Compact Almost Automorphic Solutions to Poisson’s and Heat Equations

In the present work, we revisit several structural properties of almost automorphic and compact almost automorphic functions from the Euclidean space ℝ^m ( m ≥ 1 ) with values in a Banach space 𝕏 . When 𝕏 is a Banach algebra, it is proven that the spaces formed by these functions are also Banach algebras. As applications, first we prove regularity of almost automorphic solution of Poisson’s equation; that is, we prove that bounded continuous functions with weak (distributional) almost automorphic Laplacian are compact almost automorphic; then, we prove the compact almost automorphy in space variable of classical solutions to heat equation with almost automorphic initial datum.