## 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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### References:

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