Global attractor for a class of doubly nonlinear abstract evolution equations

In this paper we consider the Cauchy problem for the abstract nonlinear evolution equation in a Hilbert space $\H$ $\A(u'(t))+ \B(u(t))-\lambda u(t)$ ∋ $f \mbox{in } \H \mbox{ for a.e. }t\in (0,+\infty)$ $u(0)=u_{0},$ where $\A$ is a maximal (possibly multivalued) monotone operator from the Hilbert space $\H$ to itself, while $\B$ is the subdifferential of a proper, convex and lower semicontinuous function φ:$\H\rightarrow (-\infty,+\infty]$ with compact sublevels in $\H$ satisfying a suitable compatibility condition. Finally, $\lambda$ 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 Semiflow in the sense of John M. Ball [8] in the phase space given by the domain of the potential φ. This process is shown to be point dissipative and asymptotically compact; moreover the global attractor, 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.

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