Stepanov-like pseudo almost periodic mild solutions to nonautonomous neutral partial evolution equations

We obtain new existence and uniqueness theorems of pseudo almost periodic mild solutions to nonautonomous neutral partial evolution equations ddt[u(t)+f(t,u(t))]=A(t)[u(t)+f(t,u(t))]+g(t,u(t)),t∈R,ddt[u(t)+f(t,Bu(t))]=A(t)[u(t)+f(t,Bu(t))]+g(t,Cu(t)),t∈R, assuming that A(t)A(t) satisfy “Acquistapace–Terreni” conditions, the evolution family generated by A(t)A(t) has exponential dichotomy, R(λ0,A(⋅))R(λ0,A(⋅)) is almost periodic, B,CB,C are densely defined closed linear operators, f,gf,g are Lipschitz with respect to the second argument uniformly in the first argument, ff is pseudo almost periodic in the first argument, gg is Stepanov-like pseudo almost periodic in the first argument for p>1p>1 and jointly continuous. To illustrate our abstract results, two examples are given.