Positive solutions with a singular set for semilinear parabolic equations

We study the existence and large time behavior of positive solutions for the semilinear parabolic equation $\frac{\partial u}{\partial t}(x,t) =\Delta u(x,t)+ V(x)u(x,t) +f(x,u(x,t))$ with initial Dirichlet boundary conditions on $(D\setminus E)\times (0,\infty)$, where $D$ is a bounded Lipschitz domain in $\mathbb{R}^n$, $ n\geq 3$, $E$ is a prescribed compact set of $D$, and $V$ and $f$ are real-valued functions satisfying some general conditions. Our results cover various types of nonlinearities and extend known results proved for the power nonlinearity $f(x,u)=W(x)u^p$ and a singular one-point set $E$.