Metastability for small random perturbations of a PDE with blow-up

TL;DR

This paper investigates how small random noise affects the behavior of a reaction-diffusion PDE with blow-up, revealing metastability and convergence of explosion times under perturbations.

Contribution

It demonstrates the metastable behavior of solutions under small noise and characterizes the convergence of explosion times in perturbed systems.

Findings

Metastability persists until an abrupt transition to explosion occurs.

Time averages remain stable around the equilibrium before explosion.

Explosion times of perturbed solutions converge to deterministic explosion times.

Abstract

We study small random perturbations by additive space-time white noise of a reaction-diffusion equation with a unique stable equilibrium and solutions which blow up in finite time. We show that for initial data in the domain of attraction of the stable equilibrium the perturbed system exhibits metastable behavior: its time averages remain stable around this equilibrium until an abrupt and unpredictable transition occurs which leads to explosion in a finite (but exponentially large) time. On the other hand, for initial data in the domain of explosion we show that the explosion time of the perturbed system converges to the explosion time of the deterministic solution.