Stochastic Wave Equations with Polynomial Nonlinearity

Abstract

This paper is concerned with a class of nonlinear stochastic wave equations in $\mathbb{R}^d$ with $d \leq 3$, 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 $H_1$ are proven with the degree of polynomial $m \leq 3$ for $d = 3$, and $m \geq 2$ for $d = 1$ or 2.