Asymptotic behavior for a class of viscoelastic equations with memory lacking instantaneous damping

: In this paper, we mainly investigate long-time behavior for viscoelastic equation with fading memory


Introduction
In this paper, we mainly study the following initial-boundary value problem for viscoelastic equation with hereditary memory: where Ω ⊂ R 3 is a bounded smooth domain, ν > 0, and the forcing term g = g(x) ∈ L 2 (Ω) is given.Next, we establish the following hypotheses for the kernel function k(s) (H 1 ) Let µ(s) = −k (s), and assume and there exists δ > 0, such that µ (s) + δµ(s) 0, ∀s ∈ R + , ( and let (H 2 ) The nonlinearity f ∈ C 1 satisfies f (0) = 0 and also fulfills the following conditions and lim where c, λ 1 are positive constants and λ 1 is the first eigenvalue of −∆ in H 1 0 (Ω) with Dirichlet boundary condition.From (1.5), it's easy to get that there exist λ(0 < λ < λ 1 ) and c f ≥ 0; such that f (s)s ≥ −λs The equation associated with Eq (1.1) is as follows which mainly describes a pure dispersion wave process, such as the motion equation of strain-arc wave of linear elastic rod considering transverse inertia and ion-acoustic wave propagation equation in space transformation with weak nonlinear effects (see e.g., [1][2][3][4]).
Qin et al. [19] proved the existence of uniform attractors in non-autonomous case by improving the result of [18] when −∆u t was still included.Recently, Conti et al. [20] obtained the existence of global attractors and optimal regularity of global attractors for the following equation when the nonlinearity f meets critical growth with ρ ∈ [0, 4] and ρ < 4, respectively.Therefore, based on the above existing research, we devote to obtain the existence of global attractors in higher regular space for the problem (1.1) which doesn't contain the strong damping −∆u t in this paper.
Firstly, because the Eq (1.1) doesn't contain strong dissipative term −∆u t , which makes that the Eq (1.1) is different from usual viscoelastic equations.Next, for the Eq (1.1), its dissipation is only generated by memory term with weaker dissipation rather than the strong dissipative term −∆u t , which leads to the need of higher regularity to ensure compactness, so the multiplier A κ u t will be used to obtain our result.We use new analytical techniques to obtain the upper semicontinuity of global attractors.Thus, our results complement the existing conclusions because we only use the memory dissipation to prove the existence and the semicontinuity of global attractors.
In addition, to the best of our knowledge, the key point for proving the existence of global attractors is to verify the existence of bounded absorbing set and the compactness of the semigroup in some sense.However, the absence of term −∆u t causes that energy dissipation of Eq (1.1) is lower than usual viscoelastic equation, and its dissipation only is presented by the memory term.Hence, this will lead to two main difficulties.On the one hand, the absence of term −∆u t makes the equation lacks strong structural damping.On the other hand, to ensure strong convergence of the solution in L 2 (0, T ; H 1 0 (Ω)), how to obtain higher regularity of solutions.Thereby, for obtaining dissipative and compactness of semigroup, we will use analysis techniques and the ideas in [21,22] to overcome these difficulties.
The plan of this paper is as follows.In Section 2, we recall some basic concepts and useful results that will be used later.In Section 3, firstly, the bounded absorbing set is obtained.Secondly, we verify asymptotic compact of semigroup by contractive function method [23,24].Finally, the existence of global attractors A is proved in In section 4, we obtain the upper semicontinuity of global attractors.

Preliminaries
Following the Dafermos' idea of introducing an additional variable η t , the past history of u, whose evolution is ruled by a first-order hyperbolic equation (see e.g., [25] and references therein).Thus the original problem (1.1) can be translated into a dynamical system on a phase space with two components (see [26]).In particular, in the following, we introduce the past history of u in the, i.e.
Provided that let η t t = ∂ ∂t η t , η t s = ∂ ∂s η t , then we have Thus, if we assume ν − m 0 = 1, then the system (1.1) can be rewrite as , (2.6) with initial-boundary condition In the whole paper, unless otherwise stated, z(t) = (u(t), u t (t), η t ) is the solution of systems (2.6), (2.7) with initial value z 0 = (u 0 , u 1 , η 0 ).For conveniences, hereafter let | • | p be the norm of L p (Ω)(p ≥ 1).Let •, • be the inner product of L 2 (Ω), • 2 0 be the equivalent norm ) and its inner product and norm are Then phase spaces of the Eq (2.6) are and their corresponding norms are )) and let • κ be the norm of V κ .Then we can also define phase space of the Eq (1.1) is and the corresponding norm is And there exists the following compact embedding (2.8) Definition 2.1.Let X be a Banach spaces and X be a family of operators defined on it.We say that {S (t)} t≥0 is a continuous semigroup on X if {S (t)} t≥0 fulfills S (t) : X → X, ∀t ≥ 0. and satisfies (i)S (0) = Id(Identity operator); (ii)S (t + s) = S (t)S (s), ∀t, s ≥ 0.
The main results of this paper (the existence of global attractors) can be obtained by the following definitions and theorem.Next, let's talk about it (it's similar to [14,23,26]).Definition 2.2.Let X, Y be two Banach spaces and B be a bounded subset of X × Y.We call a function φ(•, We denote the set of all contractive functions on B × B by E(B).
Lemma 2.3.Let X, Y be two Banach spaces and B be a bounded subset of X × Y, {S (t)} t≥0 is semigroup with a bounded absorbing set B 0 on X × Y.Moreover, assume that for any ε > 0 there exist where φ T depends on T .Then the semigroup {S (t)} t≥0 is asymptotically compact in X × Y.
In the following theorem, we give a method to verify the asymptotically compactness of a semigroup generated by the Eq (1.1), which will be used in our later discussion.
Theorem 2.4.Let X, Y be two Banach spaces and {S (t)} t≥0 be a continuous semigroup on X × Y. Then {S (t)} t≥0 has a global attractor in X × Y. Provided that the following conditions hold: Lemma 2.5.Let X ⊂⊂ H ⊂ Y be Banach spaces, with X reflexive.Suppose that u n is a sequence that is uniformly bounded in L 2 (0, T ; X) and du n /dt is uniformly bounded in L p (0, T ; Y), for some p > 1.Then there is a subsequence of u n that converges strongly in L 2 (0, T ; H).

Global attractors in M 1
Throughout the paper, we assume that Ω ⊂ R n (n = 3) be bounded smooth domain, the kernel function µ and the nonlinearity satisfy (H 1 ) and (H 2 ) respectively, and g ∈ L 2 (Ω).
. By Lemma 3.1, the semigroup {S (t)} t≥0 in M 1 will be defined as the following: and it is a strongly continuous semigroup on M 1 .Lemma 3.2.For some R > 0 and z 0 M 1 ≤ R, then there exists a constant R 1 = R 1 (R), such that for any t ≥ 0 , the following estimate holds: Multiplying the first equation of (2.6) by u t , and integrating over Ω, we obtain that 1 2 then by (H 2 ), H ölder inequality and Young inequality, we can get that , and hold for any t ≥ 0. In addition, it's easy to obtain that Integrating (3.4) about t from 0 to t, and combining with (3.2), (3.3), we have where Lemma 3.3.For any T > 0, z 0 ∈ M 1 and z 0 M 1 ≤ R, then there exists a constant holds for any t ∈ [0, T ].
Proof.Multiplying the first equation of (2.6) by u tt , and integrating over Ω, we obtain that Using Lemma3.2,H ölder inequality and Young inequality, then By Lemma 3.2, we have (3.9) Combining with (3.8) and t ∈ [0, T ], we get (1 + T ), then (3.6) holds.Lemma 3.4.Provided that (u(t), u t (t), η t ) is a sufficiently regular solution of (2.6), (2.7).Then, for the functional And we can also obtain where Proof.First of all, by Hölder inequality and Young inequality, it's easy to get that Next, taking the derivative about t for Λ 0 (t), we have Now, we sequentially deal with the two terms on the right of (3.13).
The estimate for the first term is as follows The estimate for the second term, by concerning the second equation of (2.6), we obtain (3.17) And we can obtain differential inequality where k is a positive constant and ε ∈ (0, 1).
Proof.Using H ölder inequality, Young inequality and Poincaré inequality, it's easy to get Furthermore, taking the time-derivative for N(t) and combining with the first equation (2.6), it follows that where k 1 ≥ 1 λ 1 + 1. Next, we dispose each term on the right side of (3.20), it follows that Theorem 3.6.There exists a constant R 0 , such that, for any T 0 = T 0 ( z 0 M 1 ) > 0, whenever then for all t ≥ T 0 , we have Proof.According to the definition of H(t), we can obtain In addition, the following functional can be defined By Lemma 3.4 and Lemma 3.5, we get Let its perturbation be small enough and C be sufficiently large, and combining with (H 2 ), then it yields However, combining with (H 2 ), (3.11), (3.18) and (3.23), we have i.e.
Thus, when δ is fixed, then we can choose appropriate l, , C, such that  , we obtain where R 0 = 12γ 0 γ (|g| 2 2 + 1) + 8 c f C |Ω|.Therefore, we can know that the set Corollary 3.7.There exists a constant C R 0 , such that, for all t ≥ T 0 , we have t+1 t Proof.Integrating (3.29) about t from t to t + 1, and combining with (3.25) and Lemma 3.7, the above estimate is easily obtained.Lemma 3.8.For any T > 0, there exists a constant R 3 > 0, such that, whenever Proof.Multiplying the first equation of (2.6) by A κ u t , and integrating over Ω, we obtain that where and by (H 2 ) and Lemma 3.2, we obtain Then by (3.33)-(3.35),we have where h, h 1 are positive constant.Hence, using Gronwall lemma, we can obtain that holds for any t ∈ [0, T ].This proof is finished.
Lemma 3.9.For any t ∈ [0, T ], there exists a constant R 5 > 0, such that, whenever Proof.Firstly, the first equation of the system (2.6), it can be rewritten Next, let v = u t + δ 1 u, and multiplying (3.38) by A κ v, and integrating over Ω, we obtain that 1 2 In addition, we deal with each term on the right of (3.39), it yields by using Minkowski inequality
next, we deal with the nonlinear term by using Hölder inequality and Sobolev embedding theorem, it yields which, together with (3.39)-(3.42),obtains 1 2

.44)
Where h 0 , h 2 are positive constant.Let δ 1 be small enough, such that Then combining Lemma 3.1, Lemma 3.2, Lemma 3.8 and Gronwall lemma, we get Moreover, integrating (3.41) about t from t to t+1, then we have where In order to prove the existence of global attractor for {S (t)} t≥0 on M 1 , we have to verify some compactness for the semigroup {S (t)} t≥0 .For further purpose, we will give asymptotically compact theorem of the semigroup on M 1 .
Combining with (3.51) and (3.52), we have Secondly, we can define the following functional By (3.48) and (3.50), we get Next, let its perturbation be small enough and C 1 be sufficiently large, then yields Therefore, we can also deduce easily that In the same way, let ε > 0 be small enough and C 1 is sufficiently large, such that Using Gronwall lemma, we can deduce that holds for any T > 3C 2β ln 2L ω (0) Cε .
Proof.Let z = (u, u t , η t ) and z ω = (v, v t , ξ t ) are unique solutions of Eqs (2.6) and (4.1) with initial value z 0 ∈ A ω respectively.Setting w = u − v, ζ t = η t − ξ t , then (w, w t , ζ t ) is a unique solution of the following equations  Proof.For any ε > 0, since A ω is an universal bounded subset of M 1 for any ω ∈ [0, 1], and A is a compact attracting set for {S (t)} t≥0 on M 1 .So there exists T > 0 such that S (t)(T )A ω ⊂ N(A, ε 2 ).On the other hand, associating with the invariance of A ω and Lemma (4.2), for any t ≥ T , we have ), as ω small enough.Setting ε = ω C , so we have here ν ≥ C is a constant, which completes the proof of the desired results.