Local bounds for stochastic reaction diffusion equations

TL;DR

This paper establishes uniform space-time bounds for solutions of stochastic reaction diffusion equations with super-linear damping, linking noise regularity to solution integrability and enabling large-scale analysis.

Contribution

It introduces a novel a priori bound that depends on the noise realization, controlling solutions uniformly and facilitating analysis of solutions on unbounded domains.

Findings

Bounds depend only on the noise realization and boundary data.

The method reveals the relationship between noise regularity and solution integrability.

Provides tools for constructing solutions on full space without noise decay at infinity.

Abstract

We prove a priori bounds for solutions of stochastic reaction diffusion equations with super-linear damping in the reaction term. These bounds provide a control on the supremum of solutions on any compact space-time set which only depends on the specific realisation of the noise on a slightly larger set and which holds uniformly over all possible space-time boundary values. This constitutes a space-time version of the so-called 'coming down from infinity' property. Bounds of this type are very useful to control the large scale behaviour of solutions effectively and can be used, for example, to construct solutions on the full space even if the driving noise term has no decay at infinity. Our method shows the interplay of the large scale behaviour, dictated by the non-linearity, and the small scale oscillations, dictated by the rough driving noise. As a by-product we show that there is a close relation between the regularity of the driving noise term and the integrability of solutions.