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We study large time behaviour of solutions of the Cauchy problem forequations 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 respectto the capacity determined by ${\mathcal{E}}$. We show that under themonotonicity and some integrability assumptions on $f,g$ as well as someassumptions 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 someelliptic equation associated with our parabolic equation. We also provide therate convergence. Some examples illustrating the utility of our general resultsare given.

Large time behaviour of solutions to parabolic equations with Dirichlet operators and nonlinear dependence on measure data