  We study large time behaviour of solutions of the Cauchy problem for
equations of the form $\partial_tu-L u+\lambda u=f(x,u)+g(x,u)\cdot\mu$, where
$L$ is the operator associated with a regular lower bounded semi-Dirichlet form
${\mathcal{E}}$ and $\mu$ is a nonnegative bounded smooth measure with respect
to the capacity determined by ${\mathcal{E}}$. We show that under the
monotonicity and some integrability assumptions on $f,g$ as well as some
assumptions on the form ${\mathcal{E}}$, $u(t,x)\rightarrow v(x)$ as
$t\rightarrow\infty$ for quasi-every $x$, where $v$ is a solution of some
elliptic equation associated with our parabolic equation. We also provide the
rate convergence. Some examples illustrating the utility of our general results
are given.
