Evolution inclusions governed by subdifferentials in reflexive Banach spaces

Abstract.

The existence, uniqueness and regularity of strong solutions for Cauchy problem and periodic problem are studied for the evolution equation: $du(t)/dt + \partial \varphi (u(t)) \mathrel\backepsilon f(t),\,t \in ]0,\,T[,$ where ∂φ is the so-called subdifferential operator from a real Banach space V into its dual V*. The study in the Hilbert space setting (V = V* = H: Hilbert space) is already developed in detail so far. However, the study here is done in the V−V* setting which is not yet fully pursued. Our method of proof relies on approximation arguments in a Hilbert space H. To assure this procedure, it is assumed that the embeddings $V \subset H \subset V^*$ are both dense and continuous.