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+β(x)u-Δu&=f(x,u),&&(t,x)\in[0,+\infty[\timesΩ, u&=0,&&(t,x)\in[0,+\infty[\times\partialΩ} {equation*} in $H^1_0(Ω)$ has finite Hausdorff dimension. Here $Ω$ 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 $Ω$ 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.

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