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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

John M. Ball 1,

1. 

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

Received  February 2003 Revised  March 2003 Published  October 2003

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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