Dimension of attractors and invariant sets in reaction diffusion equations

Abstract

Under fairly general assumptions, we prove that every compact invariant set $\mathcal I$ of the semiflow generated by the semilinear reaction diffusion equation $$ \alignat 2 u_t+\beta(x)u-\Delta u&=f(x,u),&\quad &(t,x)\in[0,+\infty\mathclose[\times\Omega,\\ u&=0,&\quad &(t,x)\in[0,+\infty\mathclose\times\partial\Omega \endalignat $$ in $H^1_0(\Omega)$ has finite Hausdorff dimension. Here $\Omega$ is an arbitrary, possibly unbounded, domain in $\mathbb{R}^3$ 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 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.