Time--space white noise eliminates global solutions in reaction diffusion equations

TL;DR

This paper demonstrates that adding time-space white noise to a reaction-diffusion equation causes solutions to almost surely explode, contrasting with the deterministic case where solutions remain bounded and stationary.

Contribution

It proves that stochastic perturbation with white noise leads to explosion of solutions in reaction-diffusion equations, showing a fundamental difference from deterministic models.

Findings

Solutions explode with probability one under white noise perturbation

Deterministic model admits positive stationary solutions

Stochastic noise prevents the existence of bounded solutions

Abstract

We prove that perturbing the reaction--diffusion equation $u_t=u_{xx} + (u_+)^p$ ($p>1$), with time--space white noise produces that solutions explodes with probability one for every initial datum, opposite to the deterministic model where a positive stationary solution exists.