Stochastic Wave Equations with Polynomial Nonlinearity

This paper is concerned with a class of nonlinear stochastic wave equations in Rd with d.3, for which the nonlinear terms are polynomial of degree m. As an example of the nonexistence of a global solution in general, it is shown that there exists an explosive solution of some cubically nonlinear wave equation with a noise term. Then the existence and uniqueness theorems for local and global solutions in Sobolev space H1 are proven with the degree of polynomial m.3 for d=3, and m.2 for d=1 or 2.