Dimension of attractors and invariant sets in reaction diffusion equations

TL;DR

This paper proves that all compact invariant sets of certain reaction-diffusion equations in possibly unbounded domains have finite Hausdorff dimension, providing explicit bounds under additional conditions.

Contribution

It establishes finite Hausdorff dimension for invariant sets without requiring dissipativeness or attractor conditions, extending previous results to more general settings.

Findings

All compact invariant sets have finite Hausdorff dimension.

Explicit bounds are provided for the Hausdorff dimension of the global attractor.

Results apply to unbounded domains and non-dissipative nonlinearities.

Abstract

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