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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 |
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.
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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.
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Global behavior of the solutions to a class of nonlinear, singular second order ODE, Nonlinear Analysis, 96 (2014), 18-37.
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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.
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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. |
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T. Bárta, Sharp and optimal decay estimates for solutions of gradient-like systems, preprint. |
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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. |
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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. |
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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. |
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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. |
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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. |
show all references
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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