# American Institute of Mathematical Sciences

September  2013, 2(3): 461-470. doi: 10.3934/eect.2013.2.461

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

 1 Laboratoire Jacques-Louis Lions, U.M.R C.N.R.S. 7598, Université Pierre et Marie Curie, Boite courrier 187, 75252 Paris Cedex 05 2 Université de Carthage, Institut Préparatoire aux Etudes Scientifiques et Techniques, B.P. 51, 2070 La Marsa, Tunisia

Received  May 2013 Revised  June 2013 Published  July 2013

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.
Citation: Alain Haraux, Mohamed Ali Jendoubi. Asymptotics for a second order differential equation with a linear, slowly time-decaying damping term. Evolution Equations & Control Theory, 2013, 2 (3) : 461-470. doi: 10.3934/eect.2013.2.461
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##### References:
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