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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

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, 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.
Keywords: asymptotic behaviour, gradient system., asymptotically small dissipation, Dissipative dynamical system.
Mathematics Subject Classification: 34A12, 34A40, 34Dx.
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
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

A. Haraux, "Systèmes Dynamiques Dissipatifs et Applications," Masson, Paris, 1991, xii+132 pp.  Google Scholar

[9]

A. Haraux, Positively homogeneous functions and the Łojasiewicz gradient inequality, Ann. Polon. Math., 87 (2005), 165-174. doi: 10.4064/ap87-0-13.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

A. Haraux and T. S. Pham, On the Łojasiewicz exponents of quasi-homogeneous functions, Universitatis Iagellonicae Acta Mathematica, 49 (2011), 45-57. Google Scholar

[16]

S. Łojasiewicz, Ensembles semi-analytiques, I. H. E. S. Notes, (1965). Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

A. Haraux, "Systèmes Dynamiques Dissipatifs et Applications," Masson, Paris, 1991, xii+132 pp.  Google Scholar

[9]

A. Haraux, Positively homogeneous functions and the Łojasiewicz gradient inequality, Ann. Polon. Math., 87 (2005), 165-174. doi: 10.4064/ap87-0-13.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

A. Haraux and T. S. Pham, On the Łojasiewicz exponents of quasi-homogeneous functions, Universitatis Iagellonicae Acta Mathematica, 49 (2011), 45-57. Google Scholar

[16]

S. Łojasiewicz, Ensembles semi-analytiques, I. H. E. S. Notes, (1965). Google Scholar

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