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

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

Takeshi Taniguchi

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. $
Keywords: Wave equation, exponential behavior of solutions, nonlinear boundary condition.
Mathematics Subject Classification: Primary: 35L05; Secondary: 35L20, 35B40.
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics Springer-Verlarg, Berlin, 1989. doi: 10.1007/978-1-4684-0313-8.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics Springer-Verlarg, Berlin, 1989. doi: 10.1007/978-1-4684-0313-8.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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