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 thatand 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 spaceswith the standard inner product and norms, respectively,here and mean inner product and norm, respectively. Then, we know thatThe following interpolation inequality also holds: given , for any , there exists a positive constant such that
For convenience, we set, for ,then, , , and .
Set and use the assumption ; (1) can be transformed into the following system:The associated initial-boundary conditions arewhere satisfies the condition as follows: there exist two positive constants and ; here is the first eigenvalue of , such thathere 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 satisfywhere 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 fulfilsis said to be a weak solution of problem (17)-(18) in the time interval with initial data , provided thatfor 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 haveThen, using Poincaré inequality and Cauchy inequality, and taking small enough, we can get thatApplying Lemma 1, we haveand, from (8), we seeTherefore,From (2), we know that there exist and such thathere, . ThenEvidently, we can getChoose to be sufficiently small, such thatthus,Substituting (32)–(40) into (31), we can obtain thatwhere and .
Hence, from (36)–(38), there exist positive constants , , such thatAccording to Gronwall lemma,Assuming that , as , we haveWe 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 thatthe 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
Setwhere 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 haveFor the forcing term, we knowAbout the nonlinearity, from (48), we havewhere .
In view of (49), (13), (14), and (44), we knowhere the constant being very small depends not only on and , but also on the bound in (44). Choosing to be sufficiently small, we haveThanks to Lemma 1 and using (7), we get thatThen,here, .
Obviously, we see thatsoThen, combining the above estimates with (59), we see thatwhere .
Obviously, there exist constants , , such thatthus, using Gronwall lemma and taking , as , we obtainThe 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 constantsuch that the solutions of (56) satisfywhere .
Proof. Multiplying (56) by , we get thatUsing Lemma 1, we can obtainSimilarly, we haveChoosing to be sufficiently small, such that and , we can obtain thatSince , we haveWe deal with the terms in (77) one by one.
First, by virtue of (3), we deduce thatSecond, since , we havewhere , and we have used the embedding , ,
Third, from (53), we obtain thatwhere , and we have used the embedding , ,
Finally, by virtue of (51), we getwhere we have employed the inequality and used the embedding , and .
Obviously, we seeand, noting that , we know thatCombining above estimates, we can deduce thatwhere , .
Applying the Gronwall inequality and integrating over , we gain thatClearly, there exist constants , , such thatsoThat 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. SetUnder 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 thatwhich, combined with Lemma 8, implies that (i) holds.
After that, we can easily know thatBy 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 thatSimilar to (35), we haveAbout the nonlinearity, from (5), (6), and (44), these yieldwhere (or ) and .
For the forcing term, we see easilyand since , thenTaking small enough, we then obtain thatIn combination with the above estimates, we can deduce that, for all ,where and .
From (102), there exist constants , , such thattherefore, applying Gronwall lemma, we havePresuming that , as , we get thatThe 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 thatFor the forcing term, we haveAbout the nonlinearity, similar to (100) and from (48), (50), and (70), as , we seeSimilarly,here, .
Then, substituting the above estimates into (110), this yieldswhere , .
Obviously, there exist constants , , such thatFrom Gronwall lemma, for all , we obtainThe 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 constantsuch that the solutions of (56) satisfywhere .
Proof. Multiplying (56) by , we obtain thatSimilarly, choosing to be sufficiently small, we can obtain thatClearlyWe deal with the terms in (121) one by one.
First, from (4), we obtainSecond, for , we havewhere , and we have used the embedding
Third, from (53), we obtain thatwhere , and we have used the embedding , ,
Finally, by virtue of (51), we getwhere we have employed the inequality and used the embedding , , and .
At the same time, due to , we haveCombining the above estimates, we can deduce thatwhere , , and the constant only depends on ,
Applying the Gronwall inequality and integrating over , we deduce thatSo, there exist constants , , such thatthenThat is,where
The proof is complete.
Let be the bounded absorbing set obtained from Theorem 14, andThanks 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).