Almost periodic and almost automorphic solutions to semilinear parabolic boundary differential equations

Abstract

In this paper, we study the existence and uniqueness of almost periodic and almost automorphic solutions to the semilinear parabolic boundary differential equations ( SBDE ) { x ′ ( t ) = A m x ( t ) + h ( t , x ( t ) ) , t ∈ R , L x ( t ) = ϕ ( t , x ( t ) ) , t ∈ R , where A ≔ A m | ker L generates a hyperbolic analytic semigroup on a Banach space X . The functions h and ϕ are defined on some intermediate subspaces X β , 0 < β < 1 , and take values in X and in a boundary space ∂ X respectively.