Mathematical Problems in Engineering
Volume 2017 (2017), Article ID 1410408, 16 pages
https://doi.org/10.1155/2017/1410408

Research Article

Attractors for a Class of Abstract Evolution Equations with Fading Memory

Xuan Wang, Fenxia Duan, and Didi Hu

College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China

Correspondence should be addressed to Xuan Wang; wangxuan@nwnu.edu.cn

Received 1 January 2017; Accepted 27 April 2017; Published 14 June 2017

Academic Editor: Kishin Sadarangani

Copyright © 2017 Xuan Wang 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.

In this paper, we study the dynamics of an abstract evolution equation with fading memory with a critical growing nonlinearity. By use of some new methods and asymptotic estimate techniques, we first verify the asymptotic compact of solution semigroup and then prove the existence of global attractors in weak topological space and strong topological space, while the forcing term only belongs to or , respectively. The results are new and appear to be optimal.

1. Introduction

In this paper, we consider the asymptotic behaviors of solutions for the following abstract evolution equation with fading memory in a bounded domain :

where .

About the forcing term , we consider two cases: the weak solutions case and the strong solutions case . For the nonlinearity, we assume that with and satisfies the following conditions.

Dissipation condition:

where is the first eigenvalue of in .

Growth condition:

Since the nonlinearities critical exponent is in the strong topological space, we need to satisfy the following condition instead of (3):

And there exists a constant , such that

and assumption (6) about will be used to prove the existence of absorbing set in the strong topological space and that it has nothing to do with the testing of asymptotic compactness.

There are many functions satisfying the above assumption (6). For example, when , satisfies this condition. Without loss of generality, we also assume .

The effects of fading memory in this equation are shown through the linear time convolution of the function and memory kernel . As that in [1], we also assume that the memory kernel , for every . Besides, we also presume that the function satisfies the following hypotheses:

where is a positive constant. Obviously, it follows that the kernel decays to zero with exponential rate. This behavior displays the fading memory of the far history in model (1).

If and , (1) represents a semilinear hyperbolic equation in viscoelasticity (cf. [2, 3]), and this model is originally derived from the theory of isothermal viscoelasticity and describes a process of energy dissipation of a homogeneous and isotropic viscoelastic solid (cf. [4]). Furthermore, if , (1) reduces to semilinear wave equation, where denotes some displacement-dependant body's force density (cf. [5]); and, under these conditions, if , (1) can be transformed into a model of Sine-Gordon equation (cf. [6]); if , (1) can be turned into the relative quantum mechanical equation (cf. [6]). If , , and , it expresses a floating beam equation; denotes bent movement crosswise of beam (cf. [7, 8]).

For the semilinear hyperbolic equation in viscoelasticity, the asymptotic behaviors of the solution have been studied in [1, 9–13]. In [1, 9, 10], the authors investigated the asymptotic behaviors of solutions when nonlinearity is subcritical growth. In [11], Sun et al. considered large-time behaviors of the solutions and gained the existence of global attractors for nonautonomous strongly damped wave-type evolution equation with the critical nonlinearity and linear memory. In [12], Cavalcanti et al. studied the long-time dynamics of a semilinear wave equation with degenerate viscoelasticity defined in a bounded domain of , with Dirichlet boundary condition and nonlinear forcing term with critical growth. In [13], Zhou and Zhao proved the existence of random attractors for the continuous random dynamical systems generated by stochastic damped nonautonomous wave equations with linear memory and additive white noise when the nonlinearity has a critically growing exponent and studied the upper semicontinuity of random attractors.

For the floating beam equation, the authors considered the dynamics and obtained the existence of a global attractor for the deterministic floating beam in [8] and deeply studied the existence of a compact random attractor for the random dynamical system generated by a model for nonlinear oscillations in a floating beam equation with strong damping and white noise in [14].

The above equations are all special cases of an abstract evolution equation with fading memory that we will study in this paper. Until now, we find that no one else has studied the long-time behavior of the solutions about the problem with weak damping and critical nonlinearity, which just causes our strong research motivation and interest. Based on this, we will study the asymptotic behaviors and regularity of solutions for this problem in this paper.

As we know, if we want to prove the existence of global attractors, the key point is to obtain the compactness of the semigroup in some sense. However, there exist some essential difficulties to test the continuity and compactness (asymptotic compactness) of the semigroup of solutions for (1). First, due to the memory terms, we cannot use or as test function to verify the asymptotic compactness of the solution semigroup; second, the critical nonlinearity leads to the fact that some obstacles are hard to avoid in the process of energy estimation. Furthermore, since is an abstract operator, it brings about more substantive barriers in the process of proof. Despite all this, we still overcome the above bottlenecks in the process of estimation and proof by applying semigroup theory and decomposition technology. Finally, we test the continuity and compactness (asymptotic compactness) of the solution semigroup of solutions and obtain the existence of global attractors of (1) in the weak topological space and the strong topological space .

This paper is organized as follows. Some preliminaries, including some notations that we will use, the assumption on nonlinearity, and some general abstract results about dynamical system are presented in Section 2. Then, the proof of our main result about compactness testing and the existence of global attractor for the dynamical system generated by the solution of (1) are given in Sections 3 and 4. The main results are Theorems 11 and 18.

2. Preliminaries

Let and be a bilinear continuous form on which is symmetric and coercive. With this form, we associate the linear operator in by setting , and can be considered as a self-adjoint unbounded operator in with domain .

We assume that , are eigenvalue and eigenvectors of , so can form a group of orthonormal bases of ; then

Using this group of bases we can easily define the powers of (see [6]), , with domain . we can consider a family of Hilbert spaces

with the standard inner product and norms, respectively,

here and mean inner product and norm, respectively. Then, we know that

The following interpolation inequality also holds: given , for any , there exists a positive constant such that

For convenience, we set, for ,

then, , , and .

As in [1, 9], we denote

Set and use the assumption ; (1) can be transformed into the following system:

The associated initial-boundary conditions are

where satisfies the condition as follows: there exist two positive constants and ; here is the first eigenvalue of , such that

here means norm.

In view of (7), we let be a family of Hilbert spaces of -valued functions on and be a family of Hilbert spaces of -valued functions on , endowed with the following inner product and norm, respectively:

So, we introduce the family of Hilbert spaces:

and endow this family of spaces with norm, respectively,

In order to estimate conveniently, using assumptions (7)–(9), we can obtain the preliminary result as follows (cf. [9, 15]).

Lemma 1. Set . If memory kernel satisfies (7)–(9), then, for any , , , there exists a constant , such that

The following results will be used to prove the compactness of the memory term and the existence of strong global attractors.

Lemma 2 (see [9, 15, 16]). Assume that is a nonnegative function and satisfies the following: if there exists , such that , then for all holds. Moreover, let , , be Banach space, where , are reflexive and satisfy

where the embedding is compact. Let and satisfy

(i) in ;(ii).

Then is relatively compact in .

Lemma 3 (see [6]). Let be a complete metric space, is a semigroup in , and has a bounded absorbing set in . If, for every , the operator allows the decomposition and satisfies that

(i)the semigroup is uniformly compact, as is increasing sufficiently;(ii)the operator is continuous and, for any bounded set , as ,

then -limit set of absorbing set is global attractors of .

3. Global Attractors in

3.1. The Existence and Uniqueness of Weak Solution

At first, we define the concept of weak solution for dynamical systems (17)-(18) (cf. [15] about the corresponding definition of solutions for wave equations).

Definition 4. Define . Let and . A ternary form which fulfils

is said to be a weak solution of problem (17)-(18) in the time interval with initial data , provided that

for all . .

Then, we start with the following general existence and unique result. The proof is based on the normal Faedo-Galerkin methods introduced in [9, 15].

Theorem 5 (existence and uniqueness of solutions). Let satisfy (2)-(3), , and (7)–(9) hold. Then for any given , and any , there is a unique solution for problem (17)-(18), satisfying

According to Theorem 5 above, we can define the solution operator; that is,

Obviously, can form a solution semigroup.

In the remainder of this section, we denote by the semigroup associated with the solution of (17)-(18).

3.2. The Existence of Bounded Absorbing Set

Theorem 6 (bounded absorbing in ). Let be any bounded subset of and be a solution of (17)-(18) with initial data . If the nonlinearity satisfies (2), , (7)–(9) hold. Then there is a positive constant such that, for any bounded subset , there exists such that

Proof. Taking , multiplying (17) by , and integrating over , we have

Then, using Poincaré inequality and Cauchy inequality, and taking small enough, we can get that

Applying Lemma 1, we have

and, from (8), we see

Therefore,

From (2), we know that there exist and such that

here, . Then

Evidently, we can get

Choose to be sufficiently small, such that

thus,

Substituting (32)–(40) into (31), we can obtain that

where and .
Hence, from (36)–(38), there exist positive constants , , such that

According to Gronwall lemma,

Assuming that , as , we have

We complete the proof.

3.3. The Existence of Global Attractor in

Now we will establish the necessary asymptotic smoothness, similar to that in [17–19]; we also need to make some decompositions about nonlinearity, forcing term, solution, and solution semigroup.

About forcing term , we think about two cases: the weak solutions case : for every and any , there is which depends on and , such that

the strong solutions case : for every and any , there is which depends on and , such that

For the nonlinearity , we take into account two cases: for the weak solutions case, critical exponent ; for the strong solutions case, critical exponent . Similar to that in [5, 18, 19], we know that allows the following decomposition:

where with , and satisfy

Set

where is defined by (51).

Analogous to that in [17], we decompose the solution of (17) corresponding to initial data as follows:

here and satisfy the following equations:

According to the general theorem about the existence of global attractors of infinite-dimensional dynamical systems (see [6, 15]), we also need to prove the asymptotic compactness of in .

Similar to the proof of Theorem 5, we can gain the corresponding existence and uniqueness of solutions for (55) and (56); furthermore, we know that the solutions of (55) also form a semigroup. For the sake of convenience, we denote the solution operators of (55) and (56) by and , respectively. Then, for every , we have

Hereafter, we will test the necessary condition of asymptotic smoothness.

At first, akin to the proof of Theorem 6, we can obtain the following result about the solution of (55) in .

Lemma 7. Assume that satisfy (48) and (49), , and (6)–(8) hold. Then, for any , there is a constant , such that the solutions of (55) satisfy the following estimates:

where is an increasing function on and only depends on , , and .

Proof. Taking the inner product of (70) with , we have

For the forcing term, we know

About the nonlinearity, from (48), we have

where .
In view of (49), (13), (14), and (44), we know

here the constant being very small depends not only on and , but also on the bound in (44). Choosing to be sufficiently small, we have

Thanks to Lemma 1 and using (7), we get that

Then,

here, .
Obviously, we see that

so

Then, combining the above estimates with (59), we see that

where .
Obviously, there exist constants , , such that

thus, using Gronwall lemma and taking , as , we obtain

The proof is complete.

Lemma 8. Let the nonlinearity satisfy (2)-(3) and (51)-(52), , and (7)–(9) hold. Then, for every given and , there is a position constant

such that the solutions of (56) satisfy

where .

Proof. Multiplying (56) by , we get that

Using Lemma 1, we can obtain

Similarly, we have

Choosing to be sufficiently small, such that and , we can obtain that

Since , we have

We deal with the terms in (77) one by one.
First, by virtue of (3), we deduce that

Second, since , we have

where , and we have used the embedding , ,
Third, from (53), we obtain that

where , and we have used the embedding , ,
Finally, by virtue of (51), we get

where we have employed the inequality and used the embedding , and .
Obviously, we see

and, noting that , we know that

Combining above estimates, we can deduce that

where , .
Applying the Gronwall inequality and integrating over , we gain that

Clearly, there exist constants , , such that

so

That is,

where
The proof is complete.

In addition, for any , Cauchy problem (see [9, 15, 16])

has a unique solution and explicit expression:

Let be the bounded absorbing set in obtained from Theorem 6; we can obtain the following.

Lemma 9. Set

Under the assumption of Lemma 8, for every given , there is a positive constant , such that

(i) is bounded in ;(ii),

where . is the solution operators of (56), and is a projection operator.

Proof. Due to the explicit expression (90), we can deduce that

which, combined with Lemma 8, implies that (i) holds.
After that, we can easily know that

By virtue of (88), we get that (ii) holds.
We complete the proof.

Moreover, applying Lemma 2, we can obtain that is relatively compact in . Then, using the compact embedding once again, we can get the following.

Lemma 10. Let be the corresponding solution operator of (56). Under the assumption of Lemma 8, for any , is relatively compact in .

By applying the classical theory (e.g., see [6, 20]), from Lemma 3, Theorem 6, and Lemmas 7 and 10, we can gain the main result.

Theorem 11 (global attractor in ). Let the nonlinearity satisfy (2)-(3), , (7)–(9) hold, and be the solution semigroup of (17)-(18) in . Then has a global attractor in ; namely, is compact and invariant in and attracts every bounded subset of with respect to the -norm.

4. Global Attractors in

4.1. The Existence and Uniqueness of Strong Solution

First of all, we define the concept of strong solution for dynamical systems (17)-(18).

Definition 12. Define . Let and . A ternary form is said to be the strong solution of problem (17)-(18) in the time interval with initial data , if satisfies (27), and

Hereafter, similar to Theorem 5, we can obtain the following.

Theorem 13 (existence and uniqueness of strong solutions). Assume that satisfy (2) and (4), , and (7)–(9) hold. Then, for any given , and any , there exists a unique solution for problem (17)-(18), in the sense that

By Theorem 13, we can define the strong solution operator as follows:

Evidently, is a solution semigroup.

4.2. The Existence of Bounded Absorbing Set

At first, we will prove the existence of bounded absorbing set in .

Theorem 14 (bounded absorbing in ). Let the nonlinearity satisfy (2), (5), and (6), , (7)–(9) hold, and be bounded subset of . If is a solution of (14) and (17) with initial data , then there is a positive constant such that, for any bounded subset , there exists such that

Proof. Taking the inner product of (17) with , we get that

Similar to (35), we have

About the nonlinearity, from (5), (6), and (44), these yield

where (or ) and .
For the forcing term, we see easily

and since , then

Taking small enough, we then obtain that

In combination with the above estimates, we can deduce that, for all ,

where and .
From (102), there exist constants , , such that

therefore, applying Gronwall lemma, we have

Presuming that , as , we get that

The proof is complete.

4.3. The Existence of Global Attractor in

According to the general theorem about the existence of global attractors of infinite-dimensional dynamical systems (see [6, 15]), we also need to test the asymptotic compactness of in .

Similarly, we also gain the corresponding existence and uniqueness of solutions for (55) and (56); moreover, we can deduce that the solutions of (55) also form a semigroup. For the sake of convenience, we denote the solution operators of (55) and (56) by and , respectively. Then, for every , we have

After that, the necessary condition of asymptotic smoothness will be verified.

At first, akin to the proof of Lemma 7, the following results about the solution of (55) in can be obtained.

Lemma 15. Let satisfy (48)–(50), , and (7)–(9) hold. Then, for any , there is a constant and , such that the solutions of (55) satisfy the following estimates:

here is an increasing function on and only depends on , , and .

Proof. Multiplying (55) by , we get that

For the forcing term, we have

About the nonlinearity, similar to (100) and from (48), (50), and (70), as , we see

Similarly,

here, .
Then, substituting the above estimates into (110), this yields

where , .
Obviously, there exist constants , , such that

From Gronwall lemma, for all , we obtain

The proof is complete.

Lemma 16. Let the nonlinearity satisfy (2)–(6) and (51)-(52), , and (7)–(9) hold. Then, for any given and , there exists a position constant

such that the solutions of (56) satisfy

where .

Proof. Multiplying (56) by , we obtain that

Similarly, choosing to be sufficiently small, we can obtain that

Clearly

We deal with the terms in (121) one by one.
First, from (4), we obtain

Second, for , we have

where , and we have used the embedding
Third, from (53), we obtain that

where , and we have used the embedding , ,
Finally, by virtue of (51), we get

where we have employed the inequality and used the embedding , , and .
At the same time, due to , we have

Combining the above estimates, we can deduce that

where , , and the constant only depends on ,
Applying the Gronwall inequality and integrating over , we deduce that

So, there exist constants , , such that

then

That is,

where
The proof is complete.

Let be the bounded absorbing set obtained from Theorem 14, and

Thanks to Lemma 2, we can obtain that is relatively compact in . Thus, using the compact embedding once again, we can get the following.

Lemma 17. Let be the corresponding solution operator of (56). Under the assumption of Lemma 16, for any , is relatively compact in .

By applying the classical theory (e.g., see [6, 20]), from Lemma 3, Theorem 14, and Lemmas 15 and 17, we can gain the main result.

Theorem 18 (global attractor in ). Let the nonlinearity satisfy (2)–(6), , (7)–(9) hold, and be the solution semigroup of (17)-(18) in . Then has a global attractor in ; namely, is compact and invariant in and attracts every bounded subset of with respect to the -norm.

Conflicts of Interest

The authors declare that they have no competing interests.

Authors’ Contributions

The authors read and approved the final manuscript.

Acknowledgments

This work was partly supported by the NSFC (Grants 11361053, 11561064, 11401479, and 11661071) and the Young Teachers Scientific Research Ability Promotion Plan of Northwest Normal University (NWNU-LKQN-14-6).

References

C. Giorgi, J. E. M. Rivera, and V. Pata, “Global attractors for a semilinear hyperbolic equation in viscoelasticity,” Journal of Mathematical Analysis and Applications, vol. 260, no. 1, pp. 83–99, 2001.

B. D. Coleman and W. Noll, “Foundations of linear viscoelasticity,” Reviews of Modern Physics, vol. 33, pp. 239–249, 1961.

C. M. Dafermos, “Asymptotic stability in viscoelasticity,” Archive for Rational Mechanics and Analysis, vol. 37, pp. 297–308, 1970.

M. Fabrizio and A. Morro, Mathematical problems in linear viscoelasticity, vol. 12 of SIAM Studies in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992.

J. Arrieta, A. N. Carvalho, and J. K. Hale, “A damped hyperbolic equation with critical exponent,” Communications in Partial Differential Equations, vol. 17, no. 5-6, pp. 841–866, 1992.

R. Temam, Infinit Dimensional Dynamical System in Mechanichs and Physics, Springer, New York, NY, USA, 2nd edition, 1997.

A. C. Lazer and P. J. McKenna, “Large-amplitude periodic oscillations in suspension bridges: some new connections with nonlinear analysis,” SIAM Review. A Publication of the Society for Industrial and Applied Mathematics, vol. 32, no. 4, pp. 537–578, 1990.

Q. Z. Ma, “Asymptotic behavior of the solution for the floating beam,” Journal of Northwest Normal University (Natural Science), vol. 41, no. 6, pp. 4–7, 2005.

V. Pata and A. Zucchi, “Attractors for a damped hyperbolic equation with linear memory,” Journal of Advanced Mathematics and Applications, vol. 11, no. 2, pp. 505–529, 2001.

Q. Ma and C. Zhong, “Existence of strong global attractors for hyperbolic equation with linear memory,” Applied Mathematics and Computation, vol. 157, no. 3, pp. 745–758, 2004.

C. Sun, D. Cao, and J. Duan, “Non-autonomous wave dynamics with memory-asymptotic regularity and uniform attractor,” Discrete and Continuous Dynamical Systems. Series B. A Journal Bridging Mathematics and Sciences, vol. 9, no. 3-4, pp. 743–761, 2008.

M. M. Cavalcanti, L. H. Fatori, and T. F. Ma, “Attractors for wave equations with degenerate memory,” Journal of Differential Equations, vol. 260, no. 1, pp. 56–83, 2016.

S. F. Zhou and M. Zhao, “Random attractors for damped non-autonomous wave equations with memory and white noise,” Nonlinear Analysis, vol. 120, Article ID 10484, pp. 202–226, 2015.

L. Xu and Q. Ma, “Existence of random attractors for the floating beam equation with strong damping and white noise,” Boundary Value Problems, 2015:126, 13 pages, 2015.

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. Gatti, A. Miranville, V. Pata, and S. Zelik, “Attractors for semi-linear equations of viscoelasticity with very low dissipation,” The Rocky Mountain Journal of Mathematics, vol. 38, no. 4, pp. 1117–1138, 2008.

C. Sun and M. Yang, “Dynamics of the nonclassical diffusion equations,” Asymptotic Analysis, vol. 59, no. 1-2, pp. 51–81, 2008.

X. Wang, W. C. Ju, and C. K. Zhong, “Strong attractors for the non-autonomous nonclassical diffusion equations with fading memory,” Chinese Annals of Mathematics, vol. 34A, no. 6, pp. 671–688, 2013.

X. Wang, L. Yang, and C. Zhong, “Attractors for the nonclassical diffusion equations with fading memory,” Journal of Mathematical Analysis and Applications, vol. 362, no. 2, pp. 327–337, 2010.

J. W. Cholewa and T. Dlotko, Global Attractors in Abstract Parabolic Problems, vol. 278, Cambridge University Press, 2000.