Under fairly general assumptions, we prove that every compact invariant set $\mathcal I$ of the semiflow generated
by the semilinear reaction diffusion equation
\begin{equation*}
\begin{aligned}
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
\end{aligned}\end{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 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.
