Initial value problems for diffusion equations with singular potential

Abstract

Let $V$ be a nonnegative locally bounded function defined in $Q_\infty:=\BBR^n\times(0,\infty)$. We study under what conditions on $V$ and on a Radon measure $\gm$ in ${\mathbb{R}}^d$ does it exist a function which satisfies $\partial_t u-\xD u+ Vu=0$ in $Q_{\infty }$ and $u(.,0)=\xm$. We prove the existence of a subcritical case in which any measure is admissible and a supercritical case where capacitary conditions are needed. We obtain a general representation theorem of positive solutions when $tV(x,t)$ is bounded and we prove the existence of an initial trace in the class of outer regular Borel measures.

To appear in Contemporary Mathematics