  In this paper, we establish a necessary and sufficient condition for the
existence and regularity of the density of the solution to a semilinear
stochastic (fractional) heat equation with measure-valued initial conditions.
Under a mild cone condition for the diffusion coefficient, we establish the
smooth joint density at multiple points. The tool we use is Malliavin calculus.
The main ingredient is to prove that the solutions to a related stochastic
partial differential equation have negative moments of all orders. Because we
cannot prove $u(t,x)\in \mathbb{D}^\infty$ for measure-valued initial data, we
need a localized version of Malliavin calculus. Furthermore, we prove that the
(joint) density is strictly positive in the interior of the support of the law,
where we allow both measure-valued initial data and unbounded diffusion
coefficient. The criteria introduced by Bally and Pardoux [1] are no longer
applicable for the parabolic Anderson model. We have extended their criteria to
a localized version. Our general framework includes the parabolic Anderson
model as a special case.
