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

We study large time behaviour of solutions of the Cauchy problem for equations of the form ${\mathrm{\partial }}_{t}u-Lu+\lambda u=f(x,u)+g(x,u)\cdot \mu$, 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 $\mathcal{E}$, $u(t,x)\to v(x)$ as $t\to \mathrm{\infty }$ 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.