American Institute of Mathematical Sciences

Advanced
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

Tomáš Bárta

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.

Keywords: Rate of convergence to equilibrium, second order equation with damping, non-autonomous decay.
Mathematics Subject Classification: Primary: 34D05, 34C10.
Citation: Tomáš Bárta. Exact rate of decay for solutions to damped second order ODE's with a degenerate potential. Evolution Equations & 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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[5]

T. Bárta, Sharp and optimal decay estimates for solutions of gradient-like systems, preprint. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[5]

T. Bárta, Sharp and optimal decay estimates for solutions of gradient-like systems, preprint. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[1]

Ahmed Y. Abdallah, Rania T. Wannan. Second order non-autonomous lattice systems and their uniform attractors. Communications on Pure & Applied Analysis, 2019, 18 (4) : 1827-1846. doi: 10.3934/cpaa.2019085
[2]

Maurizio Grasselli, Morgan Pierre. Convergence to equilibrium of solutions of the backward Euler scheme for asymptotically autonomous second-order gradient-like systems. Communications on Pure & Applied Analysis, 2012, 11 (6) : 2393-2416. doi: 10.3934/cpaa.2012.11.2393
[3]

Yanan Li, Zhijian Yang, Fang Da. Robust attractors for a perturbed non-autonomous extensible beam equation with nonlinear nonlocal damping. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 5975-6000. doi: 10.3934/dcds.2019261
[4]

Tomás Caraballo, David Cheban. On the structure of the global attractor for non-autonomous dynamical systems with weak convergence. Communications on Pure & Applied Analysis, 2012, 11 (2) : 809-828. doi: 10.3934/cpaa.2012.11.809
[5]

Yuri V. Rogovchenko, Fatoş Tuncay. Interval oscillation of a second order nonlinear differential equation with a damping term. Conference Publications, 2007, 2007 (Special) : 883-891. doi: 10.3934/proc.2007.2007.883
[6]

Mirelson M. Freitas, Alberto L. C. Costa, Geraldo M. Araújo. Pullback dynamics of a non-autonomous mixture problem in one dimensional solids with nonlinear damping. Communications on Pure & Applied Analysis, 2020, 19 (2) : 785-809. doi: 10.3934/cpaa.2020037
[7]

Fang Li, Bo You. Pullback exponential attractors for the three dimensional non-autonomous Navier-Stokes equations with nonlinear damping. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 55-80. doi: 10.3934/dcdsb.2019172
[8]

Jun-Ren Luo, Ti-Jun Xiao. Decay rates for second order evolution equations in Hilbert spaces with nonlinear time-dependent damping. Evolution Equations & Control Theory, 2019, 0 (0) : 0-0. doi: 10.3934/eect.2020009
[9]

Olivier Goubet, Wided Kechiche. Uniform attractor for non-autonomous nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2011, 10 (2) : 639-651. doi: 10.3934/cpaa.2011.10.639
[10]

Abdelaziz Rhandi, Roland Schnaubelt. Asymptotic behaviour of a non-autonomous population equation with diffusion in $L^1$. Discrete & Continuous Dynamical Systems - A, 1999, 5 (3) : 663-683. doi: 10.3934/dcds.1999.5.663
[11]

Denis Mercier, Virginie Régnier. Decay rate of the Timoshenko system with one boundary damping. Evolution Equations & Control Theory, 2019, 8 (2) : 423-445. doi: 10.3934/eect.2019021
[12]

Tomás Caraballo, David Cheban. On the structure of the global attractor for infinite-dimensional non-autonomous dynamical systems with weak convergence. Communications on Pure & Applied Analysis, 2013, 12 (1) : 281-302. doi: 10.3934/cpaa.2013.12.281
[13]

Alexandre N. Carvalho, José A. Langa, James C. Robinson. Non-autonomous dynamical systems. Discrete & Continuous Dynamical Systems - B, 2015, 20 (3) : 703-747. doi: 10.3934/dcdsb.2015.20.703
[14]

Tomás Caraballo, Antonio M. Márquez-Durán, Rivero Felipe. Asymptotic behaviour of a non-classical and non-autonomous diffusion equation containing some hereditary characteristic. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1817-1833. doi: 10.3934/dcdsb.2017108
[15]

Felipe Rivero. Time dependent perturbation in a non-autonomous non-classical parabolic equation. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 209-221. doi: 10.3934/dcdsb.2013.18.209
[16]

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
[17]

Abdelaziz Soufyane, Belkacem Said-Houari. The effect of the wave speeds and the frictional damping terms on the decay rate of the Bresse system. Evolution Equations & Control Theory, 2014, 3 (4) : 713-738. doi: 10.3934/eect.2014.3.713
[18]

Xue-Li Song, Yan-Ren Hou. Pullback $\mathcal{D}$-attractors for the non-autonomous Newton-Boussinesq equation in two-dimensional bounded domain. Discrete & Continuous Dynamical Systems - A, 2012, 32 (3) : 991-1009. doi: 10.3934/dcds.2012.32.991
[19]

Xin Li, Chunyou Sun, Na Zhang. Dynamics for a non-autonomous degenerate parabolic equation in $\mathfrak{D}_{0}^{1}(\Omega, \sigma)$. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 7063-7079. doi: 10.3934/dcds.2016108
[20]

Zhaojuan Wang, Shengfan Zhou. Random attractor for stochastic non-autonomous damped wave equation with critical exponent. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 545-573. doi: 10.3934/dcds.2017022

2018 Impact Factor: 1.048

Tools

Metrics

  • PDF downloads (131)
  • HTML views (408)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences