C(n)C(n)-almost periodic and C(n)C(n)-almost automorphic solutions for a class of partial functional differential equations with finite delay

This work deals with the existence of C(n)C(n)-almost periodic and C(n)C(n)-almost automorphic solutions for a class of partial functional differential equations with finite delay. We suppose that the homogeneous part without delay is the infinitesimal generator of an analytic semigroup and that the delayed part is continuous with respect to fractional powers of the generator. We use the variation of constants formula and the reduction method developed in Adimy et al. (2009) [13] to prove the existence of C(n)C(n)-almost periodic and C(n)C(n)-almost automorphic solutions when there is at least one bounded solution in R+R+. When the solution semigroup of the homogenous linear equation has an exponential dichotomy, we prove the existence and uniqueness of C(n)C(n)-almost periodic and C(n)C(n)-almost automorphic solutions of the following equation. ddtu(t)=−Au(t)+L(ut)+f(t)for t≥σ.