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
R
d
does it exist a function which satisfies $\partial_t u-\xD u+ Vu=0$ in
Q
∞
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.