Initial value problems for diffusion equations with singular potential

Konstantinos Gkikas (1) , Laurent Veron (1)

1 LMPT - Laboratoire de Mathématiques et Physique Théorique

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.

Mots clés: Heat kernel; Harnack inequality; Radon measures; initial trace; representation

Domaines: Equations aux dérivées partielles [math.AP]

Konstantinos Gkikas, Laurent Veron. Initial value problems for diffusion equations with singular potential. 2012. ⟨hal-00736712v5⟩