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 & Control Theory, 2018, 7 (4) : 531-543. doi: 10.3934/eect.2018025
References:
[1]

M. AbdelliM. 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. 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.

[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.

show all references

References:
[1]

M. AbdelliM. 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. 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.

[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.

[1]

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

[2]

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

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[15]

Shengfan Zhou, Jinwu Huang, Xiaoying Han. Compact kernel sections for dissipative non-autonomous Zakharov equation on infinite lattices. Communications on Pure & Applied Analysis, 2010, 9 (1) : 193-210. doi: 10.3934/cpaa.2010.9.193

[16]

Shengfan Zhou, Min Zhao. Fractal dimension of random attractor for stochastic non-autonomous damped wave equation with linear multiplicative white noise. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2887-2914. doi: 10.3934/dcds.2016.36.2887

[17]

Karsten Matthies, George Stone. Derivation of a non-autonomous linear Boltzmann equation from a heterogeneous Rayleigh gas. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3299-3355. doi: 10.3934/dcds.2018143

[18]

Bo You, Yanren Hou, Fang Li, Jinping Jiang. Pullback attractors for the non-autonomous quasi-linear complex Ginzburg-Landau equation with $p$-Laplacian. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1801-1814. doi: 10.3934/dcdsb.2014.19.1801

[19]

Zhaojuan Wang, Shengfan Zhou. Existence and upper semicontinuity of random attractors for non-autonomous stochastic strongly damped wave equation with multiplicative noise. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2787-2812. doi: 10.3934/dcds.2017120

[20]

Xinguang Yang, Baowei Feng, Thales Maier de Souza, Taige Wang. Long-time dynamics for a non-autonomous Navier-Stokes-Voigt equation in Lipschitz domains. Discrete & Continuous Dynamical Systems - B, 2019, 24 (1) : 363-386. doi: 10.3934/dcdsb.2018084

2017 Impact Factor: 1.049

Metrics

  • PDF downloads (101)
  • HTML views (198)
  • Cited by (0)

Other articles
by authors

[Back to Top]