# American Institute of Mathematical Sciences

June  2011, 1(2): 251-265. doi: 10.3934/mcrf.2011.1.251

## Decay of solutions of the wave equation with localized nonlinear damping and trapped rays

 1 Yangtze Center of Mathematics, Sichuan University, Chengdu 610064

Received  November 2010 Revised  April 2011 Published  June 2011

We prove some decay estimates of the energy of the wave equation governed by localized nonlinear dissipations in a bounded domain in which trapped rays may occur. The approach is based on a comparison with the linear damped wave equation and an interpolation argument. Our result extends to the nonlinear damped wave equation the well-known optimal logarithmic decay rate for the linear damped wave equation with regular initial data.
Citation: 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
##### References:
 [1] C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary,, SIAM J. Control Optim., 30 (1992), 1024. doi: 10.1137/0330055. Google Scholar [2] M. Bellassoued, Decay of solutions of the wave equation with arbitrary localized nonlinear damping,, J. Differential Equations, 211 (2005), 303. doi: 10.1016/j.jde.2004.12.010. Google Scholar [3] N. Burq, Décroissance de l'énergie locale de l'équation des ondes pour le problème extérieur et abscence de résonnance au voisinage du réel, (French) [Decay of the local energy of the wave equation for the exterior problem and absence of resonance near the real axis],, Acta. Math., 180 (1998), 1. doi: 10.1007/BF02392877. Google Scholar [4] N. Burq and M. Hitrik, Energy decay for damped wave equations on partially rectangular domains,, Math. Res. Lett., 14 (2007), 35. Google Scholar [5] M. Daoulatli, Rate of decay of solutions of the wave equation with arbitrary localized nonlinear damping,, Nonlinear Anal., 73 (2010), 987. doi: 10.1016/j.na.2010.04.026. Google Scholar [6] X. Fu, Logarithmic decay of hyperbolic equations with arbitrary small boundary damping,, Comm. Partial Differential Equations, 34 (2009), 957. doi: 10.1080/03605300903116389. Google Scholar [7] G. Lebeau, Équation des ondes amorties, (French) [Damped wave equation],, in, 19 (1996), 73. Google Scholar [8] G. Lebeau and L. Robbiano, Contrôle exact de l'équation de la chaleur, (French) [Exact control of the heat equation],, Comm. Partial Differential Equations, 20 (1995), 335. doi: 10.1080/03605309508821097. Google Scholar [9] G. Lebeau and L. Robbiano, Stabilisation de l'équation des ondes par le bord,, Duke Math. J., 86 (1997), 465. doi: 10.1215/S0012-7094-97-08614-2. Google Scholar [10] J.-L. Lions, "Quelques Méthodes de Résolution des Probl\emes aux Limites Non Linéaires,", Dunod, (1969). Google Scholar [11] J.-L. Lions and W. Strauss, Some non-linear evolution equations,, Bull. Soc. Math. France, 93 (1965), 43. Google Scholar [12] Z. Liu and B. Rao, Characterization of polynomial decay rate for the solution of linear evolution equation,, Z. Angew. Math. Phys., 56 (2005), 630. doi: 10.1007/s00033-004-3073-4. Google Scholar [13] M. Nakao, Decay of solutions of the wave equation with a local nonlinear dissipation,, Math. Ann., 305 (1996), 403. doi: 10.1007/BF01444231. Google Scholar [14] H. Nishiyama, Polynomial decay rate for damped wave equations on partially rectangular domains,, Math. Res. Lett., 16 (2009), 881. Google Scholar [15] K. D. Phung, Polynomial decay rate for the dissipative wave equation,, J. Differential Equations, 240 (2007), 92. doi: 10.1016/j.jde.2007.05.016. Google Scholar [16] K. D. Phung, Boundary stabilization for the wave equation in a bounded cylindrical domain,, Discrete Contin. Dyn. Syst., 20 (2008), 1057. doi: 10.3934/dcds.2008.20.1057. Google Scholar [17] L. Tcheugoué Tébou, Stabilization of the wave equation with localized nonlinear damping,, J. Differential Equations, 145 (1998), 502. doi: 10.1006/jdeq.1998.3416. Google Scholar

show all references

##### References:
 [1] Mohammad A. Rammaha, Daniel Toundykov, Zahava Wilstein. Global existence and decay of energy for a nonlinear wave equation with $p$-Laplacian damping. Discrete & Continuous Dynamical Systems - A, 2012, 32 (12) : 4361-4390. doi: 10.3934/dcds.2012.32.4361 [2] 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 - A, 2008, 20 (3) : 459-509. doi: 10.3934/dcds.2008.20.459 [3] Aníbal Rodríguez-Bernal, Enrique Zuazua. Parabolic singular limit of a wave equation with localized boundary damping. Discrete & Continuous Dynamical Systems - A, 1995, 1 (3) : 303-346. doi: 10.3934/dcds.1995.1.303 [4] Takeshi Taniguchi. Exponential boundary stabilization for nonlinear wave equations with localized damping and nonlinear boundary condition. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1571-1585. doi: 10.3934/cpaa.2017075 [5] Moez Daoulatli. Energy decay rates for solutions of the wave equation with linear damping in exterior domain. Evolution Equations & Control Theory, 2016, 5 (1) : 37-59. doi: 10.3934/eect.2016.5.37 [6] Dalibor Pražák. On the dimension of the attractor for the wave equation with nonlinear damping. Communications on Pure & Applied Analysis, 2005, 4 (1) : 165-174. doi: 10.3934/cpaa.2005.4.165 [7] 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 [8] Serge Nicaise, Cristina Pignotti. Stability of the wave equation with localized Kelvin-Voigt damping and boundary delay feedback. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 791-813. doi: 10.3934/dcdss.2016029 [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] Tae Gab Ha. On viscoelastic wave equation with nonlinear boundary damping and source term. Communications on Pure & Applied Analysis, 2010, 9 (6) : 1543-1576. doi: 10.3934/cpaa.2010.9.1543 [11] Claudianor O. Alves, M. M. Cavalcanti, Valeria N. Domingos Cavalcanti, Mohammad A. Rammaha, Daniel Toundykov. On existence, uniform decay rates and blow up for solutions of systems of nonlinear wave equations with damping and source terms. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 583-608. doi: 10.3934/dcdss.2009.2.583 [12] Moez Daoulatli. Rates of decay for the wave systems with time dependent damping. Discrete & Continuous Dynamical Systems - A, 2011, 31 (2) : 407-443. doi: 10.3934/dcds.2011.31.407 [13] Nguyen Thanh Long, Hoang Hai Ha, Le Thi Phuong Ngoc, Nguyen Anh Triet. Existence, blow-up and exponential decay estimates for a system of nonlinear viscoelastic wave equations with nonlinear boundary conditions. Communications on Pure & Applied Analysis, 2020, 19 (1) : 455-492. doi: 10.3934/cpaa.2020023 [14] 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 - A, 2004, 11 (2&3) : 731-743. doi: 10.3934/dcds.2004.11.731 [15] Le Thi Phuong Ngoc, Nguyen Thanh Long. Existence and exponential decay for a nonlinear wave equation with nonlocal boundary conditions. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2001-2029. doi: 10.3934/cpaa.2013.12.2001 [16] Marcelo M. Cavalcanti, Valéria N. Domingos Cavalcanti, Irena Lasiecka, Flávio A. Falcão Nascimento. Intrinsic decay rate estimates for the wave equation with competing viscoelastic and frictional dissipative effects. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 1987-2011. doi: 10.3934/dcdsb.2014.19.1987 [17] Karen Yagdjian, Anahit Galstian. Fundamental solutions for wave equation in Robertson-Walker model of universe and $L^p-L^q$ -decay estimates. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 483-502. doi: 10.3934/dcdss.2009.2.483 [18] Belkacem Said-Houari, Salim A. Messaoudi. General decay estimates for a Cauchy viscoelastic wave problem. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1541-1551. doi: 10.3934/cpaa.2014.13.1541 [19] Brenton LeMesurier. Modeling thermal effects on nonlinear wave motion in biopolymers by a stochastic discrete nonlinear Schrödinger equation with phase damping. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 317-327. doi: 10.3934/dcdss.2008.1.317 [20] 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

2018 Impact Factor: 1.292