Acta mathematica scientia, Series B >
ON A SECOND ORDER DISSIPATIVE ODE IN HILBERT SPACE WITH AN INTEGRABLE SOURCE TERM
Received date: 2011-10-10
Online 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.
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 h ∈ L (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.
Alain Haraux , Mohamed Ali Jendoubi . ON A SECOND ORDER DISSIPATIVE ODE IN HILBERT SPACE WITH AN INTEGRABLE SOURCE TERM[J]. Acta mathematica scientia, Series B, 2012 , 32(1) : 155 -163 . DOI: 10.1016/S0252-9602(12)60009-5
[1] Alvarez F. On the minimizing property of a second order dissipative system in Hilbert space. SIAM J Control Optim, 2000, 38(4): 1102–1119
[2] Attouch H,Goudou X,Redont P.Theheavy ballwith frictionmethod, I.The continuous dynamical system: global exploration of the local minima of a real-valued function asymptotic by analysis of a dissipative dynamical system. Commun Contemp Math, 2000, 2: 1–34
[3] Baillon J B, Haraux A. Comportement `a l’infini pour les ′equations d’′evolution avec forcing p′eriodique. Arch Ration Mech Anal, 1977, 67(1): 101–109
[4] Br′ezis H. Op′erateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. North-Holland Mathematics Studies, No 5. Notas de Matematica (50). Amsterdam-London, New York:
North-Holland Publishing Co, 1973
[5] Bruck R. Asymptotic convergence of nonlinear contraction semigroups in Hilbert space. J Funct Anal, 1975, 18: 15–26
[6] Chill R, Jendoubi M A. Convergence to steady states in asymptotically autonomous semilinear evolution
equations. Nonlinear Anal, 2000, 53(7/8): 1017–1039
[7] Haraux A. Equations d’′evolution non lin′eaires: solutions born′ees et p′eriodiques. Ann Inst Fourier (Greno-ble), 1978, 28(2): 201–220
[8] Haraux A. Nonlinear evolution equations-global behavior of solutions//Lecture Notes in Mathematics, 841. Berlin-New York: Springer-Verlag, 1981
[9] Opial Z. Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull
Amer Math Soc, 1967, 73: 591–597
[10] Schatzman M. Le syst`eme diff′erentiel (d2u/dt 2)+∂φ(u) ξ f avec conditions initiales. (English summary)
C R Acad Sci Paris A-B, 1977, 284(11): A603–A606
/
| 〈 |
|
〉 |