Global attractors for damped semilinear wave equations

John M. Ball

doi:10.3934/dcds.2004.10.31

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2004, Volume 10, Issue 1&2: 31-52. Doi: 10.3934/dcds.2004.10.31

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Global attractors for damped semilinear wave equations

John M. Ball^1,

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

Received: February 2003

Revised: March 2003

Published: January 2004

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Abstract

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.

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Mathematics Subject Classification: 37L30, 37L05, 35L70.

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