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The existence of global attractors for a class nonlinear evolution equation. (English) Zbl 1132.35019

The authors deal with the long-time behaviour of the solutions for the following class of nonlinear evolution equations
\[ \begin{gathered} u_{tt}- \Delta_x u-\mu\Delta_x u_t- \Delta_x u_{tt}= f(u)\quad\text{in }\Omega,\\ u|_{t=0}= u_0,\quad u_t|_{t=0}= u_1\quad\text{in }\Omega,\\ u= 0\quad\text{on }\partial\Omega,\end{gathered}\tag{1} \] where \(\Omega\) is a bounded domain in \(\mathbb{R}^d\) with smooth boundary \(\partial\Omega\) and \(f\in C^0(\mathbb{R}, \mathbb{R})\) satisfying some natural growth assumptions. Due to the presence of the term \(\Delta_x u_{tt}\) and some critical growth exponent for the nonlinearity \(f\), (1) cannot be directly reformulated and studied in the framework of the wave equation. The key idea of the authors is to use the asymptotic a priori estimate method, which leads to an existence of a global attractor for (1) in \(H^1_0(\Omega)\times H^1_0(\Omega)\).

MSC:

35B41 Attractors
35L82 Pseudohyperbolic equations
35L20 Initial-boundary value problems for second-order hyperbolic equations
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