Linearized Stability of Non-Isolated Equilibria of Quasilinear Parabolic Problems in Interpolation Spaces

TL;DR

This paper proves the stability of non-isolated equilibria in quasilinear parabolic equations within interpolation spaces, extending previous results by relaxing regularity assumptions and applying to problems like Hele-Shaw and fractional mean curvature flow.

Contribution

It introduces a flexible approach to stability analysis in interpolation spaces, broadening applicability to low-regularity semilinear problems.

Findings

Established stability of non-isolated equilibria in interpolation spaces.

Extended previous maximal regularity results to more general settings.

Applied results to Hele-Shaw and fractional mean curvature flow problems.

Abstract

The stability of non-isolated equilibria to quasilinear parabolic problems of the form $u' = A(u)u + f(u)$ is established in interpolation spaces (and thus extending previous results relying on maximal regularity). The approach allows full flexibility in choosing the interpolation methods and requires only low regularity assumptions on the semilinear part $f$. Applications to concrete problems are presented, including the capillarity-driven Hele--Shaw problem and the fractional mean curvature flow.