Higher regularity of homeomorphisms in the Hartman-Grobman theorem for semilinear evolution equations

TL;DR

This paper improves the regularity results of the conjugacy in the Hartman-Grobman theorem for semilinear evolution equations, showing that under certain conditions, the conjugacy can be Lipschitz continuous.

Contribution

It establishes higher regularity (Lipschitz continuity) of the conjugacy in the Hartman-Grobman theorem for semilinear evolution equations, which was previously only known to be Hölder continuous.

Findings

Conjugacy is Lipschitz continuous when solutions are bounded.

Inverse conjugacy remains Hölder continuous.

First demonstration of higher regularity of homomorphisms in this context.

Abstract

Hein and Pr\"{u}ss [J. Differential Equations, 261(2016)4709-4727] presented a version of Hartman-Grobman type $C^{0}$ linearization result for semilinear hyperbolic evolution equations. They showed that the linearising map (homomorphism) and its inverse are H\"{o}lder continuous. An important question: is it possible to improve the regularity of the homomorphisms? In the present paper, we prove that if the mild solutions of semilinear system are bounded, then the regularity of the homomorphisms is Lipchitzian, but the inverse is merely H\"{o}lder continuous. We also give a generalized local linearization result in this paper. Finally, some applications end the paper. As pointed out by Backes [J. Differential Equations, 297 (2021) 536-574], even if the diffeomorphism $F$ is $C^{\infty}$, the homomorphism can fail to be locally Lipschitz. The homomorphisms are in general only locally H\"older continuous. However, by establishing two effective dichotomy integral inequalities, we prove that the conjugacy is Lipchitzian, but the inverse is H\"{o}lder continuous. Our result is the first one to observe the higher regularity of homomorphisms in the Hartman-Grobman theorem.