Global solutions for the stochastic reaction-diffusion equation with super-linear multiplicative noise and strong dissipativity

Abstract

A condition is identified that implies that solutions to the stochastic reaction-diffusion equation $\frac{\partial u}{\partial t} = \mathcal{A} u + f(u) + \sigma(u) \dot{W}$ on a bounded spatial domain never explode. We consider the case where $\sigma$ grows polynomially and $f$ is polynomially dissipative, meaning that $f$ strongly forces solutions toward finite values. This result demonstrates the role that the deterministic forcing term $f$ plays in preventing explosion.