# 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 & 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. 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, 6 (2001), 539. 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. 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. 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. 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. 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. doi: 10.4064/ap87-0-5. Google Scholar [8] A. Haraux, "Systèmes Dynamiques Dissipatifs et Applications,", Masson, (1991). Google Scholar [9] A. Haraux, Positively homogeneous functions and the Łojasiewicz gradient inequality,, Ann. Polon. Math., 87 (2005), 165. 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. 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. 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. 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. Google Scholar [14] A. Haraux, P. Martinez and J. Vancostenoble, Asymptotic stability for intermittently controlled second-order evolution equations,, Asymptotic Analysis, 43 (2006). 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. 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. 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, 6 (2001), 539. 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. 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. 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. 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. 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. doi: 10.4064/ap87-0-5. Google Scholar [8] A. Haraux, "Systèmes Dynamiques Dissipatifs et Applications,", Masson, (1991). Google Scholar [9] A. Haraux, Positively homogeneous functions and the Łojasiewicz gradient inequality,, Ann. Polon. Math., 87 (2005), 165. 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. 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. 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. 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. Google Scholar [14] A. Haraux, P. Martinez and J. Vancostenoble, Asymptotic stability for intermittently controlled second-order evolution equations,, Asymptotic Analysis, 43 (2006). 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. Google Scholar [16] S. Łojasiewicz, Ensembles semi-analytiques,, I. H. E. S. Notes, (1965). Google Scholar
 [1] Jianfeng Feng, Mariya Shcherbina, Brunello Tirozzi. Dynamical behaviour of a large complex system. Communications on Pure & Applied Analysis, 2008, 7 (2) : 249-265. doi: 10.3934/cpaa.2008.7.249 [2] Tomás Caraballo, Francisco Morillas, José Valero. Asymptotic behaviour of a logistic lattice system. Discrete & Continuous Dynamical Systems - A, 2014, 34 (10) : 4019-4037. doi: 10.3934/dcds.2014.34.4019 [3] María Anguiano, Tomás Caraballo. Asymptotic behaviour of a non-autonomous Lorenz-84 system. Discrete & Continuous Dynamical Systems - A, 2014, 34 (10) : 3901-3920. doi: 10.3934/dcds.2014.34.3901 [4] Giuseppe Viglialoro, Thomas E. Woolley. Eventual smoothness and asymptotic behaviour of solutions to a chemotaxis system perturbed by a logistic growth. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3023-3045. doi: 10.3934/dcdsb.2017199 [5] Radosław Kurek, Paweł Lubowiecki, Henryk Żołądek. The Hess-Appelrot system. Ⅲ. Splitting of separatrices and chaos. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 1955-1981. doi: 10.3934/dcds.2018079 [6] Zaki Chbani, Hassan Riahi. Existence and asymptotic behaviour for solutions of dynamical equilibrium systems. Evolution Equations & Control Theory, 2014, 3 (1) : 1-14. doi: 10.3934/eect.2014.3.1 [7] Tomás Caraballo, María J. Garrido–Atienza, Björn Schmalfuss, José Valero. Asymptotic behaviour of a stochastic semilinear dissipative functional equation without uniqueness of solutions. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 439-455. doi: 10.3934/dcdsb.2010.14.439 [8] Sébastien Court. Stabilization of a fluid-solid system, by the deformation of the self-propelled solid. Part II: The nonlinear system.. Evolution Equations & Control Theory, 2014, 3 (1) : 83-118. doi: 10.3934/eect.2014.3.83 [9] Sébastien Court. Stabilization of a fluid-solid system, by the deformation of the self-propelled solid. Part I: The linearized system.. Evolution Equations & Control Theory, 2014, 3 (1) : 59-82. doi: 10.3934/eect.2014.3.59 [10] Kolade M. Owolabi. Dynamical behaviour of fractional-order predator-prey system of Holling-type. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 823-834. doi: 10.3934/dcdss.2020047 [11] Kosuke Ono. Global existence and asymptotic behavior of small solutions for semilinear dissipative wave equations. Discrete & Continuous Dynamical Systems - A, 2003, 9 (3) : 651-662. doi: 10.3934/dcds.2003.9.651 [12] Rana D. Parshad. Asymptotic behaviour of the Darcy-Boussinesq system at large Darcy-Prandtl number. Discrete & Continuous Dynamical Systems - A, 2010, 26 (4) : 1441-1469. doi: 10.3934/dcds.2010.26.1441 [13] Dominique Blanchard, Nicolas Bruyère, Olivier Guibé. Existence and uniqueness of the solution of a Boussinesq system with nonlinear dissipation. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2213-2227. doi: 10.3934/cpaa.2013.12.2213 [14] Paweł Lubowiecki, Henryk Żołądek. The Hess-Appelrot system. I. Invariant torus and its normal hyperbolicity. Journal of Geometric Mechanics, 2012, 4 (4) : 443-467. doi: 10.3934/jgm.2012.4.443 [15] P.K. Newton. The dipole dynamical system. Conference Publications, 2005, 2005 (Special) : 692-699. doi: 10.3934/proc.2005.2005.692 [16] Benedetta Lisena. Dynamic behaviour of a periodic competitive system with pulses. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 723-729. doi: 10.3934/dcdss.2013.6.723 [17] J. K. Krottje. On the dynamics of a mixed parabolic-gradient system. Communications on Pure & Applied Analysis, 2003, 2 (4) : 521-537. doi: 10.3934/cpaa.2003.2.521 [18] Dorota Bors, Robert Stańczy. Dynamical system modeling fermionic limit. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 45-55. doi: 10.3934/dcdsb.2018004 [19] Xiangnan He, Wenlian Lu, Tianping Chen. On transverse stability of random dynamical system. Discrete & Continuous Dynamical Systems - A, 2013, 33 (2) : 701-721. doi: 10.3934/dcds.2013.33.701 [20] Aicha Balhag, Zaki Chbani, Hassan Riahi. Existence and continuous-discrete asymptotic behaviour for Tikhonov-like dynamical equilibrium systems. Evolution Equations & Control Theory, 2018, 7 (3) : 373-401. doi: 10.3934/eect.2018019

2018 Impact Factor: 1.048

## Metrics

• PDF downloads (8)
• HTML views (0)
• Cited by (5)

## Other articlesby authors

• on AIMS
• on Google Scholar

[Back to Top]