Paper detail

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

Under fairly general assumptions, we prove that every compact invariant set $\mathcal I$ of the semiflow generated by the semilinear damped wave equation u_{tt}+αu_t+β(x)u-\Deltau = f(x,u), (t,x)\in[0,+\infty[\timesΩ, u = 0, (t,x)\in[0,+\infty[\times\partialΩin $H^1_0(Ω)\times L^2(Ω) has finite Hausdorff and fractal dimension. Here $Ω$ is a regular, possibly unbounded, domain in $\R^3$ 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.