Existence of the global attractor for the plate equation with nonlocal nonlinearity in R^{n}

We consider Cauchy problem for the semilinear plate equation with nonlocal nonlinearity. Under mild conditions on the damping coefficient, we prove that the semigroup generated by this problem possesses a global attractor.


Introduction
In this paper, we study the long-time behavior of the solutions for the following plate equation with localized damping and nonlocal nonlinearity in terms of global attractors: Plate equations have been investigated for many years due to their importance in some physical areas such as vibration and elasticity theories of solid mechanics. For instance, in the case when f (·) is identically constant, equation (1.1) becomes an equation with local polynomial nonlinearity which arises in aeroelasticity modeling (see, for example, [7], [8]), whereas in the case when p = 2, the nonlinearity f ( u L p (R n ) ) |u| p−2 u appears in the models of Kerr-like medium (see [15], [21]).
The study of the long-time dynamics of evolution equations has become an outstanding area during the recent decades. As it is well known, the attractors can be used as a tool to describe the long-time dynamics of these equations. In particular, there have been many works on the investigation of the attractors for the plate equations over the last few years. For the attractors of the plate equations with local and nonlocal nonlinearities in bounded domains, we refer to [3], [5], [14], [16][17][18][19] and [22]. In the case of unbounded domains, owing to the lack of Sobolev compact embedding theorems, there are difficulties in applying the methods given for bounded domains. In order to overcome these difficulties, the authors of [9][10], [13] and [23] established the uniform tail estimates for the plate equations with local nonlinearities and then used the weak continuity of the nonlinear source operators.
In the case when the domain is unbounded and the equation includes nonlocal nonlinearity, an additional obstacle occurs. For equation (1.1), this obstacle is caused by the operator defined by F (u) := f ( u L p (R n ) ) |u| p−2 u , because the operator F , besides being not compact, is not also weakly continuous from H 2 (R n ) to L 2 (R n ). This situation does not allow us to apply the standard splitting method and the energy method devised in [2]. Recently in [1], the obstacle mentioned above is handled for the nonlinearity f ( ∇u L 2 (R n ) )∆u by using compensated compactness method introduced in [11]. In that paper, the strictly positivity condition on the damping coefficient α (·) is critically used. In the present paper, we replace this condition with the weaker conditions (see (2.3), (2.4)), and by using effectiveness of the dissipation for large enough x, we prove the existence of the global attractor which equals the unstable manifold of the set of stationary points. The paper is organized as follows: In Section 2, we give the statement of the problem and the main result. In Section 3, we firstly prove two auxiliary lemmas and then establish the asymptotic compactness of the solution, which together with the presence of the strict Lyapunov function leads to the existence of the global attractor.

Statement of the problem and the main result
We consider the following initial value problem where λ > 0, h ∈ L 2 (R n ) , p ≥ 2, p (n − 4) ≤ 2n− 4 and the functions α (·), f (·) satisfy the following conditions: Applying the semigroup theory (see [4, p.56-58]) and repeating the arguments done in the introduction of [1], one can prove the following well-posedness result.
where c : R + × R + → R + is a nondecreasing function with respect to each variable and r = max Thus, according to Theorem 2.1, by the formula (u(t), u t (t)) = S (t) (u 0 , u 1 ), problem (2.1)-(2.2) generates a strongly continuous semigroup is the weak solution of (2.1)-(2.2), determined by Theorem 2.1, with initial data (u 0 , u 1 ). Now, we are in a position to state our main result.

Proof of Theorem 2.2
We start with the following lemmas.
Lemma 3.1. Assume that the condition (2.5) holds. Also, assume that the sequence {v m } ∞ m=1 is weakly star convergent in L ∞ 0, ∞; is convergent, for all t ≥ 0. Then, for all r > 0 for ε > 0. Then, we have Let us estimate the first term on the right hand side of (3.1).
For the first two terms on the right hand side of (3.2), we have is convergent, by continuity of f ε , it follows that the se- also converges for all t ∈ [0, ∞). Moreover, by the conditions of the lemma and the definition of f ε , we obtain that the sequence converges weakly star in where q < 2n (n−4) + . Then, considering (3.3) and (3.4), we get For the last two terms on the right hand side of (3.2), by using (3.3), we have Hence, taking into account (3.5)-(3.6) and passing to limit in (3.2), we obtain Then, by (3.1) and (3.7), for all r > 0, we have Thus, passing to the limit in the above inequality as ε → 0, we obtain the claim of the lemma.
On the other hand, for the first term on the right hand side of (3.8), we get Now, let us estimate the first term on the right hand side of (3.10).
Proof. To get the claim of the theorem, it is sufficient to prove the following sequential limit estimate We obtain (3.18) by means of the sequential limit estimate of the energy of v m − v l which is proved in the following three steps. In the first step, we get the tail estimates, by using the effect of the damping term. In the second step, we obtain the interior estimates. Finally, in the last step, we get the sequential limit estimate of the energy in R n , by considering the results obtained in the previous steps. Note that we establish these estimates for the smooth solutions of (2.1)-(2.2) with the initial data in H 4 (R n ) × H 2 (R n ) , for which the estimates in the following text are justified. These estimates can be extended to the weak solutions with the initial data in H 2 (R n ) × L 2 (R n ) by the standard density arguments.
Passing to limit as r → ∞ in the last inequality, we get lim sup m→∞ lim sup Multiplying (3.21) by 2t (v mt − v lt ), integrating over (0, T ) × R n , using integration by parts and considering (2.4), we find Multiplying (3.21) by tη r (v m − v l ), integrating over (0, T ) × R n and using integration by parts, we get Then, considering (2.3), we obtain x)) dxdt, ∀T ≥ 0 and ∀r ≥ r 0 .
Then, for sufficiently small γ and δ, we obtain Applying Gronwall inequality and considering (2.6) and (3.19) in the above inequality, we get As a consequence, from the above sequential limit inequality, we get (3.18) which completes the proof.