American Institute of Mathematical Sciences

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 & 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.  Google Scholar [2] C. W. de Silva, "Vibration and Shock Handbook," Mechanical Engineering, CRC Press, 2005. doi: doi:10.1201/9781420039894.  Google Scholar [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.  Google Scholar [4] P. Freitas and E. Zuazua, Stability results for the wave equation with indefinite damping, J. Differential Equations, 132 (1996), 338-352.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [11] A. Marino and D. Mugnai, Asymptotically critical points and their multiplicity, Topol. Methods Nonlinear Anal., 19 (2002), 29-38.  Google Scholar [12] A. Marino and D. Mugnai, Asymptotical multiplicity and some reversed variational inequalities, Topol. Methods Nonlinear Anal., 20 (2002), 43-62.  Google Scholar [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.  Google Scholar [14] D. Mugnai, On a "reversed" variational inequality, Topol. Methods Nonlinear Anal., 17 (2001), 321-358.  Google Scholar [15] P. Pucci and J. Serrin, Asymptotic stability for intermittently controlled nonlinear oscillators, SIAM J. Math. Anal., 25 (1994), 815-835.  Google Scholar [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.  Google Scholar [17] G. Somieski, Shimmy analysis of a simple aircraft nose landing gear model using different mathematical methods, Aerosp. Sci. Technol., 1 (1997), 545-555. Google Scholar

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References:
 [1] A. Benaddi and B. Rao, Energy decay rate of wave equations with indefinite damping, J. Differential Equations, 161 (2000), 337-357.  Google Scholar [2] C. W. de Silva, "Vibration and Shock Handbook," Mechanical Engineering, CRC Press, 2005. doi: doi:10.1201/9781420039894.  Google Scholar [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.  Google Scholar [4] P. Freitas and E. Zuazua, Stability results for the wave equation with indefinite damping, J. Differential Equations, 132 (1996), 338-352.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [11] A. Marino and D. Mugnai, Asymptotically critical points and their multiplicity, Topol. Methods Nonlinear Anal., 19 (2002), 29-38.  Google Scholar [12] A. Marino and D. Mugnai, Asymptotical multiplicity and some reversed variational inequalities, Topol. Methods Nonlinear Anal., 20 (2002), 43-62.  Google Scholar [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.  Google Scholar [14] D. Mugnai, On a "reversed" variational inequality, Topol. Methods Nonlinear Anal., 17 (2001), 321-358.  Google Scholar [15] P. Pucci and J. Serrin, Asymptotic stability for intermittently controlled nonlinear oscillators, SIAM J. Math. Anal., 25 (1994), 815-835.  Google Scholar [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.  Google Scholar [17] G. Somieski, Shimmy analysis of a simple aircraft nose landing gear model using different mathematical methods, Aerosp. Sci. Technol., 1 (1997), 545-555. Google Scholar
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