In this paper, we discuss the regularity of global attractors for the following Kirchhoff wave equation
where is a bounded domain in with the smooth boundary , , and are nonlinear functions and is an external force term which is independent of time.
G. Kirchhoff  introduced the Equation (1.1) in without dissipation and nonlinear perturbations and , and described the oscillation of an elastic stretched string. Furthermore, if the string is made up of the viscoelastic material of rate-type, the equation with the strong damping appeared . Since , the Equation (1.1) became the following strongly damped semi-linear wave equation
which described the thermal evolution and denoted a source term depending nonlinearly on displacement, denoted a nonlinearly temperature-dependent internal source term . With different conditions about the growth exponents q and p of the nonlinearities and , some scholars   analyzed the longtime behaviour of solutions of (1.3)-(1.2) by the global and exponential attractors in a bounded region of . When the nonlinearities are of fully supercritical growth, which lead to that the weak solutions of the equation lose their uniqueness. Z. J. Yang and Z. M. Liu  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  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  proved the existence of a finite-dimensional global attractor in the natural energy space. And H. L. Ma and C. K. Zhong  proved that global attractors for the Kirchhoff equations with strong nonlinear damping attracted -bounded set with respect to the norm.
Since , the following quasi-linear wave equation of Kirchhoff type
was studied by M. Nakao, and the author proved the existence and absorbing properties of attractors in a local sense . Replacing with , Y. H. Wang and C. K. Zhong  proved the upper semicontinuity of pullback attractors in non-autonomous case. Then Z. J. Yang and Y. Q. Wang  studied the longtime behavior of the Kirchhoff type equation with a strong dissipation and proved that the continuous semigroup possessed global attractors in the phase spaces with low regularity. As for the Kirchhoff wave equation with strong damping and critical nonlinearities, Z. J. Yang and F. Da  also studied the stability for the Kirchhoff wave equation with strong damping and critical nonlinearities and proved the existence of global attractors and exponential attractors. Comparing with many researches about the longtime dynamic behavior of solutions for the Kirchhoff wave equation with different types of dissipations  - , 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 .
Based on these, the purpose of this paper is to prove the global attractor of problem (1.1)-(1.2), which attracts every -bounded set that is compacted in by the way in ( , Theorem 3.1). And we also establish the asymptotic compactness of the global attractor by operator decomposition technique ( , Theorem 1.1). So these jobs provide a way to research the longtime dynamic behaviour of such Kirchhoff wave equations, and also reflect the strong damped properties of to some extent.
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.
Let on with , and A strictly positive on . We define the spaces are Hilbert spaces with the following scalar products and the norms
Let be the first eigenvalue of A, then with .
We define the phase space with usual graph norm. Let , then problem (1.1)-(1.2) becomes
For any , we have the continuous embeddings ,
and the following inequalities hold true:
Interpolation inequality: if , where and , then there exists a constant such that
The Generalized Poincare inequality:
where is the first eigenvalue of A.
The Young’s inequality with : Let , and , then
especially, when , then
The Gronwall inequality (differential form): let is nonnegative continuous differentiable function (or nonnegative absolutely continuous function), and satisfy
here are nonnegative integrable functions, then
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:
1) , and
where if .
Definition 2.2. Let be a semigroup on a metric space . A subset A of E is called a global attractor for the semigroup, if A is compact and enjoys the following properties:
1) A is invariant, i.e. ;
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  :
Lemma 2.3. Let (2.11)-(2.13) be valid. Then problem (2.2)-(2.3) admits a unique weak solution u, with , . Moreover, this solution possesses the following properties:
where k denotes a small positive constant, and are positive constants.
Lemma 2.4. Let (2.11)-(2.13) be valid and when . Then
Actually, by exploiting (2.11) and (2.14), we can get are respectively bounded in .
3. Existence of Global Attractors in
For every fixed , we split the solution into the sum , where and solve the Cauchy problems
Having set , and satisfying
From now on, and will denote generic constants and a generic function, respectively, depending only on .
Theorem 3.1. Let (2.11)-(2.13) be valid, then the solution semigroup possesses a global attractor in X.
Proof. Estimate (2.14) shows
such that the ball is an absorbing set of the semigroup in X for .
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 and integrating over , we get
By using and the generalized Poincare inequality, then
By , we know
where is small enough such that
Actually, noting that , and by exploiting (2.8) and (2.12), we deduce that
From (3.5)-(3.7), we get
where is small enough such that ( ) is negative. Furthermore, by the Gronwall inequality, we can get
Next multiplying the first equation of (3.2) by and integrating over , we get
where is small enough. Then we define the energy functional
At the same time, by the interpolation inequality, we have
and by the embedding , then
By exploiting (2.8) and the generalized Poincare inequality, from (3.9)-(3.11), we get
where is small enough and by , we get are negative. Then from the Gronwall inequality and noting that , we get
which provides the following estimate
From (3.8) and (3.12), we obtain that the evolution semigroup is asymptotically compact in X, so the solution semigroup possesses a global attractor in , which
where is chosen such that for .
4. Regularity of Global Attractors
Now we are in a position to state and prove the main result:
Theorem 4.1. The attractor of the semigroup on X is bounded in .
Proof. Having set . For , we split the solution into the sum
where and solve the following equations with initial data ,
Multiplying the first equation of (4.1) by and integrating over , by we get
where is small enough such that
By and the generalized Poincare inequality, we deduce that
then by the Gronwall inequality, we get
Next multiplying the first equation of (4.2) by and integrating over, exploiting (2.8) and the Hölder’s inequality, the right side becomes
where is small enough, we know is bounded by (2.13) and lemma 2.3. At the same time, the left side becomes
then we define the energy functional
where is small enough such that. By combining (4.5)-(4.7) and the embedding, we get
where is small enough and by, we get are negative. From the Gronwall inequality, we get
which provides the estimate
From (4.4) and (4.8), for every bounded set, we get
Then we finish the proof.
In this paper, we first prove that the Kirchhoff wave equation with strong damping and critical nonlinearities possesses a global attractor in. Then we split the solution into two parts, one part decays exponentially and the other part satisfies asymptotic behaviour in spaces with higher regularity. By the operator decomposition technique, we get the global attractor which is compactly bounded in
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