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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 308616, 14 pages
http://dx.doi.org/10.1155/2013/308616
Research Article

Asymptotic Periodicity for Strongly Damped Wave Equations

1Departamento de Matemática, Universidade Federal de Pernambuco 50540-740 Recife, PE, Brazil
2Departamento de Matemática, Universidad de Santiago de Chile, Casilla 307, Correo 2, Santiago, Chile
3Departamento de Matemática y Estadística, Universidad de La Frontera, Casilla 54-D, Temuco, Chile

Received 22 April 2013; Accepted 23 June 2013

Academic Editor: Nasser-Eddine Tatar

Copyright © 2013 Claudio Cuevas et al. 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

This work deals with the existence and uniqueness of asymptotically almost-periodic mild solutions for a class of strongly damped semilinear wave equations.

1. Introduction

Let be a reflexive Banach space and let be a closed densely defined operator and . Consider the Cauchy problem where is the fractional power space associated with as in [1]. Equations like (1) appear in the literature under the name of strongly damped wave equations. An example of mathematical model represented in form (1) is the wave equation with structural damping (see [25]). The strongly damped wave equations has been investigated in several contexts by many authors in the last years, for example, existence [6, 7], global classical solution [6, 8], long-time asymptotic behavior [911], attractor [2, 1216], well-posedness [17], decay estimates [18], blowup [8, 19, 20], controllability [19], bootstrapping, and regularity [21]. Another important aspect of the qualitative study of the solutions of strongly damped wave equations is their asymptotic periodicity. In recent years, the study of periodicity and its various extensions for evolution equations has attracted a great deal of attention of many mathematicians (see [2232] and references therein).

To the best of our knowledge, the study of the existence of asymptotically almost-periodic solutions for strongly damped wave equations of type (1) is a topic not yet considered in the literature.

Problem (1) can be written as a first order in time Cauchy problem in : where is defined by

Definition 1. The pair is said to be an admissible pair if there exist and such that for all in the sector and .
If is an admissible pair, by [21, Proposition 2.1] is a closed operator with . Indeed, has the inverse expressed in the matrix form as Moreover, the operator has compact resolvent whenever has compact resolvent. By [21, Theorem 2.3] the operator is sectorial in . The semigroup generated by in is exponentially decaying analytic. That is, there are constants and such that Throughout this paper, we always assume that is an admissible pair.
This paper has four sections. In the next section, we consider some definitions, technical aspects, and basic properties related to asymptotically almost-periodic functions. In the third section, we obtain general results on the existence of asymptotically almost-periodic (mild) solutions to the problem (1). The main abstract results are Theorems 12, 16, 17, and 19. Finally, in the fourth section we consider several applications. In particular, we consider the following class of partial differential equations: in a bounded smooth domain and where and satisfy certain growth conditions. We prove that if , , and is small enough, then the previous problem has an asymptotically almost periodic mild solution. The same type of conclusion is derived for the wide class where and .

2. Preliminaries

In this section, we present some concepts and properties needed to develop the following sections. Let be a reflexive Banach space. For an interval , denotes the space formed by the bounded continuous functions from into , endowed with the norm of uniform convergence. When , we denote instead of . The notation stands for the subspace of consisting of functions that vanish at infinity. We denote by the Banach algebra of bounded linear operators defined on . For , the notation stands for the closed ball . For a linear operator with domain and range in , we represent by (resp., ) the spectrum (resp., the resolvent set) of . For , we denote by the resolvent operator of . When is closed, we denote by the domain of endowed with the graph norm .

Next, we present a brief summary of the main properties of asymptotically almost-periodic functions.

Definition 2 (see [33]). A continuous function is called almost-periodic if for each there exists such that for every interval of length it contains a number with the property that for each .
The previous number is called an -translation number for . We denote by the space formed by the almost periodic functions . We note that each almost-periodic function is bounded and uniformly continuous. It is well known that the range of an almost periodic function is relatively compact. The space is a Banach space endowed with the norm of uniform convergence.
Let be a Banach space. We have the following concept of parameter-dependent almost-periodic function.

Definition 3. A continuous function is called almost-periodic in uniformly for in compact subsets of if for every compact subset of and each there exists such that every interval of length contains a number with the property that for all , .
Henceforth, we abbreviate the terminology by calling almost-periodic from into to those functions that are almost-periodic in uniformly for in compact subsets of , and we denote by the set formed by the almost-periodic functions from into .
The proof of the following result is similar to the proof of [22, Lemma 2.12] and therefore omitted.

Lemma 4. If is an almost-periodic function and then is almost-periodic.

It is well known that the study of composition of two functions with special properties is important and basic for deep investigations. The following result has been established in [34].

Lemma 5. Let be an almost-periodic function and let be an almost-periodic function. Then the function is almost-periodic.

We will need the following definition.

Definition 6. A continuous function is called asymptotically almost-periodic if there exist two functions and such that The function is called the almost-periodic part of . We denote by the space formed by the asymptotically almost-periodic functions . The space is a Banach space endowed with the norm of uniform convergence. Furthermore, .
In what follows denotes the space consisting of continuous functions such that uniformly for in compact subsets of .

Definition 7. A continuous function is called asymptotically almost-periodic if there are two functions and such that We denote by the set consisting of all asymptotically almost-periodic functions from into .
Let be an interval. We have the following concept of function uniformly continuous on compacts sets.

Definition 8. A continuous function is called uniformly continuous on compact sets if for all compact set and all there is such that for all and with .

Lemma 9 (see [22]). Let be an asymptotically almost-periodic and uniformly continuous on compact sets function. Let be an asymptotically almost-periodic function. Then the function , is asymptotically almost-periodic.

The proof of the following result is similar to the proof of [22, Lemma 2.13]. Therefore, we will omit it.

Lemma 10. If is an asymptotically almost-periodic function and Then, is asymptotically almost-periodic.

3. Asymptotically Almost-Periodic Mild Solutions

We recall the following definition that will be essential for us.

Definition 11. Let be in . We say that is a mild solution to (2) (or to (1)) if it satisfies the Cauchy integral formula:

Theorem 12. Let be an asymptotically almost-periodic function and assume that there exists a locally integrable function satisfying for all , and each . If where and are given in (7). Then, (2) has a unique asymptotically almost-periodic mild solution.

Proof. We define the map on the space by the expression where is an asymptotically almost-periodic function; that is, Since ; then there are two functions and so that .
We set for For , we have the following decomposition: where and denotes the remained terms of the previous decomposition.
Next, let us show that . By (7) we have that We observe that are relatively compact in . Since , for each there exists a constant such that for all .
We deduce On the other hand, since by [35, Appendix] we get Next, we estimate the first term of (21). For , we choose big enough so that for all , and for all and all . We have the following estimates: From (23) to (29), we deduce that ; that is, .
Since and , we get from Lemma 5 that Now, by Lemma 4, we obtain that , and hence is well defined. It suffices to show that the operator has a unique fixed point in .
For this, we consider that . We can deduce that by the contraction principle, has a unique fixed point in . This completes the proof.

Remark 13. We wish to emphasize that condition (15) is optimal in the sense that the function is locally integrable. This is the largest possible class of Lipschitz constant for which the conclusion of Theorem 12 holds true. However, this condition in makes our analysis much more harder, because to prove Theorem 12 we cannot use the standard composition lemma for asymptotically almost-periodic functions (see Lemma 9). To overcome this difficulty we need to use a suitable decomposition for the natural operator associated with the mild solution (see (14) and (21)). In contrast, we note that in the more restrictive case of to be an integrable bounded function we can use Lemma 9 directly.

Corollary 14. Let be an asymptotically almost-periodic function that satisfies the Lipschitz condition (15) with . If then problem (2) has a unique asymptotically almost-periodic mild solution.

Remark 15. Let be an asymptotically almost-periodic function that satisfies the Lipschitz condition (15). We can avoid the condition (32) by using the fixed-point iteration method. Indeed, we consider two cases.

Case 1. . We consider the following space endowed with the norm of the uniform convergence. We define the map on the space by Combining Lemmas 9 and 10, we know that is a continuous function from into . Moreover, for , we have With the notation and the previous estimate yields Since is a nondecreasing function, proceeding inductively, we can show that where denotes the -fold convolution of with itself.
On the other hand, the map , given by is a bounded linear map. Moreover, it follows from [36, Theorem] that , which implies that as . This shows that is a contraction for sufficiently large. As a consequence has a fixed point in .
We note that the function is an asymptotically almost-periodic mild solution to problem (2).

Case 2. in (15) is an integrable bounded function. By Lemmas 9 and 10 the space is invariant under (see (17)). The fixed point iteration method and the following estimate are responsible for the fact that has a unique fixed point in . This concludes the discussion of Remark 15.
We have the following results of the existence of local type.

Theorem 16. Let be an asymptotically almost-periodic function that satisfies the Lipschitz condition for each and such that , where is nondecreasing continuous function such that and for all . Then, there exists such that, for each satisfying , there is a unique asymptotically almost-periodic mild solution of problem (2).

Proof. We choose small enough such that and . Assume that . We consider the space endowed with the norm of the uniform convergence. We consider the map given by (17) on . By Lemma 9 We next prove that .
In fact, if , we have which permit us to infer that .
On the other hand, for , we have that which shows that is a contraction from into itself. Therefore, the assertion holds for .

Theorem 17. Let be a function that satisfies the local Lipschitz condition (42) with a nondecreasing function. Assume that there is a constant such that where , , and are the constant given in (7). Then, there is an asymptotically almost-periodic mild solution of problem (2).

Proof. We define the map on the closed ball by means of the expression (34).
If , we have the estimate
Moreover, for , , we have
Using (50), we get that is a contraction on .

In many concrete situations, the operator has compact resolvent, which in turn implies that the semigroup generated by the operator is compact for . To exploit this property of compactness, we need to introduce some preliminaries.

Let be an arbitrary Banach space and let be a nondecreasing continuous function such that when . In next, we denote by the space endowed with the norm

For reference purposes, we state the following property.

Lemma 18 (see [26]). A set is relatively compact in if the following conditions are fulfilled: (C1) For all , the set is relatively compact in . (C2) uniformly for .

To establish our next result we introduce the following condition.

There is a continuous nondecreasing function such that for all and all .

We next denote We have the following result.

Theorem 19. Assume that the operator in (1) has compact resolvent. Suppose, in addition, that the following conditions are fulfilled:(a) The function is uniformly continuous on compact sets and satisfies the condition .(b) For each , , and ,
as uniformly for all so that .(c) For each , (d) For each , there is such that for every , with
implies that
for each .(e).
Then, problem (2) has an asymptotically almost-periodic mild solution.

Proof. Let be the space consisting of the functions in such that , . It is clear that is a closed subspace of . We define the operator on by (34). It follows from conditions and that where Thus, we have that .
We divide the rest of the proof into several steps.(i) The map is continuous. For each there is such that for , with implies that which shows the assertion.(ii) The map is completely continuous. We take and we set .For , we set We first show that is a relatively compact set in for each . It follows from the mean value theorem that , where denotes the convex hull of where , and is a constant given in (7).Since and taking into account that is compact for , we infer that is a relatively compact set in and consequently is also relatively compact. Let now fixed and the set formed by the functions restricted to the interval . We affirm that the set is equicontinuous.In fact, if For , we obtain the following estimate: where . It is immediate that the first term on the right hand side converges to zero when and, using condition we obtain the second term on the right-hand side also converges to zero when and the convergence is independent of the function .We now show that independent of . This assertion is a direct consequence of the following estimate and the condition Combining these assertions with Lemma 18, we get that is a relatively compact set in . Since was chosen arbitrary, this proves that is completely continuous.(iii) There is such that . In fact, if we assume that the assertion is false; then for all we can choose such that for all . Then which contradicts condition and establishes the assertion.(iv) If , then the function given by is in . Since is uniformly continuous on compact sets, we have from Lemma 9 that Applying Lemma 10, we obtain that . Consequently, combining with we infer that where denotes the closure of in . Using Schauder fixed point theorem, we deduce that has a fixed point .(v) Finally, we show that . Let be a sequence in that converges to for the topology in . For , let be the constant in condition ; there is so that for all and all . Therefore, for Hence Since we get that and completes the proof.

Remark 20. Note that in Theorem 19 we do not need to assume that the operator in (1) has compact resolvent if the following condition holds.(f) For all and the set is relatively compact in .

4. Applications

Suppose that , () is a bounded continuous function, , and . In a bounded smooth domain , we consider the following partial differential equation: where and satisfy the following growth conditions: where and are positive constants. Here, we describe the asymptotically almost-periodic behavior of solutions of problem (79) in the -setting. To model this problem in the abstract form (1) we set that , , the operator is defined in by ( is the Dirichlet Laplacian in ) on the domain where is the standard Sobolev space (see [37]). With this specification, problem (79) will fall into the abstract formulation (1). Since , we can choose the angle for the sector as small as needed and therefor (see [21, Example 4.3]) will be an admissible pair for any . From [21, Section 3] we get that We define by where is the Nemytski operator associated with , and represents the divergence of . For and . Using Minkowski's inequality and Sobolev embedding, we have the estimate Whence is well defined. We claim that satisfies (42) with