Evolution Equations and Control Theory (EECT)

Asymptotics for a second order differential equation with a linear, slowly time-decaying damping term

Pages: 461 - 470, Volume 2, Issue 3, September 2013      doi:10.3934/eect.2013.2.461

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Alain Haraux - Laboratoire Jacques-Louis Lions, U.M.R C.N.R.S. 7598, Université Pierre et Marie Curie, Boite courrier 187, 75252 Paris Cedex 05, France (email)
Mohamed Ali Jendoubi - Université de Carthage, Institut Préparatoire aux Etudes Scientifiques et Techniques, B.P. 51, 2070 La Marsa, Tunisia (email)

Abstract: A gradient-like property is established for second order semilinear conservative systems in presence of a linear damping term which is asymptotically weak for large times. The result is obtained under the condition that the only critical points of the potential are absolute minima. The damping term may vanish on large intervals for arbitrarily large times and tends to $0$ at infinity, but not too fast (in a non-integrable way). When the potential satisfies an adapted, uniform, Łojasiewicz gradient inequality, convergence to equilibrium of all bounded solutions is shown, with examples in both analytic and non-analytic cases.

Keywords:  Dissipative dynamical system, asymptotically small dissipation, asymptotic behaviour, gradient system.
Mathematics Subject Classification:  34A12, 34A40, 34Dxx.

Received: May 2013;      Revised: June 2013;      Available Online: July 2013.