American Institute of Mathematical Sciences

Advanced
2009, 2009(Special): 744-752. doi: 10.3934/proc.2009.2009.744

Equivalence between observability and stabilization for a class of second order semilinear evolution

Louis Tcheugoue Tebou 1,

1. 

Department of Mathematics, Florida International University, University Park, Miami, Florida 33199, United States

Received  August 2008 Revised  February 2009 Published  September 2009

We consider an abstract second order semilinear evolution equation with a bounded dissipation. We establish an equivalence between the stabilization of this system and the observability of the corresponding undamped system. Our technique of proof relies on an appropriate decomposition of the solution, and the energy method. Our result generalizes an earlier one by Haraux [5] who studied the same type of problem for linear systems. Some applications of our result are provided, and the paper ends with a few open problems.
Keywords: localized damping, stabilization, observability, hyperbolic equation, elasticity equations, bounded dissipation, semilinear equation, second order evolution equation, Euler-Bernoulli equation.
Mathematics Subject Classification: Primary: 93D15; Secondary: 35L70, 35L05, 37L15.
Citation: Louis Tcheugoue Tebou. Equivalence between observability and stabilization for a class of second order semilinear evolution. Conference Publications, 2009, 2009 (Special) : 744-752. doi: 10.3934/proc.2009.2009.744
[1]

Louis Tebou. Well-posedness and stabilization of an Euler-Bernoulli equation with a localized nonlinear dissipation involving the $p$-Laplacian. Discrete & Continuous Dynamical Systems, 2012, 32 (6) : 2315-2337. doi: 10.3934/dcds.2012.32.2315
[2]

Fathi Hassine. Asymptotic behavior of the transmission Euler-Bernoulli plate and wave equation with a localized Kelvin-Voigt damping. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1757-1774. doi: 10.3934/dcdsb.2016021
[3]

Marcelo Moreira Cavalcanti. Existence and uniform decay for the Euler-Bernoulli viscoelastic equation with nonlocal boundary dissipation. Discrete & Continuous Dynamical Systems, 2002, 8 (3) : 675-695. doi: 10.3934/dcds.2002.8.675
[4]

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
[5]

Ruy Coimbra Charão, Juan Torres Espinoza, Ryo Ikehata. A second order fractional differential equation under effects of a super damping. Communications on Pure & Applied Analysis, 2020, 19 (9) : 4433-4454. doi: 10.3934/cpaa.2020202
[6]

Igor Chueshov, Irena Lasiecka, Daniel Toundykov. Long-term dynamics of semilinear wave equation with nonlinear localized interior damping and a source term of critical exponent. Discrete & Continuous Dynamical Systems, 2008, 20 (3) : 459-509. doi: 10.3934/dcds.2008.20.459
[7]

Zhijian Yang, Zhiming Liu, Na Feng. Longtime behavior of the semilinear wave equation with gentle dissipation. Discrete & Continuous Dynamical Systems, 2016, 36 (11) : 6557-6580. doi: 10.3934/dcds.2016084
[8]

Kim Dang Phung. Boundary stabilization for the wave equation in a bounded cylindrical domain. Discrete & Continuous Dynamical Systems, 2008, 20 (4) : 1057-1093. doi: 10.3934/dcds.2008.20.1057
[9]

Louis Tebou. Energy decay estimates for some weakly coupled Euler-Bernoulli and wave equations with indirect damping mechanisms. Mathematical Control & Related Fields, 2012, 2 (1) : 45-60. doi: 10.3934/mcrf.2012.2.45
[10]

Kim Dang Phung. Decay of solutions of the wave equation with localized nonlinear damping and trapped rays. Mathematical Control & Related Fields, 2011, 1 (2) : 251-265. doi: 10.3934/mcrf.2011.1.251
[11]

Aníbal Rodríguez-Bernal, Enrique Zuazua. Parabolic singular limit of a wave equation with localized boundary damping. Discrete & Continuous Dynamical Systems, 1995, 1 (3) : 303-346. doi: 10.3934/dcds.1995.1.303
[12]

Jong Yeoul Park, Sun Hye Park. On uniform decay for the coupled Euler-Bernoulli viscoelastic system with boundary damping. Discrete & Continuous Dynamical Systems, 2005, 12 (3) : 425-436. doi: 10.3934/dcds.2005.12.425
[13]

Maja Miletić, Dominik Stürzer, Anton Arnold. An Euler-Bernoulli beam with nonlinear damping and a nonlinear spring at the tip. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 3029-3055. doi: 10.3934/dcdsb.2015.20.3029
[14]

Joseph A. Iaia. Localized radial solutions to a semilinear elliptic equation in $\mathbb{R}^n$. Conference Publications, 1998, 1998 (Special) : 314-326. doi: 10.3934/proc.1998.1998.314
[15]

Min Zhu. On the higher-order b-family equation and Euler equations on the circle. Discrete & Continuous Dynamical Systems, 2014, 34 (7) : 3013-3024. doi: 10.3934/dcds.2014.34.3013
[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]

Jiacheng Wang, Peng-Fei Yao. On the attractor for a semilinear wave equation with variable coefficients and nonlinear boundary dissipation. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021043
[18]

Jeong Ja Bae, Mitsuhiro Nakao. Existence problem for the Kirchhoff type wave equation with a localized weakly nonlinear dissipation in exterior domains. Discrete & Continuous Dynamical Systems, 2004, 11 (2&3) : 731-743. doi: 10.3934/dcds.2004.11.731
[19]

Jiann-Sheng Jiang, Kung-Hwang Kuo, Chi-Kun Lin. Homogenization of second order equation with spatial dependent coefficient. Discrete & Continuous Dynamical Systems, 2005, 12 (2) : 303-313. doi: 10.3934/dcds.2005.12.303
[20]

Jiayun Lin, Kenji Nishihara, Jian Zhai. Critical exponent for the semilinear wave equation with time-dependent damping. Discrete & Continuous Dynamical Systems, 2012, 32 (12) : 4307-4320. doi: 10.3934/dcds.2012.32.4307

 Impact Factor: 

Tools

Metrics

  • PDF downloads (74)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy