The heat equation for local Dirichlet forms: Existence and blow up of nonnegative solutions

Abstract

We establish conditions ensuring either existence or blow up of nonnegative solutions for the following parabolic problem: \begin{equation} \begin{cases} Hu -Vu + ({\partial u}/{\partial t}) =0 & \mbox {in } X\times (0,T), \\ u(x,0)=u_{0}(x) & \mbox {in } X, \end{cases} \end{equation} where $T>0$, $X$ is a locally compact separable metric space, $H$ is a selfadjoint operator associated with a regular Dirichlet form $\mathcal E$; the initial value $u_{0}\in L^{2}(X,m)$, where $m$ is a positive Radon measure on Borel subset $U$ of $X$ such that $m(U)>0$ and $V$ is a Borel locally integrable function on $X$.