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

Tomáš Bárta

doi:10.3934/eect.2018025

Article Contents

2018, Volume 7, Issue 4: 531-543. Doi: 10.3934/eect.2018025

This issue Previous Article Next Article Observability of wave equation with Ventcel dynamic condition

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

Tomáš Bárta

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

Received: May 31, 2017

Revised: April 30, 2018

Published: September 2018

Abstract Full Text(HTML) Related Papers Cited by

Abstract

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.

Keywords:

Mathematics Subject Classification: Primary: 34D05, 34C10.

Citation:

Full Text(HTML)

Related Papers

Cited by

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.