Stochastic wave equation with critical nonlinearities: Temporal regularity and uniqueness

Abstract

We consider a nonlinear wave equation u t t = Δ u + f ( u ) + g ( u ) W ˙ on R d driven by a spatially homogeneous Wiener process W with a finite spectral measure and with nonlinear terms f, g of critical growth. We study pathwise uniqueness and norm continuity of paths of ( u , u t ) in H 1 ( R d ) ⊕ L 2 ( R d ) under the hypothesis that there exists a local solution u such that ( u , u t ) has weakly continuous paths in H 1 ( R d ) ⊕ L 2 ( R d ) .