Pullback Attractors for a Class of Semilinear Second-Order Nonautonomous Evolution Equations with Hereditary Characteristics

In this paper, we investigate the long-time behavior for the nonautonomous semilinear second-order evolution equation with some hereditary characteristics, where domain 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 U ( t, τ ) { } on C H 10 ( Ω ) ,H 10 ( Ω ) , we obtain the existence of pullback attractors in the Banach spaces C H 10 ( Ω ) ,H 10 ( Ω ) for the multivalued process generated by a class of second- order nonautonomous evolution equations with hereditary characteristics and ill-posedness.


Introduction
e study of nonlinear dynamics is a fascinating question which is at the very heart of understanding of many important problems of the natural sciences. e 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. e study of pullback attractor for in nite dimensional dynamical systems has attracted much attention and has made fast progress in recent decades [7][8][9][10][11][12][13].
In this paper, we consider the following nonautonomous semilinear second-order evolution equation with delays: where Ω is an open-bounded domain of R N (N ≥ 3) with smooth boundary zΩ, τ is the initial time, and ϕ is the initial data on the interval [τ − h, τ] with h > 0. e nonlinearity f(·) and the external force g(t, x) satisfy the following conditions, respectively.
For each fixed ] > 0 and without variable delays, Equation (1) becomes It is a special form of the so-called improved Boussinesq equation (see [21][22][23][24]) with damped term − Δu t , 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. e long-time behavior of, especially the global attractor, exponential attractors has been extensively studied by several authors [26][27][28][29] . 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 wellposedness of the solution operator S(t) is not known or does not hold true [11][12][13][31][32][33] .
To the best of our knowledge, the long-time dynamics of Equation (1) with hereditary characteristics has not been considered by predecessors.
ere are some barriers encountered. On the one hand, Equation (1) contains the term − Δu tt , and it is essentially different from the usual wave equation in Refs. [1,7,12,[14][15][16][17][18][19][20]. 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 ϕ ∈ C H 1 0 (Ω),H 1 0 (Ω) , then the solution u t (·) is always in C H 1 0 (Ω),H 1 0 (Ω) and has no higher regularity because of − Δu tt , and it will cause some difficulties [26][27][28][29]. On the other hand, suppose that (A 1 ) − (A 2 ) hold true, then g ∈ L 2 loc (R; H) and ϕ ∈ C H 1 0 (Ω),H 1 0 (Ω) , 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.
is 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 ϕ ∈ C H 1 0 (Ω),H 1 0 (Ω) . e existence of pullback attractor for the multivalued process U(t, τ) { } corresponding to Equation (1) is proved in Section 4.
Let X be a complete metric space with metric d X (·, ·), and let P(X) be the class of nonempty subsets of X. Denoted by H semi X (·, ·) the Hausdorff semidistance between two nonempty subsets of a complete metric space X can be defined as Definition 1. A family of mappings U(t, τ): X ⟶ P(X) and t ≥ τ, τ ∈ R is called to be a multivalued process if Let D be a nonempty class of parametrized sets

Journal of Mathematics
Definition 2. A collection D of some families of nonempty closed subsets of X is said to be inclusion-closed if for each is a non empty subset ofD(t), ∀t ∈ R . (12) which also belongs to D.
} be a multivalued process on X, then we get those as follows: (2) A attracts every member of D, i.e., for every B � B(t) { } t∈R ∈ D and any fixed t ∈ R, we get Suppose that U can be written as and for any fixed t ∈ R, then we get those as follows:

Theorem 3. Let D be an inclusion-closed collection of some families of nonempty closed subsets of X and U(t, τ)
{ } be a multivalued process on X. Also, U has a closed values and let { } t∈R is unique for each t ∈ R and is given by Let H � L 2 (Ω) and V � H 1 0 (Ω), which are Hilbert spaces for the usual inner products and associated norms.
Let X be a Banach space with norm ‖ · ‖ X . Let h > 0 be a given positive number, which will denote the delay time, and let C X denote the Banach space C 0 ([− h, 0]; X) with the supnorm, then we get We can denote by C X,X the Banach spaces Given τ ∈ R, T > τ and u: Denote the function defined on by Without the loss of generality, we assume that ] � 1 in the following discussion.

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 eorem 4 in Ref. [2], Sec. XV.3 and the arguments in [6] Sec. IV. 4.4. Proof Since A is self-adjoint, positive operator and has a compact inverse, and there exists a complete set of eigenvectors ω i ∞ i�1 in H, and the corresponding eigenvalues Setting V m � span ω 1 , ω 2 , . . . , ω m and P m is the orthogonal projection onto H m , then we get We consider the approximate solutions of Equation (1) in the form en u m (t) satisfies Let v m � u m ′ + ηu m (0 < η < 1), and we write Equation Multiplying Equation (27) by v m in L 2 (Ω), we infer that Noting (2), using Young's inequality, we get that and By the Poincáre inequality λ 1 ‖u‖ 2 ≤ ‖∇u‖ 2 , we get that Choosing η � min 1/3, λ 1 /λ 1 + 6 , we infer that Note that for any η > 0, we have Now integrating (32) from τ to t, we get In view of ρ(s) ∈ [0, h] and the fact for all s ∈ R, setting r � s − ρ(s), we arrive at 4 Journal of Mathematics us, we obtain that By the integral form of Gronwall lemma, we infer that en, (39) us, we can extract a subsequence, still denoted as m, such that and Furthermore, and Note that f ∈ C(R × R N ; R), then We then pass the limit in Equation (26), and we can find that u is a solution of Equation (1) such that can be established with the methods indicated in Section II.3 and II.4 in the research by Temam [6] (e.g., eorem 3.1 and 3.2). is completes the proof.
□ Remark 1. According to eorem 4, we can define a family of multivalued mappings U(t, τ) corresponding to Equation (1) by

Pullback Attractors in C V,V
We denote by R the set of all functions r: where δ > 0 is defined in (50) and is denoted by D C V,V in the class of all families D � D(t) { } t∈R ⊂ P(C V,V ) such that D(t) ⊂ N(0, r D (t)), for some r D ∈ R, where P(C V,V ) denotes the family of all nonempty subsets of C V,V , and N(0, r D (t)) denotes the closed ball in C V,V centered at zero with radius r D (t).

Lemma 1. (Existence of D-pullback absorbing set) Suppose
loc (R; H) and ϕ ∈ C V,V , and there exists a constant δ satisfying where δ * satisfies λ 1 is the positive constant in the Poincáre inequality. en the multivalued process U(t, τ) Proof. Let v � u ′ + δu, and we write Equation (1) as (53) Multiplying Equation (1) by v in L 2 (Ω), we infer that Noting (2), using Young's inequality, for ϵ 1 , ϵ 2 > 0, we infer that and Applying the Poincáre inequality, we get that Let δ ′ > 0 be determined later, then we infer that Now integrating (58) from τ to t, we get Note that ρ(s) ∈ [0, h] and the fact for all s ∈ R.