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

TL;DR

This paper proves that compact invariant sets of damped wave equations in unbounded domains have finite Hausdorff and fractal dimensions, even without dissipativity assumptions, and provides explicit bounds when the set is a global attractor.

Contribution

It establishes finite-dimensionality of invariant sets for damped wave equations in unbounded domains under general conditions, extending previous results to non-dissipative and non-attracting sets.

Findings

Invariant sets have finite Hausdorff and fractal dimensions.

Explicit bounds are provided for global attractors.

Results apply to equations with critical growth nonlinearities.

Abstract

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.