Abstract and Applied Analysis

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

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

Research Article

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

Alberto Favaron and Angelo Favini

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 , . In fact, the embedding

is a consequence of the property of the functional entering the definition of the interpolation spaces through the “-method”, and in particular of its monotonicity (see the proof of [27, Theorem ]). With embedding (49) at hands, to derive (20) it thus suffices to prove that , (see the proof of [27, Theorem ] taking there and using ). If we try to repeat the proof of (49) for the spaces , we will be faced with two problems. The first is that we do not know if the function , , , is monotone decreasing, which would allow us to prove , , . For if was monotone decreasing, then for every and , , , we would find

where . Taking the supremum with respect to in the latter inequality, we would get , proving , , . The second problem is that the function is not necessarily bounded for , , , precluding us to prove , , . Indeed, from (35) we can only find , and when , the right-hand side of this inequality goes to infinity as goes to infinity. On the contrary, if were bounded, then for every we would obtain

If now in addition were also monotone decreasing, in order that , from the latter inequality we would get , completing the proof of , , . Due to the former computations, we can thus conclude that in the case the quoted problems are the main obstacles which prevent us to extend (49) and, as its consequence, (20) to the spaces .

Remark 5. Let be fixed and for every and let us set and . We thus have the two families of sets and . Let first . In this case, since , from (20) we deduce that the sets and are related by the following inclusions in which :

Now let . As observed in Remark 4, in this case the embedding , , may be not satisfied and the chain of inclusions (52) could not take place. However, (38) and (39) hold true and for every , and we have and .

We have already pointed out that may be not strongly continuous in the -norm on . On the contrary, the following result (cf. [24, Proposition 5.2] for the proof) shows thatthe things are finer on and . Later, we will need this fact.

Proposition 6. Let be as in Proposition 2. If ; then is strongly continuous in the -norm on for every .

We conclude the section listing some estimates for the operators defined by (11) with respect to the spaces and . First, in [19, Lemma 3.1] it is shown that for every and that the estimate is satisfied. Hence, using (14), we get

Combining (14) and (53) with (25) and letting , it thus follows (cf. [19, Proposition 3.1]) that for every and the following estimate holds:

Remark 7. We stress that if , then we can not derive an estimate for the -norm of simply by replacing with in (54). This is for two reasons. First, when , we are not assured that for every . For if , then the space may be smaller than the domain to which belongs by virtue of [19, Lemma 3.1]. The second reason is that, even limiting to in order that , from (31) we only get , , and we do not know if the right-hand side can be bounded from above by some constant times . Of course, we can employ (32), but in this way all that we can reach is the estimate

where , and . Letting , (55) can be rewritten equivalently as

where and . When , there are good motivations to believe that estimate (56) is not the best one. In fact, for instance, when , (56) leads us to an estimate which is rougher than the estimate

as shown in [2, Proposition 3.2], with being a positive constant depending on , , and . Also, (57) ensures that , , belongs to for every and not only for as (56) suggests. Furthermore, due to (31), estimate (57) yields (54) with . This leads us to believe that (57) can be improved and that estimate (54) holds the same if is taken in place of .

Now let , , . As far as the estimates for the -norm of operators are concerned, instead, at the moment only the following estimates for the case are available (cf. [24, Lemma 5.1]):

with being a positive constant depending on , , , and . Estimates (58) are successfully applied in [24, Corollary 5.4] to prove that if , then the map is Hölder continuous from to , , , with Hölder exponent . In Section 3 we will extend (58), proving some estimates for the -norm of , , which reduce to (58) in the case .

Remark 8. Observe that an estimate for the norm , , , , , , can be obtained combining (14), (15), and (58). Indeed, using (15), for every , and , we have

Therefore, due to (14) and (58), from (59) we deduce that

where , and . As we will see in the next section estimate (60) is not optimal, in the sense that the negative exponent can be refined; of course, unless . The main reason to believe that (60) can be improved is that its derivation consists of two steps: the first in which is decomposed with the help of (15), and the second in which (60) is obtained combining estimates of very different nature, such as (14) and (58). It is thus to be expected that in this double step derivation some regularity goes missing and that a better result can be reached analyzing more detailedly for .

3. Behaviour of in and

According to Remark 7 we begin by improving (54), showing that the same estimate holds with being replaced by if and by if . Throughout this and the next section, will be an m. l. operator in having nonempty domain and satisfying the resolvent condition of Section 2.

Proposition 9. Let , and let . Then, there exist positive constants , , depending on , , , , and such that

Proof. If , then and (61) and (62) with , , follow by taking in (32) and (54). Therefore, without the loss of generality, we assume that is such that if . Let , , , and be fixed and let be an arbitrary element of . Then, for every we have

Of course, from estimate (54) we find

with being such that , , . It thus suffices to investigate only the second terms on the right-hand side of (63) and (64). We begin by proving (61). First, using the second identity in (6), for every we get

Here we have used twice the equality , , which follows from Cauchy’s formula after having enclosed on the left with an arc of the circle , , and letting to infinity. From (66), using , , it follows that

Now, since for every and since , we have

Therefore, for every and the following inequality holds:

where we have used the fact that the function , , , attains its maximum value at the point . Coming back to (67) and setting , we thus find (here we use also that on it holds , so that ):

where . Finally, taking the supremum with respect to in (70) and performing the transformation in the integral on the right, we obtain

where , , , being the Euler gamma function . Then, summing up (65) and (71), from (63) it follows that

Since was arbitrary, this completes the proof of (61) with . Let us now prove (62). For every we write

where , , . First, (35) with yields

Therefore, since for every , where or according that or , from (54), we deduce that

with . As far as is concerned, exploiting (71) and recalling that we have assumed , we obtain

where . Summing up (73)–(76), it thus follows that

where . Finally, (65) and (77) lead us to

Since was arbitrary, this completes the proof of (62) with .

Remark 10. If , then (61) is precisely the estimate (57). In this sense our result improves [2] and shows that (54) holds the same with being replaced with if and and if . Also, when , (61) and (62) are in two aspects better than the estimate (55) deduced from (54) with the help of (32). First, here we do not need to restrict to . Further, despite limiting to , (61) and (62) show that , , , , enjoys more regularity than that predicted by (55). For, since when it holds , from (38) and (39) it follows , .

Remark 11. We recall that when the spaces , , , are intermediate spaces between and for , but they may be contained in for . Therefore, whereas (61) is satisfied for spaces eventually smaller than , for (62) to hold we have to consider only spaces , , bigger than . In fact, letting , we have for every .

In accordance with Remark 8 we now improve estimate (58).

Proposition 12. Let , , and let . Then, there exists a positive constant depending on , , , , and such that

Proof. First, using the identity , , for every , we rewrite , , in the following way:

Here we have used , which follows from the Cauchy formula applied to after having enclosed on the left with an arc of the circle , , and letting to infinity. Let now , , , and be fixed and let be an arbitrary element of . From (35) it then follows that

where . Now, recalling that for every , we have , . As a consequence, the following inequality holds:

where or according that or ( if ). Therefore, setting , (81) and (82) yield

with being as in (70). Finally, the transformation in the last integral leads us to the following estimate:

where , , , is the Euler’s gamma function. Notice that here implies for every and , so that makes sense. Since (84) is satisfied for every arbitrary element , the proof is complete with .

Remark 13. Estimate (79) is better than (60) obtained in Remark 8 using (14), (15), and (58). In fact, for every , , and , the following inequality holds:

Then, , , and (79) is more accurate than (60) for small values of .

Estimate (79) with yields the following result which we will need in Section 5 to prove the equivalence between problem (170) and the fixed-point equation (179).

Corollary 14. Let in (H1). Then, for every the following equalities hold:

Proof. The assertion is obvious for . Let and let . Commuting with the integral sign, from (9) and the resolvent equation, we have , which proves the first equality in (86). To prove the second equality, we first write

and we show that the latter integral is convergent. Indeed, since , we may consider as an element of , where and . With this choice for , from (79) with and (25) we obtain (here we use also , due to . Then, ):

where . We now recall that (cf. [24, formula ])

with being the negative fractional powers of defined by (cf. [24, Section 3]) , . To complete the proof it thus suffices to apply (89) with to (87) and to recall that , . Notice that the integral on the right-hand side of (86) is convergent, too. In fact, from (14), it follows that .

Remark 15. In particular, from (86) it follows that if , then for every and . This extends to m. l. operators satisfying the well-known result for sectorial single-valued linear operators (see, for instance, [9, Proposition 2.1.4(ii)] and [11, Proposition 1.2(ii)]).

With the help of (54) and Proposition 12, we can now derive the following interpolation estimates (90) for the operators , , which are considered as operators from to . As we will see in the proof of Proposition 16, here the fact that the spaces are real interpolation spaces between and plays a key role. For it allows us to exploit the interpolation inequality (24) in the derivation of our estimates in the case .

Proposition 16. Let , , and . Then, there exist positive constants , , depending on , , , , , and such that for every

Proof. For brevity, we will use the shortenings , , . We begin by proving the first estimate in (90). Let , , and be fixed and let be an arbitrary element of . Moreover, let and be two arbitrary complex numbers such that and whose real parts satisfy and . From the decomposition formula (15) it then follows for every :

Therefore, using (54) and (79) with the triplet being equal to and , respectively, from (91) and , we deduce that

where . This completes the proof of the first estimate in (90), due to the arbitrariness of . Let us now prove the second estimate in (90). Let , , , , and be fixed. Using , we fix , and we let . Clearly, since , we have . In addition, it holds:

Due to (93), we now set , so that and . From (24) with it thus follows that

where , . Applying (54) and (79) with the pair being replaced with and , respectively, from (94) we finally obtain

This completes the proof of the second estimate in (90) with .

Remark 17. We stress that if and , then the first estimate in (90) is rougher than the second one for small values of , which justify our special attention to the case . Indeed, if , then for every the following inequality holds:

so that for . In other words, if and are both less than one, then the second estimate in (90) establishes that the norm , , may blow up as goes to , but with an order of singularity lower than that predicted by the first estimate. In this sense, though less general, the second estimate in (90) is better than the first one.

Remark 18. The reason why the second estimate in (90) yields a better exponent than the first one is the same mentioned in Remark 8. That is, while the first estimate is obtained in two steps: decomposing through (15) and then applying (54) and (79), the second estimate is essentially derived in a single step, using (24).

The following Remark 19 points out why, with the exception of the case when and is single-valued, to prove (90) we can not proceed as in [9, Proposition 2.2.9].

Remark 19. In the optimal case , the exponents in both estimates (90) coincide equals to . Hence, in this special case, the assumption does not give any enhancement. Also, if we further assume that , then we restore the same estimates as in [9, Proposition ]. In this respect, our result extends [9] to the m. l. case, even though our proof really differs from that in [9]. For, there, the norms in the spaces are replaced with the norms in the spaces , with the latter being the spaces of all such that , where . It is well known that if and is single-valued, then (cf. [31, Theorem 3], [9, Proposition ] and [27, Theorem ]). On the contrary, if and/or is really an m. l. operator, such equivalence is no longer true and we have

Differently from the spaces and as a consequence of , the spaces contain . It can thus be shown that if , then for every and (here , since ) the following embeddings hold:

with being endowed with the norm of . Obviously, due to (29), it suffices to prove the embeddings on the right of (97) and on the left of (98). It is out of the aims of this paper to go into the details of these proofs, and for them we refer the readers to [24, Proposition 6.3]. Here we want only to make clear that, with the exception of the case when and is single-valued, embeddings (97) and (98) prevent us from carrying out the proof of estimates (90) simply by repeating the computations in [9]. Notice that, due to the property , from the second embeddings in (97) and (98) it follows that if and , then

Since (indeed, implies ), (99) agrees with (38) for . In addition, if and , then the first embeddings in (97) and (98) yield for every the following:

Since , (100) agrees with (38) for . Furthermore, if , then from (29), (30), and (99) it follows that , . This confirms that in the real m. l. case the equivalence between , and does not hold even when .

Using Propositions 9 and 12, we now obtain estimates for the operators , , considered as operators from to  . Clearly, since the spaces may be not real interpolation spaces between and , we can not proceed as in the proof of the second estimate in (90) and a weaker result has to be expected.

Proposition 20. Let , , and . Then, there exist positive constants , , depending on , , , , , and such that

Moreover, if and are such that , then

Proof. Due to (61) and (79), in order to prove (101) and (102) it suffices to repeat the same computations as in (91) and (92), with the pair being replaced with or with provided that or . In this way we derive (101) and (102) with , . As far as (103) is concerned, we recall that if , , are four Banach spaces such that , , and , then with , and being the positive constants such that , , . Applying this result to with , from (29)–(32) and the second estimate in (90) we deduce (103) with . This completes the proof.

Remark 21. The assumption with and implies that . Therefore (cf. Remark 11), we conclude that for (103) to hold we have to consider , , as an operator between the intermediate spaces and , where , , , .

4. Hölder Regularity of Some Operator Functions

Here, we study the Hölder regularity of those operator functions that we will need in Section 5. From now on, with being a complex Banach space, and , , , denote, respectively, the spaces of all continuous and -Hölder continuous functions from into endowed with the norms and , where is the seminorm . We endow the subspace , with the norm . Further, for and we set , (), and , . Recall that if , then and , . Finally, given three complex Banach spaces , , and a bilinear bounded operator from to with norm , that is, and , we denote by the convolution operator

where , . Of course, if and if is the scalar multiplication in , that is, , , , then and reduces to the usual convolution operator . As usual, for every , we will denote by the conjugate exponent of .

Now let and introduce the following linear operators , , where , , , , , , , , , and , as follows:

with being the function from to defined by . We will find conditions on , , , , in order that , and for opportunely chosen . We emphasize of the presence of the increment inside the integral defining . As we will see, and differently from , it is just this presence which makes well-defined for smooth enough functions . This is the reason why the operator as it was defined in [20, formula ] can make no sense and has to be replaced with that defined by the present (106) (cf. the appendix below). We begin our analysis on the ’s with the following result proven in [20, Lemma 4.1]. Since we will need it later, here, removing some misprints in [20], we report its short proof for the reader’s convenience.

Lemma 22. Let in (H1). Then, for every , the operator defined by (105) maps into , and for every satisfies the following estimate, where as follows:

Here is a nondecreasing function of depending also on , , , and .

Proof. Let , , and . From (14) and the Hölder inequality with , for any , we deduce that

where . Here , since . For . passing to the supremum with respect to in (112) we thus find

Now let (since , the case follows from (112) with ) . The change of variable in (105) leads us to , where and . Reasoning as in (112) and using the inequality , , we get

Similarly, but taking advantage from , we obtain

Thus, letting from (114) and (115) it follows that

Finally, summing up (113) and (116) and using , we derive (111) with . This completes the proof.

Remark 23. We stress that if we renounce to its Hölder regularity, then for to be well-defined it suffices that and are as in Lemma 22 and that is merely in . In fact (see the last part of the proof of Corollary 14, replacing there with ), , .

Lemma 24. Let in (H1). Then, for every , the operator defined by (106) maps into , , and for every it satisfies the following estimate:

Here is a nondecreasing function of depending also on , , and .

Proof. Denote by the number . In particular, since implies , we have . Let , , , and . We notice that and . Then, using (14) with , for every we obtain

where . Hence

Now let (since , the case follows from (118) with ) . We have , where for a function we set

First, using (13) with , , and (14) with , and letting , we get

Let us turn to . We first observe that the integral is convergent. For, , where is equal to if and to if . Thus, we may rewrite it as . Consequently,

where we have used and . As far as is concerned, instead, reasoning as in the derivation of (118) we find

Then, summing up (121)–(123) and letting , we obtain

Finally, (119) and (124) yield (117) with .

Remark 25. In particular, Lemma 24 establishes that, with the exception of the case in which , produces a loss of regularity equal to .

As Corollary 14, the next result will be needed to prove the equivalence between problem (170) and the fixed-point equation (179). From now on, if and , , with we will always mean the function in defined by . Notice that , .

Corollary 26.

(i)Let in (H1). Then, for every , ,

(ii)Let in (H1). Then, for every

Proof. Of course, it suffices to assume that . Let us first prove (i). So, let , , , and , and we observe that both sides of (125) are well defined. Indeed, replacing the pair with , from (118) we get

On the other side, satisfies

where . Then, commuting with the integral signs, using (80) with , and taking into account (7), we find

Since , the proof of (125) is complete. We now prove (ii). Let , and . Then, for every , the same reasonings made to derive (88), except for replacing with , yield

Hence, being meaningful, we obtain (126) simply applying to it formula (89) with and then using , . In particular, a better estimate than (130) holds. For, satisfies

where . The proof is complete.

Let us now examine the operator defined by (107). To this purpose we need the following result which is proved in [20, Corollary 3.2].

Lemma 27. Let , , be such that , . Then the convolution operator defined by (104) maps into , and for every satisfies the following estimate:

Here is a nondecreasing function of depending also on and . Further, in the cases , , and , the following estimates hold, respectively, as follows:

From Lemmas 24 and 27 we obtain the following Lemma 28.

Lemma 28. Let and be as in Lemma 24. Then, for every and such that , , the operator defined by (107) maps into , , and for every satisfies the following estimate:

Proof. First, if and , then . Consequently, the assumption , , makes sense. Now, Lemma 27 yields for any pair . Then, recalling that , the assertion follows from Lemma 24, with and being replaced by and , respectively. Finally, (134) follows from (117) and (132).

We can now restore the loss of regularity produced by .

Proposition 29. Let in (H1). Then, for every , the operator defined by (107) maps into , and for every satisfies the following estimate, where and :

Proof. Let and let . Then, . We are thus in position to apply Lemma 28 with from which we deduce that maps into , . But, since our choice for implies , we a fortiori have the fact that maps into . Finally, (135) follows from (134) and the estimate , , .

The next Lemma 30 concerns the operator . Its proof is similar to that of Lemma 24, but with the essential difference that the presence of allows us to use estimate (79) in place of (14). As a consequence and provided to choose large enough, we will achieve a better result in which any loss of regularity is observed.

Lemma 30. Let in (H1) and . Then, for every and the operator defined by (108) maps , , into , and for every satisfies the following estimate:

Here is a nondecreasing function of depending on , , , and .

Proof. Let , , , and , , . As in the proof of Lemma 24 we set and we observe that, since implies , here . Furthermore, we denote by the number , so that the exponents and appearing in (79) with and may be rewritten, as and , respectively. Then, using (79) with , we obtain

where . Hence, taking the supremum with respect to , one has

Now, let (since , the case follows from (137) with ) . We have , the ’s, , being as in (120). Using (13) with , , and (79) with , and letting , we get

Now, let us examine , . First, using (79) with , we find

Instead, the same computations made to derive (137) yield

From (139)–(141) and , it follows that

where . Finally, summing up (138) and (142) we get (136) with . The proof is complete.

Remark 31. Notice that if , then in order to be sure that the conclusions of Lemma 30 hold with which really belongs to some intermediate space between and we have to choose . This is possible, provided that the stronger assumption is satisfied. Otherwise, if , , then and may be contained in .

Finally, for the operator we have the following result. Again a loss of regularity is exhibited, even though of an amount smaller than that in Lemma 24 (cf. Remark 33).

Lemma 32. Let in (H1). Then, for every , the operator defined by (109) maps into , , and for every satisfies the following estimate:

Here is a nondecreasing function of depending also on , , and .

Proof. Let , , and . We still let and as in Lemma 30 we have . Further, observe that . Let . Then, using (14) and , we get

Now, let (since , the case follows from (144) and ) . We have , where for a function we let

First, since for every , we deduce that

As far as is concerned, instead, rewriting as and using both and , it follows that

Then, since , from (146) and (147) we find

where . Summing up (144) and (148) we obtain (143) with . This completes the proof.

Remark 33. Thus, with the exception of , produces a loss of regularity equal to . In this sense behaves better than .

Remark 34. Notice that, under the weaker assumptions and , (86) with , , yields .

Similarly as we have done in Proposition 29 for restoring the loss of regularity produced by , we now show how Lemma 27 allows to restore that produced by . We begin with the following version of Lemma 28 relative to , and which is obtained combining Lemma 27 with Lemma 32 instead of Lemma 24.

Lemma 35. Let and be as in Lemma 32. Then, for every and such that , , the operator defined by (110) maps into , , and for every satisfies the following estimate:

Proof. First, if and , then . Consequently, the assumption makes sense, provided to choose small enough. Lemma 27 then yields for any pair . Then, since , the assertion follows from Lemma 32, with the pair being replaced by . Finally, (149) follows from (143) and (132).

From Lemma 35 we obtain the analogous of Proposition 29 for .

Proposition 36. Let in (H1). Then, for every , the operator defined by (110) maps into , and for every satisfies the following estimate, where and :

Proof. Let and . Then, and we can apply Lemma 35 with , . We thus deduce that maps into , . But, since implies , we a fortiori have the fact that maps into . Finally, (150) follows from (149) and .

In Section 6 we will also encounter acting on functions which enjoy some space regularity, that is, functions which are Hölder continuous in time with values on . In this case Lemma 32 can be refined, and the loss of regularity produced by is naturally restored by the additional condition of space regularity on . In some sense, the forthcoming Corollary 38 is the analogous of Lemma 30, where the function involved in the definition of (cf. (108)) was of class .

Lemma 37. Let in (H1) and , , . Then, for every , the operator defined by (109) maps into , and for every satisfies the following estimate:

Here is a positive constant depending on , , , and .

Proof. Let and let be the number , so that the exponent in (79) with is equal to . Let , , . Since , we assume that and we observe that, due to Propositions 6 and 12, is rewritten as follows:

Indeed, for every and , (79) with yields

where . From (152) and (153) with we thus get

Now, let . From (152) it follows that , where for every function we have set

Hence, using (153) with the triplet being replaced by and , respectively, we deduce that

As a consequence, since ,

Summing up (154) and (157), we obtain (151). The proof is complete.

Since in Lemma 37 it is not required that , the special case of the constant function , , is admissible, and we obtain the following result.

Corollary 38. Let , , and be as in Lemma 37, and let , , and . Then, for every , the function belongs to , and for every satisfies the estimate

Proof. Let in the proof of Lemma 37, and observe that reduces to the zero element of . Estimate (158) then follows from (154) and the second estimate in (156).

For later purposes, we conclude the section with the following remark.

Remark 39. The condition in required in Proposition 29 is the strongest among the conditions for the pair required in Corollary 14 and the other results of this section. Indeed,

Hence, if , then Corollary 14 and all the results from Lemma 22 to Corollary 38 are applicable. Next we will make large usage of this fact, but we warn the reader that, for brevity and regarding it as acquired, we will not mention it anymore.

5. Application to Maximal Time Regularity

The results of the previous sections are here applied to correct, refine, and extend the results in [20] concerning the maximal time regularity of the solutions to a class of degenerate abstract evolution equations. Let and be two complex Banach spaces, and consider the following degenerate first-order integrodifferential Cauchy problem for , where , , and :

Here is the convolution operator (104) in which , whereas , , and , , are closed single-valued linear operators from to itself, whose domains fulfill the relation . Further, we assume that

whereas we allow to have no bounded inverse. Hence, in general, is only the m. l. operator defined by

Therefore, problem (160) can not be reduced, via the change of unknown , to an integrodifferential problem related to single-valued linear operators. On the contrary, due to (161) and the closed graph theorem, , . As far as the data vector is concerned, at the moment, we only assume , , , , , , , and , in order that (160) makes sense in . This minimal assumptions will be refined later. In general, only strict solutions to (160) shall be investigated, where (cf. [22, 23]) by a strict solution to (160) we mean that, being endowed with the graph norm , , , and (160) holds. Clearly, if is really a m. l. operator, then does not necessarily mean , but only . As we will see below, if and the data , , and , , , satisfy suitable assumptions, then for a strict solution to (160) it just holds . Throughout the section, , , , will always denote one between the spaces and , being defined by (162). That is, . To avoid confusion, if more than a single is involved in some statement, that is, if we write , , , then it is understood that the same choice has been made for all the in the sense that or for every .

According to [2, Section 1.6], we recall that the -modified resolvent set of is defined to be the set . The bounded operator is called the modified resolvent of . It is easy to prove that and that , (cf. [2, Theorem 1.14]). With the notion of -modified resolvent of at hand, we assume that

(H2) contains a region , , , and for some exponent and constant the estimate holds for every .

Before we proceed with our analysis we remark that, due to the wide range of choices for the data vector, problem (160) contains many subcases at its interior. So, in spite of the case when at least one between the ’s is different from zero and problem (160) is really an integrodifferential one, the choice , , yields to consider also various nonintegrodifferential degenerate problems. For instance, those corresponding to and , , , respectively:

Although (164) differs from (163) only in the fact that is replaced with ; nevertheless a very different result is achieved when the ’s are assumed to belong to , at least for opportunely chosen , . As we will see (cf. Remark 51 and Theorem 56), in this situation the loss of time regularity for the pair with respect to that of , typical of the case in (see [21, Theorem 9], [2, Theorem 3.26], and [22, Theorem 7.2]), can be restored in order that possesses the maximal time regularity which is the minimal between the time regularities of the ’s. The same phenomenon is carried over into the integrodifferential case for the following problems, corresponding to , , and :

. When , the loss of time regularity for the pair with respect to that of the vector in problem (165) (cf. [22, Theorem 7.1] and [23, Theorem 2.1] for ) can be restored in problem (166) assuming that , . In this context (cf. Remark 51 and Theorem 53) the pair has the maximal time regularity which is the minimal between the time regularities of the ’s and ’s.

We stress that, if , then no loss of time regularity is observed and all the quoted results agree with the well-known theory of maximal regularity in spaces of continuous functions for the nondegenerate version of (160), corresponding to the case when and generates an analytic semigroup. Hence, roughly speaking, one can verify the consistency of any result on problem (160) with condition simply by letting on its statement, and then checking if it is compatible with those for the nondegenerate case. To this purpose, we recall that the question of maximal regularity for the nondegenerate (possibly nonautonomous) version of (160) has been deeply investigated by several authors. See, for instance, [4, 6–8, 10, 32] for problem (165) with and [9, 11] for problem (163) with .

Finally, assumption (161) excludes the case of in (160), so that our results cannot be compared with those in [5, 33, 34]. There, assuming that the bilinear bounded operator underlying the definition of is the scalar multiplication in , the problem

is treated under the following assumptions: (i) is a closed densely defined linear operator generating an analytic semigroup; (ii) is absolutely Laplace transformable. Observe that, if , then problem (167) reduces to the abstact wave equation , , , whereas when and , , , problem (160) reduces to the abstract heat equation , . In other words, whereas [5, 33, 34] are concerned with the hyperbolic case, here we are concerned with the parabolic one.

Let us now come back to problem (160). Of course, assumption implies that the operator defined by (162) satisfies , so that it generates a semigroup defined by and (9) and satisfying (14). Assuming that , we let

Then, by setting

where , , , and , we see that is a strict solution to (160) if and only if satisfies (indeed, if , then as , , that is, . Conversely, if , then and as , , that is, . Finally, since , we have if and only if ) , , and solves to the following problem:

Now let , and assume that , , and , where , , . Then, if is a solution to (170) such that , the function satisfies

Indeed, being the smallest Hölder exponent, for every , , we have and , (cf. Lemma 27 for the case with the pair being replaced by (in fact, since , , if , then , whereas the constant functions , , , obviously belong to ) and , resp.). Consequently (cf. [2, Theorem 3.7 and Remark p. 54] with ), the solution to the multivalued evolution problem , is necessarily of the form

with being the operator defined by (105). Further (cf. [2, Remark p. 55] with , and where stands for ) the derivative of is given by

with being the operator in (106). Notice that is well defined by virtue of (127) with . Now let and where , , and . Since , , from (169) it thus follows that is independent on and

Indeed (cf. (20) or (38)), we have , , and , the embeddings being equalities for those between the numbers and which are equal to . Then, under these assumptions on the data, formula (173) for can be extended until . For, we have and the differential equation in (170) is satisfied even at . To see this, we observe that

where , , and . First, from Proposition 6 we get . On the other side, using , , we obtain

so that . Finally, (127) with yields , too. Formula (173) thus holds at with .

Remark 40. In [2, Remark p. 55], formula (173) was extended until only under the more restrictive assumption , . Indeed [24, Proposition 5.2] was not available at the time of [2] and only the strong continuity of in the -norm on the spaces , , was known (cf. [2, Theorem 3.3]). Notice that in the case of problem (163) the element reduces to , so that in the nondegenerate case we get back the classical assumption , , (see, for instance, [9, Theorem 4.3.1(iii)] and [11, Theorem 4.5]).

Since (170) implies that , from (173) we thus find that

where, according to the notation in Corollary 38, we have set . In particular, . We conclude that, under the previous assumptions on the pair and on the data vector , if solves (170), then necessarily . As a consequence (cf. (168)), the strict solution to (160) satisfies the initial condition just in the sense .

Introduce the functions and , , defined by

Then, replacing with the right-hand side of (169), using (174), and recalling the definitions of the operators , , in (106)–(110), from (177) we deduce that solves the fixed-point equation

the functions , , and the operator being defined by

Conversely, let be a solution to the fixed-point equation (179), and assume that the pair and the data vector satisfy the assumptions below (170) and (173). Then, as before, and , , , being as in (171). We apply to both sides of (179), and we show that satisfies (172) with as in (169), so that is a solution to problem (170). To this purpose, we take into account Corollaries 14 and 26. Let . First (cf. Remark 34 and recall that ), using (86), (174), and (178), we get

Instead, due to the definition of and , using (125) we obtain

Therefore, from (183), (184), and the definition (105) of it follows that

the left-hand side being well-defined due to Remark 23. As far as is concerned, we first observe that, being in , from formula (126) and Remark 34 it follows that and are both well defined and equal to and , respectively. Consequently

Hence, commuting with both the integral sign and the semigroup, one has

Similarly, since Remark 34 and formula (125) yield

we find that

. In conclusion, from (187) and (189) it follows that

Summing up (185) and (190), we finally obtain , being as in (169). This completes the proof of the equivalence between problem (170) and the fixed point equation (179), provided that the data satisfy the mentioned assumptions.

Remark 41. We can summarize the previous reasonings as follows: problem (160) has been reduced to the fixed-point equation (179) for the new unknow , . This fixed-point argument is similar to that first successfully applied in [4, 7, 8, 32] to problem (165) with and then generalized in [23] to the degenerate case. A different approach has been followed in [6, 10] for the nondegenerate case and in [22] for the degenerate one. There, assuming that is absolutely Laplace transformable (cf. [6, 22]) or of bounded variation (cf. [10]), problem (165) with is solved by constructing its relative resolvent operator. We quote also [35] where the method of constructing the fundamental solution for the equation without the integral term is applied to a class of concrete degenerate integrodifferential equations.

From now on, for , , , and , will denote the interval defined by

Clearly, if , , then .

Lemma 42. Assume (161), and let in (H2). Assume that , , , and let . Then, for every fixed , the operator defined by (182) maps continuously into , and for every satisfies the following estimate, where :

Here is a positive constant depending only on , , , , , , , and , .

Proof. Let , , , and let us fix an arbitrary number , where . In particular, since , we have with , . Now let and . First, formula (186) being applicable, we rewrite (182) as

Now, we notice that implies that

Since , from (194) it follows that , and, consequently,

We conclude (cf. Remark 39) that Lemma 22 and Propositions 29 and 36 are applicable with and . Then, using estimates (111), (135), and (150) with the pair and the quintuplets , , being replaced, respectively, by and (indeed, since , if , then with , ) ,   , from (193) we finally obtain

Here we have set , where , , are the values at of the functions in Lemma 22 and Propositions 29 and 36. This completes the proof.

Remark 43. Assume that in Lemma 42 the Hölder exponents are such that belongs to . In this case (cf. (191)), the choice is admissible, and the meaning of Lemma 42 is that the operator defined by (182) preserves the minimal of the time regularities of .

Corollary 44. Let the assumptions of Lemma 42 be satisfied, and let and be as there. Then, for every fixed , the sequence (, , ) satisfies the following estimates, where and :

Proof. Reasoning as in [23, p. 468], we prove (197) by induction. Since for every fixed the operator maps in , replacing with in (192) and introducing the sequence of scalar nonnegative nondecreasing functions defined by , , from (192) we obtain

Then, applying to (198) an induction argument in which the first step of the induction follows from (192), we immediately deduce the following estimates:

The proof is complete.

Lemma 45. Let in (H2) and . Assume that , , and , where , , , , and . Let . Then, for every fixed , the function defined by (181) belongs to , provided that , , .

Proof. Let us fix , . Of course, and , , . Then, Proposition 29 and Lemma 30 applied with the quintuplets and the quadruplet being replaced, respectively, by (the constant functions , , , being obviously of class ) and , imply that , , . Now, since , the number satisfies

and assumption , , is meaningful. Lemma 24 with then yields , . Since , we get , too. Summing up, we get the assertion.

Before considering the function in (180), we introduce the following notation. In the sequel, for , , , and , will denote the interval

Notice that, since , if the stronger condition is satisfied, then (191) and (201) yield for every fixed . The introduction of the intervals is justified by Lemma 46, which requires a weaker condition on the pair than the one in Lemmas 42 and 45.

Lemma 46. Let in (H2), and let . Assume that , , , and , where , , , , , and . Let and , where . Then, for every fixed , the function defined by (180) belongs to , provided that , .

Proof. Observe that (cf. (159)) all the results from Lemma 32 to Corollary 38 will be applicable. First, since , the choice , , is meaningful. Moreover, since , the number satisfies . Hence, , too, and is well defined. Now, let be fixed. Due to (20) or (38), the element defined by (174) belongs to , whereas the functions defined by (178) are of class . Then, since , from Lemma 37 and Corollary 38 applied with the pairs and being replaced by and , respectively, we deduce that , . In addition, since the ’s and the constant functions belong to , from Proposition 36 applied with , it follows that , . Finally, since , the number satisfies

and the assumption , , makes sense. Then, the function being of class , Lemma 32 applied with yields , . Since , we conclude that , too. Summing up, we get the assertion.

Remark 47. We stress that, if in (H2), then , so that in both Lemmas 45 and 46 we have to assume that with . This is necessary in order to restore the loss of regularity produced by the operators and .

We can now prove the main results of the section.

Theorem 48. Assume (161) and , and let in (H2). Assume that , , , and , where , , , , , and . Let and , where . Then, for every fixed problem (160) admits a unique strict solution satisfying and such that , provided that , , .

Proof. Of course, due to (159), the assumption , , makes sense. In addition, since , we have . Therefore, by virtue of the choice of the Hölder exponents and , the number belongs to too, and the interval is well defined. Further, the numbers , , and being as in the statements of Lemmas 42, 45, and 46, respectively, we have . As a consequence, since and , all the mentioned lemmas are applicable with . To this purpose, we stress that since and , the conditions for the applicability of both Lemmas 45 and 46 are fulfilled. Hence, now let being fixed. First, due to Lemma 42, the operator , , , a fortiori maps into itself. Then, being endowed with the same norm of , from (197) we obtain the estimates

In particular, (203) yields that converges in . From generalized Neumann’s Theorem it thus follows that , the inverse being precisely . Since Lemmas 45 and 46 (both applied with (observe here that if , then the exponent in the last part of the proof of Lemma 46 satisfies . For, ), ) imply that , we conclude that the fixed-point equation (179) admits the unique solution

Observe now that the data vector satisfies all the assumptions which were needed to show the equivalence between the fixed-point equation (179) and problem (170). Indeed, and imply, respectively, that , and ,,, whereas, as in Lemma 46, implies that . Therefore, since , if is the solution to the fixed-point equation (179), then , too, and the function defined by (169) satisfies

where , , and . Consequently, recalling (168), we have proved that problem (160) has a unique strict global solution satisfying and such that . As far as the regularity of is concerned, instead, it suffices to observe that (168), (170), , and yield

The proof is complete.

Remark 49. Theorem 48 improves the faulty Thereoms 5.6 and 5.7 in [20] in two aspects. First, the assumption is weakened to . In fact, implies that . Hence, in the special case , the constraint in [20] reduces to the definitely weaker . Second, in [20], only for and opportunely chosen , the data and were assumed to belong to the intermediate spaces , whereas here, removing the assumption and considering the general case , we allow and to belong also to the interpolation spaces . To emphasize how much these aspects are decisive, let in Theorem 48. Then, if and the choice is understood for , we have , and the spaces and , , may be smaller than . However, the choice being admissible, in this situation too we can solve problem (160) with the data in spaces larger than . Further, since , in this case the results in [20] would not be applicable. These observations lead us to conclude that the more delicate approach followed in this paper with respect to that in [20, Sections 4 and 5], and especially the sharper results of the present Sections 3 and 4, yield a valuable refinement in the treatment of questions of maximal time regularity for the strict solutions to (160); of course, unless that the not too much significant case is assumed in .

Remark 50. The assumption in implies that and . In particular, if , then Theorem 48 holds with , , , , , and . Hence, , , and with , where

Clearly, if , then is redundant, and Theorem 48 holds with ,,,,,,,, and , , where if and if .

Remark 51. Observe that, if the ’s and ’s are assumed to vary in the smaller interval , then and the ’s can be chosen such that . To this purpose, letting , it suffices to take , , where . Then and . In other words, provided that the data vector is smooth enough, the pair has the maximal time regularities which is the minimal between the time regularities of the ’s and ’s.

We conclude with the results which follow from Theorem 48 for problems (163)–(166).

Theorem 52. Assume (161) and , and let in (H2). Assume that and , where , , , and . Let , where . Then, for every fixed problem (165) admits a unique strict solution satisfying and such that , provided that , , .

Proof. Repeat the proofs of Lemmas 42, 45, and 46, Corollary 44, and Theorem 48, letting there , . To this purpose, observe that (169) and (174) reduce to and . Consequently, (180)–(182) change to , , and .

Theorem 53. Assume (161) and , and let in (H2). Assume that , , , and , where , , , , and . Let and , where . Then, for every fixed problem (166) admits a unique strict solution satisfying and such that .

Proof. Let in the proofs of Lemmas 42, 45, and 46, Corollary 44, and Theorem 48. In this case, (169) and (174) reduce to and . Hence, (180)–(182) change to ,, and .

Let us now turn to the degenerate differential problems (163) and (164).

Theorem 54. Assume (161) and , and let in (H2). Assume that , , , and let . Then, for every fixed problem (163) admits a unique strict global solution satisfying and such that , provided that , , .

Proof. Let ,,, in problem (160) and formulae (169), (174) and, (179)–(182). Then, , and . Consequently, Lemma 42 and Corollary 44 are unneeded, and the proof of Theorem 48 simplifies as follows. First, due to we have , and the interval is well defined. Hence, let being fixed. Since (cf. (200)) , , reasoning as in the last part of the proof of Lemma 45 we get . Moreover (see the proof of Lemma 46), since , and , , , Corollary 38 and Lemma 32 applied with and yield . Summing up, we find that . The assertion then follows from and (cf. (206)) .

Remark 55. We refer to [19, Theorem 5.3] for a result of both time and space regularity for problem (163). There, provided that and are opportunely chosen and the data satisfy assumptions similar to those in Theorem 54, it is shown that , and that the higher is the order of the interpolation space where we look for space regularity, the lower is the Hölder exponent of regularity in time. Notice that has no space regularity, unless has too.

Theorem 56. Assume (161) and , and let in (H2). Assume that , , and , where , , , and . Let and , where . Then, for every fixed , problem (164) admits a unique strict global solution satisfying and such that .

Proof. Let , , in problem (160) and formulae (169), (174), and (179)–(182). Then, , and . Therefore, as in Theorem 54, we do not need Lemma 42 and Corollary 44, and the proof of Theorem 48 simplifies as follows. Again, implies that , so that , and the interval is well defined. Let be fixed. First (see the proof of Lemma 45), since , Lemma 30 applied with yields , . On the other side (see the proof of Lemma 46), since and , , from Lemma 37 and Corollary 38 applied with and we deduce that ,,. Summing up, we find that , and the assertion again follows from and (cf. (206)) .

6. An Application to a Concrete Case

Theorem 48 is here applied to determine the right functional framework where to search for the solution of an inverse problem arising in the theory of heat conduction for materials with memory. To this purpose, let ,, be a bounded domain with boundary of class (cf. [36, p. 94]). If represents a rigid thermal body with memory, then the linearized theory of heat flow yields the following equations linking the internal energy , the heat flux , and the temperature (cf. [32, 37–40]):

Here , , , , , , and , whereas the ’s represent the first-order linear differential operators

where and , , . According to the terminology of [39, 40], the functions , , , and are called, respectively, the energy-temperature relaxation function, the heat conduction relaxation functions, and the heat supply function and we assume that they satisfy the following conditions:

Notice that, different from [32, 37–40], here the energy-temperature relaxation function is assumed to depend also on the spatial variable . In physical terms, this is equivalent to say that represents a rigid inhomogeneous material with memory. Furthermore, in contrast with the quoted papers where only the cases and are treated, here we have assumed that the history record of is kept by an arbitrary number of heat conduction relaxation functions and that the ’s are the more general first-order differential operators defined in (209).

By setting

from (208) and (209), it thus follows that the temperature must satisfy the following equation:

Let us now assume that is of the following special form:

where the functions and , , satisfy the following conditions (cf. (210)):

Here, ,, is the usual space with norm (cf. [36, Chapter 7]). Using , for and we now set

Then, since (214)–(216) yield and , , if we multiply both sides of (213) by and use (218)–(223), we are led to the following basic differential equation for the temperature , where :

We endow this differential equation with the initial condition , and the Dirichlet boundary condition , , .

We now suppress the dependence on , and we transform (224) in a degenerate integrodifferential Cauchy problem in a Banach space . To this purpose, for every fixed and observing that implies that for every , , we set

Here (cf. [36, Chapter 7]), , , , denotes the usual Sobolev space endowed with the norm (), whereas denotes the completion of in , being the set of all real-valued infinitely differentiable functions having compact support in . We further assume that there exists positive constant , , such that for every the following inequalities hold:

where . Therefore, from (212), (218), and (230) we get

From (225)–(231) it follows that , and , , are closed linear operators from to itself, and the relation holds. In addition, due to (212), (217), (218), and (231), from [36, Theorem 9.15 and Lemma 9.17], it follows that for every fixed the operator admits an inverse operator . Hence, a fortiori, and so condition (161) is satisfied (observe also that implies that the norms and are equivalent on . In fact, if , then  , being a positive constant depending on ). The closed graph theorem then yield . Moreover (cf. [19, formula (77)], and [41, formula (2.16)]), the following estimate holds (of course, here is replaced with the more general ):

where , being a suitable positive constant depending on and . Hence, condition is satisfied with and . Notice that, since may have zeros in , is in general a m. l. operator, so that is determined by (cf. (162)):

Using the convolution operator in (104) in which for the bilinear operator we take the scalar multiplication in , from (224)–(229) we finally obtain that the temperature solves the following degenerate integrodifferential Cauchy problem in :

Now, assume for a moment that we are interested in solving the inverse problem of recovering both the temperature and the memory kernels in (234). Clearly, due to (222), if we recover , then the heat conduction relaxation functions will be known too, unless of the arbitrary constants , . Indeed, , . To solve such an inverse problem, we need additional informations other than the initial condition , which, in general, suffices only to guarantee the well-posedness of the direct problem of recovering in (234). Suppose then that the following additional pieces of information are given:

where and , . We will search for a solution vector of the inverse problem (234) and (235) such that and , , with the Hölder exponents and ,, to be made precise in the sequel. We stress that here we will not solve completely the mentioned inverse problem. For, its detailed treatment would lead us out of the aims of this paper. Our intention here is only to highlight how the main results of Section 5 allow to determine the correct functional framework in which the solution of the inverse problem has to be searched. However, a complete treatment of the inverse problem will be the object of a future paper.

Assuming that solves (234), we introduce the new unknown

Then, differentiating (234) with respect to time and using

we find that solves the following degenerate integrodifferential problem:

where ,, and (indeed, since is the multiplication operator by the function independent of , from the differential equation in (234) with we get ). Of course, (238) is the special case , , , of problem (160).

Conversely, assume that solves (238). Then, the function defined by (236) belongs to and solves (234). Indeed, using the fact that does not depend on time and that , , and , , are closed, we obtain

Now, observe that

whereas an application of Fubini’s theorem combined with the changes of variables , and easily yields for every the following:

Therefore, replacing (240)–(242) in (239), it follows for every that

and the latter integral is equal to zero by virtue of (238). Since from (236) it follows that , we have thus shown that (234) and (238) are equivalent. Such an equivalence is the first step in solving the mentioned inverse problem of recovering the vector with the help of the additional information (235).

Let us now apply the linear functional , , to (238). Using

we thus find the following system of equations for the unknown :

where we have set (recall that , , are known)

Therefore, if the matrix has determinant , then from Cramer’s formula it follows that the solution of (245) is given by

with , , being the cofactor of the element of (with the convention that in the case of ). We have thus found a system of fixed-point equations for the unknown .

Now, let , , , where is as in (233). Assume that in the initial condition belongs to and that

where and is the conjugate exponent of . Then (cf. (179) with , , and , ), problem (238) is equivalent to the fixed-point equation

where and

Here, the ’s, , are defined by (106)–(110), , and the functions , and are defined by , , and , respectively, where (cf. (174)) .

Then, since , if we set , , and

from (247) and (249) we deduce that to solve the inverse problems (234) and (235) for the unknown vector , it suffices to show that the fixed-point equation

has a unique solution. In general, this will be done by proving that is a contraction map in the Banach space

at least for opportunely chosen Hölder exponents and , , and, eventually, sufficiently small values of . It is just in the choice of and the ’s that the main result of Section 5 plays a key role. The Hölder exponents have to be chosen so that the direct problem (234) in which the ’s are assumed to be known is well posed. Due to the shown equivalence between problems (234) and (238), the well-posedness of the direct problem (234) is then a consequence of Theorem 48 and formula (236). More precisely, recalling Remark 50 for the case , an application of that theorem yields the following maximal time regularity result for the solution of (234).

Theorem 57. Let , , , and , , , , be defined by (225) and (226) with . Let , and , , be defined by (227)–(229) through (209), (212), and (215)–(221), and let (230) and (231) be satisfied. Further, let be defined by (233), and let , , . Let and , , and assume that

where , and are defined by (222) and (223) through (211) and (216), whereas . Let and , and let be the interval defined by (cf. (207) with )

Then, for every fixed problem (234), or, equivalently, problem (224), admits a unique strict solution satisfying and such that , provided that , .

Proof. Apply Theorem 48 with , , and , , to the equivalent problem (238). Since is the multiplication operator by the function independent of , the assertion then follows from , , and .

Larger values of in Theorem 57 can be obtained assuming more smoothness and some order of vanishing for the function . In fact, let be such that the following estimate holds for some positive constant :

Then (232) holds with being replaced by (cf. [41, formulae (3.23) and (4.41)]):

(precisely, in [41, formula (3.23)] it is shown that , where and . Using (cf. [41, formula (2.15)]) , we thus find that ; that is, ). Under (256) we thus find the following better result, where may be greater than two.

Theorem 58. Let (256) holds, and let , , , , , be as in Theorem 57, but with . Let (254) be fulfilled, but with and , , where is as in (257). Let and , and let be as in (207). Then, for every fixed problem (234), or, equivalently, problem (224), admits a unique strict solution satisfying and such that , provided that , .

Proof. It suffices to observe that for every and , the number in (257) satisfies . Hence, proceeding as in the proofs of Theorem 57, except for replacing there with as in (257), we get the assertion.

Appendix

Here we clarify why the definition of in [20] has to be modified in accordance to that in this paper. To avoid confusion with the present notation, we will denote the operator in [20] with . Precisely, in [20, formula (4.12)], was defined as follows:

and considered as acting on functions , , . Even though , formula (A.1) may have no sense, since

and the integral on the right is not convergent, the exponent being less or equal than . It is for this reason that in (A.1) has to be replaced with the increment as in formula (106) (see inequality (118)) and to introduce the operator as in (109). Of course, as a consequence, the definitions of and in [20, Lemmas 4.6 and 4.8] as and , respectively, have to be changed too in accordance with the present formulae (107) and (108) containing the increments and . To this purpose, we want to make clear that, contrarily to [20, Lemma 4.4], the statement and the proof of [20, Lemma 4.8] is correct, since there the function inside the integral on the right-hand side of (A.1) takes its values in an opportune intermediate space . However, the correctness of that lemma does not suffice to proceed as in [20, Section 5] to solve problem (160) with .

For the reader’s convenience we thus now indicate how to change the definitions of the functions , , and the operator in [20, formulae (5.8)–(5.10)], and we state the amended version of [20, Theorems 5.6 and 5.7]. First, according to [20] where only this case was treated, let in problem (160), and write , , in place of , and , respectively. Then, under the same assumptions on the vector as those in the present Section 5, it can be shown that problem (160) with is equivalent to the fixed-point equation (179), where (cf. (180)–(182))

Here, , , is the value at of the function defined by (169) with , and are defined, respectively, by , and , is the operator , and the ’s, , are as in (106)–(110). Formulae (A.3) replace the definitions of , and in [20, formulae (5.8)–(5.10)]. Therefore, from Lemmas 42, 45, and 46 and Corollary 44 with we obtain the following version of Theorem 48.

Theorem A.1. Assume (161) and , and let in (H2). Assume that , , , and , where , , and . Let and . Then, for every fixed the problem

admits a unique strict solution satisfying and such that , provided that , .

Theorem A.1 substitutes [20, Theorem 5.6 and 5.7]. Notice that, differently than [20], here only one statement occurs. In fact, the more suitable procedure followed in this paper makes the separation in [20] of two distinct intervals in which may vary totally unneeded. Finally, letting in Theorems 52, 5.14, 54, and 56, we obtain the correct versions of [20, Theorems 5.11, 53, and 5.16] for the subcases of (A.4) corresponding to the choices , , , and , respectively. For saving space, we leave this easy task to the reader.

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