# American Institute of Mathematical Sciences

June  2011, 31(2): 407-443. doi: 10.3934/dcds.2011.31.407

## Rates of decay for the wave systems with time dependent damping

 1 Department of Mathematics and Informatics, ISSATS, University of Sousse, Sousse, 4003, Tunisia

Received  April 2010 Revised  February 2011 Published  June 2011

We study the rate of decay of solutions of the wave systems with time dependent nonlinear damping. The damping is modeled by a continuous monotone function without any growth restrictions imposed at the origin and infinity. The decay rate of the energy functional is obtained by solving a nonlinear non-autonomous ODE.
Citation: Moez Daoulatli. Rates of decay for the wave systems with time dependent damping. Discrete & Continuous Dynamical Systems - A, 2011, 31 (2) : 407-443. doi: 10.3934/dcds.2011.31.407
##### References:
 [1] F. Alabau-Boussouira, Convexity and weighted integral inequalities for energy decay rates of nonlinear dissipative hyperbolic systems,, Appl. Math. Optim., 51 (2005), 61. doi: 10.1007/s00245. [2] K. Ammari and M. Tucsnak, Stabilization of second order evolution equations by a class of unbounded feedbacks,, ESAIM, 6 (2001), 361. doi: 10.1051/cocv:2001114. [3] V. Barbu, "Analysis and Control of Nonlinear Infinite-dimensional Systems,", Academic Press, (1993). [4] M. Bellassoued, Energy decay for the elastic wave equation with a local time-dependent nonlinear damping,, Acta Math. Sin., 7 (2008), 1175. doi: 10.1007/s10114-007-6468-2. [5] L. Bociu and I. Lasiecka, Uniqueness of weak solutions for the semilinear wave equations with supercritical boundart/interior sources and damping,, Discrete Contin. Dyn. Syst., 22 (2008), 835. doi: 10.3934/dcds.2008.22.835. [6] M. M. Cavalcanti, V. N. Cavalcanti and I. Lasiecka, Well-posedness and optimal decay rates for the wave equation with nonlinear boundary damping-source interaction,, J. of Diff Equa, 236 (2007), 407. doi: 10.1016/j.jde.2007.02.004. [7] I. Chueshov and I. Lasiecka, Long-time dynamics of a semilinear wave equation with nonlinear interior/boundary damping and sources of critical exponents,, in, (2005), 153. [8] M. Daoulatli, I. Lasiecka and D. Toundykov, Uniform energy decay for a wave equation with partially supported nonlinear boundary dissipation without growth restrictions,, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 67. doi: 10.3934/dcdss.2009.2.67. [9] A. Elbert, Stability of some difference equations,, Advances in Difference Equations, (1995), 165. [10] G. Fragnelli and D. Mugnai, Stability of solutions for some classes of nonlinear damped wave equations,, SIAM J. Control Optim., 47 (2008), 2520. doi: 10.1137/070689735. [11] L. Hatvani and T. Krisztin, Necessary and sufficient conditions for intermittent stabilization of linear oscillators by large damping,, Differential Integral Equations, 10 (1997), 265. [12] A. Haraux, Semi-linear hyperbolic problems in bounded domains,, Math. Rep., 3 (1987). [13] A. Haraux, Une remarque sur la stabilisation de certains systémes du deuxiéme ordre en temps,, Portugal Math., 46 (1989), 245. [14] H. Koch and I. Lasiecka, Hadamard wellposedness of weak solutions in nonlinear dynamic elasticity-full von Karman system,, Progress in Nonlinear PDE's, 50 (2002), 197. [15] V. Komornik, "Exact Controllability and Stabilization,", The Multiplier Method, (1994). [16] I. Lasiecka and D. Tataru, Uniform boundary stabilization of semi-linear wave equation with nonlinear boundary dissipation,, Differential Integral Equations, 6 (1993), 507. [17] I. Lasiecka and D. Toundykov, Energy decay rates for the semilinear wave equation with nonlinear localized damping and source terms,, Nonlinear Anal., 64 (2006), 1757. doi: 10.1016/j.na.2005.07.024. [18] I. Lasiecka and D. Toundykov, Stability of higher-level energy norms of strong solutions to a wave equation with localized nonlinear damping and a nonlinear source term,, Control and Cybernetics, 36 (2007), 681. [19] J. L. Lions and W. A. Strauss, Some non-linear evolution equations,, Bull SMF, 93 (1965), 43. [20] W. Liu and E. Zuazua, Decay rates for dissipative wave equations,, Ric. Mat., 48 (1999), 61. [21] P. Martinez, Precise decay rate estimates for time-dependent dissipative systems,, Isr. J. Math., 119 (2000), 291. doi: 10.1007/BF02810672. [22] P. Martinez and J. Vancostenoble, Optimality of energy estimates for the wave equation with nonlinear boundary velocity damping,, SIAM J. Control Optim., 39 (2000), 776. doi: 10.1137/S0363012999354211. [23] M. Nakao, On the decay of solutions of the wave equation with a local time-dependent nonlinear dissipation,, Adv. Math. Sci. Appl., 7 (1997), 317. [24] P. Pucci and J. Serrin, Asymptotic stability for non-autonomous damped wave systems,, Comm. Pure Appl. Math., 49 (1996), 177. doi: 10.1002/(SICI)1097-0312(199602)49:2<177::AID-CPA3>3.0.CO;2-B. [25] M. Slemrod, Weak asymptotic decay via a Relaxed invariance principle for a wave equation with nonlinear, monotone damping,, Proc. Royal Soc. Edinberg Sect. A, 113 (1989), 87. [26] E. Zuazua, Stability and decay for a class of nonlinear hyperbolic problems,, Asym Ana, 1 (1988), 161.

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##### References:
 [1] F. Alabau-Boussouira, Convexity and weighted integral inequalities for energy decay rates of nonlinear dissipative hyperbolic systems,, Appl. Math. Optim., 51 (2005), 61. doi: 10.1007/s00245. [2] K. Ammari and M. Tucsnak, Stabilization of second order evolution equations by a class of unbounded feedbacks,, ESAIM, 6 (2001), 361. doi: 10.1051/cocv:2001114. [3] V. Barbu, "Analysis and Control of Nonlinear Infinite-dimensional Systems,", Academic Press, (1993). [4] M. Bellassoued, Energy decay for the elastic wave equation with a local time-dependent nonlinear damping,, Acta Math. Sin., 7 (2008), 1175. doi: 10.1007/s10114-007-6468-2. [5] L. Bociu and I. Lasiecka, Uniqueness of weak solutions for the semilinear wave equations with supercritical boundart/interior sources and damping,, Discrete Contin. Dyn. Syst., 22 (2008), 835. doi: 10.3934/dcds.2008.22.835. [6] M. M. Cavalcanti, V. N. Cavalcanti and I. Lasiecka, Well-posedness and optimal decay rates for the wave equation with nonlinear boundary damping-source interaction,, J. of Diff Equa, 236 (2007), 407. doi: 10.1016/j.jde.2007.02.004. [7] I. Chueshov and I. Lasiecka, Long-time dynamics of a semilinear wave equation with nonlinear interior/boundary damping and sources of critical exponents,, in, (2005), 153. [8] M. Daoulatli, I. Lasiecka and D. Toundykov, Uniform energy decay for a wave equation with partially supported nonlinear boundary dissipation without growth restrictions,, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 67. doi: 10.3934/dcdss.2009.2.67. [9] A. Elbert, Stability of some difference equations,, Advances in Difference Equations, (1995), 165. [10] G. Fragnelli and D. Mugnai, Stability of solutions for some classes of nonlinear damped wave equations,, SIAM J. Control Optim., 47 (2008), 2520. doi: 10.1137/070689735. [11] L. Hatvani and T. Krisztin, Necessary and sufficient conditions for intermittent stabilization of linear oscillators by large damping,, Differential Integral Equations, 10 (1997), 265. [12] A. Haraux, Semi-linear hyperbolic problems in bounded domains,, Math. Rep., 3 (1987). [13] A. Haraux, Une remarque sur la stabilisation de certains systémes du deuxiéme ordre en temps,, Portugal Math., 46 (1989), 245. [14] H. Koch and I. Lasiecka, Hadamard wellposedness of weak solutions in nonlinear dynamic elasticity-full von Karman system,, Progress in Nonlinear PDE's, 50 (2002), 197. [15] V. Komornik, "Exact Controllability and Stabilization,", The Multiplier Method, (1994). [16] I. Lasiecka and D. Tataru, Uniform boundary stabilization of semi-linear wave equation with nonlinear boundary dissipation,, Differential Integral Equations, 6 (1993), 507. [17] I. Lasiecka and D. Toundykov, Energy decay rates for the semilinear wave equation with nonlinear localized damping and source terms,, Nonlinear Anal., 64 (2006), 1757. doi: 10.1016/j.na.2005.07.024. [18] I. Lasiecka and D. Toundykov, Stability of higher-level energy norms of strong solutions to a wave equation with localized nonlinear damping and a nonlinear source term,, Control and Cybernetics, 36 (2007), 681. [19] J. L. Lions and W. A. Strauss, Some non-linear evolution equations,, Bull SMF, 93 (1965), 43. [20] W. Liu and E. Zuazua, Decay rates for dissipative wave equations,, Ric. Mat., 48 (1999), 61. [21] P. Martinez, Precise decay rate estimates for time-dependent dissipative systems,, Isr. J. Math., 119 (2000), 291. doi: 10.1007/BF02810672. [22] P. Martinez and J. Vancostenoble, Optimality of energy estimates for the wave equation with nonlinear boundary velocity damping,, SIAM J. Control Optim., 39 (2000), 776. doi: 10.1137/S0363012999354211. [23] M. Nakao, On the decay of solutions of the wave equation with a local time-dependent nonlinear dissipation,, Adv. Math. Sci. Appl., 7 (1997), 317. [24] P. Pucci and J. Serrin, Asymptotic stability for non-autonomous damped wave systems,, Comm. Pure Appl. Math., 49 (1996), 177. doi: 10.1002/(SICI)1097-0312(199602)49:2<177::AID-CPA3>3.0.CO;2-B. [25] M. Slemrod, Weak asymptotic decay via a Relaxed invariance principle for a wave equation with nonlinear, monotone damping,, Proc. Royal Soc. Edinberg Sect. A, 113 (1989), 87. [26] E. Zuazua, Stability and decay for a class of nonlinear hyperbolic problems,, Asym Ana, 1 (1988), 161.
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