doi:
10.3934/dcds.2004.10.31 
Full text:
(246.2K)
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
Published: October 2003
