The aim of this paper is to investigate the existence of bounded and compact almost automorphic solutions to the following evolution equation in a Banach space X:
where is the infinitesimal generator of a -semigroup of bounded linear operators on and is an almost automorphic function in uniformly with respect to the second argument. The existence of almost automorphic solutions to this equation has been extensively studied when the semigroup generated by the operator has an exponential dichotomy and is almost automorphic in and Lipschitzian with respect to the second variable with a small Lipschitz constant.
In this paper, the authors give sufficient conditions for the existence of compact almost automorphic solutions when the equation admits at least a bounded solution on . Then they apply the results of the paper to nonlinear partial differential equations and give sufficient conditions ensuring the existence of compact almost automorphic solutions to some heat and wave equations.