    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:
  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), 1024-1065. doi: 10.1137/0330055.  Google Scholar  M. Bellassoued, Decay of solutions of the wave equation with arbitrary localized nonlinear damping, J. Differential Equations, 211 (2005), 303-332. doi: 10.1016/j.jde.2004.12.010.  Google Scholar  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), 1-29. doi: 10.1007/BF02392877.  Google Scholar  N. Burq and M. Hitrik, Energy decay for damped wave equations on partially rectangular domains, Math. Res. Lett., 14 (2007), 35-47. Google Scholar  M. Daoulatli, Rate of decay of solutions of the wave equation with arbitrary localized nonlinear damping, Nonlinear Anal., 73 (2010), 987-1003. doi: 10.1016/j.na.2010.04.026.  Google Scholar  X. Fu, Logarithmic decay of hyperbolic equations with arbitrary small boundary damping, Comm. Partial Differential Equations, 34 (2009), 957-975. doi: 10.1080/03605300903116389.  Google Scholar  G. Lebeau, Équation des ondes amorties, (French) [Damped wave equation], in "Algebraic and Geometric Methods in Mathematical Physics" (Kaciveli, 1993), Math. Phys. Stud., 19, Kluwer Acad. Publ., Dordrecht, (1996), 73-109. Google Scholar  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), 335-356. doi: 10.1080/03605309508821097.  Google Scholar  G. Lebeau and L. Robbiano, Stabilisation de l'équation des ondes par le bord, Duke Math. J., 86 (1997), 465-491. doi: 10.1215/S0012-7094-97-08614-2.  Google Scholar  J.-L. Lions, "Quelques Méthodes de Résolution des Probl\emes aux Limites Non Linéaires," Dunod, Gauthier-Villars, Paris, 1969. Google Scholar  J.-L. Lions and W. Strauss, Some non-linear evolution equations, Bull. Soc. Math. France, 93 (1965), 43-96. Google Scholar  Z. Liu and B. Rao, Characterization of polynomial decay rate for the solution of linear evolution equation, Z. Angew. Math. Phys., 56 (2005), 630-644. doi: 10.1007/s00033-004-3073-4.  Google Scholar  M. Nakao, Decay of solutions of the wave equation with a local nonlinear dissipation, Math. Ann., 305 (1996), 403-417. doi: 10.1007/BF01444231.  Google Scholar  H. Nishiyama, Polynomial decay rate for damped wave equations on partially rectangular domains, Math. Res. Lett., 16 (2009), 881-894. Google Scholar  K. D. Phung, Polynomial decay rate for the dissipative wave equation, J. Differential Equations, 240 (2007), 92-124. doi: 10.1016/j.jde.2007.05.016.  Google Scholar  K. D. Phung, Boundary stabilization for the wave equation in a bounded cylindrical domain, Discrete Contin. Dyn. Syst., 20 (2008), 1057-1093. doi: 10.3934/dcds.2008.20.1057.  Google Scholar  L. Tcheugoué Tébou, Stabilization of the wave equation with localized nonlinear damping, J. Differential Equations, 145 (1998), 502-524. doi: 10.1006/jdeq.1998.3416.  Google Scholar

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##### References:
  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), 1024-1065. doi: 10.1137/0330055.  Google Scholar  M. Bellassoued, Decay of solutions of the wave equation with arbitrary localized nonlinear damping, J. Differential Equations, 211 (2005), 303-332. doi: 10.1016/j.jde.2004.12.010.  Google Scholar  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), 1-29. doi: 10.1007/BF02392877.  Google Scholar  N. Burq and M. Hitrik, Energy decay for damped wave equations on partially rectangular domains, Math. Res. Lett., 14 (2007), 35-47. Google Scholar  M. Daoulatli, Rate of decay of solutions of the wave equation with arbitrary localized nonlinear damping, Nonlinear Anal., 73 (2010), 987-1003. doi: 10.1016/j.na.2010.04.026.  Google Scholar  X. Fu, Logarithmic decay of hyperbolic equations with arbitrary small boundary damping, Comm. Partial Differential Equations, 34 (2009), 957-975. doi: 10.1080/03605300903116389.  Google Scholar  G. Lebeau, Équation des ondes amorties, (French) [Damped wave equation], in "Algebraic and Geometric Methods in Mathematical Physics" (Kaciveli, 1993), Math. Phys. Stud., 19, Kluwer Acad. Publ., Dordrecht, (1996), 73-109. Google Scholar  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), 335-356. doi: 10.1080/03605309508821097.  Google Scholar  G. Lebeau and L. Robbiano, Stabilisation de l'équation des ondes par le bord, Duke Math. J., 86 (1997), 465-491. doi: 10.1215/S0012-7094-97-08614-2.  Google Scholar  J.-L. Lions, "Quelques Méthodes de Résolution des Probl\emes aux Limites Non Linéaires," Dunod, Gauthier-Villars, Paris, 1969. Google Scholar  J.-L. Lions and W. Strauss, Some non-linear evolution equations, Bull. Soc. Math. France, 93 (1965), 43-96. Google Scholar  Z. Liu and B. Rao, Characterization of polynomial decay rate for the solution of linear evolution equation, Z. Angew. Math. Phys., 56 (2005), 630-644. doi: 10.1007/s00033-004-3073-4.  Google Scholar  M. Nakao, Decay of solutions of the wave equation with a local nonlinear dissipation, Math. Ann., 305 (1996), 403-417. doi: 10.1007/BF01444231.  Google Scholar  H. Nishiyama, Polynomial decay rate for damped wave equations on partially rectangular domains, Math. Res. Lett., 16 (2009), 881-894. Google Scholar  K. D. Phung, Polynomial decay rate for the dissipative wave equation, J. Differential Equations, 240 (2007), 92-124. doi: 10.1016/j.jde.2007.05.016.  Google Scholar  K. D. Phung, Boundary stabilization for the wave equation in a bounded cylindrical domain, Discrete Contin. Dyn. Syst., 20 (2008), 1057-1093. doi: 10.3934/dcds.2008.20.1057.  Google Scholar  L. Tcheugoué Tébou, Stabilization of the wave equation with localized nonlinear damping, J. Differential Equations, 145 (1998), 502-524. doi: 10.1006/jdeq.1998.3416.  Google Scholar
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