Compact almost automorphic weak solutions for some monotone differential inclusions: Applications to parabolic and hyperbolic equations

We study the existence of compact almost automorphic weak solutions for the differential inclusion u′(t)+Au(t)∋f(t) for t∈R, where A:D(A)⊂H⟶2H is maximal monotone and the forcing term f is compact almost automorphic. We prove that the existence of a uniformly continuous weak solution on R+ having a relatively compact range over R+ implies the existence of a compact almost automorphic weak solution. For that goal, we use Amerio's principle. We prove also the existence, uniqueness, and global attractivity of a compact almost automorphic weak solution where A is strongly maximal monotone. For illustration, some applications are provided for parabolic and hyperbolic equations.