Comportement à l'infini pour certains systèmes dissipatifs non linéaires

Abstract

SynopsisIn the Hilbert space framework, we give some results concerning the behaviour when t goes to infinity for solutions of equations of the form:A is assumed to be a maximal monotone operator and F(t) is a periodic function.When F = 0, under a compactness assumption for trajectories of (1), we give the complete description of the asymptotic behaviour, e.g. every trajectory is asymptotic to an almost-periodic solution of (1). When F ≠ cst, the compactness hypothesis being too restrictive, we concentrate our efforts on the case of the equation:with Dirichlet boundary condition) and get weak convergence to particular solutions of the equation when β is either univalued or strictly monotone. The methods used in these cases seem of general interest for hyperbolic equations of dissipative type with periodic forcing term.