We prove the existence and uniqueness, for any time, of a real-valued process solving a nonlinear stochastic wave equation driven by a Gaussian noise white in time and correlated in the two-dimensional space variable. We prove that the solution is regular in the sense of the Malliavin calculus. We also give a decay condition on the covariance function of the noise under which the solution has Hölder continuous trajectories and show that, under an additional ellipticity assumption, the law of the solution at any strictly positive time has a smooth density.
Annie Millet. Marta Sanz-Solé. "A Stochastic Wave Equation in Two Space Dimension: Smoothness of the Law." Ann. Probab. 27 (2) 803 - 844, April 1999. https://doi.org/10.1214/aop/1022677387