Initial value problems for diffusion equations with singular potential

TL;DR

This paper investigates the existence of solutions to diffusion equations with singular potentials, identifying conditions under which solutions exist for various initial measures and characterizing their properties.

Contribution

It establishes criteria for existence of solutions in subcritical and supercritical cases, and provides a general representation theorem for positive solutions.

Findings

Existence of solutions in the subcritical case for any initial measure.

Capacitary conditions are necessary in the supercritical case.

A representation theorem for positive solutions when $tV(x,t)$ is bounded.

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 $t V(x,t)$ is bounded and we prove the existence of an initial trace in the class of outer regular Borel measures.