Global Attractors of the Extensible Plate Equations with Nonlinear Damping and Memory

We prove in this paper the existence of a global attractor for the plate equations of Kirchhoff type with nonlinear damping and memory using the contraction function method.


Introduction
Let us consider the long-time behavior for the following Kirchhoff plate equations with fading memory and nonlinear damping: where Ω ⊂ R  is a bounded domain with smooth boundary Ω; , , and  are positive constants,  ∈ R, ℎ ∈  2 (Ω); ] is the unit outer normal on Ω;  0 : Ω × (−∞, 0] → R is the prescribed past history of . Problem (1) arises from the isothermal viscoelastic theory; it describes a fourth-order viscoelastic plate with a lower order perturbation and also models transversal vibrations of a thin extensible elastic plate in a history space, which is established based on the framework of elastic vibration by Woinowsky-Krieger [1] and Berger [2], and can be seen as an elastoplastic flow equation with some kind of memory effects (see [3,4] for details).The convolution term means that the stress at any instant  depends on the whole history of strains which the material has undergone and produced a weak damping mechanism (see [5,6]).
In the case where () ≡ 0,  =  = 0, (1) becomes the normal plate equation which has been treated in many papers such as [7][8][9][10][11][12][13][14].For instance, the authors investigated the existence of the compact attractor for the plate equation on both the bounded domain [8,10,13] and the unbounded domain in [7,11,12], respectively.Yue and Zhong [9] proved the existence of global attractors to the plate equations when the nonlinear function satisfies the critical exponent in a locally uniform space.In [14], the authors studied the existence of the random attractor for the stochastic strongly damping plate equations with additive noise and critical nonlinearity.
The case of  =  ≡ 0 in problem (1) has been studied by several authors (see [2,5,[15][16][17][18][19][20][21][22][23] and references therein).For instance, Wu [15] scrutinized the existence of global attractors for the nonlinear plate equation with thermal memory effects due to non-Fourier heat flux laws when (  ) = −Δ  .Recently, Conti and Geredeli [5] paid attention to the existence of a smooth global attractor for the nonlinear viscoelastic equations with memory, in which they required the nonlinear damping (  ) to be the polynomial growth.Shen and Ma [21] studied the existence of the random attractor for plate equations with memory and additive white noise.On the other hand, the asymptotic behavior of solutions for the extensible plate equations without memory affection was studied by several authors in [24][25][26][27].
We focus on the existence of the extensible plate equations with nonlinear damping and history memory in the present paper.To prove the existence of a compact global attractor, the key goal is to establish the compact property of the semigroup associated with the dynamical system.Regarding problem (1), we need to overcome the following difficulties.One difficulty is caused by the critical nonlinearity and nonlinear damping.In order to overcome these obstacles, we apply the contraction function method into our problem.Another difficulty is brought about by the memory kernel, because there is no compact embedding in the history space; besides, we cannot use the finite rank method.We solve this term by introducing a new variable and defining an extending phase space (see [20] for details).In addition, the terms ( −  ∫ Ω |∇| 2 )Δ ( ∈ R) make the estimates more complex, so we have to deal with them through accurate computation.Our main result is Theorem 12.

Assumptions
The following conditions are necessary for our main result.
Concerning the nonlinear term  ∈  1 (R), there exists a constant  0 > 0 such that        ()      ≤  0 (1 where Condition (9) implies that  2 0 (Ω) →  2(+1) (Ω).Also, we say that  = 4/(−4) is a critical exponent for the growth of () when  ≥ 5.In addition, we assume that lim inf where  1 > 0 is the principal eigenvalue of Δ 2 in  2 0 (Ω).The nonlinear damping function  ∈  1 (R) satisfies With respect to the memory kernel , we assume that and that there exists a constant  2 > 0 such that Now, we consider the Hilbert spaces that will be used in our analysis.Let equipped with the respective inner products and norms, (, V)  = (Δ, ΔV) , where (⋅, ⋅) is  2 -inner product and we use ‖ ⋅ ‖  to denote   -norms.We define the following weighted  2 -space: which is a Hilbert space endowed with inner product and norm Finally, we introduce the phase space equipped with the norm In order to obtain the global attractor of problems ( 4)-( 6), we need the following theorem.Under our hypotheses, we can derive an existence result by standard Faedo-Galerkin method (see [12,28,29]).For arguments involving the memory term, we follow Giorgi et al. [19,20].

Some Abstract Results
In this section, we will recall some basic theories of infinite dimensional dynamical systems; we refer to [30] for more details.
Definition 3. The global attractor A is the maximal compact invariant set and the minimal set that attracts all bounded sets:

Existence of Attractors
In this section, we will use the abstract results presented in Section 3 to prove our main result.
In line with (52), the Hölder and Young inequalities, and the Sobolev embedding  2 0 →  +1 , similar to the progress of [31] where  is a small positive constant and   is an embedding constant.
In order to prove that the dynamical system (H, ()) is asymptotically smooth, we need the following lemma.