June  2011, 4(3): 615-622. doi: 10.3934/dcdss.2011.4.615

Stability of solutions for nonlinear wave equations with a positive--negative damping

1. 

Dipartimento di Ingegneria dell’Informazione, Università degli Studi di Siena, Via Roma 56, 53100 Siena

2. 

Dipartimento di Matematica e Informatica, Università di Perugia, Via Vanvitelli 1, 06123 Perugia, Italy

Received  March 2009 Revised  February 2010 Published  November 2010

We prove a stability result for damped nonlinear wave equations, when the damping changes sign and the nonlinear term satisfies a few natural assumptions.
Citation: Genni Fragnelli, Dimitri Mugnai. Stability of solutions for nonlinear wave equations with a positive--negative damping. Discrete and Continuous Dynamical Systems - S, 2011, 4 (3) : 615-622. doi: 10.3934/dcdss.2011.4.615
References:
[1]

A. Benaddi and B. Rao, Energy decay rate of wave equations with indefinite damping, J. Differential Equations, 161 (2000), 337-357.

[2]

C. W. de Silva, "Vibration and Shock Handbook," Mechanical Engineering, CRC Press, 2005. doi: doi:10.1201/9781420039894.

[3]

G. Fragnelli and D. Mugnai, Stability of solutions for some classes of nonlinear damped wave equations, SIAM J. Control Optim., 47 (2008), 2520-2539.

[4]

P. Freitas and E. Zuazua, Stability results for the wave equation with indefinite damping, J. Differential Equations, 132 (1996), 338-352.

[5]

A. Haraux, P. Martinez and J. Vancostenoble, Asymptotic stability for intermittently controlled second order evolution equations, SIAM J. Control and Opt., 43 (2005), 2089-2108.

[6]

L. Hatvani and T. Krisztin, Necessary and sufficient conditions for intermittent stabilization of linear oscillators by large damping, Differential Integral Equations, 10 (1997), 265-272.

[7]

S. Konabe and T. Nikuni, Coarse-grained finite-temperature theory for the bose condensate in optical lattices, J. Low Temp. Phys., 150 (2008), 12-46. doi: doi:10.1007/s10909-007-9517-4.

[8]

A. C. Lazer and P. J. McKenna, Large-amplitude periodic oscillations in suspension bridges: Some new connections with nonlinear analysis, SIAM Review, 32 (1990), 537-578.

[9]

H. A. Levine, S. R. Park and J. Serrin, Global existence and global nonexistence of solutions of the Cauchy problem for a nonlinearly damped wave equation, J. Math. Anal. Appl., 228 (1998), 181-205.

[10]

K. Liu, B. Rao and X. Zhang, Stabilization of the wave equations with potential and indefinite damping, J. Math. Anal. Appl., 269 (2002), 747-769.

[11]

A. Marino and D. Mugnai, Asymptotically critical points and their multiplicity, Topol. Methods Nonlinear Anal., 19 (2002), 29-38.

[12]

A. Marino and D. Mugnai, Asymptotical multiplicity and some reversed variational inequalities, Topol. Methods Nonlinear Anal., 20 (2002), 43-62.

[13]

P. Martinez and J. Vancostenoble, Stabilization of the wave equation by on-off and positive-negative feedbacks, ESAIM Control Optim. Calc. Var., 7 (2002), 335-377.

[14]

D. Mugnai, On a "reversed" variational inequality, Topol. Methods Nonlinear Anal., 17 (2001), 321-358.

[15]

P. Pucci and J. Serrin, Asymptotic stability for intermittently controlled nonlinear oscillators, SIAM J. Math. Anal., 25 (1994), 815-835.

[16]

P. Pucci and J. Serrin, Precise damping conditions for global asymptotic stability for nonlinear second order systems. II, J. Differential Equations, 113 (1994), 505-534.

[17]

G. Somieski, Shimmy analysis of a simple aircraft nose landing gear model using different mathematical methods, Aerosp. Sci. Technol., 1 (1997), 545-555.

show all references

References:
[1]

A. Benaddi and B. Rao, Energy decay rate of wave equations with indefinite damping, J. Differential Equations, 161 (2000), 337-357.

[2]

C. W. de Silva, "Vibration and Shock Handbook," Mechanical Engineering, CRC Press, 2005. doi: doi:10.1201/9781420039894.

[3]

G. Fragnelli and D. Mugnai, Stability of solutions for some classes of nonlinear damped wave equations, SIAM J. Control Optim., 47 (2008), 2520-2539.

[4]

P. Freitas and E. Zuazua, Stability results for the wave equation with indefinite damping, J. Differential Equations, 132 (1996), 338-352.

[5]

A. Haraux, P. Martinez and J. Vancostenoble, Asymptotic stability for intermittently controlled second order evolution equations, SIAM J. Control and Opt., 43 (2005), 2089-2108.

[6]

L. Hatvani and T. Krisztin, Necessary and sufficient conditions for intermittent stabilization of linear oscillators by large damping, Differential Integral Equations, 10 (1997), 265-272.

[7]

S. Konabe and T. Nikuni, Coarse-grained finite-temperature theory for the bose condensate in optical lattices, J. Low Temp. Phys., 150 (2008), 12-46. doi: doi:10.1007/s10909-007-9517-4.

[8]

A. C. Lazer and P. J. McKenna, Large-amplitude periodic oscillations in suspension bridges: Some new connections with nonlinear analysis, SIAM Review, 32 (1990), 537-578.

[9]

H. A. Levine, S. R. Park and J. Serrin, Global existence and global nonexistence of solutions of the Cauchy problem for a nonlinearly damped wave equation, J. Math. Anal. Appl., 228 (1998), 181-205.

[10]

K. Liu, B. Rao and X. Zhang, Stabilization of the wave equations with potential and indefinite damping, J. Math. Anal. Appl., 269 (2002), 747-769.

[11]

A. Marino and D. Mugnai, Asymptotically critical points and their multiplicity, Topol. Methods Nonlinear Anal., 19 (2002), 29-38.

[12]

A. Marino and D. Mugnai, Asymptotical multiplicity and some reversed variational inequalities, Topol. Methods Nonlinear Anal., 20 (2002), 43-62.

[13]

P. Martinez and J. Vancostenoble, Stabilization of the wave equation by on-off and positive-negative feedbacks, ESAIM Control Optim. Calc. Var., 7 (2002), 335-377.

[14]

D. Mugnai, On a "reversed" variational inequality, Topol. Methods Nonlinear Anal., 17 (2001), 321-358.

[15]

P. Pucci and J. Serrin, Asymptotic stability for intermittently controlled nonlinear oscillators, SIAM J. Math. Anal., 25 (1994), 815-835.

[16]

P. Pucci and J. Serrin, Precise damping conditions for global asymptotic stability for nonlinear second order systems. II, J. Differential Equations, 113 (1994), 505-534.

[17]

G. Somieski, Shimmy analysis of a simple aircraft nose landing gear model using different mathematical methods, Aerosp. Sci. Technol., 1 (1997), 545-555.

[1]

Lan Qiao, Sining Zheng. Non-simultaneous blow-up for heat equations with positive-negative sources and coupled boundary flux. Communications on Pure and Applied Analysis, 2007, 6 (4) : 1113-1129. doi: 10.3934/cpaa.2007.6.1113

[2]

A. Kh. Khanmamedov. Global attractors for strongly damped wave equations with displacement dependent damping and nonlinear source term of critical exponent. Discrete and Continuous Dynamical Systems, 2011, 31 (1) : 119-138. doi: 10.3934/dcds.2011.31.119

[3]

Jie Yang, Sen Ming, Wei Han, Xiongmei Fan. Lifespan estimates of solutions to quasilinear wave equations with damping and negative mass term. Mathematical Control and Related Fields, 2022  doi: 10.3934/mcrf.2022022

[4]

Hiroshi Takeda. Global existence of solutions for higher order nonlinear damped wave equations. Conference Publications, 2011, 2011 (Special) : 1358-1367. doi: 10.3934/proc.2011.2011.1358

[5]

Sandra Lucente. Global existence for equivalent nonlinear special scale invariant damped wave equations. Discrete and Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021159

[6]

Fengjuan Meng, Meihua Yang, Chengkui Zhong. Attractors for wave equations with nonlinear damping on time-dependent space. Discrete and Continuous Dynamical Systems - B, 2016, 21 (1) : 205-225. doi: 10.3934/dcdsb.2016.21.205

[7]

A. Kh. Khanmamedov. Long-time behaviour of wave equations with nonlinear interior damping. Discrete and Continuous Dynamical Systems, 2008, 21 (4) : 1185-1198. doi: 10.3934/dcds.2008.21.1185

[8]

Abdullah Özbekler, A. Zafer. Second order oscillation of mixed nonlinear dynamic equations with several positive and negative coefficients. Conference Publications, 2011, 2011 (Special) : 1167-1175. doi: 10.3934/proc.2011.2011.1167

[9]

Salim A. Messaoudi, Ala A. Talahmeh. Blow up of negative initial-energy solutions of a system of nonlinear wave equations with variable-exponent nonlinearities. Discrete and Continuous Dynamical Systems - S, 2022, 15 (5) : 1233-1245. doi: 10.3934/dcdss.2021107

[10]

Takeshi Taniguchi. Exponential boundary stabilization for nonlinear wave equations with localized damping and nonlinear boundary condition. Communications on Pure and Applied Analysis, 2017, 16 (5) : 1571-1585. doi: 10.3934/cpaa.2017075

[11]

Yoshihiro Ueda, Tohru Nakamura, Shuichi Kawashima. Stability of planar stationary waves for damped wave equations with nonlinear convection in multi-dimensional half space. Kinetic and Related Models, 2008, 1 (1) : 49-64. doi: 10.3934/krm.2008.1.49

[12]

Varga K. Kalantarov, Edriss S. Titi. Global stabilization of the Navier-Stokes-Voight and the damped nonlinear wave equations by finite number of feedback controllers. Discrete and Continuous Dynamical Systems - B, 2018, 23 (3) : 1325-1345. doi: 10.3934/dcdsb.2018153

[13]

Björn Birnir, Kenneth Nelson. The existence of smooth attractors of damped and driven nonlinear wave equations with critical exponent , s = 5. Conference Publications, 1998, 1998 (Special) : 100-117. doi: 10.3934/proc.1998.1998.100

[14]

Zhiqing Liu, Zhong Bo Fang. Global solvability and general decay of a transmission problem for kirchhoff-type wave equations with nonlinear damping and delay term. Communications on Pure and Applied Analysis, 2020, 19 (2) : 941-966. doi: 10.3934/cpaa.2020043

[15]

Claudianor O. Alves, M. M. Cavalcanti, Valeria N. Domingos Cavalcanti, Mohammad A. Rammaha, Daniel Toundykov. On existence, uniform decay rates and blow up for solutions of systems of nonlinear wave equations with damping and source terms. Discrete and Continuous Dynamical Systems - S, 2009, 2 (3) : 583-608. doi: 10.3934/dcdss.2009.2.583

[16]

Jingyu Wang, Yejuan Wang, Lin Yang, Tomás Caraballo. Random attractors for stochastic delay wave equations on $ \mathbb{R}^n $ with linear memory and nonlinear damping. Discrete and Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021141

[17]

Chao Yang, Yanbing Yang. Long-time behavior for fourth-order wave equations with strain term and nonlinear weak damping term. Discrete and Continuous Dynamical Systems - S, 2021, 14 (12) : 4643-4658. doi: 10.3934/dcdss.2021110

[18]

Vando Narciso. On a Kirchhoff wave model with nonlocal nonlinear damping. Evolution Equations and Control Theory, 2020, 9 (2) : 487-508. doi: 10.3934/eect.2020021

[19]

Dalibor Pražák. On the dimension of the attractor for the wave equation with nonlinear damping. Communications on Pure and Applied Analysis, 2005, 4 (1) : 165-174. doi: 10.3934/cpaa.2005.4.165

[20]

John M. Ball. Global attractors for damped semilinear wave equations. Discrete and Continuous Dynamical Systems, 2004, 10 (1&2) : 31-52. doi: 10.3934/dcds.2004.10.31

2020 Impact Factor: 2.425

Metrics

  • PDF downloads (113)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]