Stabilization by Multiplicative It\^o Noise for Chafee-Infante Equation in Perforated Domains

TL;DR

This paper demonstrates that appropriate multiplicative Itô noise can stabilize unstable parabolic equations in perforated domains with small holes, providing explicit estimates for the parameters involved.

Contribution

It introduces a quantitative method to stabilize the Chafee-Infante equation in perforated domains using multiplicative noise, with explicit bounds on hole size and diffusion coefficients.

Findings

Stability achieved with sufficiently small holes and specific noise parameters

Explicit estimates for hole size and diffusion coefficients for stabilization

Asymptotic analysis of eigenvalues as holes shrink

Abstract

The stabilization by noise for parabolic equations in perforated domains, i.e. domains with small holes, is investigated. We show that when the holes are small enough, one can stabilize the unstable equations using suitable multiplicative It\^o noise. The results are quantitative, in the sense that we can explicitly estimate the size of the holes and diffusion coefficients for which stabilization by noise takes place. This is done by using the asymptotic behaviour of the first eigenvalue of the Laplacian as the hole shrinks to a point.