# American Institute of Mathematical Sciences

December  2018, 7(4): 531-543. doi: 10.3934/eect.2018025

## Exact rate of decay for solutions to damped second order ODE's with a degenerate potential

 Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic

Received  June 2017 Revised  May 2018 Published  September 2018

We prove exact rate of decay for solutions to a class of second order ordinary differential equations with degenerate potentials, in particular, for potential functions that grow as different powers in different directions in a neigborhood of zero. As a tool we derive some decay estimates for scalar second order equations with non-autonomous damping.

Citation: Tomáš Bárta. Exact rate of decay for solutions to damped second order ODE's with a degenerate potential. Evolution Equations and Control Theory, 2018, 7 (4) : 531-543. doi: 10.3934/eect.2018025
##### References:
 [1] M. Abdelli, M. Anguiano and A. Haraux, Existence, uniqueness and global behavior of the solutions to some nonlinear vector equations in a finite dimensional Hilbert space, Nonlinear Analysis, 161 (2017), 157-181.  doi: 10.1016/j.na.2017.06.001. [2] M. Abdelli and A. Haraux, Global behavior of the solutions to a class of nonlinear, singular second order ODE, Nonlinear Analysis, 96 (2014), 18-37.  doi: 10.1016/j.na.2013.10.023. [3] M. Balti, Asymptotic behavior for second-order differential equations with nonlinear slowly time-decaying damping and integrable source, Electron. J. Differential Equations, 2015 (2015), 1-11. [4] T. Bárta, Rate of convergence to equilibrium and Lojasiewicz-type estimates, J. Dynam. Differential Equations, 29 (2017), 1553-1568.  doi: 10.1007/s10884-016-9549-z. [5] T. Bárta, Sharp and optimal decay estimates for solutions of gradient-like systems, preprint. [6] I. Ben Hassen and L. Chergui, Convergence of global and bounded solutions of some nonautonomous second order evolution equations with nonlinear dissipation, J. Dynam. Differential Equations, 23 (2011), 315-332.  doi: 10.1007/s10884-011-9212-7. [7] 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. [8] L. Chergui, Convergence of global and bounded solutions of a second order gradient like system with nonlinear dissipation and analytic nonlinearity, J. Dynam. Differential Equations, 20 (2008), 643-652.  doi: 10.1007/s10884-007-9099-5. [9] A. Haraux, Sharp decay estimates of the solutions to a class of nonlinear second order ODE's, Anal. Appl. (Singap.), 9 (2011), 49-69.  doi: 10.1142/S021953051100173X. [10] A. Haraux and M. A. Jendoubi, Asymptotics for a second order differential equation with a linear, slowly time-decaying damping term, Evol. Equ. Control Theory, 2 (2013), 461-470.  doi: 10.3934/eect.2013.2.461.

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##### References:
 [1] M. Abdelli, M. Anguiano and A. Haraux, Existence, uniqueness and global behavior of the solutions to some nonlinear vector equations in a finite dimensional Hilbert space, Nonlinear Analysis, 161 (2017), 157-181.  doi: 10.1016/j.na.2017.06.001. [2] M. Abdelli and A. Haraux, Global behavior of the solutions to a class of nonlinear, singular second order ODE, Nonlinear Analysis, 96 (2014), 18-37.  doi: 10.1016/j.na.2013.10.023. [3] M. Balti, Asymptotic behavior for second-order differential equations with nonlinear slowly time-decaying damping and integrable source, Electron. J. Differential Equations, 2015 (2015), 1-11. [4] T. Bárta, Rate of convergence to equilibrium and Lojasiewicz-type estimates, J. Dynam. Differential Equations, 29 (2017), 1553-1568.  doi: 10.1007/s10884-016-9549-z. [5] T. Bárta, Sharp and optimal decay estimates for solutions of gradient-like systems, preprint. [6] I. Ben Hassen and L. Chergui, Convergence of global and bounded solutions of some nonautonomous second order evolution equations with nonlinear dissipation, J. Dynam. Differential Equations, 23 (2011), 315-332.  doi: 10.1007/s10884-011-9212-7. [7] 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. [8] L. Chergui, Convergence of global and bounded solutions of a second order gradient like system with nonlinear dissipation and analytic nonlinearity, J. Dynam. Differential Equations, 20 (2008), 643-652.  doi: 10.1007/s10884-007-9099-5. [9] A. Haraux, Sharp decay estimates of the solutions to a class of nonlinear second order ODE's, Anal. Appl. (Singap.), 9 (2011), 49-69.  doi: 10.1142/S021953051100173X. [10] A. Haraux and M. A. Jendoubi, Asymptotics for a second order differential equation with a linear, slowly time-decaying damping term, Evol. Equ. Control Theory, 2 (2013), 461-470.  doi: 10.3934/eect.2013.2.461.
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