Dimension of attractors and invariant sets of damped wave equations in unbounded domains

Abstract

Under fairly general assumptions, we prove that every compact invariant set $\mathcal I$ of the semiflow generated by the semilinear damped wave equation $$ \begin{alignat}{2} u_{tt}+\alpha u_t+\beta(x)u-\Delta u& =f(x,u), &\quad&(t,x)\in[0,+\infty[\times\Omega, \\ u&=0,&\quad &(t,x)\in[0,+\infty[\times\partial\Omega, \end{alignat} $$ in $H^1_0(\Omega)\times L^2(\Omega)$ has finite Hausdorff and fractal dimension. Here $\Omega$ is a regular, possibly unbounded, domain in $\mathbb R^3$ and $f(x,u)$ is a nonlinearity of critical growth. The nonlinearity $f(x,u)$ needs not to satisfy any dissipativeness assumption and the invariant subset $\mathcal I$ needs not to be an attractor. If $f(x,u)$ is dissipative and $\mathcal I$ is the global attractor, we give an explicit bound on the Hausdorff and fractal dimension of $\mathcal I$ in terms of the structure parameters of the equation.