数学物理学报(英文版) ›› 2012, Vol. 32 ›› Issue (1): 155-163.doi: 10.1016/S0252-9602(12)60009-5

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ON A SECOND ORDER DISSIPATIVE ODE IN HILBERT SPACE WITH AN INTEGRABLE SOURCE TERM

Alain Haraux1, Mohamed Ali Jendoubi2   

  1. 1.UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, Bo^?te courrier 187, 75252 Paris Cedex 05, France|2.Facult′e des Sciences de Bizerte, D′epartement de Math′ematiques, 7021 Jarzouna Bizerte, Tunisie
  • 收稿日期:2011-10-10 出版日期:2012-01-20 发布日期:2012-01-20
  • 基金资助:

    The authors gratefully acknowledge support by the France-Tunisia cooperation under the auspices of the CNRS/DGRSRT agreement No. 08/R 15-06: Syst`emes dynamiques et ′equations d’′evolution. Part of this work was done during a sojourn of the second author at Laboratoire Jacques-Louis Lions under the auspices of the Fondation Sciences Math′ematiques de Paris.

ON A SECOND ORDER DISSIPATIVE ODE IN HILBERT SPACE WITH AN INTEGRABLE SOURCE TERM

Alain Haraux1, Mohamed Ali Jendoubi2   

  1. 1.UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, Bo^?te courrier 187, 75252 Paris Cedex 05, France|2.Facult′e des Sciences de Bizerte, D′epartement de Math′ematiques, 7021 Jarzouna Bizerte, Tunisie
  • Received:2011-10-10 Online:2012-01-20 Published:2012-01-20
  • Supported by:

    The authors gratefully acknowledge support by the France-Tunisia cooperation under the auspices of the CNRS/DGRSRT agreement No. 08/R 15-06: Syst`emes dynamiques et ′equations d’′evolution. Part of this work was done during a sojourn of the second author at Laboratoire Jacques-Louis Lions under the auspices of the Fondation Sciences Math′ematiques de Paris.

摘要:

Asymptotic behaviour of solutions is studied for some second order equations including the model case x¨(t)+γx˙(t)+?Φ(x(t)) = h(t) with γ > 0 and hL (0,+∞; H), Φ being continuouly di?erentiable with locally Lipschitz continuous gradient and bounded from below. In particular when Φ is convex, all solutions tend to minimize the potential Φ as time tends to infinity and the existence of one bounded trajectory implies the weak convergence of all solutions to equilibrium points.

关键词: dissipative dynamical system, asymptotic behaviour, gradient system, heavy ball with friction

Abstract:

Asymptotic behaviour of solutions is studied for some second order equations including the model case x¨(t)+γx˙(t)+?Φ(x(t)) = h(t) with γ > 0 and hL (0,+∞; H), Φ being continuouly di?erentiable with locally Lipschitz continuous gradient and bounded from below. In particular when Φ is convex, all solutions tend to minimize the potential Φ as time tends to infinity and the existence of one bounded trajectory implies the weak convergence of all solutions to equilibrium points.

Key words: dissipative dynamical system, asymptotic behaviour, gradient system, heavy ball with friction

中图分类号: 

  • 34A12