On the principle of linearized stability for quasilinear evolution equations in time-weighted spaces

TL;DR

This paper develops a framework using time-weighted spaces to establish the principle of linearized stability for quasilinear parabolic equations, allowing analysis in intermediate, potentially critical, function spaces.

Contribution

It introduces a novel approach employing time-weighted spaces to prove stability in intermediate domains, enhancing the analysis of quasilinear evolution equations.

Findings

Established linearized stability in intermediate spaces.

Flexibility in phase space analysis for quasilinear problems.

Applicable to critical spaces with scaling invariance.

Abstract

Quasilinear (and semilinear) parabolic problems of the form $v'=A(v)v+f(v)$ with strict inclusion $\mathrm{dom}(f)\subsetneq \mathrm{dom}(A)$ of the domains of the function $v\mapsto f(v)$ and the quasilinear part $v\mapsto A(v)$ are considered in the framework of time-weighted function spaces. This allows one to establish the principle of linearized stability in intermediate spaces lying between $\mathrm{dom}(f)$ and $\mathrm{dom}(A)$ and yields a greater flexibility with respect to the phase space for the evolution. In applications to differential equations such intermediate spaces may correspond to critical spaces exhibiting a scaling invariance. Several examples are provided to demonstrate the applicability of the results.