Global attractor for a class of doubly nonlinear abstract evolution equations, Discrete Cont

In this paper we consider the Cauchy problem for the abstract nonlinear evolution equa-tion in a Hilbert space H A (u0(t)) + B(u(t)) u(t) 3 f in H for a.e. t 2 (0; T) u(0) = u0; where A is a maximal (possibly multivalued) monotone operator from the Hilbert space H to itself, while B is the subdierential of a proper, convex and lower semicontinuous func-tion ' : H! (1; +1] with compact sublevels in H satisfying a suitable compatibility condition. Finally, is a positive constant. The existence of solutions is proved by using an approximation-a priori estimates-passage to the limit procedure. The main result of this paper is that the set of all the solutions generates a Generalized Semi ow in the sense of John M. Ball [Bal97] in the phase space given by the domain of the potential '. This process is shown to be point dissipative and asymptotically compact; moreover the global attrac-tor, which attracts all the trajectories of the system with respect to a metric strictly linked to the constraint imposed on the unknown, is constructed. Applications to some problems involving PDEs are given