Long time existence for the wave equation with a noise term

Abstract

.1 2 We consider the equation $u_{tt} = \Delta u + a(u) \mathsf{N}$ for $x \epsilon \mathbf{R}^1$ or $R^2$. $\mathsf{N}$ is a Gaussian noise term, which is white noise if $x \epsilon \mathbf{R}^1$. If $a(u)$ grows no faster than $u (\log u)^{1/2-\varepsilon}$, then there is a unique solution valid for all time.