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Journal of Mathematical Sciences, Vol. xxx, No. y, 2003

DIFFERENTIAL EQUATIONS IN BANACH SPACES I I. THEORY OF COSINE OPERATOR FUNCTIONS

V. V. Vasil'ev and S. I. Piskarev

UDC 517.986.7; 517.983.6 Dedicated to Vasil'eva Aleksandra Vladimirovna and Piskareva Lidiya Ivanovna, our mothers.

INTRODUCTION More than 13 years have passed since the fundamental survey [16] was prepared, which, as the author intended, should b e the first part of a large work devoted to abstract differential equations and methods for solving them. However, the troubles b eing in the Russian science during the whole this p eriod have influenced also on the authors, and instead of two years supp osed, the preparation of the second part has occupied considerably more time. During the last 10 years, the work in the field of differential equations in abstract spaces was very active (in foreign countries), and every year several b ooks and a heavy numb er of pap ers devoted to this direction app ear in the world (of course, the most of them are not available for the Russian reader). At the same time, only two b ooks [33, 75] of such a typ e app eared b eing translated by the authors of the present survey and [20], which were edited by Yu. A. Daletskii. Therefore, the work whose second part is prop osed to the reader will b e undoubtedly useful for the Russian reader. Its style coincides with that of [16], i.e., the material is often presented without proofs, and the main attention is paid to the structure of presentation, although we present certain proofs from foreign sources that are almost inaccessible for Russian readers. From our viewp oint, this allows us to demonstrate clearly the philosophy, to describ e the results obtained, and to indicate the main directions of the development of the theory in the framework of a limited volume of the survey.
Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory. Vol. 113, Functional Analysis, 2002. 1072­3374/03/xxxy­0059 $ 27.00 c 2003 Plenum Publishing Corporation 59


Moreover, the authors have prepared a separate edition of the bibliographical index [18], which can serve as a sufficiently complete source of information ab out the theory of differential equations in abstract spaces during the recent years. The main ob ject of the study in this part are second-order differential equations that are presented very little in Russian literature up to now. Here, we can only mention the pap er [20] written in accordance with the own interests of the author, which does not pretend to the exhausting description of all asp ects of the theory. Moreover, the material of the present survey includes the presentation of the abstract Cauchy problem for first- and second-order equations that is not considered in [17]. As was already mentioned in [16], the philosophy of the theory of C0 -cosine op erator functions is very close to the op erator semigroup theory and often is develop ed in parallel to it. Therefore, the reader easily draws analogies b etween the material presented here and that presented in [16]. At the same time, the theory of C0 -cosine op erator functions considerably differs from the op erator C0 -semigroup theory. First of all, these distinctions concern with the prop erties inherent to the corresp onding parab olic and hyp erb olic partial differential equations. We now present the main notation, a certain part of which was already introduced in [17] and which is also used here. The set of natural numb ers is denoted by N, N0 := N {0}, the set of integers by Z, the set of reals by R, and the set of complex numb ers by C. A tuple of numb ers 1, 2, ..., m, m N, is denoted by 1,m, the real semiaxis (0, ) by R+ , and [0, ) by R+ . We denote by E a Banach space over the field of complex numb ers with the norm · . For a Hilb ert space with the inner product (·, ·), we use the symb ol H . The b oundary of a set is denoted by , the interior of the set by int(), the closure in the strong top ology by , and, for example, the closure in the weak top ology by w-cl-(). As usual, the space dual to E is denoted by E , with elements x , y , ..., and the value of a functional x E at an element x E is written as x, x . The domain and range of an op erator A will b e written as D(A) and R(A), resp ectively, and the null-space (kernel) as N (A). The set of linear op erators acting from D(A) E into E is denoted by L(E ), and the set of linear continuous op erators by B (E ). Closed linear op erators with dense domain (D(A) = E ) in E are distinguished into the set C (E ) L(E ). In the case where op erators act from one space E into another F , we write L(E, F ) and B (E, F ), resp ectively. The linear variety D(A) endowed with the norm x denote it by D(A).
A

:= x + Ax in the case of a closed linear op erator b ecomes a Banach space; we

60


We use the traditional notation for the resolvent set (A) and sp ectrum (A) of an op erator A; as usual, the latter is divided into the p oint sp ectrum P (A), the continuous sp ectrum C (A), and the residual sp ectrum R (A). Sections 2.2, 2.4, 6.1, 6.3, 7.1­7.2, 9.2, 10.2, 10.3, 12.1­12.4, 12.6­12.10, 13.3­13.6 and Chapter 14 were written by S. I. Piskarev, the other part of the text was prepared by the authors in collab oration.

Chapter 1
CAUCHY PROBLEM AND RESOLVING FAMILIES Before considering the theory of C0 -cosine op erator functions, we describ e the general picture of the statement of the well-p osed Cauchy problem in a Banach space. As is easily noted, a natural generalization of the concept of solution leads to more general families: integrated semigroups and C -semigroups. These families will b e considered in a forthcoming survey in more detail. 1.1. Cauchy Problem for a Complete Differential Equation Let E b e a Banach space, and let A0 ,A1 , ..., Am-1 b e closed linear op erators on E , i.e., Ak C (E ),k 0,m - 1. In the Banach space E , let us consider the following abstract Cauchy problem of order m: m-1 (m) u (t) = Ak u(k) (t), t R+ , (1.1) k =0 (k) u (0) = u0 , k 0,m - 1, m 2. k Definition 1.1.1. A function u(·) C m (R+ ; E ) is called a classical solution of problem (1.1) if u(k) (t) D(Ak ), Ak u(k) (·) C (R+ ; E ) for t R+ , k 0,m - 1, and relations (1.1) hold. As in [17], we define the propagators Pj (t), j = 0,m - 1, which give a solution of the Cauchy problem (1.1) with initial conditions uj (0) = jk u0 (jk is the Kronecker symb ol), i.e., uj (t) = Pj (t)u0 . j j Definition 1.1.2. The Cauchy problem (1.1) is said to b e uniformly wel l-posed if Pk (·)x C k R+ ; E ,
(k -1) Pm-1 (k )



(t)x D(Ak ),t R+ , and

(k -1) Ak Pm-1

x C (R+ ; E ) for any x E and k 0,m - 1.

In the general case, in the Banach space E , problem (1.1) has b een studied incompletely. In particular cases, some concepts are introduced, which will b e considered in the next chapters in more detail. Definition 1.1.3. We say that an op erator A generates times integrated semigroup with 0 if (, ) (A) for a certain R and there exists a strongly continuous function S (·) : [0, ) B (E ) such that S (t) Met , t R+ , with a certain constant M 0 and (I - A)-1 =
0

e-t S (t) dt 61


for all > max{, 0}. The family S (·) itself is called an times integrated semigroup. Theorem 1.1.1 ([118]). Let m 2, and let Am-1 generate an r times integrated semigroup. Assume
m that D(Ai ) D(Am-1 ) for al l i 0,m - 2, and, moreover, Ai D(Am-1 ) D(Ai--1+r m m-1 j =0 +2

) for i m -

r - 1. Then problem (1.1) has a unique exponential ly bounded solution for u0 -1 D(Ar+11 ), u0 m m- k D(Ar -1 Ai ), k 0,m - 2, and for certain constants c, > 0, we have the fol lowing estimate for m

t R+ :
r +1 m-2 m-2 r m-2

u(t) + Am-1 u(t) cet
l=0

Al -1 u0 m n

-1

+
k =0 i=0 l=0

Al -1 Ai u0 + m k
k =0

( u0 + Am-1 u0 ) . k k

In [193], this theorem was slightly changed by extending the set of initial data and by the absence of the exp onential b oundedness.
m

Denote P () :=
i=0

i Ai with the domain D(P ) :=
(k )

m-1 i=0

D(Ai ).

Theorem 1.1.2 ([194]). The propagators Pk (·), k 0,m - 1, are norm-continuous for t R+ iff there exists 0 R+ such that lim (0 + i )m-1 P
-1 -1

|| ||

(0 + i ) = 0, k 0,m - 1.

(1.2) (1.3)

lim

(0 + i )k

P

-1

(0 + i )Ak = 0,

Corollary 1.1.1. Let conditions (1.2)­(1.3) hold. Then for each k 0,m - 1, the operator Pk (t) is norm-continuous for t R+ . The case of time-dep endent Ak = Ak (t), t R+ , was considered, e.g., in [227, 228]. In [295], the conditions for the existence of a unique entire solution of problem (1.1) were presented. Consider problem (1.1) with Ak C (E ) for all k 0,m - 1; let D(A0 ) D(Ak ) for k 1,m - 1. Theorem 1.1.3 ([226]). Under the assumptions described above, the fol lowing conditions are equivalent for the Cauchy problem (1.1): (i) the operator A0 generates a C0 -semigroup; (ii) for any u0 ,u1 , ..., um-1 D(A0 ), the Cauchy problem (1.1) has a unique solution u(·) C
(m-1)

(R+ , D(A0 )).

The following theorem on the uniform stability of problem (1.1) holds. 62


Theorem 1.1.4. Let an operator A0 generate a C0 -semigroup, and let u(·) be a solution of the Cauchy problem (1.1) with initial conditions uk (k 1,m - 1; l ), uk D(A) for k 1,m - 1, uk 0 in E . l l l Then ul (·) 0 uniformly on any compact set. Theorem 1.1.5 ([226]). Let A0 generate a C0 -semigroup, Ak C (E ), D(A) D(Ak ), and let be such that for Re > , there exists a generalized resolvent (pencil resolvent) R := R(; A0 , ..., Am-1 ) = (m I - m-1 Am-1 - ... - A1 - A0 )-1 (such an always exists!). Also, on D(A), let the relation Ak R = R Ak (k 1,m - 1; Re > ) hold. Then problem (1.1) is uniformly wel l-posed, and its solution has the form
m-1

u(t) =
k =0

Qm-1-k,
m-1

m-1

(t)uk ,

(1.4)

where the operator-valued functions Qm-1-k, ator semigroup

(t) are strongly continuous families composing the oper Q0,m-1 (t) . . . Qm-1,m-1 (t) I 0 0 I ... 0 , t R+ ,

Q (t) ... 0,0 . .. . G(t) = . . Qm,0 (t) ... with the generator = A0 A1 . . . . . .

... .. .. . . 0 . .. .. . . . . 0 ...

Am-1 0

0 0 . I 0

Now let us consider the Cauchy problem for the following equation of order m having the sp ecial form:
m j =1

d - Aj u(t) dt

d - Am ... dt

d - A1 u(t) = 0 dt

(1.5)

with initial conditions u(k) (0) = u0 , k and op erators Aj C (E ),j 1,m. Definition 1.1.4. The Cauchy problem (1.5) is said to b e uniformly wel l-posed if the following conditions hold: 63 k 0,m - 1, m 2,


(i) there exists a solution of the Cauchy problem (1.5) for u0 , ..., um-1 taken from a certain dense set D in E ; (ii) for u0 , ..., um-1 D, the solution of the Cauchy problem (1.5) has the prop erty
k j =1

d - Aj u(t) C dt

m-k

(R+ ,E )

(1.6)

for k 1,m - 1; (iii) the uniform stability of the solution of (1.5) is complemented by the following condition on any compact set: the convergence
k j =1

d - Aj up (0) 0 dt

implies the convergence
k j =1

d - Aj up (t) 0 dt

uniformly on each compact set in R+ (here, k 1,m - 1; p ). Theorem 1.1.6 ([20]). In the Cauchy problem (1.5), let Aj C (E ) (j 1,m), let the intersection of the resolvent sets (Aj ) of the operators Aj be nonempty, and let the set ~ D {D(Ai1 ...Aim ) : ik 1,m} (1.7)

be dense in E . Then problem (1.5) is uniformly wel l-posed iff Aj generates a C0 -semigroup for each j 1,m. Theorem 1.1.7 ([20]). Under the conditions of Theorem 1.1.6, let the operators Aj generate C0 semigroups for j 1,m, and, moreover, let these semigroups commute: exp(tAi )exp(sAj ) = exp(sAj )exp(tAi ), Then problem (1.5) is wel l posed. Theorem 1.1.8. For the Cauchy problem (1.5), let the conditions of Theorem 1.1.7 hold, and let 0 m d (Ai - Aj ) for al l i = j . Then the condition w(t) N for t R implies the relation - Ai i=1 dt m d - Ai , t R. wi (t), where wi (t) N w(t) = dt i=1 Condition (1.8) in Theorem 1.1.7 can b e replaced by a numb er of conditions imp osed on the domains R(Ai - Aj ) for i = j . Let 1,1 ,2 , ..., p 64
-1

t, s R ,

i, j 1,m.

(1.8)

be roots of pth degree of the unity, i.e., k = e

2k p

i

.


Definition 1.1.5. A C0 -function of the Mittag-Leffler type with a parameter p is a function M : C B (E ) having the following prop erties:
p-1

(i)
k,l=0

M(k t + l s) = p2 M(t)M(s) for any t, s R;

(ii) M(0) = I ; (iii) the family of op erators T (t) M(k t + l s), k, l 0,p - 1, with a fixed s R is strongly continuous in t R. For the Mittag-Leffler C0 -function with a parameter p, the p-generator A is defined by the relation Ax = s- lim p!
t0

M(t) - I x tp

for those x at which the limit exists. It is known that the generator of the Mittag-Leffler C0 -function with a parameter p is a linear closed densely defined op erator, and the following relation holds for any x D(A): dp M(t)x = AM(t)x = M(t)Ax, dtp (0) = 0 for k 0,p - 1 .

and, moreover, M(k

)

For the Mittag-Leffler C0 -function with a parameter p, the p erturbation theorems of the Philips­ Miyadera typ e hold (see [17]). Theorem 1.1.9 ([32]). Let A generate a Mittag-Leffler C0 -function with a parameter p, and let M(t) Met , t R. Then for any B B (E ), the operator A + B generates a Mittag-Leffler C0 -function with the parameter p. Proposition 1.1.1. The Mittag-Leffler C0 -function with a parameter p is a C0 -group of operators in the case of p = 1, and in the case of p = 2, it is a C0 -cosine operator-valued function. A Mittag-Leffler C0 -function with a parameter p has a bounded generator A B (E ) for p 3. In the simplest case m = 2, for example, the following theorems hold for problem (1.1). Theorem 1.1.10 ([226]). Let the Cauchy problem (1.1) be uniformly wel l-posed for m = 2, and let P1 (t)E D(A1 ) for t R+ . Then A0 generates a C0 -cosine operator-valued function on E . Theorem 1.1.11 ([226]). Let A1 B (E ). Then the Cauchy problem (1.1) is uniformly wel l posed for m = 2 iff A0 generates a C0 -cosine operator-valued function on E . However, in the general case, even for m = 2, the Cauchy problem (1.1) turns out to b e very complicated. First, in [135], H. O. Fattorini has presented an example of the Cauchy problem (1.1) that has a solution for m = 2, but this solution is not exp onentially b ounded. Second, in contrast to the 65


Cauchy problem for m = 1, the case m = 2 admits more flexibility in the sense of well-p osedness of its statement. As one of the variants, let us present the approach coming back to H. O. Fattorini. The constructions used in proving these theorems practically completely rep eat the techniques used in proving the assertions concerning C0 -cosine and C0 -sine op erator-valued functions (see also [30]). Consider the Cauchy problem u (t)+ Bu (t)+ Au(t) = 0, with A, B C (E ). Definition 1.1.6. We say that the op erators A and B generate M, N -families of operators on E if the following conditions hold: (i) M (t)and BN (t) are strongly continuous in t R+ , and the function N (t)x is strongly continuously differentiable in t R+ for any x E ; ^ (ii) the set E = {x E : M (t)x is strongly differentiable in t R+ , and BM (t)x is continuous in t R+ } is dense in E ; ^ (iii) the op erator A = -M (0) is B -closed, and Bx = -N (0)x for all x E ; (iv) M (0) = N (0) = I and N (0) = 0; ^ (v) M (t + s)x = M (t)M (s)x + N (t)M (s)x for all x E and t, s R+ ; (vi) N (t + s) = M (t)N (s)+ N (t)N (s) for all t, s R+ . Theorem 1.1.12 ([30]). Let A and B generate M, N -families. Then ^ (i) A is closed, D(A) D(B ) E D(B ), and D(A) D(B ) is dense in E ; (ii) the families M and N are uniquely defined by the operators A and B ; (iii) M (0)x = 0 for al l x D(M (0)); (iv) M (t)x = -N (t)Ax for al l x D(A) and t R+ ; ^ (v) N (t)x = M (t)x - N (t)Bx for al l x E and t R+ ; (vi) N (t)x + N (t)Bx + N (t)Ax = 0 for al l x D(A) D(B ) and t R+ ; ^ ^ (vii) for al l x E and t R+ , the element N (t)x E , N (t)x D(A), and N (t)x - x + BN (t)x + AN (t)x = 0; (viii) for al l x E and t R+ , the element A
t 0 t 0

t R+ ,

u(0) = u0 ,

u (0) = u1 ,

(1.9)

N (s)xds D(A) and N (t)x - x + BN (t)x +

N (s)xds = 0; (ix) for al l x E and t R+ , the element
t 0

^ N (s)xds E , M (t)x D(A) and M (t)x + BM (t)x +

AM (t)x = 0; 66


(x) for al l x D(A)
t

^ ^ E and t R+ , the element M (t)x - x E ,

t 0

M (s)xds D(A), and

M x + B (M - I )x + A M (s)xds = 0; (xi) there exist constants C, 0 such that M (t) , N (t) , BN (t) , N (t) Cet , ^ and for al l x E , there exist constants C, 0 such that M (t)x , BM (t)x C (x)et , (xii) the operator 2 + B + A is closable for al l C; (xiii) there exists a constant R+ such that (A, B ) for al l with Re > and ()x := (2 I + B + A)-1 x =
0 0

t R+ ,

t R+ ;

e-t N (t)xdt for

for

x E;

()(I + B )x =
0

e-t M (t)xdt

^ x E;

(xiv) 2 ()x x as for al l x E . The following analog of Theorem 2.1.1 from [17] holds. Theorem 1.1.13 ([294]). Operators A and B generate M, N -families iff the fol lowing conditions hold: (i) the operators A and B are closed, and D(A) D(B ) is dense in E ; (ii) there exist constant C, 0 such that (A, B ), and for Re > , the operator ()A is closable and (())(k) , (B ())(k) , (()B )(k
)



Ck! (Re - )k

+1

for

k N,

Re > ,

(1.10)

where ()B is a bounded extension of the operator ()B with the domain D(A) D(B ) and (·)(k) is the derivative of order k in . In the case where A and B commute, instead of the estimate with the op erator B in (1.10), it can b e, e.g., dk (I - A)() dk (see [30]). If A = 0, then A and B generate M, N -families iff B generates a C0 -semigroup. Let D(B ) D(A), and let (B ) = . If (0 I - B )-1 A has a b ounded extension for a certain p oint 0 (B ), then A and B generate M, N -families iff B generates a C0 -semigroup. 67 Mk! (Re - )k
+1

,kN

0


Proposition 1.1.2 ([294]). Let B be dominated by A with exponent 0 1, i.e., D(A) D(B ) and Bx C x
1-

Ax



for al l x D(A), and let A and B commute. If -A generates a C0 -cosine

operator-valued function and (2 I + A)-1 C ||-2 for Re > , then A and B generate M and N families. Now let us consider an analytic extension of a solution of Eq. (1.9) to the sector ( ) = {z C : z = 0, | arg z | < }. Theorem 1.1.14 ([294]). For given , 0, the fol lowing conditions are equivalent: (i) the Cauchy problem (1.9) is uniformly wel l posed, the families M, N can be analytical ly extended to the sector ( ) = {z C : z = 0, | arg z | < }, for any z we have the embedding N (z )E D(B ), and BN (·) is analytic in ( ). Moreover, for each (0, ), x E , lim N (z )x = 0, lim BN (z )x = 0, lim M (z )x = x, lim N (z )x = 0,
z z 0 z z 0 z z 0

z z 0

and there exists a constant C > 0 such that N (z ) , BN (z ) , M (z ) C e
Re z

for al l

z ( );

(ii) the set D(A) D(B ) is dense in E . For each (0, ), there exists M > 0 such that for ( , ) = C : = , | arg( - )| < + 2 ,

the operator () := (2 + B + A)-1 B (E ) exists, the operator ()A is closable, and () M , | - | B () M , | - | ()B0 M , | - |

where B0 B with D(B0 ) = D(A) D(B ). Moreover, in this case, we have N (z )+ BN (z )+ AN (z ) = 0, M (z )+ BM (z )+ AM (z ) = 0,

where AN (z ) and AM (z ) are analytic in ( ). For each (0, ), lim M (z )x = 0 for any x D(A).

z z 0

The existence and uniqueness of solutions of Eq. (1.9) under certain "hyp erb olic" conditions is considered in [230]. Problem (1.9) in the case of nonlinear B was considered in [188]. 68


1.2. Cauchy Problems for Differential Equations of the 1st and 2nd Orders In this section, we present certain statements of the Cauchy problem for equations of the first and second orders. Equations of the first order were already considered in the pap er [17] but, however, only in connection with C0 -semigroups on the space E . As was already noted, different statements of the Cauchy problem are p ossible. We now present certain arguments that show that a solution is given not by C0 -families on the whole space E . Definition 1.2.1. An integrated solution of the Cauchy problem u (t) = Au(t), u(0) = x, (1.11)

is a continuously differentiable function v (·) : R+ E such that dv (t) = Av (t)+ x, v (0) = 0. (i) v (·) C ([0, ); D(A)) and (ii) dt Definition 1.2.2. Denote by Z (A) the resolving set of the op erator A, i.e., the set of all x E for which the Cauchy problem (1.11) has an integrated solution. Proposition 1.2.1. Let Z (A) be the resolving subspace endowed with the family of seminorms x
a,b

= sup u(t) ,
t[a,b]

a, b R+ .

(1.12)

Then Z (A) is a Frech´ space and T (t)x = u(t) is a local ly equicontinuous semigroup generated by et the operator A|Z
(A)

.

Recall the definition of entire vectors, which is equivalent to [17, Definition 3.1.3]. Definition 1.2.3. Denote by Uc (A) the set of entire vectors of an op erator A, i.e., the set of x D(A ) such that for any t R+ ,


Ak x
k =0

tk < . k!

Proposition 1.2.2 ([117]). Any linear closed operator on E with the resolvent set (A) containing the semiaxis (, ) generates a C0 -semigroup on a certain maximal subspace in E . Proposition 1.2.3 ([117]). For any closed linear operator A, there exists a maximal Frech´ space Z (A) et such that Z (A) E and the Cauchy problem (1.11) is automatical ly wel l posed on Z (A). As is seen from this prop osition, the requirement of existence of a C0 -semigroup is very restrictive. At the same time, namely for C0 -families of op erators, the technical tools for studying approximate methods are most well elab orated. 69


Theorem 1.2.1 ([117]). Let A C (E ). Then Uc (A) = {x E : problem (1.11) has an entire solution} and Uc (A) Z (A). Theorem 1.2.2 ([85]). Let A generate an analytic C0 -semigroup on E . Then Uc (A) = Z (A), and, moreover, the equality holds topological ly and algebraical ly. As is known, a self-adjoint op erator A = A 0 on a Hilb ert space H generates an analytic C0 semigroup, as well as a C0 -cosine op erator function. Moreover, in this case, by the Stone theorem, the op erator iA generates an unitary C0 -group on H . At the same time, the practical problems often require the omitting of the self-adjointness of the op erator and Hilb ert prop erty of the initial space. Therefore, to reveal whether a concrete op erator generates a C0 -semigroup or not is not a simple but often a complicated indep endent problem. Here, we present examples showing when the verification of generation of C0 -families on the Banach space E is p ossible. Theorem 1.2.3 ([43]). For A C (E ) to be a generator of an analytic C0 -semigroup, it is necessary and sufficient that there exist numbers , , and > 1 such that the fol lowing inequality holds for al l M || ; (Re ) +-1 (Re - ) moreover, the fol lowing representation holds for this semigroup: ( I - A)-1 exp(zA) = - for z z C : Im z < Re z cot 2 . 2i
+i -i

Re > 0 :

ezµ µ-1 (µ I - A)-1 dµ


Let Rd b e a certain domain. Denote by C() the space of uniformly continuous b ounded functions on with the norm v (·)
C()

= sup |v (x)|,
x C ()

and let C () = {v (·) : (·)v (·) C(), (t) 0}, v (·)

= (·)v (·)

C ()

.

Example 1.2.1 ([43]). Let = [0, 1], and let Av = v (·) with D(A) = {v (·) : v C (),Av in C ([0, 1]). C (),v (0) = v (1) = 0}. Then A H , 2 At the same time, for the op erator A0 v = v (·) with D(A0 ) = {v (·) : v C ([0, 1]),v(0) = v (1) = 0}, we have A0 H(0, ) on C0 ([0, 1]) = {v (·) : v C ([0, 1]),v(0) = v (1) = 0}. Finally, the op erator A v = v with D(A ) = {v (·) : (·)v (·) C ([0, 1]), A v C ([0, 1])} generates an analytic C0 -semigroup with the estimate exp(tA ) 70
C ([0,1])

e- t ,
2

t R+ .


d

Example 1.2.2 ([43, 268]). The Laplace op erator v =
j =1

generator of an analytic C0 -semigroup on E = W

2,p

(Rd ).

2 v (x) , x Rd , for 1 < p < gives a x2 j

Here, it is appropriate to recall (see [167]) that the op erator i does not generate a C0 -semigroup on Lp (Rd ) for p = 2. Moreover, the op erator generates a C0 -cosine op erator function iff p = 2 or d = 1. Also, we note (see [126]) that the op erator -(i)1/2 does not generate a C0 -semigroup on L1 (R1 ). Example 1.2.3 ([126]). The Laplace op erator on Lp (Rd ), 1 p < , generates an times integrated 11 cosine op erator function for > (d - 1) - . 2p ~ Example 1.2.4 ([125]). Let A b e a strongly elliptic op erator on Rd . Denote by Tr (·) the C0 ~ semigroup generated by the op erators A with the Dirichlet or Neumann conditions on the b oundary in Lr (). Then there exists an analytic C0 -semigroup Tp (·) with the angle /2 in Lp () such that Tp (t)x = Tr (t)x for all x Lp () Lr (). ~ Example 1.2.5 ([126]). Let 1 < p < , let the op erator Ap generate a semigroup Tp (·), and let µ() < ~ . Then Ap generates an times integrated cosine op erator function on Lp (Rd ) for > d1 1 1 - +. 22 p 2

Example 1.2.6 ([35]). Let = R+ . The op erator (Av )(x) = v (x) +

a c v (x) + v (x) generates an x x analytic C0 -semigroup for a, c R and D(A) = {v (·) : v C(R+ ),Av C (R+ )} iff c 0. dv(x) d (1 - x2 ) dx dx with D(A) =

Example 1.2.7 ([43]). Let = [-1, 1]. Then the op erator (Av )(x) = {v (·) : v C(), Av C()} generates a C0 -semigroup.

Example 1.2.8 ([43]). Let (Av)(x) = v (x) + q (x)v (x), x R. Denote by Sp the Banach space of Stepanov functions, i.e., the space of functions on R with the norm v (·)
p,l
1 p

1 = su p l xR

x+l x

|f (s)| ds
p

, l > 0, p 1.

It is known that for different l, the norms are equivalent. For the op erator A H(, ) on C (R), it suffices, and in the case q (x) c > -, it is necessary that q (·) S1 . Denote H
-1 0 -1

2 (z ) =

e-s

2

-2sz

ds and 1 =- 2i
+i

H (t)(µ I - A)
2

H
-i

-1

( )et (2 I - A)-1 d.

71


Theorem 1.2.4 ([35]). For A C (M, ), it is necessary and sufficient that this operator be the generator of an analytic C0 -semigroup, and for each t [0,T ], the estimate H (t)(µ2 I - A)-1 M (t) hold uniformly in (0,), > 0. In this case, C (t, A) = s- lim (H (t)+ H (-t))(µ2 I - A)-1 ,
0

(1.13)

t R+ .

Example 1.2.9 ([35]). Let the op erator A f (x) = we have H (1)(2 I - A)-1 f >

b e given as in Example 1.2.6. Then for the function 0 x-1 2 1 if x [0, 1), if x [1, 1+ 2 ), if x [1+2, )

M , and, therefore, by Theorem 1.2.4, such an op erator A does not generate a C0 -cosine op erator function.


Example 1.2.10 ([43]). Consider the op erator A from Example 1.2.6 but on the space C (R+ ) with (x) = xex , x R+ , R. Then H (t)(2 I - A)-1 v
C

Me|

µt|

v

C

, and, therefore, A C (M, ).

Example 1.2.11 ([43]). Let A b e given as in Example 1.2.8. For A C (M, ) on the space C (R) it is sufficient, and in the case q (x) c > -, it is necessary that q (·) S1 . Consider the problem 2 u(t, x) 2 u(t, x) u(t, x) = xm + xm-1 2 2 t x x where m > 0, x > 0, and initial conditions lim u(t, x) = (x), lim , C
t0 (2) t0

(1.14)

u(t, x) = (x) for any x R+ , where t (R+ ) E , and E is the Banach space of functions C (R+ ) such that lim (x) = lim (x) =
x0 x xR+

0 with the norm = sup |(x)|.

Definition 1.2.4. Problem (1.14) is said to b e uniformly wel l-posed if for any compact set J R+ , we have max |u(t, x)| M ( + ).
tJ

Example 1.2.12 ([43]). For the op erator (Av )(x) = xm v (x)+ xm-1 v (x) with D(A) = {v E : v 2 - (1 + )m < 1. C (2) (R+ ) E, Av E } on the space E just describ ed, condition (1.13) holds for 0 2-m m 2 - (1 + ) For 1, the op erator A does not generate a C0 -cosine op erator function. 2-m 72


2 d 2k 2 defined on the space C (R ), the op erator i=1 xi on C (Rd ) generates a C0 -cosine op erator function iff d 4k +1.
d

Example 1.2.13 ([43]). For the op erator =

+1

Here, in connection with Example 1.2.13, it is relevant to note that for any A C (M, ), every polynomial P (A) = A2m+1 +
2m k =0

ck Ak , ck R, generates an analytic semigroup.

Moreover, in the case of an even m, the op erator (-1)m+1 Am does not necessarily generate a C0 cosine op erator function. Proposition 1.2.4 ([177]). Let A C (M, 0). Then for any k N, the operator (-1)k A2 generates an times integrated cosine operator function for a certain > 0. Moreover, (-1)k A2 H(, /2). Theorem 1.2.5 ([231]). Let A H 0, (-1)m+1 Am + B1 Am-1 + ... + Bm-1 , and let m N. Let Bi B (E ), i 1,m. Then the operator 2 A + Bm generates an analytic C0 -semigroup with the angle . 2
k k

An analogous assertion is not true for C0 -cosine op erator functions! Theorem 1.2.6 ([178]). Let {Aj }m be resolvent commuting operators, and let Aj C (M, 0), j 1,m, j =1
m m

be given on E . Define A0 =
j =1

Aj , D(A0 ) =
j =1

D(Aj ). Then the operator A0 is closable and A0

m-1 . 2 Moreover, this times integrated semigroup satisfies the estimate S (t) M t , t R+ for certain m-1 . M > 0 and 2 generates an times integrated cosine operator function for Theorem 1.2.7 ([177]). Under the conditions and notation of Theorem 1.2.6, the operator iA0 generates a times integrated semigroup for > m/2. Proposition 1.2.5 ([178]). Under the conditions of Theorem 1.2.6 and an additional assumption that the space E = H is a Hilbert space, A0 generates a C0 -cosine operator function. Proposition 1.2.6 ([178]). Let the conditions of Theorem 1.2.6 hold and, additional ly, let E = H be a Hilbert space and a Banach lattice, and, moreover, let C (t, Aj )H+ H+ , t R, j 1,m. Define Ck (t) =
t 0

C (t, A ) 0

(t - s)k-1 C (s, A0 ) ds for k 1, (k - 1)! for k = 0.

Then Ck (·) are positive for k

m . 2 73


Example 1.2.14 ([162]). Let E = Lp (Rd ), 1 < p < . Then the Laplace op erator with D() = 11 W 2,p (Rd ) generates an times integrated cosine op erator function iff (d - 1) - . 2p In [296], concrete differential op erators are studied for revealing whether or not they generate a well-p osed Cauchy problem for a complete second-order equation. 1.3. Resolvent Families For functions k(·) Lp (R+ ) and g(·) W loc
t 1,1

([0,T ]; E ), let us consider the Volterra equation t [0,T ]. (1.15)

u(t) = g(t)+
0

k(t - s)Au(s) ds,

Definition 1.3.1. A strongly continuous family of b ounded linear op erators {R(t) : t R+ } on E is called a resolvent fam