Global attractors for damped semilinear wave equations doi:10.3934/dcds.2004.10.31 Abstract Full Text (246.2K) Related Articles
John M. Ball  Mathematical Institute, University of Oxford, 2429 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
$1\leq\gamma\leq\frac{n}{n2}$.
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.
Keywords: Global attractor, semilinear wave equation, damped, generalized semiﬂow, Kneser's property, Lyapunov function, weakly continuous, weak solution, nonuniqueness,
asymptotically compact, asymptotically smooth.
Received: February 2003; Revised: March 2003; Published: October 2003. 
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