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

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

2011-07-13

arxiv.org/abs/1107.2589v1

Mathematics

Analysis of PDEs

17 pages

Scientific paper

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}+\alpha u_t+\beta(x)u-\Deltau = f(x,u), (t,x)\in[0,+\infty[\times\Omega, u = 0, (t,x)\in[0,+\infty[\times\partial\Omega in $H^1_0(\Omega)\times L^2(\Omega) has finite Hausdorff and fractal dimension. Here $\Omega$ 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.

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