# 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 and Continuous Dynamical Systems, 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), 61-105. doi: 10.1007/s00245. [2] K. Ammari and M. Tucsnak, Stabilization of second order evolution equations by a class of unbounded feedbacks, ESAIM, Control Optim. Calc. Var., 6 (2001), 361-386. doi: 10.1051/cocv:2001114. [3] V. Barbu, "Analysis and Control of Nonlinear Infinite-dimensional Systems," Academic Press, Inc., Boston, MA, 1993. [4] M. Bellassoued, Energy decay for the elastic wave equation with a local time-dependent nonlinear damping, Acta Math. Sin., Engl. Ser. 24, 7 (2008), 1175-1192, 1439-7617. 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), 835-860. 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), 407-459. 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 "Control Methods in PDE-Dynamical Systems" (Snowbird, Utah 2005), Contemp. Math., vol. 426, AMS, Providence, RI, 2007, 153-193. [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), 67-94. doi: 10.3934/dcdss.2009.2.67. [9] A. Elbert, Stability of some difference equations, Advances in Difference Equations, (Veszprém, 1995), Gordon and Breach, Amsterdam, 1997, pp. 165-187. [10] G. Fragnelli and D. Mugnai, Stability of solutions for some classes of nonlinear damped wave equations, SIAM J. Control Optim., 47 (2008), 2520-2539. 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), 265-272. [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), 245-258. [14] H. Koch and I. Lasiecka, Hadamard wellposedness of weak solutions in nonlinear dynamic elasticity-full von Karman system, Progress in Nonlinear PDE's, and their Applications, 50 (2002), 197-206. [15] V. Komornik, "Exact Controllability and Stabilization," The Multiplier Method, Collection RMA, Masson-Wiley, Paris, 1994. [16] I. Lasiecka and D. Tataru, Uniform boundary stabilization of semi-linear wave equation with nonlinear boundary dissipation, Differential Integral Equations, 6 (1993), 507-533. [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), 1757-1797. 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), 681-710. [19] J. L. Lions and W. A. Strauss, Some non-linear evolution equations, Bull SMF, 93 (1965), 43-96. [20] W. Liu and E. Zuazua, Decay rates for dissipative wave equations, Ric. Mat., 48 (1999) Suppl., 61-75. [21] P. Martinez, Precise decay rate estimates for time-dependent dissipative systems, Isr. J. Math., 119 (2000), 291-324. 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), 776-797. 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), 317-331. [24] P. Pucci and J. Serrin, Asymptotic stability for non-autonomous damped wave systems, Comm. Pure Appl. Math., 49 (1996), 177-216. 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), 87-97. [26] E. Zuazua, Stability and decay for a class of nonlinear hyperbolic problems, Asym Ana, 1 (1988), 161-185.

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), 61-105. doi: 10.1007/s00245. [2] K. Ammari and M. Tucsnak, Stabilization of second order evolution equations by a class of unbounded feedbacks, ESAIM, Control Optim. Calc. Var., 6 (2001), 361-386. doi: 10.1051/cocv:2001114. [3] V. Barbu, "Analysis and Control of Nonlinear Infinite-dimensional Systems," Academic Press, Inc., Boston, MA, 1993. [4] M. Bellassoued, Energy decay for the elastic wave equation with a local time-dependent nonlinear damping, Acta Math. Sin., Engl. Ser. 24, 7 (2008), 1175-1192, 1439-7617. 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), 835-860. 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), 407-459. 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 "Control Methods in PDE-Dynamical Systems" (Snowbird, Utah 2005), Contemp. Math., vol. 426, AMS, Providence, RI, 2007, 153-193. [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), 67-94. doi: 10.3934/dcdss.2009.2.67. [9] A. Elbert, Stability of some difference equations, Advances in Difference Equations, (Veszprém, 1995), Gordon and Breach, Amsterdam, 1997, pp. 165-187. [10] G. Fragnelli and D. Mugnai, Stability of solutions for some classes of nonlinear damped wave equations, SIAM J. Control Optim., 47 (2008), 2520-2539. 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), 265-272. [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), 245-258. [14] H. Koch and I. Lasiecka, Hadamard wellposedness of weak solutions in nonlinear dynamic elasticity-full von Karman system, Progress in Nonlinear PDE's, and their Applications, 50 (2002), 197-206. [15] V. Komornik, "Exact Controllability and Stabilization," The Multiplier Method, Collection RMA, Masson-Wiley, Paris, 1994. [16] I. Lasiecka and D. Tataru, Uniform boundary stabilization of semi-linear wave equation with nonlinear boundary dissipation, Differential Integral Equations, 6 (1993), 507-533. [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), 1757-1797. 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), 681-710. [19] J. L. Lions and W. A. Strauss, Some non-linear evolution equations, Bull SMF, 93 (1965), 43-96. [20] W. Liu and E. Zuazua, Decay rates for dissipative wave equations, Ric. Mat., 48 (1999) Suppl., 61-75. [21] P. Martinez, Precise decay rate estimates for time-dependent dissipative systems, Isr. J. Math., 119 (2000), 291-324. 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), 776-797. 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), 317-331. [24] P. Pucci and J. Serrin, Asymptotic stability for non-autonomous damped wave systems, Comm. Pure Appl. Math., 49 (1996), 177-216. 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), 87-97. [26] E. Zuazua, Stability and decay for a class of nonlinear hyperbolic problems, Asym Ana, 1 (1988), 161-185.
 [1] Moez Daoulatli, Irena Lasiecka, Daniel Toundykov. Uniform energy decay for a wave equation with partially supported nonlinear boundary dissipation without growth restrictions. Discrete and Continuous Dynamical Systems - S, 2009, 2 (1) : 67-94. doi: 10.3934/dcdss.2009.2.67 [2] Nadjat Doudi, Salah Boulaaras, Nadia Mezouar, Rashid Jan. Global existence, general decay and blow-up for a nonlinear wave equation with logarithmic source term and fractional boundary dissipation. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022106 [3] Jun Zhou. Global existence and energy decay estimate of solutions for a class of nonlinear higher-order wave equation with general nonlinear dissipation and source term. Discrete and Continuous Dynamical Systems - S, 2017, 10 (5) : 1175-1185. doi: 10.3934/dcdss.2017064 [4] Moez Daoulatli. Energy decay rates for solutions of the wave equation with linear damping in exterior domain. Evolution Equations and Control Theory, 2016, 5 (1) : 37-59. doi: 10.3934/eect.2016.5.37 [5] Vanessa Barros, Carlos Nonato, Carlos Raposo. Global existence and energy decay of solutions for a wave equation with non-constant delay and nonlinear weights. Electronic Research Archive, 2020, 28 (1) : 205-220. doi: 10.3934/era.2020014 [6] Jiacheng Wang, Peng-Fei Yao. On the attractor for a semilinear wave equation with variable coefficients and nonlinear boundary dissipation. Communications on Pure and Applied Analysis, 2022, 21 (6) : 1857-1871. doi: 10.3934/cpaa.2021043 [7] 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 [8] John A. D. Appleby, Alexandra Rodkina, Henri Schurz. Pathwise non-exponential decay rates of solutions of scalar nonlinear stochastic differential equations. Discrete and Continuous Dynamical Systems - B, 2006, 6 (4) : 667-696. doi: 10.3934/dcdsb.2006.6.667 [9] Kim Dang Phung. Decay of solutions of the wave equation with localized nonlinear damping and trapped rays. Mathematical Control and Related Fields, 2011, 1 (2) : 251-265. doi: 10.3934/mcrf.2011.1.251 [10] Le Thi Phuong Ngoc, Nguyen Thanh Long. Existence and exponential decay for a nonlinear wave equation with nonlocal boundary conditions. Communications on Pure and Applied Analysis, 2013, 12 (5) : 2001-2029. doi: 10.3934/cpaa.2013.12.2001 [11] Mohammad A. Rammaha, Daniel Toundykov, Zahava Wilstein. Global existence and decay of energy for a nonlinear wave equation with $p$-Laplacian damping. Discrete and Continuous Dynamical Systems, 2012, 32 (12) : 4361-4390. doi: 10.3934/dcds.2012.32.4361 [12] Olivier Goubet, Wided Kechiche. Uniform attractor for non-autonomous nonlinear Schrödinger equation. Communications on Pure and Applied Analysis, 2011, 10 (2) : 639-651. doi: 10.3934/cpaa.2011.10.639 [13] Jeong Ja Bae, Mitsuhiro Nakao. Existence problem for the Kirchhoff type wave equation with a localized weakly nonlinear dissipation in exterior domains. Discrete and Continuous Dynamical Systems, 2004, 11 (2&3) : 731-743. doi: 10.3934/dcds.2004.11.731 [14] Barbara Kaltenbacher, Irena Lasiecka. Global existence and exponential decay rates for the Westervelt equation. Discrete and Continuous Dynamical Systems - S, 2009, 2 (3) : 503-523. doi: 10.3934/dcdss.2009.2.503 [15] Kangsheng Liu, Xu Liu, Bopeng Rao. Eventual regularity of a wave equation with boundary dissipation. Mathematical Control and Related Fields, 2012, 2 (1) : 17-28. doi: 10.3934/mcrf.2012.2.17 [16] Zhijian Yang, Zhiming Liu, Na Feng. Longtime behavior of the semilinear wave equation with gentle dissipation. Discrete and Continuous Dynamical Systems, 2016, 36 (11) : 6557-6580. doi: 10.3934/dcds.2016084 [17] Zhong-Jie Han, Enrique Zuazua. Decay rates for $1-d$ heat-wave planar networks. Networks and Heterogeneous Media, 2016, 11 (4) : 655-692. doi: 10.3934/nhm.2016013 [18] Ryo Ikehata, Shingo Kitazaki. Optimal energy decay rates for some wave equations with double damping terms. Evolution Equations and Control Theory, 2019, 8 (4) : 825-846. doi: 10.3934/eect.2019040 [19] Petronela Radu, Grozdena Todorova, Borislav Yordanov. Higher order energy decay rates for damped wave equations with variable coefficients. Discrete and Continuous Dynamical Systems - S, 2009, 2 (3) : 609-629. doi: 10.3934/dcdss.2009.2.609 [20] Yongqin Liu, Shuichi Kawashima. Decay property for a plate equation with memory-type dissipation. Kinetic and Related Models, 2011, 4 (2) : 531-547. doi: 10.3934/krm.2011.4.531

2020 Impact Factor: 1.392