Stability of solutions fornonlinear wave equations with a positive--negative damping

Genni Fragnelli; Dimitri Mugnai

doi:10.3934/dcdss.2011.4.615

Article Contents

2011, Volume 4, Issue 3: 615-622. Doi: 10.3934/dcdss.2011.4.615

This issue Previous Article Regular boundary value problems for ordinary differential-operator equations of higher order in UMD Banach spaces Next Article Kolmogorov equations perturbed by an inverse-square potential

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

Genni Fragnelli^1, and
Dimitri Mugnai^2,

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

Received: February 28, 2009

Revised: January 31, 2010

Published: May 2011

Abstract Related Papers Cited by

Abstract

We prove a stability result for damped nonlinear wave equations, when the damping changes sign and the nonlinear term satisfies a few natural assumptions.

Keywords:

Damped nonlinear wave equations,
positive-negative damping.

Mathematics Subject Classification: Primary:35L70; Secondary: 93D20, 25B35.

Citation:

Related Papers

Cited by

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.