Regularity of Global Attractors for the Kirchhoff Wave Equation

In this paper, we mainly use operator decomposition technique to prove the global attractors which in 1 2 0 H L × for the Kirchhoff wave equation with strong damping and critical nonlinearities, are also bounded in 2 10 H H × .


Introduction
In this paper, we discuss the regularity of global attractors for the following Kirchhoff wave equation where Ω is a bounded domain in  [6] established the existence of global attractor for the subclass of limit solutions of (1.3)-(1.2) by using J. Ball's attractor theory on the generalized semiflow. Recently, I. Chueshov [7] founded that the Kirchhoff wave equation with strong nonlinear damping was still well-posed and the related evolution semigroup had a finite-dimensional global attractor in in the sense of "partially strong topology". Without "partially strong topology", P. Y. Ding, Z. J. Yang [8] proved the existence of a finite-dimensional global attractor in the natural energy space. And H. L. Ma and C. K. Zhong [9] proved that global attractors for the Kirchhoff equations with strong nonlinear damping attracted -bounded set with respect to the was studied by M. Nakao, and the author proved the existence and absorbing properties of attractors in a local sense [10]. Replacing t u with t u −∆ , Y. H. Wang and C. K. Zhong [11] proved the upper semicontinuity of pullback attractors in non-autonomous case. Then Z. J. Yang and Y. Q. Wang [12] [14]- [23], there are few researches about problem of (1.1)-(1.2). And the attractor is a key point for studying these properties, we introduce readers to see the classical book [24].
Based on these, the purpose of this paper is to prove the global attractor of problem (1.1)-(1.2), which attracts every ( ) ( ) The paper is arranged as follows. In Section 2, we verify some preliminaries. In Section 3, we prove the existence of the global attractor. In Section 4, we prove the regularity of the global attractor. We define the spaces

Preliminaries
 are Hilbert spaces with the following scalar products and the norms We define the phase space For any s r > , we have the continuous embeddings and the following inequalities hold true: The Generalized Poincare inequality: especially, when 2 p q = = , then The Gronwall inequality (differential form): let ( ) η ⋅ is nonnegative continuous differentiable function (or nonnegative absolutely continuous function), Throughout this paper, we will denote by C a positive constant which is various in different line or even in the same line and use the following abbreviations: subset A of E is called a global attractor for the semigroup, if A is compact and enjoys the following properties: 2) A attracts all bounded set of E. That is, for any bounded subset B of E, Next we only formulate the following results, which is proved in [

Existence of Global Attractors in H L
solve the Cauchy problems From now on, 0 0 , 0 c υ > and 0 J will denote generic constants and a generic function, respectively, depending only on 0 B .
In order to prove the existence of the global attractors, now we need to prove the asymptotic compactness.
Multiplying the first equation of (3.1) by ˆt v v γ + and integrating over Ω , we get ( ) ( ) and the generalized Poincare inequality, then ( ) Actually, noting that ( ) 0ˆt v C ϕ′ ≥ , and by exploiting (2.8) and (2.12), we deduce that and ( ) where 0 γ > is small enough such that ( where 0 γ > is small enough. Then we define the energy functional ( ) At the same time, by the interpolation inequality, we have

Regularity of Global Attractors
Now we are in a position to state and prove the main result: , t t Z t z w t w t ζ = = solve the following equations with initial data ( ) ( )