Dimension of attractors and invariant sets in reaction diffusion equations

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 $H_0^1(\mathrm{\Omega })$ has finite Hausdorff dimension. Here $\mathrm{\Omega }$ 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 an attractor. If $\mathrm{\Omega }$ 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.