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

Alain Haraux; Mohamed Ali Jendoubi

doi:10.3934/eect.2013.2.461

Article Contents

2013, Volume 2, Issue 3: 461-470. Doi: 10.3934/eect.2013.2.461

This issue Previous Article Carleman Estimates and null controllability of coupled degenerate systems Next Article Traction, deformation and velocity of deformation in a viscoelastic string

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

Alain Haraux^1, and
Mohamed Ali Jendoubi^2,

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

Received: April 30, 2013

Revised: May 31, 2013

Published: August 2013

Abstract Related Papers Cited by

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:

Mathematics Subject Classification: 34A12, 34A40, 34Dxx.

Citation:

Related Papers

Cited by

References

[1]	F. Alvarez, On the minimizing property of a second order dissipative system in Hilbert space, SIAM J. Control Optim., 38 (2000), 1102-1119.doi: 10.1137/S0363012998335802.
[2]	F. Alvarez and H. Attouch, Convergence and asymptotic stabilization for some damped hyperbolic equations with non-isolated equilibria, ESAIM, Control Optim. Calc. Var., 6 (2001), 539-552.doi: 10.1051/cocv:2001100.
[3]	H. Attouch, X. Goudou and P. Redont, The heavy ball with friction method, 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., 2 (2000), 1-34.doi: 10.1142/S0219199700000025.
[4]	A. Cabot and P. Frankel, Asymptotics for some semilinear hyperbolic equations with non-autonomous damping, J. Differential Equations, 252 (2012), 294-322.doi: 10.1016/j.jde.2011.09.012.
[5]	A. Cabot, H. Engler and S. Gadat, On the long time behavior of second order differential equations with asymptotically small dissipation, Trans. Amer. Math. Soc., 361 (2009), 5983-6017.doi: 10.1090/S0002-9947-09-04785-0.
[6]	L. Chergui, Convergence of global and bounded solutions of a second order gradient like system with nonlinear dissipation and analytic nonlinearity, J. Dyn. Differ. Equations, 20 (2008), 643-652.doi: 10.1007/s10884-007-9099-5.
[7]	D. D'Acunto and K. Kurdyka, Explicit bounds for the Łojasiewicz exponent in the gradient inequality for polynomials, Ann. Polon. Math., 87 (2005), 51-61.doi: 10.4064/ap87-0-5.
[8]	A. Haraux, "Systèmes Dynamiques Dissipatifs et Applications," Masson, Paris, 1991, xii+132 pp.
[9]	A. Haraux, Positively homogeneous functions and the Łojasiewicz gradient inequality, Ann. Polon. Math., 87 (2005), 165-174.doi: 10.4064/ap87-0-13.
[10]	A. Haraux, Some applications of the Łojasiewicz gradient inequality, Comm. Pure and Applied Analysis, 11 (2012), 2417-2427.doi: 10.3934/cpaa.2012.11.2417.
[11]	A. Haraux and M. A. Jendoubi, Convergence of solutions of second-order gradient-like systems with analytic nonlinearities, J. Differential Equations, 144 (1998), 313-320.doi: 10.1006/jdeq.1997.3393.
[12]	A. Haraux and M. A. Jendoubi, Convergence of bounded weak solutions of the wave equation with dissipation and analytic nonlinearity, Calc. Var. Partial Differential Equations, 9 (1999), 95-124.doi: 10.1007/s005260050133.
[13]	A. Haraux and M. A. Jendoubi, Decay estimates of the solutions to some evolution problems with an analytic nonlinearity, Asymptotic Analysis, 26 (2001), 21-36.
[14]	A. Haraux, P. Martinez and J. Vancostenoble, Asymptotic stability for intermittently controlled second-order evolution equations, Asymptotic Analysis, 43 (2006), 2089–-2108. .doi: 10.1137/S0363012903436569.
[15]	A. Haraux and T. S. Pham, On the Łojasiewicz exponents of quasi-homogeneous functions, Universitatis Iagellonicae Acta Mathematica, 49 (2011), 45-57.
[16]	S. Łojasiewicz, Ensembles semi-analytiques, I. H. E. S. Notes, (1965).