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

TL;DR

This paper establishes conditions under which solutions to a stochastic reaction-diffusion equation with super-linear noise remain finite, highlighting the importance of the dissipative deterministic term in preventing explosion.

Contribution

It identifies a specific condition ensuring non-explosion of solutions in stochastic reaction-diffusion equations with polynomially growing noise and dissipative forcing.

Findings

Solutions do not explode under the identified condition.

The dissipative term $f$ plays a crucial role in preventing blow-up.

Polynomial growth of noise $\sigma$ is manageable with the right conditions.

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.