On global attractor for parabolic partial differential inclusion and its time semidiscretization

Abstract

In this article we study the operator version of a first order in time partial differential inclusion as well as its time discretization obtained by an implicit Euler scheme. This technique, known as the Rothe method, yields the semidiscrete trajectories that are proved to converge to the solution of the original problem. While both the time continuous problem and its semidiscretization can have nonunique solutions we prove that, as times goes to infinity, all trajectories are attracted towards certain compact and invariant sets, so-called global attractors. We prove that the semidiscrete attractors converge upper-semicontinuously to the global attractor of time continuous problem.