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Abstract and Applied Analysis

Volume 2013 (2013), Article ID 275494, 37 pages

http://dx.doi.org/10.1155/2013/275494

## On the Behaviour of Singular Semigroups in Intermediate and Interpolation Spaces and Its Applications to Maximal Regularity for Degenerate Integro-Differential Evolution Equations

Dipartimento di Matematica, Università degli Studi di Bologna, Piazza di Porta San Donato 5, 40126 Bologna, Italy

Received 6 May 2013; Accepted 25 June 2013

Academic Editor: Rodrigo Lopez Pouso

Copyright © 2013 Alberto Favaron and Angelo Favini. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

For those semigroups, which may have power type singularities and whose generators are abstract multivalued linear operators, we characterize the behaviour with respect to a certain set of intermediate and interpolation spaces. The obtained results are then applied to provide maximal time regularity for the solutions to a wide class of degenerate integro- and non-integro-differential evolution equations in Banach spaces.

#### 1. Introduction

Let be a complex Banach space and let be a semigroup of operators on , which is generated by a multivalued linear operator and which may have a power type singularity at the origin , that is, for some nonnegative constant and nonpositive exponent , where denotes the Banach algebra of all endomorphisms of endowed with the uniform operator norm. In this context our aim here is twofold. The first is to characterize the behaviour of with respect to some intermediate and interpolation spaces between and the domain of . The second is to investigate how this behaviour reflects on the question of maximal time regularity for the solutions to a class of degenerate integro- and non-integrodifferential initial value problems in .

The class of operators we will deal with consists precisely of those multivalued linear operators whose single-valued resolvents satisfy the following estimate: Here, is the identity operator, is a positive constant, , and is the complex region , , . It thus happens (cf. [1–3]) that is the infinitesimal generator of a semigroup of linear bounded operators in satisfying (1) with , where .

To outline the motivations of our research, let us assume for a moment that is a single-valued linear operator satisfying (2). It is well known that if , then is the infinitesimal generator of a bounded analytic semigroup. For this case, an extensive literature exists concerning the behaviour of with respect to the real interpolation spaces , , , and its application to questions of maximal regularity for the solutions to nondegenerate (possibly nonautonomous) integro- and non-integrodifferential abstract Cauchy problems. See, for instance, [4–11]. Due to (1) with , the case of and is definitely worsened and the literature for it is considerably less conspicuous, although estimate of type (2), with in place of , goes back even to [12, Remark p. 383] in the ambit of Abel summable semigroups admitting uniform derivatives of all orders. One of the main problems with the case is that some equivalent characterizations of begin to fail (cf. [13]), so that some spaces which were just real interpolation spaces between and in the case become only intermediate spaces in the case . However, avoiding questions of interpolation theory and of maximal regularity, a quite satisfactorily semigroup theory for the single-valued case with and its application to the unique solvability of some concrete partial (non-integro-) differential equations have been developed in [14–18]. Since the multivalued case embraces the single-valued one, our contribution in this field is to fill this gap, supplying a theory for the behaviour of singular semig intermediate and interpolation spaces which, in the case , reduces to that in [9, 11]. As an effect of this theory, there is the possibility of investigating questions of maximal time regularity for an entire class of nondegenerate evolution equations which does not fall within the case .

The case when is really a multivalued linear operator arises naturally when we shift our attention to degenerate evolution equations of the type considered in [1–3]. There, a semigroup theory for multivalued linear operators was introduced as a tool to handle degenerate equations by means of analogous techniques of the nondegenerate ones. Such a theory has been then successfully applied to questions of maximal regularity for the solutions to a wide class of degenerate integro- and non-integrodifferential equations. We quote [2, 19–23] where, in general and unless , it is shown that the time regularity of the solutions decreases with respect to that of the data. In this respect, we mention the recent results in [20] where, under an additional condition of space regularity on the data and provided that and are large enough, the loss of time regularity is restored. Regrettably (cf. the appendix below), we have found some inaccuracies in [20, Section 4], and for this reason we must indicate some changes to that paper. On the other side, fortunately, the basic idea in [20] is correct and remedy can be applied to all the inappropriate items. Furthermore, unexpectedly, we will see that the more delicate approach followed in this paper not only corrects the mistakes in [20], but also gives rise to an effective improvement of the achievable results. In fact, here, we will straighten out, refine, and extend [20], enlarging the class of the admissible spaces to which the data may belong, weakening the assumption for the pair , and complicating the structure of the underlying equations. This is why we will first analyze the behaviour of the semigroup generated by with respect to some intermediate and interpolation spaces which turn out to be equivalent only in the case . Indeed, the phenomena exhibited in [13] for the single-valued case extend to the multivalued one (cf. [24]), and, until now, for the mentioned behaviour there exist no more than some partial results obtained in [2, 19, 24].

We now give the detailed plan of the paper. In Section 2, for a multivalued linear operator having domain and satisfying (2), we introduce the corresponding generated semigroup . This leads us to define also the linear bounded operators , , , () and to recall the fundamental estimates for their -norm. For the operators a semigroup type property is proven in Proposition 1. We then introduce the spaces we will deal with in this paper, that is, the interpolation spaces and the spaces, , . Special attention is given to the embeddings linking these two classes of spaces which, in general, are equivalent only in the case . Some relations existing between the spaces for different values of and are proven in Proposition 2 and discussed in Remarks 3–5. We conclude the section recalling the estimates proven in [19, 24] for the norms , , and , . In Remarks 7 and 8 we explain why, unless we renounce to optimality, in the case these estimates can not be directly extended to the norms and , , respectively.

In Section 3, we investigate the behaviour of the operators with respect to both of the spaces and . First, in Proposition 9, we deal with the norms , , and we show that, except for replacing with if and with if , the * same* estimates of [19] for the norms continue to hold. The second significant result is Proposition 12 where, extending those in [24] to values of other than one, we establish estimates for the norms , , . As a byproduct we deduce the basic Corollary 14, which in Section 5 will be a key tool in proving the equivalence between the following problem (3) and the fixed-point equation (179). The estimates in Proposition 12 are then merged together with those in [19] to achieve estimates for the norms , . In particular, two different estimates are obtained, if or not. For if , then (cf. the proof of Proposition 16) we can take advantage of the reiteration theorem for interpolation spaces and obtain estimates that, unless , are better than those rougher estimates derived in the general case (see Remarks 17 and 18). We stress that if , and is single-valued, then we restore the estimates in [9]. Finally, in Proposition 20, a combination of Propositions 9 and 12 yields the estimate for the norms , . Since , the spaces are, in general, only intermediate spaces between and for ; here the reiteration theorem does not apply and a weaker result is obtained (cf. (101)–(103)).

The estimates of Section 3 are applied in Section 4 to study the time regularity of those operator functions , , that we will need in Section 5. In particular (cf. formula (106)), we modify the definition of in [20, Section 4] in order that it is well defined, at least when acting on functions , (cf. Corollary 26). Consequently, operators and in [20] change too, and the new and should be introduced (cf. formulae (107)–(110)). The Hölder in time regularity of the ’s is characterized in Lemmas 22, 24, 30, and 32 and Propositions 29 and 36. The main feature of these results is to show that the loss of regularity produced by and can be restored, in and respectively, employing the regularization property established in [20, Section 3] for a wide range of general convolution operators.

In Section 5 we analyze the maximal time regularity of the strict solutions to the following class of degenerate integrodifferential equations in a complex Banach space :
Here, , , , , , , whereas, being another complex Banach space and being a bilinear bounded operator, , and , . Of course, if , then may be the scalar multiplication in . As , , and , , we take closed single-valued linear operators from to itself, whose domains fulfill the relation , and we require to have a bounded inverse, allowing to be *not *invertible. Hence, in general, is only a multivalued linear operator in having domain . Assuming that satisfies (2) and that the data , , and , , , are suitably chosen, problem (3) is then reduced to an equivalent fixed point-equation for the new unknown , . It is here that the results of Sections 3 and 4 play their role, leading us to Theorem 48. In that theorem, provided that , we will prove that if , , , , and for opportunely chosen , , , and , , , then problem (3) has a unique strict solution satisfying and , where (cf. Remark 51). Section 5 concludes with applications of Theorem 48 to integral and nonintegral subcases of (3), (cf. Theorems 52–54 and 56). We stress that Theorem 48 repairs, generalizes, and improves [20, Theorems 5.6 and 5.7], where similar results were proven only for the case and under the stronger condition .

In Section 6, we give an application of Theorem 48 to a concrete case of problem (3) arising in the theory of heat conduction for materials with memory. In particular, we show how Theorem 48 characterizes the appropriate functional framework where to search for the solution of the inverse problem of recovering both and the vector , , in (3) with and , .

Finally, in the Appendix we explain how to amend [20, Theorems 5.6 and 5.7] in accordance to Theorem 48.

#### 2. Multivalued Linear Operators, Singular Semigroups, and the Spaces and

Let be a complex Banach space endowed with norm and let be the collection of all the subsets of . For a number and elements , , and denote the subsets of defined by and , respectively. Then, a mapping from into is called a * multivalued linear operator* in if its domain is a linear subspace of and satisfies the following: (i) , for all ; (ii) , for all , for all . From now on, the shortening m. l. will be always used for multivalued linear.

The set is called the range of . If , then is said to be surjective. The following properties of a m. l. operator are immediate consequences of its definition (cf. [1, Theorems 2.1 and 2.2]): (iii) , for all; (iv) , for all, for all; (v) is a linear subspace of and for any , . In particular, is single-valued if and only if .

If is an m. l. operator in , then its inverse is defined to be the operator having domain such that , . is an m. l. operator in too, and . The set is called the kernel of and denoted by . If ; that is, if is single-valued, then is said to be injective. Observe that (v) yields if and only if .

Given , we write , so that, in particular, . If , are m. l. operators in and , then the scalar multiplication , the sum , and the product are defined by where , and are m. l. operators in and .

Let and be m. l. operators in . We write if and for every . Clearly, if and only if . If and for every , then is called an extension of . If a linear single-valued operator has domain and , that is, for every , then is called a section of . With an arbitrary section , it holds , , and , but this latter sum may or may not be direct (cf. [25, p. 14]). A method for constructing sections is provided in [25, Proposition ].

If , , are two complex Banach spaces, then the linear space of all bounded * single-valued* linear operators from to is denoted by ( if ) and it is equipped with the uniform operator norm . Then the resolvent set of a m. l. operator is defined to be the set , with being the identity operator in . The basic properties of the resolvent set of single-valued linear operators hold the same for m. l. operators. First, if , then is closed; that is, its graph is closed (cf. [25, p. 43]). Further (cf. [1, Theorem 2.6]), is an open set and the operator function is holomorphic. Finally (cf. [1, formula ]), the resolvent equation , , is satisfied, too. Unlike the single-valued case, instead, for the following inclusions hold (cf. [1, Theorem 2.7]):
Then, in general, , , is only a bounded section of the m. l. operator . Throughout this paper, we denote this bounded section by , but we warn the reader that here does not necessarily denote a section of itself. Of course, if is single-valued, then reduces to . Notice that (5) implies that , , is single-valued on and with any , . Another difference with the single-valued case is that for every it holds . Indeed, . Therefore, in the m. l. case, , . However (cf. [24, Lemma 2.1]), if , then , and, in addition, if and only if , . We also recall that for every the following slight variants of the resolvent equation hold (cf. [24, Lemma 2.2]):
In particular, if , then, since , the first in (6) with yields ; that is,
Let be a m. l. operator in satisfying the following resolvent condition: (H1) contains a region , , , and for some exponent and constant the following estimate holds:
Introduce the family defined by and
where is the contour parametrized by , . Then (cf. [1, pp. 360, 361]), is a semigroup on , infinitely many times strongly differentiable for with
where . In general, no analyticity should be expected for . For if in , then does not contain any sector , , and [15, Theorem 5.3], which extends analytically to the sector containing the positive real axis, is not applicable. We stress that (9) and , , imply for every , whereas . Hence, if is really an m. l. operator, then . From the semigroup property it also follows that for .

Now, for every such that we set
Here, for the multivalued function we choose the principal branch holomorphic in the region , where for principal branch we mean the principal determination of . We briefly recall the main properties of operators . Of course, , . As shown in [26, p. 426], , , , is a section of , so that from (10) we get
Moreover (cf. [19, formula (22)] with being replaced by ), we get
Finally, implies the following estimates (cf. [1, 24]):
where the ’s are positive constants depending on , , and . Thus, letting in (14), we see that if , then the operator function may be singular at the origin and the semigroup is not necessarily strongly continuous in the -norm on the closure of in . Notice that if , then the singularity is a * weak* one, in the sense that is integrable in norm in any interval , . Further (cf. [24, Lemma 3.9]), if , then , and if , then for every .

Observe that , , , so that , . The operators satisfy the following semigroup type property.

Proposition 1. *Let , , and let , . Then
*

*Proof. *First, the function being holomorphic for every and , and the contour in (11) with can be replaced with the contour parametrized by , , , and lies to the right of . Then, for every , from the resolvent equation we obtain
Now, after having enclosed and on the left with an arc of the circle , , we apply the residue theorem and let go to infinity. To this purpose, we observe that since the contours and both lie in the half-plane , the arc may be parametrized in polar coordinates by , , . Then, for every we have
Since and , the right-hand side of the latter inequality goes to zero as goes to infinity, so that for every and . The residue theorem together with the fact that lies to the right of thus yields and . Replacing these identities in (16) and using the equality which is satisfied for the principal branch of the function , we finally find
The right-hand side being precisely , the proof is complete.

For an m. l. operator satisfying we introduce now the spaces and . We first specify a topology on equipping it with the norm , . Since , this norm is equivalent to the graph norm and makes a complex Banach space (cf. [2, Proposition 1.11]). As and being given normed complex linear spaces, we will write if and there exists a positive constant such that for every . If , that is, if and the norms and are equivalent, then we will write . Of course, with the norm satisfies . In fact, if , then for every we have , so that . Taking the infimum with respect to , we thus find for every . If is a Banach space, we denote by the set of all continuos functions from to , and for a -valued strongly measurable function , , we set , , and . Let or let , and for define if and if . Let us set
This characterization of the spaces is that obtained by the so-called “mean-methods”, and it is equivalent to that performed by the “*K*-method" (cf. [27, Theorem and Remark 1.5.2/2]) and the “trace-method” (cf. [27, Theorem 1.8.2]). Then, due to [27, Theorem 1.3.3], for every and the space is an exact real interpolation space of exponent between and . Observe that by exchanging the role of and and performing the transformation , we get . Also, if , then (cf. [27, Theorem ]). The definition of the spaces is meaningful even for the limiting cases , , whereas , , , reduces to the zero element of . In particular (cf. [28, pp. 10–15]), denoting by the completion of relative to and endowing it with the norm in [28, p. 14], we get and . Let and let , . Then, for and , , the following chain of embeddings holds:
Let . Recall that a Banach space is said to be of class and shortened to , if is an intermediate space between and , that is, if . From (20) it thus follows that , for every and . Moreover, since , , and , we have and . Then (cf. [28, p. 12], [27, Theorem ], and [9, Section ]), for and , , the reiteration theorem yields
Finally (cf. [29, Theorem 1.II and Remark 1.III]), we recall that if and are two complex Banach spaces and is such that , , , then , , , and
As a consequence of this general result and the identity
from the third in (21) we find that if is such that and , then , , , , and the following estimate holds:
Notice that here for every . Therefore, if we let and let , then , , and . Hence, in order that the additional inequalities , , are satisfied, we have to choose . As we will see this simple observation will be the key for the proof of the second estimates (90) in the following Proposition 16.

We recall that for every fixed the map satisfies , and . Then (22) with , and yields the interpolation inequality: with being the positive constant depending on and such that .

As another application of (22) and for further needs, we also recall that if satisfies , then satisfies the estimate (cf. [24, formulae (4.16) and (4.17)]).

Consider From (26), using (22) with , , and , it then follows for every and where is the positive constant depending on and such that .

For and we now define the Banach spaces by It is a well-known fact that if is single-valued and in , then (cf. [30, Theorem 3.1] and [27, Theorem ]). On the contrary, if , then such an equivalence is no longer true, as first observed in [13, Theorem 2] for single-valued operators and, in the case , in [2, Theorem 1.12] for the m. l. ones. Recently, extending [13] to m. l. operators and [2] to , in [24, Proposition 4.3] it has been shown that the following embedding relations hold: Then, as in the single-valued case, if in (H1). More precisely (see the proof of [24, Proposition 4.3]), if , , , then whereas if , , , then with being a positive constant depending on , and .

By setting , , from (30) it follows Then, if , the spaces , , , are intermediate spaces between and only for , whereas, when , they may be smaller than . In any case, when , it is not known if the spaces , , , are only intermediate or just interpolation spaces between and .

Notice that , , . Indeed, assume that there exists such that for some and . Then, since , , we have for every and , contradicting . This property plays a key role in the proof of many of the results in [24]. Further, due to (30), it implies that , , . On the contrary, since may be a proper subset of for , , in general it is not true that . This is true, instead, if . In this case the topological direct sum is a closed subspace of , and if is reflexive, it coincides with the whole (cf. [3, Theorems 2.4 and 2.6]).

For every and from (27), (29), and (31) it follows Hence, for and we may rewrite (27) and (34) more compactly as where and is equal to or according that or .

With the exception of the case , in general it is not clear if embeddings analogous to (20) hold even for the spaces . In fact, using (20), (29), and (30) we can only prove that if and , then whereas if and , then What can be proved without invoking (20), (29), and (30) and using only the definition of the norm is instead the following result, which extends to the spaces the embeddings , and , , (cf. (20) with and ).

Proposition 2. *Let be an m. l. operator satisfying the resolvent condition (H1). Then the following embeddings hold for every and :
*

*Proof. *If in (H1), then there is nothing to prove since and both (38) and (39) follow from (20). Therefore, without loss of generality, we assume that is such that if . We begin by proving (38). Let first . For every , , we write
where
. Using the first inequality in (26) we find
where . Concerning , instead, using , we get
Summing up (40)–(43) and setting , it thus follows , completing the proof of (38) in the case . Let . For every , , we write
where , , , . Again, the first inequality in (26) yields
Instead, using , we have
Summing up (44)–(46) and setting , we thus find . This completes the proof of (38) for the case . We now prove (39). Due to (38) with , it suffices to assume that . As above, for every , , we write , where and are defined by (41). Hence, the same computations as in (42) yield
As far as is concerned, instead, we have
where . Summing up (47) and (48) and setting , we deduce . The proof is complete.

*Remark 3. *Notice that (37) with yields , , and this latter embedding is less accurate than (38).

*Remark 4. *The main problem for extending (20) to the spaces in the case is that it is not clear if it holds ,