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

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)=Amx(t)+h(t,x(t)),t∈R,Lx(t)=ϕ(t,x(t)),t∈R, where A≔Am|kerLA≔Am|kerL generates a hyperbolic analytic semigroup on a Banach space XX. The functions hh and ϕϕ are defined on some intermediate subspaces Xβ,0<β<1Xβ,0<β<1, and take values in XX and in a boundary space ∂X∂X respectively.