On nonlinear stabilization of linearly unstable maps

TL;DR

This paper investigates the phenomenon of nonlinear stabilization in unstable maps, providing examples, counterexamples, and criteria for stability and instability in finite and infinite-dimensional settings.

Contribution

It introduces a mechanism for nonlinear stabilization applicable to hyperbolic PDEs and establishes a sharp criterion for nonlinear exponential instability for certain differentiable maps.

Findings

Nonlinear stabilization can occur in linearly unstable maps.

A criterion for nonlinear exponential instability is provided for maps with normal linearized operators.

Open question remains whether Fréchet differentiability guarantees nonlinear instability at a slower rate.

Abstract

We examine the phenomenon of nonlinear stabilization, exhibiting a variety of related examples and counterexamples. For G\^ateaux differentiable maps, we discuss a mechanism of nonlinear stabilization, in finite and infinite dimensions, which applies in particular to hyperbolic partial differential equations, and, for Fr\'echet differentiable maps with linearized operators that are normal, we give a sharp criterion for nonlinear exponential instability at the linear rate. These results highlight the fundamental open question whether Fr\'echet differentiability is sufficient for linear exponential instability to imply nonlinear exponential instability, at possibly slower rate.