American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Multiple positive solutions for Schrödinger-Poisson system in $\mathbb{R}^{3}$ involving concave-convex nonlinearities with critical exponent
  • CPAA Home
  • This Issue
  • Next Article
    On uniform estimate of complex elliptic equations on closed Hermitian manifolds
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

[1]

Hongwei Zhang, Qingying Hu. Asymptotic behavior and nonexistence of wave equation with nonlinear boundary condition. Communications on Pure & Applied Analysis, 2005, 4 (4) : 861-869. doi: 10.3934/cpaa.2005.4.861
[2]

Jong-Shenq Guo. Blow-up behavior for a quasilinear parabolic equation with nonlinear boundary condition. Discrete & Continuous Dynamical Systems - A, 2007, 18 (1) : 71-84. doi: 10.3934/dcds.2007.18.71
[3]

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
[4]

Hiroshi Takeda. Large time behavior of solutions for a nonlinear damped wave equation. Communications on Pure & Applied Analysis, 2016, 15 (1) : 41-55. doi: 10.3934/cpaa.2016.15.41
[5]

Haiyang He. Asymptotic behavior of the ground state Solutions for Hénon equation with Robin boundary condition. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2393-2408. doi: 10.3934/cpaa.2013.12.2393
[6]

Stéphane Gerbi, Belkacem Said-Houari. Exponential decay for solutions to semilinear damped wave equation. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 559-566. doi: 10.3934/dcdss.2012.5.559
[7]

Tsung-Fang Wu. Multiplicity of positive solutions for a semilinear elliptic equation in $R_+^N$ with nonlinear boundary condition. Communications on Pure & Applied Analysis, 2010, 9 (6) : 1675-1696. doi: 10.3934/cpaa.2010.9.1675
[8]

Cong He, Hongjun Yu. Large time behavior of the solution to the Landau Equation with specular reflective boundary condition. Kinetic & Related Models, 2013, 6 (3) : 601-623. doi: 10.3934/krm.2013.6.601
[9]

Vyacheslav A. Trofimov, Evgeny M. Trykin. A new way for decreasing of amplitude of wave reflected from artificial boundary condition for 1D nonlinear Schrödinger equation. Conference Publications, 2015, 2015 (special) : 1070-1078. doi: 10.3934/proc.2015.1070
[10]

Linglong Du, Caixuan Ren. Pointwise wave behavior of the initial-boundary value problem for the nonlinear damped wave equation in $\mathbb{R}_{+}^{n} $. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3265-3280. doi: 10.3934/dcdsb.2018319
[11]

Muhammad I. Mustafa. On the control of the wave equation by memory-type boundary condition. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 1179-1192. doi: 10.3934/dcds.2015.35.1179
[12]

Alain Hertzog, Antoine Mondoloni. Existence of a weak solution for a quasilinear wave equation with boundary condition. Communications on Pure & Applied Analysis, 2002, 1 (2) : 191-219. doi: 10.3934/cpaa.2002.1.191
[13]

Serge Nicaise, Cristina Pignotti, Julie Valein. Exponential stability of the wave equation with boundary time-varying delay. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 693-722. doi: 10.3934/dcdss.2011.4.693
[14]

Peter V. Gordon, Cyrill B. Muratov. Self-similarity and long-time behavior of solutions of the diffusion equation with nonlinear absorption and a boundary source. Networks & Heterogeneous Media, 2012, 7 (4) : 767-780. doi: 10.3934/nhm.2012.7.767
[15]

Gen Nakamura, Michiyuki Watanabe. An inverse boundary value problem for a nonlinear wave equation. Inverse Problems & Imaging, 2008, 2 (1) : 121-131. doi: 10.3934/ipi.2008.2.121
[16]

Davit Martirosyan. Exponential mixing for the white-forced damped nonlinear wave equation. Evolution Equations & Control Theory, 2014, 3 (4) : 645-670. doi: 10.3934/eect.2014.3.645
[17]

Marek Fila, Kazuhiro Ishige, Tatsuki Kawakami. Convergence to the Poisson kernel for the Laplace equation with a nonlinear dynamical boundary condition. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1285-1301. doi: 10.3934/cpaa.2012.11.1285
[18]

Kazuhiro Ishige, Ryuichi Sato. Heat equation with a nonlinear boundary condition and uniformly local $L^r$ spaces. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2627-2652. doi: 10.3934/dcds.2016.36.2627
[19]

Makoto Nakamura. Remarks on global solutions of dissipative wave equations with exponential nonlinear terms. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1533-1545. doi: 10.3934/cpaa.2015.14.1533
[20]

Zuodong Yang, Jing Mo, Subei Li. Positive solutions of $p$-Laplacian equations with nonlinear boundary condition. Discrete & Continuous Dynamical Systems - B, 2011, 16 (2) : 623-636. doi: 10.3934/dcdsb.2011.16.623

2018 Impact Factor: 0.925

Tools

Metrics

  • PDF downloads (6)
  • HTML views (18)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences