Compact almost automorphic solutions for semilinear parabolic evolution equations

Abstract

In this paper, using the subvariant functional method due to Favard [14], 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 [9], we show that the almost automorphy of the coefficients in a weaker sense (Stepanov almost automorphy of order 1 ≤ p < ∞) 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.