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. Google Scholar

[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. Google Scholar

[3]

V. Barbu, "Analysis and Control of Nonlinear Infinite-dimensional Systems,", Academic Press, (1993). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[9]

A. Elbert, Stability of some difference equations,, Advances in Difference Equations, (1995), 165. Google Scholar

[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. Google Scholar

[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. Google Scholar

[12]

A. Haraux, Semi-linear hyperbolic problems in bounded domains,, Math. Rep., 3 (1987). Google Scholar

[13]

A. Haraux, Une remarque sur la stabilisation de certains systémes du deuxiéme ordre en temps,, Portugal Math., 46 (1989), 245. Google Scholar

[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. Google Scholar

[15]

V. Komornik, "Exact Controllability and Stabilization,", The Multiplier Method, (1994). Google Scholar

[16]

I. Lasiecka and D. Tataru, Uniform boundary stabilization of semi-linear wave equation with nonlinear boundary dissipation,, Differential Integral Equations, 6 (1993), 507. Google Scholar

[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. Google Scholar

[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. Google Scholar

[19]

J. L. Lions and W. A. Strauss, Some non-linear evolution equations,, Bull SMF, 93 (1965), 43. Google Scholar

[20]

W. Liu and E. Zuazua, Decay rates for dissipative wave equations,, Ric. Mat., 48 (1999), 61. Google Scholar

[21]

P. Martinez, Precise decay rate estimates for time-dependent dissipative systems,, Isr. J. Math., 119 (2000), 291. doi: 10.1007/BF02810672. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[26]

E. Zuazua, Stability and decay for a class of nonlinear hyperbolic problems,, Asym Ana, 1 (1988), 161. Google Scholar

show all references

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. Google Scholar

[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. Google Scholar

[3]

V. Barbu, "Analysis and Control of Nonlinear Infinite-dimensional Systems,", Academic Press, (1993). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[9]

A. Elbert, Stability of some difference equations,, Advances in Difference Equations, (1995), 165. Google Scholar

[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. Google Scholar

[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. Google Scholar

[12]

A. Haraux, Semi-linear hyperbolic problems in bounded domains,, Math. Rep., 3 (1987). Google Scholar

[13]

A. Haraux, Une remarque sur la stabilisation de certains systémes du deuxiéme ordre en temps,, Portugal Math., 46 (1989), 245. Google Scholar

[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. Google Scholar

[15]

V. Komornik, "Exact Controllability and Stabilization,", The Multiplier Method, (1994). Google Scholar

[16]

I. Lasiecka and D. Tataru, Uniform boundary stabilization of semi-linear wave equation with nonlinear boundary dissipation,, Differential Integral Equations, 6 (1993), 507. Google Scholar

[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. Google Scholar

[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. Google Scholar

[19]

J. L. Lions and W. A. Strauss, Some non-linear evolution equations,, Bull SMF, 93 (1965), 43. Google Scholar

[20]

W. Liu and E. Zuazua, Decay rates for dissipative wave equations,, Ric. Mat., 48 (1999), 61. Google Scholar

[21]

P. Martinez, Precise decay rate estimates for time-dependent dissipative systems,, Isr. J. Math., 119 (2000), 291. doi: 10.1007/BF02810672. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[26]

E. Zuazua, Stability and decay for a class of nonlinear hyperbolic problems,, Asym Ana, 1 (1988), 161. Google Scholar

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