We study the smoothness of the density of a semilinear heat equation with multiplicative spacetime white noise. Using Malliavin calculus, we reduce the problem to a question of negative moments of solutions of a linear heat equation with multiplicative white noise. Then we settle this question by proving that solutions to the linear equation have negative moments of all orders.
"Regularity of the density for the stochastic heat equation." Electron. J. Probab. 13 2248 - 2258, 2008. https://doi.org/10.1214/EJP.v13-589