Global attractors for Kirchhoff wave equation with nonlinear damping and memory

In this paper, we prove that the existence of global attractors for a Kirchhoff wave equation with nonlinear damping and memory.

This kind of wave models goes back to Kirchhoff. In 1883, Kirchhoff [1] firstly introduced the following equation to describe small vibrations of an elastic stretch string: where M(s) = a + bs. There has been much research on global attractors; Lazo studied the existence for the IBVP of the Kirchhoff equation with memory term [2] u tt -M ∇u 2 u + t 0 g(tτ ) u(x, τ ) dτ = 0.
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Chueshov [3] studied the well-posedness and the global attractors of the Kirchhoff equation with strong nonlinear damping Next, Chueshov [4] also studied the Kirchhoff equation with strong nonlinear damping in nature space H = H 1 0 (Ω) ∩ L p+1 (Ω) × L 2 (Ω) as θ = 1. For related work on the Kirchhoff wave equations with strong damping, see [5,6] and the references therein.
When M(s) = 0, Eq. (1) become the well-known wave equation. Ma and Zhong [7] showed the existence of global attractors for the hyperbolic equation with memory Recently, Park and Kang [8] studied the existence of global attractors for the semilinear hyperbolic with nonlinear damping and memory In [9], Kang and Rivera showed the existence of global attractors for the beam equation localized nonlinear damping and memory Motivated by [5,[7][8][9], we will prove the existence of global attractors for Eq. (1). Following the framework proposed in [7], we shall add a new variable η to the system, which corresponds to the relative displacement history. Let us define By differentiation, we have Let μ(s) = -k (s), k(∞) = 1, (1) transforms into the following system: with boundary condition and initial conditions This paper is organized as follows. In Sect. 2, we introduce some preliminaries. In Sect. 3, we show the existence of a bounded absorbing set in H. In Sect. 4, we give the existence of global attractors of problems (6)-(9).

Preliminaries
We first state some assumptions, which will be used in this paper.

Assumption (1)
The memory kernel μ is required to satisfy the following hypotheses: where α 0 is a constant.

Assumption (4)
The damping function g ∈ C 1 (R) satisfies g is strictly increasing, and lim inf |s|→∞ g (s) > 0, In order to consider the relative displacement η as a new variable, we introduce the weighted L 2 -space which is a Hilbert space endowed with inner product and norm respectively.
Our analysis is given on the phase space which is equipped with the norm In order to obtain the global attractors of the problems (6)-(9), we need the following theorem of existence, uniqueness of solution and continuous dependence on the initial data.
Next,we recall the simple compactness criterion stated in [9,10].
Denote all such contractive functions on B × B by C(B).
Theorem 2.2 ([9, 10]) Let {s(t)} t≥0 be a semigroup on a Banach space (X, · ) and has a bounded absorbing set B 0 . Moreover, assume that for any ε ≥ 0 there exist T = T(B 0 , ε) and Φ(·, ·) ∈ C(B) such that where Φ T depends on T . Then {s(t)} t≥0 is asymptotically compact in X, i.e., for any bounded sequence {y n } ∞ n ⊂ X and {t n } with t n → ∞, {S(t n )y n } ∞ n=1 is compact in X.

Absorbing set in H
In this section, we prove the existence of the bounded absorbing set in H. We use C i to denote several positive constants. Proof we take the scalar product in L 2 of system (6) with u t and (7) with η, respectively, we have d dt where We set Then from (17) and (18) we obtain From (10), (13) we obtain By the hypothesis (12) we know that there are λ > λ 1 > 0 and C 0 such that Using the Young inequality, we have we choose proper λ and ε small enough so that 1 2 -λ 4λ 1ε > 1 8 , and we have Taking the scalar product in L 2 of (6) with v = u t + εu, we obtain d dt Let Similarly, using (21), the Poincáre inequality and the Young inequality, choosing proper λ and ε small enough so that 1 2 -ε 2 2λ 1 -λ 4λ 1ε > 1 8 , we have It is obvious that (10) and (13) imply that there are ε > 0 and C > 0 such that Due to the Young inequality we have Using (13) and (14) yields where C is a constant which is independent of s.
Then from (29), using the Hölder inequality, the Young inequality and the Sobolev em- where a 0 = sup x∈Ω a(x), and γ is a constant. From (21), (27), (28), (30) we have we choose ε and C small enough so that 1 2 where C E(0) is a constant which depends on ε, γ , C and E(0), C ε is a constant depending on ε, C δ and C.

Existence of the global attractor in H 4.1 A priori estimate
Firstly, we use the prior estimates to obtain the asymptotic compactness following the standard energy method. In this section, C i are positive constants.

Asymptotic compactness
In this subsection, following the argument in [9,10], we will prove the asymptotic compactness of the semigroup {S(t)} t≥0 in H, which is given in the following theorem. Proof since the semigroup {S(t)} t≥0 has a bounded absorbing set, for every fixed ε > 0, we can choose that δ ≤ ε 2C 3 mes(Ω) , and then let T become so large that Hence, thanks to Theorem 2.2, we only need to verify that the function Φ T (z 1 0 , z 2 0 ) defined in (50) belongs to C(B 1 ) for each fixed T. and we claim that Here (u(t), u t (t), η) = S(t)z 1 0 and (v(t), v t (t), ξ ) = S(t)z 1 0 are the solutions of (6)-(9) with respect to initial z 1 0 , z 2 0 ∈ B 1 . Then, since C(B 1 ) is a bounded positively invariant set in H, it follows that (u n , u n t , η n ) is uniformly bounded in H. We have u n → u weakly star in L ∞ 0, T; H 1 0 (Ω) ,