Stabilisation by noise on the boundary for a Chafee-Infante equation with dynamical boundary conditions

TL;DR

This paper demonstrates that boundary noise can stabilize the trivial steady state of a Chafee-Infante equation with dynamical boundary conditions, showing a finite range of noise intensities for exponential stability, a novel finding in PDE stabilization.

Contribution

It introduces the first analysis of PDE stabilization via boundary noise, revealing a finite noise intensity range necessary for exponential stability in the Chafee-Infante equation.

Findings

Existence of a finite noise intensity range for stabilization.

Boundary noise can achieve exponential stability of the trivial state.

First demonstration of boundary noise stabilizing PDEs.

Abstract

The stabilisation by noise on the boundary of the Chafee-Infante equation with dynamical boundary conditions subject to a multiplicative It\^o noise is studied. In particular, we show that there exists a finite range of noise intensities that imply the exponential stability of the trivial steady state. This differs from previous works on the stabilisation by noise of parabolic PDEs, where the noise acts inside the domain and stabilisation typically occurs for an infinite range of noise intensities. To the best of our knowledge, this is the first result on the stabilisation of PDEs by boundary noise.