  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.
