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

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  • 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.

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


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    [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.
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    [7] A. CabotH. 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.
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    [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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