Abstract

In this paper, we investigate the long-time behavior for the nonautonomous semilinear second-order evolution equation with some hereditary characteristics, where is an open-bounded domain of with smooth boundary . Firstly, we establish the existence of solutions for the second-order nonautonomous evolution equation by the standard Faedo–Galerkin method, but without the uniqueness of solutions. Then by proving the pullback asymptotic compactness for the multivalued process on , we obtain the existence of pullback attractors in the Banach spaces for the multivalued process generated by a class of second-order nonautonomous evolution equations with hereditary characteristics and ill-posedness.

1. Introduction

The study of nonlinear dynamics is a fascinating question which is at the very heart of understanding of many important problems of the natural sciences. The long-time behavior of PDEs can be described in the terms of attractors of the corresponding semigroups, such as Babin and Vishik [1], Chepyzhov and Vishik [2], Chueshov and Lasiecka [3], Hale [4], Ladyzhenskaya [5], or Temam [6], and the references therein. The study of pullback attractor for infinite dimensional dynamical systems has attracted much attention and has made fast progress in recent decades [713].

In this paper, we consider the following nonautonomous semilinear second-order evolution equation with delays:where is an open-bounded domain of with smooth boundary , is the initial time, and is the initial data on the interval with .

The nonlinearity and the external force satisfy the following conditions, respectively.

In, there exist positive constants and such that the functions and satisfy

In , the external force belongs to the space such that

When and without variable delays, Equation (1) becomes the usual strongly damped wave equation:

Its asymptotic behavior has been studied extensively in terms of attractors [1,1420] . The long-time behavior for the strongly damped wave equation with delays has been investigated in Refs. [7,12].

For each fixed and without variable delays, Equation (1) becomes

It is a special form of the so-called improved Boussinesq equation (see [2124]) with damped term , which was used to describe ion-sound waves in plasma by Makhankov [22,25] and also known to represent other sorts of “propagation problems” of, for example, lengthways waves in nonlinear elastic rods and ion-sonic waves of space transformations by a weak nonlinear effect [21]. Carvalho and Cholewa [26] presented systematic results including the existence-uniqueness and long-time behavior of Equation (6) by using the semigroup approach. The long-time behavior of, especially the global attractor, exponential attractors has been extensively studied by several authors [2629] . For the nonautonomous semilinear second-order evolution (6) with the memory term, we get

Zhang et al. in Ref. [30] constructed the existence of robust family of exponential attractors while the nonlinearity is critical and the time-dependent external forcing term is assumed to be only translation-bounded.

Indeed, for Equation (6), in all above results, we require the solution operator given as follows:

To be well-defined and continuous in a proper phase space. However, for many interesting problems, the well-posedness of the solution operator is not known or does not hold true [1113, 3133] .

To the best of our knowledge, the long-time dynamics of Equation (1) with hereditary characteristics has not been considered by predecessors. There are some barriers encountered. On the one hand, Equation (1) contains the term , and it is essentially different from the usual wave equation in Refs. [1, 7, 12, 1420]. For example, the wave equation has some smoothing effect; for example, although the initial data only belongs to a weaker topology space, the solution will belong to a stronger topology space with higher regularity. However, for Equation (1), if the initial data , then the solution is always in and has no higher regularity because of , and it will cause some difficulties [2629]. On the other hand, suppose that hold true, then and , then the uniqueness of the weak solutions for Equation (1) are lost; that is, we need to overcome some difficulties brought by ill-posedness. In addition, the delay term also causes some difficulties to obtain the pullback attractors.

This paper is organized as follows. In Section 1, we have expounded on research progress regarding our research problem and have given some assumptions. In Section 2, we introduce some notations and functions spaces, and we recall some useful results on nonautonomous multivalued dynamical systems and pullback attractors. In Section 3, we prove the existence of solutions for Equation (1) in . The existence of pullback attractor for the multivalued process corresponding to Equation (1) is proved in Section 4.

2. Preliminaries

Next, we iterate some definitions and abstract results concerning the multivalued dynamical systems and the pullback attractor, which is necessary to obtain our main results [713,34] .

Let be a complete metric space with metric , and let be the class of nonempty subsets of . Denoted by the Hausdorff semidistance between two nonempty subsets of a complete metric space can be defined as

Definition 1. A family of mappings and , is called to be a multivalued process ifLet be a nonempty class of parametrized sets .

Definition 2. A collection of some families of nonempty closed subsets of is said to be inclusion-closed if for each , we havewhich also belongs to .

Definition 3. Let be a multivalued process on , then we get those as follows:(1) is called a -pullback absorbing set for if for any and each , there exists a such that(2) is said to be -pullback asymptotically upper-semicompact in with respect to if for each fixed , any sequence has a convergent subsequence in whenever and with

Theorem 1. A family of nonempty compact subsets of is called to be a -pullback attractor for the multivalued process if(1) is an invariant, i.e.,(2) attracts every member of , i.e., for every and any fixed , we get

Theorem 2. Let be a multivalued process on Banach space , and let be a pullback absorbing set for in . Suppose that can be written asand for any fixed , then we get those as follows:(1)(2)For any fixed , every sequence is a Cauchy sequence in Then is -pullback asymptotically upper-semicompact in .

Theorem 3. Let be an inclusion-closed collection of some families of nonempty closed subsets of and be a multivalued process on . Also, has a closed values and let be upper semicontinuous in for fixed . Suppose that is -pullback asymptotical upper-semicompact in , then has a -pullback absorbing set , and is closed for every . Then, the -pullback attractor is unique for each and is given by

Let and , which are Hilbert spaces for the usual inner products and associated norms. Let for any , where . Note that is also a Hilbert space for the norm , .

Let be a Banach space with norm . Let be a given positive number, which will denote the delay time, and let denote the Banach space with the sup-norm, then we get

We can denote by the Banach spaces with the norm that is defined by

Given and , for each , we can denote as

Denote the function defined on by

Without the loss of generality, we assume that in the following discussion.

3. Existence of Solutions

In this section, we want to prove the existence of solutions which can be obtained by the standard Faedo–Galerkin methods (see [1,6,35]), and the multivalued evolution processes corresponding to Equation (1) will be constructed. We only give the sketch of proof, and the details similar to the proof of Theorem 4 in Ref. [2], Sec. XV.3 and the arguments in [6] Sec. IV. 4.4.

Theorem 4. Suppose that holds true, and , then there exist solutions of Equation (1) such that

Proof. (Sketch)
Let for any , where . Since is self-adjoint, positive operator and has a compact inverse, and there exists a complete set of eigenvectors in , and the corresponding eigenvalues satisfySetting and is the orthogonal projection onto , then we getWe consider the approximate solutions of Equation (1) in the formThen satisfiesLet , and we write Equation (26) asMultiplying Equation (27) by in , we infer thatNoting (2), using Young’s inequality, we get thatandBy the Poincáre inequality , we get thatChoosing , we infer thatNote that for any , we haveNow integrating (32) from to , we getIn view of and the factfor all , setting , we arrive atThus, we obtain thatBy the integral form of Gronwall lemma, we infer thatThen,Thus, we can extract a subsequence, still denoted as , such thatandFurthermore,andNote that , thenWe then pass the limit in Equation (26), and we can find that is a solution of Equation (1) such thatThe continuity propertiescan be established with the methods indicated in Section II.3 and II.4 in the research by Temam [6] (e.g., Theorem 3.1 and 3.2).
This completes the proof.

Remark 1. According to Theorem 4, we can define a family of multivalued mappings on ascorresponding to Equation (1) byThen, is multivalued process on .

4. Pullback Attractors in

We denote by the set of all functions such thatwhere is defined in (50) and is denoted by in the class of all families such that , for some , where denotes the family of all nonempty subsets of , and denotes the closed ball in centered at zero with radius .

Lemma 1. (Existence of -pullback absorbing set) Suppose that holds true, then and , and there exists a constant satisfyingwhere satisfies is the positive constant in the Poincáre inequality. Then the multivalued process possesses a -pullback absorbing set in .

Proof. Let , and we write Equation (1) asMultiplying Equation (1) by in , we infer thatNoting (2), using Young’s inequality, for , we infer thatandApplying the Poincáre inequality, we get thatLet be determined later, then we infer thatNow integrating (58) from to , we getNote that and the factfor all .
Setting , we arrive atThus, we obtainChoosing , and noting thatwe getNote that satisfies if , then we have