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Under fairly general assumptions, we prove that every compact invariant set$\mathcal I$ of the semiflow generated by the semilinear reaction diffusionequation 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, possiblyunbounded, domain in $\R^3$ and $f(x,u)$ is a nonlinearity of subcriticalgrowth. The nonlinearity $f(x,u)$ needs not to satisfy any dissipativenessassumption and the invariant subset $\mathcal I$ needs not to be an anattractor. If $\Omega$ is regular, $f(x,u)$ is dissipative and $\mathcal I$ isthe global attractor, we give an explicit bound on the Hausdorff dimension of$\mathcal I$ in terms of the structure parameter of the equation.

Dimension of attractors and invariant sets in reaction diffusion equations