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.