Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

Global attractors for damped semilinear wave equations

Pages: 31 - 52, Volume 10, Issue 1/2, January/February 2004      doi:10.3934/dcds.2004.10.31

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John M. Ball - Mathematical Institute, University of Oxford, 24--29 St Giles', Oxford OX1 3LB, United Kingdom (email)

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.

Keywords:  Global attractor, semilinear wave equation, damped, generalized semi-flow, Kneser's property, Lyapunov function, weakly continuous, weak solution, nonuniqueness, asymptotically compact, asymptotically smooth.
Mathematics Subject Classification:  37L30, 37L05, 35L70.

Received: February 2003;      Revised: March 2003;      Available Online: October 2003.