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Boundedness of solutions in a class of Duffing equations with a bounded restore force
Global attractor for a class of doubly nonlinear abstract evolution equations
1.  Dipartimento di Matematica "F. Casorati", Universitá di Pavia, via Ferrata 1, P.O. Box 27100, Pavia, Italy 
$\A(u'(t))+ \B(u(t))\lambda u(t)$ ∋ $f \mbox{in } \H \mbox{ for a.e. }t\in (0,+\infty)$
$u(0)=u_{0},$
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 approximationa priori estimatespassage 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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