Stability for semilinear parabolic problems in $L_2$, $W^{1,2}$, and interpolation spaces

TL;DR

This paper establishes asymptotic stability results for semilinear parabolic problems across various function spaces, improving existing results and utilizing Amann's extrapolation scales, with insights into operators satisfying Kato's square root problem.

Contribution

It extends stability analysis to interpolation spaces and enhances known results for mixed boundary value problems using Amann's approach.

Findings

Asymptotic stability in $L_2$ and interpolation spaces is proven.

Improved stability results for $W^{1,2}$ in mixed boundary problems.

New characterizations of operators satisfying Kato's square root problem.

Abstract

An asymptotic stability result for parabolic semilinear problems in $L_2(\Omega)$ and interpolation spaces is shown. Some known results about stability in $W^{1,2}(\Omega)$ are improved for semilinear parabolic mixed boundary value problems. The approach is based on Amann's power extrapolation scales. In a Hilbert space setting, a better understanding of this approach is provided for operators satisfying Kato's square root problem; as a side result some equivalent characterizations of these operators are obtained.