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

Claudio Cuevas,¹ Carlos Lizama,² and Herme Soto³

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 [2–5]). 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 [9–11], attractor [2, 12–16], 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 [22–32] 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

Indeed, it is an easy consequence of the following estimates (here will stand for some positive constant independent of and with ).

Let be a compact subset of . For and , we set that

We have the following estimates:

where is a constant depending on .

If from (89), we get

and (90) implies that

Hence,

Applying Theorem 17, we have the following result.

Proposition 21. Under the previous conditions, if , , and is small enough, then problem (79) has an asymptotically almost-periodic mild solution.

Example 22. We consider the following partial differential equation

If is small enough by Proposition 21, problem (94) has an asymptotically almost-periodic mild solution. In fact, take , , and .

The next application is a modification of problem (79). Let be a fixed nonnegative integer and let and be two bounded continuous functions. Suppose that satisfies the following growth condition:

for each , , . We consider the following partial differential equation

We model (96) in the abstract form (1) in a similar way to problem (79). That is we set , (hence which will be useful to get some a priori estimates) and we consider the operator as defined in (81) and (82). Suppose that

We define by

We observe that is well defined. In fact, we get the following estimates:

On the other hand, for and we have the following estimate:

which means that satisfies (42) with .

Since , there are two functions

so that . Let be a compact subset of . For and , we set

If , then

where is given by (88).

We note that using Sobolev embedding we have that there is a positive constant depending only on so that

Taking into account the fact that is a compact subset of , (4.29) and (103). It follows that uniformly for . Hence,

As an immediate consequence of Definition 3 and using the fact that is bounded we have that

Since , we conclude that

Proposition 23. Under the previous conditions, if is small enough, then problem (96) has an asymptotically almost-periodic mild solution.

Proof. We argue as follows. Let us choose such that is small enough, where and are constants given in (7). Then, condition (47) is fulfilled. Now applying Theorem 17 we conclude the proof.

Let be a bounded continuous function and let be in and . In a bounded smooth domain , we consider the following partial differential equation:

where , . We model (108) in the abstract form (1); that is, we set and we consider the operator as in (81) and (82). We define by

To establish our next result, we assume the following conditions.

(H1) Let be in , with and .(H2) There is a continuous nondecreasing function such that as and

Proposition 24. Under the previous conditions, problem (108) has an asymptotically almost-periodic mild solution.

Proof. Let be given so that . We can infer the following estimate:

Hence, we can define in by . Let be a compact subset of and let be as in (88). We set

We have the following estimates:

It follows from (113) and (114) that . In addition, from the estimate (115), we obtain that is uniformly continuous on compact sets.

From , for ,, we can infer that

Therefore, conditions and of Theorem 19 are satisfied. A straightforward computation shows that holds. We can prove that the set is relatively compact in . In fact, first, we denote by the zero extension of outside ; that is,

For every number , there exists a number and a subset such that for every with

Hence,

From [38, Theorem 2.21] the desired assertion follows. Note finally that using Remark 20 we have that (108) has an asymptotically almost periodic mild solution. This ends the proof.

Acknowledgments

This work was written while the first author was visiting the Universidad de La Frontera (Temuco, Chile). He is grateful to the Applicable Analysis Group and the Department of Mathematics and Statistics, for its generous hospitality and providing a stimulating atmosphere to work. C. Cuevas has been partially supported by Programa Atracción e Inserción (PAI-MEC) Grant 80112008 (CONICYT-CHILE). C. Lizama has been partially supported by Project Anillo ACT 79 and H. Soto has been partially supported by DIUFRO DI 12-0002.

References

D. Henry, Geometric Theory of Semilinear Parabolic Equations, vol. 840 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1981.

A. N. Carvalho and J. W. Cholewa, “Attractors for strongly damped wave equations with critical nonlinearities,” Pacific Journal of Mathematics, vol. 207, no. 2, pp. 287–310, 2002.

S. P. Chen and R. Triggiani, “Proof of extensions of two conjectures on structural damping for elastic systems,” Pacific Journal of Mathematics, vol. 136, no. 1, pp. 15–55, 1989.

S. P. Chen and R. Triggiani, “Characterization of domains of fractional powers of certain operators arising in elastic systems, and applications,” Journal of Differential Equations, vol. 88, no. 2, pp. 279–293, 1990.

S. P. Chen and R. Triggiani, “Proof of two conjectures by G. Chen and D. L. Russell on structural damping for elastic systems,” in Approximation and Optimization, vol. 1354 of Lecture Notes in Mathematics, pp. 234–256, Springer, 1988.

D. Bahuguna, “Strongly damped semilinear equations,” Journal of Applied Mathematics and Stochastic Analysis, vol. 8, no. 4, pp. 397–404, 1995.

V. Georgiev and G. Todorova, “Existence of a solution of the wave equation with nonlinear damping and source terms,” Journal of Differential Equations, vol. 109, no. 2, pp. 295–308, 1994.

F. Gazzola and M. Squassina, “Global solutions and finite time blow up for damped semilinear wave equations,” Annales de l'Institut Henri Poincaré, vol. 23, no. 2, pp. 185–207, 2006.

J. M. Ghidaglia and A. Marzocchi, “Longtime behaviour of strongly damped wave equations, global attractors and their dimension,” SIAM Journal on Mathematical Analysis, vol. 22, no. 4, pp. 879–895, 1991.

P. Massatt, “Limiting behavior for strongly damped nonlinear wave equations,” Journal of Differential Equations, vol. 48, no. 3, pp. 334–349, 1983.

G. F. Webb, “Existence and asymptotic behavior for a strongly damped nonlinear wave equation,” Canadian Journal of Mathematics, vol. 32, no. 3, pp. 631–643, 1980.

S. Borini and V. Pata, “Uniform attractors for a strongly damped wave equation with linear memory,” Asymptotic Analysis, vol. 20, no. 3-4, pp. 263–277, 1999.

S. M. Bruschi, A. N. Carvalho, J. W. Cholewa, and T. Dlotko, “Uniform exponential dichotomy and continuity of attractors for singularly perturbed damped wave equations,” Journal of Dynamics and Differential Equations, vol. 18, no. 3, pp. 767–814, 2006.

A. N. Carvalho and J. W. Cholewa, “Strongly damped wave equations in ,” Discrete and Continuous Dynamical Systems, pp. 230–239, 2007.

H. Li, S. Zhou, and F. Yin, “Global periodic attractor for strongly damped wave equations with time-periodic driving force,” Journal of Mathematical Physics, vol. 45, no. 9, pp. 3462–3467, 2004.

V. Pata and M. Squassina, “On the strongly damped wave equation,” Communications in Mathematical Physics, vol. 253, no. 3, pp. 511–533, 2005.

A. N. Carvalho and J. W. Cholewa, “Local well posedness for strongly damped wave equations with critical nonlinearities,” Bulletin of the Australian Mathematical Society, vol. 66, no. 3, pp. 443–463, 2002.

R. Ikehata, “Decay estimates of solutions for the wave equations with strong damping terms in unbounded domains,” Mathematical Methods in the Applied Sciences, vol. 24, no. 9, pp. 659–670, 2001.

G. Avalos and I. Lasiecka, “Optimal blowup rates for mininal energy null control for the structurally damped wave equation,” Annali della Scuola Normale Superiore di Pisa, vol. 2, pp. 601–616, 2003.

S. A. Messaoudi, “Blow up in a nonlinearly damped wave equation,” Mathematische Nachrichten, vol. 231, pp. 105–111, 2001.

A. N. Carvalho, J. W. Cholewa, and T. Dlotko, “Strongly damped wave problems: bootstrapping and regularity of solutions,” Journal of Differential Equations, vol. 244, no. 9, pp. 2310–2333, 2008.

R. P. Agarwal, B. de Andrade, and C. Cuevas, “On type of periodicity and ergodicity to a class of fractional order differential equations,” Advances in Difference Equations, vol. 2010, Article ID 179750, 25 pages, 2010.

R. P. Agarwal, B. de Andrade, and C. Cuevas, “Weighted pseudo-almost periodic solutions of a class of semilinear fractional differential equations,” Nonlinear Analysis, vol. 11, no. 5, pp. 3532–3554, 2010.

A. Berger, S. Siegmund, and Y. Yi, “On almost automorphic dynamics in symbolic lattices,” Ergodic Theory and Dynamical Systems, vol. 24, no. 3, pp. 677–696, 2004.

C. Cuevas and M. Pinto, “Existence and uniqueness of pseudo almost periodic solutions of semilinear Cauchy problems with non dense domain,” Nonlinear Analysis A, vol. 45, no. 1, pp. 73–83, 2001.

B. de Andrade, C. Cuevas, and E. Henríquez, “Asymptotic periodicity and almost automorphy for a class of Volterra integro-differential equations,” Mathematical Methods in the Applied Sciences, vol. 35, no. 7, pp. 795–811, 2012.

T. Diagana, G. Nguerekata, and N. V. Minh, “Almost automorphic solutions of evolution equations,” Proceedings of the American Mathematical Society, vol. 132, no. 11, pp. 3289–3298, 2004.

H. Henríquez and C. Cuevas, “Almost automorphy for abstract neutral differential equations via control theory,” Annali di Matematica Pura ed Applicata, vol. 192, no. 3, pp. 393–405, 2013.

J. Kaczorowski and O. Ramaré, “Almost periodicity of some error terms in prime number theory,” Acta Arithmetica, vol. 106, no. 3, pp. 277–297, 2003.

J. H. Liu and X. Q. Song, “Almost automorphic and weighted pseudo almost automorphic solutions of semilinear evolution equations,” Journal of Functional Analysis, vol. 258, no. 1, pp. 196–207, 2010.

N. V. Minh, T. Naito, and G. Nguerekata, “A spectral countability condition for almost automorphy of solutions of differential equations,” Proceedings of the American Mathematical Society, vol. 134, no. 11, pp. 3257–3266, 2006.

W. Shen and Y. Yi, “Almost automorphic and almost periodic dynamics in skew-product semi flows,” Memoirs of the American Mathematical Society, vol. 136, no. 647, 1998.

S. Zaidman, Almost-Periodic Functions in Abstract Spaces, vol. 126 of Research Notes in Mathematics, Pitman, Boston, Mass, USA, 1985.

A. M. Fink, Almost Periodic Differential Equations, vol. 377 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1974.

S. Zaidman, “Solutions of almost-periodic abstract differential equations with relatively compact range,” Nonlinear Analysis, vol. 8, no. 9, pp. 1091–1093, 1984.

G. Gripenberg, S. O. Londen, and O. Staffans, Volterra Integral and Functional Equations, vol. 34, Cambridge University Press, Cambridge, UK, 1990.

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, vol. 68, Springer, New York, NY, USA, 2nd edition, 1997.

R. A. Adams, Sobolev Spaces, Academic Press, London, UK, 1975.