American Institute of Mathematical Sciences

Advanced
June  2012, 5(3): 559-566. doi: 10.3934/dcdss.2012.5.559

Exponential decay for solutions to semilinear damped wave equation

Stéphane Gerbi 1, and  Belkacem Said-Houari 2,

1. 

Laboratoire de Mathématiques, Université de Savoie, 73376 Le Bourget du Lac, France
2. 

Division of Mathematical and Computer Sciences and Engineering, King Abdullah University of Science and Technology (KAUST), Thuwal 23955-6900, Saudi Arabia

Received  March 2010 Revised  May 2010 Published  October 2011

This paper is concerned with decay estimate of solutions to the semilinear wave equation with strong damping in a bounded domain. Introducing an appropriate Lyapunov function, we prove that when the damping is linear, we can find initial data, for which the solution decays exponentially. This result improves an early one in [4].
Keywords: stable set, decay rate, Strong damping, positive initial energy., global existence.
Mathematics Subject Classification: Primary: 35L70, 35B4.
Citation: 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
References:
[1]

J. Ball, Remarks on blow up and nonexistence theorems for nonlinear evolutions equations,, Quart. J. Math. Oxford. Ser. (2), 28 (1977), 473.   Google Scholar

[2]

A. Benaissa and S. Messaoudi, Exponential decay of solutions of a nonlinearly damped wave equation,, NoDEA Nonlinear Differential Equations Appl., 12 (2005), 391.  doi: 10.1007/s00030-005-0008-5.  Google Scholar

[3]

J. Esquivel-Avila, Qualitative analysis of a nonlinear wave equation,, Discrete. Contin. Dyn. Syst., 10 (2004), 787.  doi: 10.3934/dcds.2004.10.787.  Google Scholar

[4]

F. Gazzola and M. Squassina, Global solutions and finite time blow up for damped semilinear wave equations,, Ann. I. H. Poincaré, 23 (2006), 185.   Google Scholar

[5]

V. Georgiev and G. Todorova, Existence of a solution of the wave equation with nonlinear damping and source terms,, J. Differential Equations, 109 (1994), 295.  doi: 10.1006/jdeq.1994.1051.  Google Scholar

[6]

A. Haraux and E. Zuazua, Decay estimates for some semilinear damped hyperbolic problems,, Arch. Rational Mech. Anal., 100 (1988), 191.  doi: 10.1007/BF00282203.  Google Scholar

[7]

R. Ikehata, Some remarks on the wave equations with nonlinear damping and source terms,, Nonlinear. Anal., 27 (1996), 1165.  doi: 10.1016/0362-546X(95)00119-G.  Google Scholar

[8]

R. Ikehata and T. Suzuki, Stable and unstable sets for evolution equations of parabolic and hyperbolic type,, Hiroshima Math. J., 26 (1996), 475.   Google Scholar

[9]

J. Esquivel-Avila, The dynamics of nonlinear wave equation,, J. Math. Anal. Appl., 279 (2003), 135.  doi: 10.1016/S0022-247X(02)00701-1.  Google Scholar

[10]

V. K. Kalantarov and O. A. Ladyzhenskaya, The occurence of collapse for quasilinear equations of parabolic and hyperbolic type,, J. Soviet. Math., 10 (1978), 53.  doi: 10.1007/BF01109723.  Google Scholar

[11]

M. Kopáčkova, Remarks on bounded solutions of a semilinear dissipative hyperbolic equation,, Comment. Math. Univ. Carolin., 30 (1989), 713.   Google Scholar

[12]

H. A. Levine, Instability and nonexistence of global solutions to nonlinear wave equations of the form $Pu_{\mathcalt\mathcal t}=-Au+\mathcal F(u)$,, Trans. Amer. Math. Soc., 192 (1974), 1.  doi: 10.2307/1996814.  Google Scholar

[13]

H. A. Levine, Some additional remarks on the nonexistence of global solutions to nonlinear wave equations,, SIAM J. Math. Anal., 5 (1974), 138.  doi: 10.1137/0505015.  Google Scholar

[14]

S. Messaoudi and B. Said Houari, Global non-existence of solutions of a class of wave equations with non-linear damping and source terms,, Math. Methods Appl. Sci., 27 (2004), 1687.  doi: 10.1002/mma.522.  Google Scholar

[15]

K. Ono, On global existence, asymptotic stability and blowing up of solutions for some degenerate non-linear wave equations of Kirchhoff type with a strong dissipation,, Math. Methods Appl. Sci., 20 (1997), 151.  doi: 10.1002/(SICI)1099-1476(19970125)20:2<151::AID-MMA851>3.3.CO;2-S.  Google Scholar

[16]

L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations,, Israel J. Math., 22 (1975), 273.  doi: 10.1007/BF02761595.  Google Scholar

[17]

G. Todorova, Cauchy problem for a nonlinear wave equation with nonlinear damping and source terms,, C. R. Acad Sci. Paris Ser., 326 (1998), 191.   Google Scholar

[18]

G. Todorova, Stable and unstable sets for the Cauchy problem for a nonlinear wave with nonlinear damping and source terms,, J. Math. Anal. Appl., 239 (1999), 213.  doi: 10.1006/jmaa.1999.6528.  Google Scholar

[19]

E. Vitillaro, Global nonexistence theorems for a class of evolution equations with dissipation,, Arch. Ration. Mech. Anal., 149 (1999), 155.  doi: 10.1007/s002050050171.  Google Scholar

[20]

Z. Yang, Existence and asymptotic behavior of solutions for a class of quasi-linear evolution equations with non-linear damping and source terms,, Math. Meth. Appl. Sci., 25 (2002), 795.  doi: 10.1002/mma.306.  Google Scholar

[21]

E. Zuazua, Exponential decay for the semilinear wave equation with locally distributed damping,, Comm. Partial. Diff. Eq., 15 (1990), 205.   Google Scholar

show all references

References:
[1]

J. Ball, Remarks on blow up and nonexistence theorems for nonlinear evolutions equations,, Quart. J. Math. Oxford. Ser. (2), 28 (1977), 473.   Google Scholar

[2]

A. Benaissa and S. Messaoudi, Exponential decay of solutions of a nonlinearly damped wave equation,, NoDEA Nonlinear Differential Equations Appl., 12 (2005), 391.  doi: 10.1007/s00030-005-0008-5.  Google Scholar

[3]

J. Esquivel-Avila, Qualitative analysis of a nonlinear wave equation,, Discrete. Contin. Dyn. Syst., 10 (2004), 787.  doi: 10.3934/dcds.2004.10.787.  Google Scholar

[4]

F. Gazzola and M. Squassina, Global solutions and finite time blow up for damped semilinear wave equations,, Ann. I. H. Poincaré, 23 (2006), 185.   Google Scholar

[5]

V. Georgiev and G. Todorova, Existence of a solution of the wave equation with nonlinear damping and source terms,, J. Differential Equations, 109 (1994), 295.  doi: 10.1006/jdeq.1994.1051.  Google Scholar

[6]

A. Haraux and E. Zuazua, Decay estimates for some semilinear damped hyperbolic problems,, Arch. Rational Mech. Anal., 100 (1988), 191.  doi: 10.1007/BF00282203.  Google Scholar

[7]

R. Ikehata, Some remarks on the wave equations with nonlinear damping and source terms,, Nonlinear. Anal., 27 (1996), 1165.  doi: 10.1016/0362-546X(95)00119-G.  Google Scholar

[8]

R. Ikehata and T. Suzuki, Stable and unstable sets for evolution equations of parabolic and hyperbolic type,, Hiroshima Math. J., 26 (1996), 475.   Google Scholar

[9]

J. Esquivel-Avila, The dynamics of nonlinear wave equation,, J. Math. Anal. Appl., 279 (2003), 135.  doi: 10.1016/S0022-247X(02)00701-1.  Google Scholar

[10]

V. K. Kalantarov and O. A. Ladyzhenskaya, The occurence of collapse for quasilinear equations of parabolic and hyperbolic type,, J. Soviet. Math., 10 (1978), 53.  doi: 10.1007/BF01109723.  Google Scholar

[11]

M. Kopáčkova, Remarks on bounded solutions of a semilinear dissipative hyperbolic equation,, Comment. Math. Univ. Carolin., 30 (1989), 713.   Google Scholar

[12]

H. A. Levine, Instability and nonexistence of global solutions to nonlinear wave equations of the form $Pu_{\mathcalt\mathcal t}=-Au+\mathcal F(u)$,, Trans. Amer. Math. Soc., 192 (1974), 1.  doi: 10.2307/1996814.  Google Scholar

[13]

H. A. Levine, Some additional remarks on the nonexistence of global solutions to nonlinear wave equations,, SIAM J. Math. Anal., 5 (1974), 138.  doi: 10.1137/0505015.  Google Scholar

[14]

S. Messaoudi and B. Said Houari, Global non-existence of solutions of a class of wave equations with non-linear damping and source terms,, Math. Methods Appl. Sci., 27 (2004), 1687.  doi: 10.1002/mma.522.  Google Scholar

[15]

K. Ono, On global existence, asymptotic stability and blowing up of solutions for some degenerate non-linear wave equations of Kirchhoff type with a strong dissipation,, Math. Methods Appl. Sci., 20 (1997), 151.  doi: 10.1002/(SICI)1099-1476(19970125)20:2<151::AID-MMA851>3.3.CO;2-S.  Google Scholar

[16]

L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations,, Israel J. Math., 22 (1975), 273.  doi: 10.1007/BF02761595.  Google Scholar

[17]

G. Todorova, Cauchy problem for a nonlinear wave equation with nonlinear damping and source terms,, C. R. Acad Sci. Paris Ser., 326 (1998), 191.   Google Scholar

[18]

G. Todorova, Stable and unstable sets for the Cauchy problem for a nonlinear wave with nonlinear damping and source terms,, J. Math. Anal. Appl., 239 (1999), 213.  doi: 10.1006/jmaa.1999.6528.  Google Scholar

[19]

E. Vitillaro, Global nonexistence theorems for a class of evolution equations with dissipation,, Arch. Ration. Mech. Anal., 149 (1999), 155.  doi: 10.1007/s002050050171.  Google Scholar

[20]

Z. Yang, Existence and asymptotic behavior of solutions for a class of quasi-linear evolution equations with non-linear damping and source terms,, Math. Meth. Appl. Sci., 25 (2002), 795.  doi: 10.1002/mma.306.  Google Scholar

[21]

E. Zuazua, Exponential decay for the semilinear wave equation with locally distributed damping,, Comm. Partial. Diff. Eq., 15 (1990), 205.   Google Scholar

[1]

Guangyu Xu, Chunlai Mu, Dan Li. Global existence and non-existence analyses to a nonlinear Klein-Gordon system with damping terms under positive initial energy. Communications on Pure & Applied Analysis, 2020, 19 (5) : 2491-2512. doi: 10.3934/cpaa.2020109
[2]

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
[3]

Bilgesu A. Bilgin, Varga K. Kalantarov. Non-existence of global solutions to nonlinear wave equations with positive initial energy. Communications on Pure & Applied Analysis, 2018, 17 (3) : 987-999. doi: 10.3934/cpaa.2018048
[4]

Kai Liu, Zhi Li. Global attracting set, exponential decay and stability in distribution of neutral SPDEs driven by additive $\alpha$-stable processes. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3551-3573. doi: 10.3934/dcdsb.2016110
[5]

Denis Mercier, Virginie Régnier. Decay rate of the Timoshenko system with one boundary damping. Evolution Equations & Control Theory, 2019, 8 (2) : 423-445. doi: 10.3934/eect.2019021
[6]

M. A. Efendiev. On the compactness of the stable set for rate independent processes. Communications on Pure & Applied Analysis, 2003, 2 (4) : 495-509. doi: 10.3934/cpaa.2003.2.495
[7]

Honglv Ma, Jin Zhang, Chengkui Zhong. Global existence and asymptotic behavior of global smooth solutions to the Kirchhoff equations with strong nonlinear damping. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 4721-4737. doi: 10.3934/dcdsb.2019027
[8]

Mohammed Aassila. On energy decay rate for linear damped systems. Discrete & Continuous Dynamical Systems - A, 2002, 8 (4) : 851-864. doi: 10.3934/dcds.2002.8.851
[9]

Bopeng Rao. Optimal energy decay rate in a damped Rayleigh beam. Discrete & Continuous Dynamical Systems - A, 1998, 4 (4) : 721-734. doi: 10.3934/dcds.1998.4.721
[10]

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
[11]

Vanessa Barros, Carlos Nonato, Carlos Raposo. Global existence and energy decay of solutions for a wave equation with non-constant delay and nonlinear weights. Electronic Research Archive, 2020, 28 (1) : 205-220. doi: 10.3934/era.2020014
[12]

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
[13]

Ryo Ikehata, Shingo Kitazaki. Optimal energy decay rates for some wave equations with double damping terms. Evolution Equations & Control Theory, 2019, 8 (4) : 825-846. doi: 10.3934/eect.2019040
[14]

Yang Liu. Global existence and exponential decay of strong solutions to the cauchy problem of 3D density-dependent Navier-Stokes equations with vacuum. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020163
[15]

Zhuangyi Liu, Ramón Quintanilla. Energy decay rate of a mixed type II and type III thermoelastic system. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1433-1444. doi: 10.3934/dcdsb.2010.14.1433
[16]

Guangyu Xu, Jun Zhou. Global existence and blow-up of solutions to a singular Non-Newton polytropic filtration equation with critical and supercritical initial energy. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1805-1820. doi: 10.3934/cpaa.2018086
[17]

Jun Zhou. Global existence and energy decay estimate of solutions for a class of nonlinear higher-order wave equation with general nonlinear dissipation and source term. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1175-1185. doi: 10.3934/dcdss.2017064
[18]

Daniela Giachetti, Maria Michaela Porzio. Global existence for nonlinear parabolic equations with a damping term. Communications on Pure & Applied Analysis, 2009, 8 (3) : 923-953. doi: 10.3934/cpaa.2009.8.923
[19]

Yongming Liu, Lei Yao. Global solution and decay rate for a reduced gravity two and a half layer model. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2613-2638. doi: 10.3934/dcdsb.2018267
[20]

Yanxia Niu, Yinxia Wang, Qingnian Zhang. Decay rate of global solutions to three dimensional generalized MHD system. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020064

2019 Impact Factor: 1.233

Tools

Metrics

  • PDF downloads (52)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences