Compact almost automorphic solutions for semilinear parabolic evolution equations

TL;DR

This paper proves the existence of unique compact almost automorphic solutions for semilinear parabolic evolution equations in Banach spaces, using a subvariant functional method and weaker assumptions on coefficient automorphy.

Contribution

It improves previous results by weakening the automorphy assumptions on coefficients and establishes the existence of strong almost automorphic solutions.

Findings

Existence of unique compact almost automorphic solutions.

Weaker Stepanov automorphy assumptions suffice.

Application to reaction-diffusion problems.

Abstract

In this paper, using the subvariant functional method due to Favard \cite{Favard}, we prove the existence of aunique compact almost automorphic solution for a class of semilinear evolution equations in Banach spaces. More specifically, we improve the assumptions in \cite{CieuEzz}, we show that the almost automorphy of the coefficients in a weaker sense (Stepanov almost automorphy of order $1\leq p <\infty$) is enough to obtain solutions that are almost automorphic in a strong sense (Bochner almost automorphy). We distinguish two cases, $ p=1 $ and $ p>1$. Moreover, we propose to study a class of reaction-diffusion problems.