# 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 and Control Theory, 2013, 2 (3) : 461-470. doi: 10.3934/eect.2013.2.461
##### 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).

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##### 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).
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