#, xED..

2) there exist an m' and an l, such that 3) there exist m", m, n, such that

The semigroup U~(t) consists of operators which are bounded in E only if for any e > 0 there exists a constant Ce, such that for each xED= one can find a number 10 = k0(e, x), such that lil"(%l--A=)-"xII.. l0 and e < n/l < ~-z. (For a regular operator, the last statement is found in [538].)

2166

The operator A generates in D~ a semigroup of class Co with the additional property

]liw(t)tnu"(t)xdtll 0, s > 0, k > - - 1 , C > 0 s u c h t h a t IIR.+,(X, A ) l l ~ < C ( f + l x l ) ~ f o r Re ~ > - - b l l m X l '/~ , t h e n

[I(~)] (tnU (~)) []• pqi (j[)*etott--~lJ+l~+l) for

some p , q , and ~.

Under certain

conditions

the converse assertion

is valid.

The s e m i g r o u p U ( t ) i s a r e a l a n a l y t i c o p e r a t o r - f u n c t i o n on t h e h a l f - l i n e (0, when t h e r e e x i s t s a number k s u c h t h a t f o r a n y e > 0 i n some s e c t o r o f t h e f o r m

=) o n l y

n J arg ( ~ - - o ) l-. m0 the distribution e-mtu generates a continuous map of the Schwartz space S(R I) of rapidly decreasing functions to the space L(E). The infinitesimal operator A0 of a regular distribution semigroup can be defined as follows [473]: if there exists a delta-sequence *This means that if fz = 0 on (--~, a), then u = 0 on (--~, a).

2168

!

on'D0, such that U(0n)X + x and U(-nOn)X + y as n + ~, then x@D(Ao) and y = A0x. The infinitesimal operator admits a closure A = A0 which is called the complete infinitesimal operator of the distribution semigroup. for any [360]).

~@D0

The set

~ D(Ao n) is dense in E and contains U ( ~ ) E

(for another equivalent definition of complete infinitesimal operator, cf.

The connection between the Cauchy problem and distribution semigroups is described by the following assertion [473]: the Cauchy problem is well-posed in the sense of distribution theory if and only if the operator A is the complete infinitesimal operator of a regular distribution semigroup. The fundamental solution of the operator P coincides with the distribution semigroup generated by the operator A [268]. Fujiwara [360] noted that closely connected with a distribution semigroup is a strongly continuous semigroup of operators in the Frechet space D=. An operator A with D= = E is the complete infinitesimal operator of an exponential distribution semigroup only if it has regular points and there exists an m0 > 0 such that the restriction of the operator A-- m01 to D= is the complete infinitesimal operator of an equicontinuous semigroup of operators U=(t) in D~. An operator A with D~ = E is the complete infinitesimal operator of a distribution semigroup only if it has regular points and its restriction to D= is the complete infinitesimal operator of a semigroup of class Co in D= (Ushijima [653, 655]). In [377] the last two assertions are refined: the semigroup U=(t) in both cases is infinitely differentiable on [0, =) in the topology of the space L(D~) and (d/dt)U=(t) = AU~(t) = Um(t)A. It is also shown in [377] that in order that the operator A, which has regular points, be the complete infinitesimal operator of a regular or exponential distribution semigroup, it is sufficient that it generate a semigroup U=(t) with the corresponding properties on some closed subspace of the space D~. Lions [473] found the following spectral characteristic of the complete infinitesimal operator of a distribution semigroup: a densely defined, closed linear operator A is the complete infinitesimal operator of an exponential distribution semigroup only if there exists a half-plane Re ~ > m, in which

I]R(~,, A)II~ m > 0 consists of regular points and the distributions U~(~) form a bounded set in D'(L(E)), then A generates a distribution semigroup U = limU%, where the limit exists in Ux(~). Cf. [279] for a generalization. Foias [359] was the first to consider nonexponential distribution semigroups of order of growth no higher than q (I < q < =), for which for some m > 0 the distribution exp (-~ • (I + t2)q/2U(~) generates a continuous map from S(R) to L(E). A densely defined, closed linear

2169

operator A is the complete infinitesimal operator of a distribution semigroup of order of growth not higher than q, only if (26) holds in a domain of the form

~-={~; Re ~>(mln[ lmXl+~) '-']+, ReL>~} U{~+R', ~> ~} (Cioranescu [278]). A closed linear operator with D~ = E is the complete infinitesimal operator of a distribution semigroup with order of growth no higher than q only if its restriction to D~ generates a locally equicontinuous semigroup U~(t), for which

IlU (t)xll~Ce*tqllxll o

(0 0 I

t

we set ~,(1)= 7 ~ ( ~ )

; a regular distribution semigroup U(~)

is said to be holomorphic in

the sector EO, if for each ~6 o the function U(~,) with values in L(E) can be extended to a function U,(~) which is holomorphic in s in the sector E o. Here it turns out that for each s6Ee the distribution U,(~) is a distribution semigroup with complete infinitesimal operator sA. A distribution semigroup which is holomorphic in a sector is exponential ([268]). A densely defined, closed linear operator A is the complete infinitesimal operator of a distribution semigroup which is holomorphic in a sector, only if there exist ~ R and an integer N such that for each e > 0

IlR(L A)II-- 3, p > I for n = 2, p = I for n = I) and (if n > I) either ImqEL~o+~(~) then A = A -- q(x) with O{A)={u~Wl2(~): qu6L~oc(f~), Au6L2(~)} A-- wl is m-dissipative for some ~ I> 0. then A = A.

If

or

(Re-'-~rn/2+e~] C~Ioc (f~),

admits a closure in L2(~) and

Imq(x)~ 3,

It is well known (cf., e.g., [63, Sec. 8 of Chap. I]), that strongly elliptic operators generate analytic semigroups in the spaces LP(I < p < ~). Recently, it was discovered that the analogous fact also holds in spaces with a sup-norm. Thus, Stewart [638, 639] proved that a uniformly strongly elliptic operator of order 2m in a (possibly unbounded) domain ~ c R n generates under general boundary conditions a semigroup which is analytic in some sector of a right half-plane in a subspace of the Banach space Co (~), obtained by taking the closure of the domain of definition of the operator. It is assumed that the highest coefficient of the operator is uniformly continuous and bounded in ~, the coefficients of the boundary operators and their derivatives up to order 2 m - - m k (mk is the order of the boundary operator) satisfy the same conditions; the lowest coefficients of the operator are continuous and bounded in ~; the boundary operators satisfy a supplementarity condition, and a~EC ~= . If = R n and the operator is homogeneous with constant coefficients, Paquet [555] independently proved analytic generation in the whole right half-plane of a semigroup in Co(R n) and established that if m > I, then 'for thissubgroup U(t),llu(t)ll = llU(s) ll > I for all t, s > 0. In these same papers Paquet showed that if A generates a semigroup in the Banach space E which i s bounded and analytic in the right half-plane, then the operator ('l)m+iAm+BtAm-l+...'.+Bin generates the same kind of semigroup for any B,OL(E) . Lumer and Paquet [490, 491] showed that if the operator A generates a contracting group of class Co in E, then A 2 + BIA + B0 generates a semigroup in E which is bounded and analytic in the right half-plane for any B0, BilL(E), and applied this fact to prove the analyticity of the semigroup generated in C0(R") by an elliptic operator of special form which is degenerate in some of the variables. SableTougeron [600] found conditions under which an operator of order 2m, which is elliptic in a bounded domain ~ c R n , which degenerates like a power near ~ , under normal boundary conditions, satisfying a supplementarity condition, generates a semigroup in L2(~) which is analytic in the right half-plane (the number of boundary conditions is determined by the order of degeneracy, and the domain of definition of the operator can be described in terms of Sobolev weight spaces). Campanato [260] found the first result as far as we know about the degeneration of an analytic semigroup in a HSlder space: a second-order uniformly elliptic operator

Au=

X I~l,

Di(a~Dku)+ ~.~ biDin, I~n

-

where

Dl= o--~l,a~6.C(~), b~L~(f~) ,

with boundary con-

l~l~n

dition ul~ ~ 0, generates a semigroup in the spaces C ~ ( ~ ) ( O < ~ < I) , which is analytic in some sector of the right half-plane if ~ c R " is bounded and OQ~CL =

As is known, in the study of the Navier--Stokes system of equations by methods of the theory of semigroups, an important role is played by analytic semigroups, generated by a Stokes operator. Solonnikov [138] (for n = 3), Giga [364, 365] for arbitrary n, proved that a Stokes operator in a bounded domain ~ c R n with smooth boundary under a Dirichlet boundary condition generates a semigroup in the Banach space obtained by taking the closure of the set of all solenoidal vectors of C~(~) n in the norm of LP(~) n, I < p < ~, which is bounded and analytic in the right half-plane. Krein and Kulikov [64] showed that the Maxwell operator under Leontovich boundary conditions is a non-self-adjoint maximal dissipative operator in the space which is the closure in L2(~) x L 2 (~) of the set of pairs of smooth solenoidal vectors. An important class of semigroups of class Co is connected with functionals and functional-differential equations. Let 0 < h ~< ~ and xt(s) = x(t + s) for--o~=/=s~[--h, O] If the operator L:W~(-h, 0; R n) § R n is linear and continuous, then the problem

dx(t) =L(xt) (t>O, x(O)=~, x(s)=~(s), dt = = 4 = s~[-h, o)), is formally equivalent with the Cauchy problem ~ = A u ( t > O ) ,

=(0)=(~,~)

in the space Mp =

R n x LP(--h, 0; Rn), where D(A)--{~.~(0),~):~6Wp' (-- h, 0; R")}, A (~(0),~)=(L~, ~'),. Delfour [335], generalizing a number of earlier results, showed that the equivalence is not only formal: the

2172

operator A generates a semigroup U(t) of class C0 in Mp, while U(t)(~, ~)=(X(t), x,) , where x is the unique generalized [and for (~, ~)~D(A) also ordinary], i.e., x~CI(R+; R n) solution of the original problem. We restrict ourselves to this example; there are analogous results (under other conditions on L), when the original problem is posed in an infinite-dimensional space (rather than Rn), for equations of neutral type, for another choice of the phase space (instead of Mp), etc. To conclude, some examples of semigroups which do not belong to the class Co. Semigroups with power singularities at zero arise in reducing initial-boundary problems for second-order parabolic equations to abstract Cauchy problems in the spaces Wp (i < p < =) (Sil'chenko and Sobolevsekii [125, 126]). The SchrUdinger operator iA does not generate a semigroup of class Co in LP(R n) for p ~ 2, but generates a regular group-distribution for I < p < ~ (Balabane and Emamirad [183]). If Q is a bounded domain with smooth boundary, P(x, D) is a formally self-adjoint elliptic operator (with smooth coefficients) of order 2m, Q(x, D) is an arbitrary linear differential operator (with smooth coefficients) of order k ~ 2m-- I and A = 0

iP(x, D) + Q(x, D) with D ( A ) = W 2 2 m ( ~ ) N W ~ m ( ~ ) , then the Cauchy problem with operator A in the space L2(~) is well-posed in the sense of Gevrey distributions (cf. Paragraph 1.18), while the corresponding exponent d = 2m/k (Chazarain [266, 268]). Cf. [31, 174, 347, 478-489, 658] for other examples. 1.20. Behavior of Solutions at Infinity, Stability. In this paragraph A is the complete infinitesimal operator of the semigroup U(t) of class Co in the Banach space E, u0 is the type of the semigroup U(t), ~s=sup{Re%:%~(A)}. For bounded A it is well-known that llu(t)ll ~ eUs t, so that u0 = ~s" For an equation with unbounded operator the disposition of the spectrum o(A) does not guarantee the corresponding estimates of solutions. An example is given in [155] of an operator A with empty spectrum, for which llu(t)l[ grows exponentially. Zabczyk [691] showed that for any two real numbers a ~ b there exists a semigroup of class Co in a Hilbert space, for which ~0 = b and ~s = a. For u0 and ~s to coincide, it is sufficient that for t > 0 the operators U(t) be compact (Zabczyk [691]) or smoothing: U(OEcD(A), i.e., U(t) is an infinitely differentiable semigroup, cf. Paragraph 1.12 (Triggiani [649]). In the Hilbert case, m0 = m s i f o ( U ( ] ) ~ exp{%:~(A)U{0}} Cf. Hille and Phillips [155], Belleni-Morante [210], and also Gearhart [362] on the connection between o(U(t)) and o(A). For spectral operators (cf. Dunford and Schwartz [35, Chap. XVIII-XX]) one has o(U(t)) = exp (to(A)). RDh [598] proved that an m-dissipative operator A is spectral if E is Hilbert and separable, the subspace KiA)={xED(A) :Re(Ax, x)=0} has finite codimension in D(A), u s < 0 [from which it follows that o(A) consists of isolated eigenvalues Xn] , the sequence X n is interpolating (for example, suplRe~nl< oo and in[{l%,--~l:m=it=n}>0) , the heights of all the associated vectors are bounded in aggregate, and the root subspaces generate E (in the absence of the last requirement we have: o(U(t)) = exp (to(A)'U {0})). This result permitted the investigation of the asymptotic behavior of solutions of the equation of the oscillations of a tubular cantilever with an incompressible fluid. Kurtz [458] showed that if U(t)K~K, where the cone K={x~E: (f, x ) ~0 Vf~IH}, M is a convex subset of the unit ball in E*, and K0={x~E: ( f , x ) > 0 , V/EJV/}, then lnf

sup "( f' Au ) ~< sup

=ED(A)NK,rE M

( f , U)

inf

tE~ "ED(A)nK*

< f' Au ) ( f , u >

~< ~o.

Both inequalities can be strict, but if M is weak* compact and normalizes K0, they become equalities. Marek [94, 494] showed that if U(1)K~K , where K is a closed normal reproducing cone in E, and m s is an isolated pole of R(X, A) with finite-dimensional root subspace N, while the operators U(t) (t > 0) are K-irreducible, then dimN = I, and

U (0 = eO" (P + IV (0),

IIUw(0 [I< M e ' P ' ,

where P~L(E) is a projector onto N, p > 0 (so that ~0----~s),W(t)P=PW(t)=O, O=/=x~K~O--/= PxEK. We apply this result to the linearized Boltzman equation. The semigroup U(t) is said to be exponentially stable, if its type m0 < 0, and strongly (weakly) asymptotically stable, if V x E ~ llm U (~)~=0 in the corresponding sense. Slemrod [628] showed that if u s < 0 and U(t) is differentiable for t > to ~ 0, then U(t) is exponentially stable. same paper that if

It is shown in the

2173.

for where N i> --2, then for some ~6(0, 6), M > 0

Re~>--6,

Ilrn*l~O,

one will have

l[ ,4-(~+~)u (0 ll--<Me-*", and if N I> --1, then there exists a

~E(0, I) , such that

ile~t (f,

A-tN+I)U (I)x ) I=dt< oo vxEE, f e e * .

Since ~o=inft-'lnfiU(t)[[, for the exponential stability of U(t) it suffices that liU(to) ll < | for some to > 0. In particular, if E is Hilbert, A is m-dlssipative, and VxED(A) Re(Ax, x) < 0, then for all t > 0 one will have ]lu(t)l[ < | (Devys and Morel [338]), so U(t) is exponentially stable (D'yachenko [41] found a weaker result). In a Hilbert space, exponential stability is equivalent with Li-stability (llU(t)xllOLi(0,co) for any x~E ) and is equivalent with the existence of a positive definite H=H*OL(E) , for which Re(Ax, Hx) 0 (analog of Lyapunov's theorem) as shown by Datko [326] and Likhtarnikov [89]. In an arbitrary Banach space, strong asymptotic stability follows from weak asymptotic stability if A has compact resolvent (cf. [212]), and exponential asymptotic stability follows from weak asymptotic stability if U(t) is quasicompact, i.e., flU(to) --GII < I for some to > 0 and some compact G~L(s . The last assertion follows from a modification of the uniform ergodic theorem of Yoshida--Kakutani (cf. [34]), given in O'Brien [535]: if U(t) is quasicompact and ilu(t)ll ~< M, then there exist a compact projector P, belonging to the uniform closure of the set {U(t)':t~0} , projectors PkEL(E)', and numbers ak~R, ~ > 0 , M l ~ l such that PkPj = 0 for j ~ k, P = PI +... + P m and

U (t) Pk=e'%'Pk,

fl u (0 (I-P)II<M,e,o'.

Hence, a u n i f o r m l y b o u n d e d q u a s i c o m p a c t s e m i g r o u p , g e n e r a t e d b y an o p e r a t o r A, w h i c h has no eigenvalues

w i t h ReX = 0, i s e x p o n e n t i a l l y

stable.

B l a k e l e y and P r i t c h a r d [225] showed t h a t i f t h e s e m i g r o u p U ( t ) s a t i s f i e s w i t h ~ > 0, and t h e o p e r a t o r B : D ( B ) e - ' E ~ E " i s s u c h t h a t f o r some ~E[0, 1)

IIU(OII-.<Me-O'

II a (0 ax II 0. 4.

If under the preceding conditions e = 8, then the assertion is valid for the operator A + EB with sufficiently small e.

There is a large literature on the stability of the property of being the complete infinitesimal operator of a contracting semigroup (the property of being an m-dissipative operator). The first result is due to Trotter [650], and later it was generalized and developed by Nelson [527], Kato [408], Gustafson [380], Chernoff [271] (cf. Davies' book [116]) and Okazawa [542-544, 547]. The most general results in this direction were found by Okazawa [547], and we shall describe them here. For each element

x6E

one defines the dual map into the subset F[x] of the dual space

E*: F [x] = {f6E*; < f , x ) = [[ x [] ~ = [[ f I[ 2}. Let A be the complete i n f i n i t e s i m a l o p e r a t o r o f a c o n t r a c t i n g s e m i g r o u p , and "B be a dissipative o p e r a t o r ([[~,l-B)xll>~,[JxH,~,>O, x6~E),, w h e r e D ( B ) ~ D ( A ) . I f f o r e a c h xfiD.(A) one c a n f i n d a f u n c t i o n a l h6,F[Ax] , s u c h t h a t

~e(h, Bx) > - a l l

Axll~-bllxll 2

9f o r some n o n n e g a t i v e c o n s t a n t s a , b w i t h a < 1, i n d e p e n d e n t o f x , t h e n t h e o p e r a t o r A + B is the complete infinitesimal operator of a contracting semigroup. From w h i c h G u s t a f s o n ' s r e s u l t [ 3 8 0 ] f o l l o w s e a s i l y : i f f o r t h e A and B d e s c r i b e d a b o v e t h e r e l a t i v e norm o f B i s l e s s t h a n 1, t h e n t h e a s s e r t i o n is valid. Let A be t h e complete i n f i n i t e s i m a l o p e r a t o r o f a c o n t r a c t i n g s e m i g r o u p and D(A*) = E*. L e t B be a d i s s i p a t i v e o p e r a t o r w i t h D(B) ~- D(A) and l e t t h e e q u a t i o n

Axo+,xn-

n=y

2

. . . .

)

for some I > 0 and each 96E have a solution x n with uniform bounds on llxnlland [[(A + B)xnll, then the closure A + B of the operator A + B is the complete infinitesimal operator of a contracting semigroup and D(A) is the core of the operator A + B. A result of Chernoff [271] and Okazawa [544] follows from this: if for the operators described above B has relative norm I, then the operator A + B generates a contracting semigroup. In [547] there are a number of theorems about perturbation of maximally dissipative operators and their concrete examples in Hilbert space. Vainikko and Shlapikiene [24] showed that the property of being the complete infinitesimal operator A of an analytic semigroup is preserved under perturbation by an operator B, for which BR(10, A) is compact for some X0, if one of the sets D(A*) or D(B*) is dense in E*. In other theorems on perturbation one imposes conditions on the operator U(t)B or BC(t) [225, 659]. Cf. [44] for some counterexamples. Cf. also [267, 285, 462, 688, 689] in connection with what has been said. Cf. [280] for theorems on perturbation of complete infinitesimal operators of semigroup-distributions. 1.22. Asymptotic Analysis. Here it is assumed that A is the complete infinitesimal operator of a contracting semigroup U(t), which has the property of strong ergodicity [i.e., (28) holds]. One considers the differential equation du = Au-~Bu,: 8--if

2176

(29)

where e is a small positive parameter, and B is a closed linear operator in E, where the set is the core of the operator A. One investigates the behavior of solutions of the equation as e § 0.

,D(A)nD(B)

Kurtz [457] assumed that the operator B is also the complete infinitesimal operator A + sB of a contracting semigroup Us(t) of class Co. On the intersection D ( B ) N K e r A = E I there is defined the operator Boo = ~A~. It is assumed that R(l/--B00)mEl for some I > 0. Then the closure of the restriction of the operator Boo to those elements of Ez, for which B00x6EL is the complete infinitesimal operator of a contracting semigroup V0(t) is defined on El, and e~0

for any xEEL. In applications to stochastic processes, frequentiy the operator Boo = 0 and V0(t) ~ I. In this case the assertion cited is developed further in [457]. The element y 6 D ( A ) N D ( B > is said to be B-adjoint to the element x6E~, if Ay = --Bx. We denote by E2 the collection of all x6E~ for which there exist B-adjoint elements. On E2 the operator Coo is properly defined by C00x = EA]Iy, where y is an arbitrary B-adjoint element to x. If for R(k -- C 0 0 ) D E2 for some ~ > 0, then the closure of the restriction of Coo to those elements of E2, for which C~xEE2 , is the complete infinitesimal operator of a contracting semigroup W0(t), defined on the space E2, and

limU,(sJ--r

for x~e~.

The semigroup U~(t/~ 2) is connected with the equation "2

s

du -g-i-=

.

-

Au +sBu.

The corresponding results are recounted in [334] for the equation

82 du = A u + s B u +s~Cu ' where B and C are bounded dissipative operators, and also a series of results of the author for the case when the operator A is the complete infinitesimal operator of the group of isometries are recounted. In this same book there are references to papers connected with applications of similar results to the theory of stochastic processes and quantum mechanics (cf. [327-332]). Kertz [418] generalized the results of [457] to the case of a semigroup with complete infinitesimal operator q(e)A + a(c)B, where q and a tend compatibly to ~ as s § 0. Turbin with coauthors clarified the asymptotic structure of the semigroup U~(t/e). In [140] it is assumed that the operators A + sB (0 ~ E < s0) generate semigroups Us(t) of class Co, and the closure of the operator Boo generates a semigroup V0(t) of class Co in KerA. Then under certain conditions on the element x6E , t

t

where t

+I 0

Fromthls, for example, under the condition of strong ergodlclty of the semigroup [U6(t)x § ~x as t + ~ and x ~ it is easy to describe the asymptotic behavior of Us(t/e). Analogous representations are obtained for the semigroup Us(t/Es) (s is an integer >I) but they have extremely complicated form. (Earlier, under more rigid restrictions than (28), the representations (30) were obtained in [61, 62], in connection with the theory of asymptotic phase enlargement of Markov processes.) In [110], under the assumption that A generates an analytic semigroup and B is subordinate to A and leaves D(A) invariant, complete decompositions of the regular and boundary-layer

2177

components of the semigroup U~(t/E) are constructed. In [139] the problem of approximation of a solution of an equation of the form (29) by solutions of regular equations lying in certain subspaces is solved. 1.23. Other B@undary Problems. Savchenko [122, 123], starting from problems of the theory of transport of radiant energy, considered the so-called semiboundary problems for the equation (I). It is assumed that the space E decomposes into the direct sum of subspaces E+ and E_, invariant with respect to a closed, linear, densely defined operator A. The restrictions of A to E• are denoted by A• and it is assumed that they generate semigroups of class Co of negative type with ~ = --I in E+ and E_; respectively. For the equation

d.--~u~-Au+Qu (0}. Effective estimates of the number of solutions are found in the case of a self-adjoint operator A in a Hilbert space. Beals [208], Kislov [59] investigated the solvability of certain inhomogeneous boundary problems for differential equations which are unsolved with respect to the derivative with self-adjoint operator coefficients in a Hilbert space, in whose boundary conditions the projection operators P+ and P- figure. Shteinvil' [157-158] considered a wide class of boundary problems for (I) with operator A, which generates an analytic semigroup. The boundary condition is given in the form

r(U(Q) =x,

(31)

where r is a linear operator, acting from some space of functions on [0, T] with values in E to the space E and having the property that for functions of the form a(t)x with scalar function a(t) and x6E one has the equation r(a(t)x) = Bx, where ~ is a number. If one assumes that the functions eltx for any X6C belong to the domain of definition of the operator r, then from the equation

t h e r e a r i s e s a f u n c t i o n ~ F ( t ) , c a l l e d the c h a r a c t e r i s t i c of the boundary o p e r a t o r F. One considers the space Fk([0, T], E) of all functions u(t) with values in E, infinitely differentiable for t6(0, ~ and k times continuously differentiable at zero. In the natural topology this space w i l l b e a Frechet space. The problem (I), (31) is said to be Hadamard proper, if for some n and k, for any x6D(A") there exists a unique solution of the problem (I), (30) in the space Fk([0, T], E) and there is continuous dependence of the solutions in the space Fk([0, T], E) on the initial data from the space D n. Suppose in the sector ~0.a=

larg(~--a)J 0) and integers k ~ 2, s ~ --I such that if a solution ~Cl[--a. a] and fECk~s[--=,~] , then ~ C ~ [ - - a ', ~'], then

][(~I-- A)x[I>C]~ [-sexp{-(a-a')lRe~]}Hx]l k--1 1.27. Equations Whlch Are Unsolvable for the Derivative. of equations of the form du

B ~-T-= Cu,

The investigation of solutions

(37)

w h e r e B and C a r e m a t r i c e s , was c a r r i e d o u t i n t h e c l a s s i c a l p a p e r s o f W e i e r s t r a s s and K r o necker. The b e h a v i o r o f s o l u t i o n s d e p e n d s on t h e p r o p e r t i e s o f t h e o p e r a t o r s h e a f XB -- C, H e r e one d i s t i n g u i s h e s t w o c a s e s : t h e o p e r a t o r XB -- C has f o r a t l e a s t one X = X0 a b o u n d e d

inverse (the regular case) and this operator is noninvertible for all ~ C (nonregular case) In the regular case the substitution u--exp{~-1~)v" {~0~=0) reduces (37) to one in which the operator C is invertible (if X0 = 0, then C is invertible at once). Osipov [104] considered the Cauchy problem for (37) with closed, linear, densely defined operators B and C, acting in the Banach space E with D(B)-~D(C) . The case when K e r B ~ 0 is interesting. One assumes that C [ K e r B ] N R ( B ) = O 9 If the function u(t) satisfies (37) for t = O, then in the regular case u(O)~R((~BmC)-tB). It is proved in [104] that if one has M II (~B-- C)-'Bx II ~ i~3-~T II x

il

for all x~O(B) and X with R e X >i ~ for initial data u'(O)=x~R((~B--C)-tB) , the Cauchy problem is solvable. Zubov and Chernyshov [53, 54] made a complete investigation of (37) when B is a closed Fredholm operator of index zero, and C is a bounded operator acting from one Banach space F to another Banach space E (cf. [60, 110] for special cases). In this case the operator X B - C is either invertible for sufficiently large X, or is not invertible for all X. In the regular case the Cauchy problem is uniquely solvable for all initial data from a certain subspace with finite defect. Its solution for all t also lies in this subspace. In the nonregular case the solution exists only for initial data satisfying a countable number of conditions, while the solution is nonunique. For the inhomogeneous equation

2182

du

B ~F=Cu+/(0

(38)

in the regular case the solution exists only for functions f(t) which have specific smoothness and satisfy some compatibility conditions with the initial data. The degree of smoothness and the number of compatibility conditions depend on the structure of the set of C-adjolnt elements to the kernel of the operator B. In the nonregular case one requires of f(t) infinite smoothness and one imposes on it a countable number of compatibility conditions. "Rutkas [120] considered (37) with arbitrary bounded operators from F to E by applying the Laplace transform and local Laplace transform (cf. [92]). He investigated the solvability questions of the Cauchy problem, the exponential boundedness of solutions, the propriety and uniform propriety, the dissipativeness of the Cauchy problem. For example, if for some integer k ~ O ~ [l(lB--C)-'Bll 0 and i n t e g r a l k ~ 0, t h e n t h e s o l u t i o n o f ( 3 9 ) - ( 4 0 ) i s u n i q u e , If the last estimate h o l d s f o r ReX ; ~ > 0, t h e n f o r y = B ( C - 1 B ) k + t z , f ~ C (t+2) (R-+; E) and f ( J ) ( 0 ) = B ( C - X B ) k - J w j , where z, w~O(C) there exists a unique solution of (39)-(40). (Without loss of generality one can assume that C is invertible.) Under the same conditions on the operators for x = (C-'B)~+2x,,/6C~ k+s~~§ /~n (O)=B(C-'B)h§ where xl, zj6D (C) , there exists a solution of (38) with initial condition u(0) = x. If (41) holds for k = 0 (analytic case) and Rel > 0, then for yEB[D(C)] and /6C(~)(R+, E) there exists a unique solution of (39)-(40), and for x = C-IBxl with x,6D(C),/EC~8)(R+,E),/(O)----B~,~D(C), there is a solution of (38) with

u(0) = x. Sauer [613] considered (38) by semigroup methods, giving up the requirement of closedhess or even closability of the operators B and C. Let F and E be two Banach spaces and B be a linear operator, acting from D(B)cF to E. A family S(t) (0 < t) of bounded linear operators from E to F is said to be a B-evolutlon, if S(O:[~cD(B) and S(t + s) = S(s)BS(t) for all s, t > 0. It follows from the definition that the operators U(t) = BS(t) form a semigroup in E. Let A be its complete infinitesimal operator. One assumes that U(t) belongs to class C0. In this case the function u(t) = S(t)y is right continuous for any g~F, and for y~[D(AB)] it is a solution of (37) with C = AB. Here the set B[D(AB)] is dense in E. Without loss of generality the subgroup U(t) can be assumed bounded, if, in addition, the operator-function S(t) is uniformly bounded, then S(t)y is continuous on (0, =) for any g%E and the limit ]iNS(t)y=Oy exists. The operator Q is a bounded linear operator from E to F, while S(t) = QU(t).

One requires in addition

B

that

the operator B have the property

;e-~'S(t)Ydt--ye-~tU (t)ydt ~O.E,

(If B is closed, then this condition holds.)

Re~ > 0).

(42)

Then one has the decomposition

D (C)==Do+ Ker B,

2183

where Do = Q[D(A)]. Let B0 and Co be the restrictions of the operators B and C to Do. Then B0 has bounded inverse operator B~ I, defined on D(A). For y6D{A) the function S(t)y is a solution of (37) with B -- B0 and C = Co. The pair of operators B0 and Co uniquely determines S(t) and is called the complete infinitesimal operator of the pair of the B-evolution of S(t). The pair is closed in the sense that from the relations x,~Do, x,-+x in F, B0x n + y and C0x n + z in E, it follows that y -- Box and z = Cox. If K e r B is closed, then the pair [defined on D(AB)] is also closed. Let be a closed pair of linear operators with common domain of definition O o c F with values in E. In order that it be the generator of a pair of uniformly bounded B-evolutions with property (42), it is necessary and sufficient that: a) B0 has bounded inverse, b) the operator CoB~ I is the complete infinitesimal operator of a uniformly bounded semigroup U(t); c) there exists a strong limit of the operators (B0 -- n-iC0) -I, which is invertible on the set E0 + R(B0), where E0 is the image of the semigroup U(t). If B0 admits a closure or U(t) is analytic, then condition c) holds automatically. An example is given in [613] of a boundary problem for the equation of thermal conductivity with the time derivative in the boundary condition, for which the operators which arise naturally do not admit closure. We note that starting from a number of physical problems, in 1973 Lagnese [459] investigated (38) in a Hilbert space with a self-adjoint operator B with nonzero kernel and an operator C for which D(~)~D(B) and D(C*)~D(B). Moreover it is assumed that K e r B is invariant with respect to C. Conditions are found for unique solvability, which include certain conditions on the smoothness of f(t) and compatibility with the initial data. Cf. also [371]. 1.28. Semigroups Which Preserve Positivity. Let E be a Banach structure with order relation ~O). Conversely, a semigroup satisfying 6) preserves positivity and is contracting, and a linear operator A which satisfies 7) is m-dissipative and satisfies 5). If R(I -- hA) = E for % > 0, then 7) is equivalent with 8) [Iz+H~0. A linear o p e r a t o r A w h i c h satisfies 8) is said to be dispersive. Thus, we arrive at Sato's theorem [603]: in order that the linear operator A be the positivity-preserving complete infinitesimal operator of a contracting semigroup in E, it is necessary and sufficient that A be dispersive and D(A) = E = R(I -- XA) for X > 0. Phillips (cf. [63]) had already obtained a different characterization in 1962. Sato [603] found explicit conditions for dispersivity in various function spaces. Thus, in the real space LP(S) with natural order relation the dispersivity of A means that

S (Ax (s))x (S)P-lds-~O VxGD(A)

x(s)>O

(here S is a space with measure, I ~< p < l). In the space of continuous real functions C(S), where S is compact or C0(S), where S is locally compact, with the natural order relation and

2184

usual norm 11 [I~ the dispersivity of A means that u at some point so such that x(s0) = ilx+li~; now if R(I -- XA) = D(A)(X > 0), then (Ax)(s0) ~ 0 at any such points. In an arbitrary Banach structure a sufficient condition for dispersivity is the condition +~0 Vx~D(A), where the functional < , >+:E • E + R is defined by (2.10) (this follows from a result of Phillips, cf. also [254, 436]). It follows from the results of Crandall and Tartar [302] that for a semigroup U(t) of class Co, the conditions U(t)E+ cE~ and IIu(t)]l < I are equivalent in the following two cases: a) E = LZ(S) and

S(Ax)(s)ds=O VxED(A); S

b) E = L=(S) o r

E=C(S),

lED(A)

and A1 = 0.

C a l v e r t [ 2 5 5 ] showed t h a t a d i f f e r e n t i a l o p e r a t o r A w i t h D ( A ) ~ C ~ ( f l ) , where f l c R n , c a n be d i s p e r s i v e i n any o f t h e s p a c e s L P ( f l ) , 1 ~ p ~ ~ o n l y i f t h e o r d e r o f A i s no h i g h e r t h a n two; f o r s u c h o p e r a t o r s s u f f i c i e n t conditions for dispersivity are found (Calvert used the term "T-accretive" for --A) in terms of the coefficients. Let E be the complex space Li(s; C). We set E§ x(s)GR+ a.e. on S} and Ixl(s) = Ix(s) l (the modulus of the complex number) for xEE ; we shall assume the relation x > y only for yELi(S; R ) = E R . If the semigroup U(t) in E preserves positivity, i.e., U ( Q E + c E + , then U(t)ER~ER and conditions I)-5) hold, and if llu(t)ll ~ I, then (for x6ER ) 6) and 7) also hold. Sato's theorem remains valid under the additional hypothesis R(L, A ) E R C E R for ~ > 0, if dispersivity is understood as a property of the restriction of A to ER. If A is self-adjoint and semibounded above, then it follows from U (1)E+~E+ that 0 ~ = U (t)xEE+ for 0~Ax~E+. For self-adjoint A with (Ax, x) ~ 0 simple tests for U(t) to preserve positivity (Beurling--Denis tests) are given in [118, Theorem XIII.50]. One of the tests is this:

x~D ( A ~/2)~IxlED (A1/2).

and

(A t/~ Ix], A '/21xl) ~ ( A , : x , A 't~x).

Another test, found by Simon [621], is that zED one has the abstract Kate inequality

(A,/~x, where (sgn z)(s) is equal to:

z(s)/Iz(s)l

(A)*IzI~D (Al/9 and for O..<x6D (A'/2).- zs (A)

A,,=Izl) > Re (x sgn

z, Az),

for z(s) ~ 0; 0 for z(s) = 0.

Some special classes of positivity-preserving semigroups were studied in [116, Theorem X.55], [118, Sec. XIII.12 and the supplement to it], [284, 327, 331, 333, 336, 337, 391, 423, 471]. In [118] in particular, self-adjoint semigroups in E = Li(S) strengthening positivity are characterized: (U(t)x)(s) > 0 a.e. on S for 0~AxEF+. Bratteli, Kishimoto, and Robinson [226, 421] showed that if the self-adjoint operator A in Li(S) generates a contracting positivity-preserving semigroup U(t), and f:R+ § R+ is such that/6C=(0, ~), (--I),/t"+')(r)>O for r > 0, n = 0, I .... and/(O)• 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>

~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)-- 0); d) (~z, ~ x ) • x ) + for ~, ~6R, ~ > ~ 0 ; e) ( z + ~ x , x > + = (z, x ) • =llxll2 for =ER ; f ) ( y , x > + ( z , x > _ ~ < 0, i.e.,

(16)

the equation z -- ~Az = x is

U N R(I--LA)"~b(A).

(17)

~>00+ 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[~[. 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~ 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 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.. 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~ 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,~)].. 0 , ~ E , we have:

(([ - ~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 -- 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-. 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-- 0 f o r any

of u to [0, t],

xEWNDo, D 0 ' = U D~

C0~})~

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.. 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