On stochastic partial differential equations with spatially correlated noise: smoothness of the law

We deal with the following general kind of stochastic partial differential equations:with null initial conditions, L a second-order partial differential operator and F a Gaussian noise, white in time and correlated in space. Firstly, we prove that the solution u(t,x) possesses a smooth density pt,x for every . We use the tools of Malliavin Calculus. Secondly, we apply this general result to two particular cases: the d-dimensional spatial heat equation, d[greater-or-equal, slanted]1, and the wave equation, d[set membership, variant]{1,2}.