Global attractors for damped semilinear wave equations

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January & February 2004, 10(1&2): 31-52. doi: 10.3934/dcds.2004.10.31

Global attractors for damped semilinear wave equations

1.	Mathematical Institute, University of Oxford, 24--29 St Giles', Oxford OX1 3LB, United Kingdom

Received February 2003 Revised March 2003 Published October 2003

Abstract
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The existence of a global attractor in the natural energy space is proved for the semilinear wave equation $u_{t t}+\beta u_t -\Delta u + f(u)=0$ on a bounded domain $\Omega\subset\mathbf R^n$ with Dirichlet boundary conditions. The nonlinear term $f$ is supposed to satisfy an exponential growth condition for $n=2$, and for $n\geq 3$ the growth condition $|f(u)|\leq c_0(|u|^{\gamma}+1)$, where $1\leq\gamma\leq\frac{n}{n-2}$. No Lipschitz condition on $f$ is assumed, leading to presumed nonuniqueness of solutions with given initial data. The asymptotic compactness of the corresponding generalized semiflow is proved using an auxiliary functional. The system is shown to possess Kneser's property, which implies the connectedness of the attractor.
In the case $n\geq 3$ and $\gamma>\frac{n}{n-2}$ the existence of a global attractor is proved under the (unproved) assumption that every weak solution satisfies the energy equation.

Keywords: Lyapunov function, semilinear wave equation, asymptotically compact, Kneser's property, damped, weak solution, asymptotically smooth., nonuniqueness, weakly continuous, Global attractor, generalized semi-ﬂow.

Mathematics Subject Classification: 37L30, 37L05, 35L7.

Citation: John M. Ball. Global attractors for damped semilinear wave equations. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 31-52. doi: 10.3934/dcds.2004.10.31

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