Instability of an equilibrium of a partial differential equation

TL;DR

This paper demonstrates that an equilibrium of a nonlinear parabolic PDE, stable under linearization, can be nonlinearly unstable due to specific small perturbations, highlighting limitations of linear stability analysis.

Contribution

It introduces a technique inspired by Fujita to show nonlinear instability of an equilibrium despite linear stability, emphasizing the importance of nonlinear effects.

Findings

Linearization indicates stability of the equilibrium.

Certain small perturbations lead to non-global solutions.

Nonlinear instability occurs despite linear stability.

Abstract

A nonlinear parabolic differential equation with a quadratic nonlinearity is presented which has at least one equilibrium. The linearization about this equilibrium is asymptotically stable, but by using a technique inspired by H. Fujita, we show that the equilibrium is unstable in the nonlinear setting. The perturbations used have the property that they are small in every $L^p$ norm, yet they result in solutions which fail to be global.