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2017, 16(5): 1571-1585. doi: 10.3934/cpaa.2017075

## Exponential boundary stabilization for nonlinear wave equations with localized damping and nonlinear boundary condition

 Division of Mathematical Sciences, Graduate School of Comparative Culture, Kurume University, Miimachi, Kurume, Fukuoka 839-8502, Japan

Received  September 2015 Revised  October 2015 Published  May 2017

Fund Project: To Shiho and Sarasa from Grandpapa. The author is partially supported by the Grant-in-Aid for Scientific Research (No.24540198) from Japan Society for the Promotion of Science

Let
 $D\subset R^{d}$
be a bounded domain in the
 $d-$
dimensional Euclidian space
 $R^{d}$
with smooth boundary $Γ=\partial D.$ In this paper we consider exponential boundary stabilization for weak solutions to the wave equation with nonlinear boundary condition:
 $\left\{ \begin{gathered}u_{tt}(t)-ρ(t)Δ u(t)+b(x)u_{t}(t)=f(u(t)), \\ u(t)=0\ \ \text{on }Γ_{0}×(0,T), \\ \dfrac{\partial u(t)}{\partialν}+γ(u_{t}(t))=0\ \ \text{on }Γ _{1}×(0,T), \\ u(0)=u_{0},u_{t}(0)=u_{1},\end{gathered} \right.$
where
 $\left\| {{u_0}} \right\| < {\lambda _\beta },$
 $E(0) < d_{β},$
where
 $λ_{β},$
 $d_{β}$
are defined in (21), (22) and
 $Γ=Γ_{0}\cupΓ_{1}$
and
 $\bar{Γ}_{0}\cap\bar{Γ}_{1}=φ.$
Citation: 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
##### References:
 [1] F. D. Araruna and A. B. Maciel, Existence and boundary stabilization of the semilinear wave equation, Nonlinear Analysis, 67 (2007), 1288-1305. doi: 10.1016/j.na.2006.07.015. [2] M. M. Cavalcanti, V. N. D. Cavalcanti and P. Martinez, Existence and decay rate estimates for the wave equation with nonlinear boundary damping and source term, J. Differential Equations, 203 (2004), 119-158. doi: 10.1016/j.jde.2004.04.011. [3] M. M. Cavalcanti and V. N. Domingos Cavalcanti, Existence and asymptotic stability for evolution problems on manifolds with damping and source terms, J. Math. Anal. Appli., 291 (2004), 109-127. doi: 10.1016/j.jmaa.2003.10.020. [4] M. M. Cavalcanti, V. N. Domingos Cavalcanti and I. Lasiecka, Well posedness and optimal decay rates for the wave equation with nonlinear damping-source interaction, J. Differential Equations, 236 (2007), 407-459. [5] M. M. Cavalcanti, V. N. D. Cavalcanti and J. A. Soriano, On existence and asymptotic stability of solutions of the degenerate wave equation with nonlinear boundary conditions, J. Math. Anal. Appli., 281 (2003), 108-124. [6] V. Georgiev and G. Todorova, Existence of a solution of the wave equation with damping and source terms, J. Differential Equations, 109 (1994), 295-308. doi: 10.1006/jdeq.1994.1051. [7] S. Gerbi and B. Said-Houari, Local existence and exponential growth for a semilinear damping wave equation with dynamic boundary conditions, Ad. Diff. Equ., 13 (2008), 1051-1074. [8] B. Guo and Z-C. Shao, On exponential stability of a semilinear wave equation with variable coefficients under the nonlinear boundary feedback, Nonlinear Analysis, 71 (2009), 5961-5978. doi: 10.1016/j.na.2009.05.018. [9] V. Komornik and E. Zuazua, A direct method for boundary stabilization of the wave equation, J. Math. Pures et appl., 69 (1990), 33-54. [10] A. T. Louredo, M. A. Ferreira and M. M. Miranda, On a nonlinear wave equation with boundary damping, Math. Meth. in Applied Sciences, 37 (2014), 1278-1302. [11] J. Malek, J. Necas, M. Rokyta and M. Ruzicka, Weak and Measure-valued Solutions to Evolutionary PDEs Chapman and Hall, 1996. doi: 10.1007/978-1-4899-6824-1. [12] S. A. Messaoudi, Blow up in a nonlinearly damped wave equation, Math Nachr, 231 (2001), 105-111. doi: 10.1002/1522-2616(200111)231:1<105::AID-MANA105>3.3.CO;2-9. [13] K. Ono, Asymptotic stability and blowing up of solutions for some degenerate non-linear wave equations of Kirchhoff type with a strong dissipation, Math. Meth. in Applied Sciences, 20 (1997), 151-177. doi: 10.1002/(SICI)1099-1476(19970125)20:2<151::AID-MMA851>3.3.CO;2-S. [14] R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics Springer-Verlarg, Berlin, 1989. doi: 10.1007/978-1-4684-0313-8. [15] E. Vitillaro, A potential well method for the wave equation with nonlinear source and boundary damping terms, Glasgow Math. J, 44 (2002), 375-395. doi: 10.1017/S0017089502030045. [16] E. Vitillaro, Global existence for the wave equation with nonlinear boundary damping and source terms, J. Differential Equations, 186 (2002), 259-298. doi: 10.1016/S0022-0396(02)00023-2. [17] Zai-yun Zhang and Xiu-jin Miao, Global existence and uniform decay for wave equation with dissipative term and boundary damping, Computers and Math. Appli., 59 (2010), 1003-1018. doi: 10.1016/j.camwa.2009.09.008.

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##### References:
 [1] F. D. Araruna and A. B. Maciel, Existence and boundary stabilization of the semilinear wave equation, Nonlinear Analysis, 67 (2007), 1288-1305. doi: 10.1016/j.na.2006.07.015. [2] M. M. Cavalcanti, V. N. D. Cavalcanti and P. Martinez, Existence and decay rate estimates for the wave equation with nonlinear boundary damping and source term, J. Differential Equations, 203 (2004), 119-158. doi: 10.1016/j.jde.2004.04.011. [3] M. M. Cavalcanti and V. N. Domingos Cavalcanti, Existence and asymptotic stability for evolution problems on manifolds with damping and source terms, J. Math. Anal. Appli., 291 (2004), 109-127. doi: 10.1016/j.jmaa.2003.10.020. [4] M. M. Cavalcanti, V. N. Domingos Cavalcanti and I. Lasiecka, Well posedness and optimal decay rates for the wave equation with nonlinear damping-source interaction, J. Differential Equations, 236 (2007), 407-459. [5] M. M. Cavalcanti, V. N. D. Cavalcanti and J. A. Soriano, On existence and asymptotic stability of solutions of the degenerate wave equation with nonlinear boundary conditions, J. Math. Anal. Appli., 281 (2003), 108-124. [6] V. Georgiev and G. Todorova, Existence of a solution of the wave equation with damping and source terms, J. Differential Equations, 109 (1994), 295-308. doi: 10.1006/jdeq.1994.1051. [7] S. Gerbi and B. Said-Houari, Local existence and exponential growth for a semilinear damping wave equation with dynamic boundary conditions, Ad. Diff. Equ., 13 (2008), 1051-1074. [8] B. Guo and Z-C. Shao, On exponential stability of a semilinear wave equation with variable coefficients under the nonlinear boundary feedback, Nonlinear Analysis, 71 (2009), 5961-5978. doi: 10.1016/j.na.2009.05.018. [9] V. Komornik and E. Zuazua, A direct method for boundary stabilization of the wave equation, J. Math. Pures et appl., 69 (1990), 33-54. [10] A. T. Louredo, M. A. Ferreira and M. M. Miranda, On a nonlinear wave equation with boundary damping, Math. Meth. in Applied Sciences, 37 (2014), 1278-1302. [11] J. Malek, J. Necas, M. Rokyta and M. Ruzicka, Weak and Measure-valued Solutions to Evolutionary PDEs Chapman and Hall, 1996. doi: 10.1007/978-1-4899-6824-1. [12] S. A. Messaoudi, Blow up in a nonlinearly damped wave equation, Math Nachr, 231 (2001), 105-111. doi: 10.1002/1522-2616(200111)231:1<105::AID-MANA105>3.3.CO;2-9. [13] K. Ono, Asymptotic stability and blowing up of solutions for some degenerate non-linear wave equations of Kirchhoff type with a strong dissipation, Math. Meth. in Applied Sciences, 20 (1997), 151-177. doi: 10.1002/(SICI)1099-1476(19970125)20:2<151::AID-MMA851>3.3.CO;2-S. [14] R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics Springer-Verlarg, Berlin, 1989. doi: 10.1007/978-1-4684-0313-8. [15] E. Vitillaro, A potential well method for the wave equation with nonlinear source and boundary damping terms, Glasgow Math. J, 44 (2002), 375-395. doi: 10.1017/S0017089502030045. [16] E. Vitillaro, Global existence for the wave equation with nonlinear boundary damping and source terms, J. Differential Equations, 186 (2002), 259-298. doi: 10.1016/S0022-0396(02)00023-2. [17] Zai-yun Zhang and Xiu-jin Miao, Global existence and uniform decay for wave equation with dissipative term and boundary damping, Computers and Math. Appli., 59 (2010), 1003-1018. doi: 10.1016/j.camwa.2009.09.008.
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