# American Institute of Mathematical Sciences

March  2017, 22(2): 491-506. doi: 10.3934/dcdsb.2017024

## Energy decay of solutions for the wave equation with a time-varying delay term in the weakly nonlinear internal feedbacks

 Laboratory ACEDP, Djillali Liabes university, 22000 Sidi Bel Abbes, Algeria

* Corresponding author: hakemali@yahoo.com

Received  October 2015 Revised  October 2016 Published  December 2016

Fund Project: The authors are supported by CNEPRU-ALGERIA.

In this paper, we investigate the nonlinear wave equation in a bounded domain with a time-varying delay term in the weakly nonlinear internal feedback
 $\left(|u_{t}|^{\gamma-2}u_{t}\right)_{t}-Lu-\int_{0}^{t}g(t-s)L u(s)ds+ \mu_{1} \psi(u_{t}(x, t))+ \mu_{2} \psi(u_{t}(x, t-\tau(t)))=0.$
The asymptotic behavior of solutions is studied by using an appropriate Lyapunov functional. Moreover, we extend and improve the previous results in the literature.
Citation: Ferhat Mohamed, Hakem Ali. Energy decay of solutions for the wave equation with a time-varying delay term in the weakly nonlinear internal feedbacks. Discrete and Continuous Dynamical Systems - B, 2017, 22 (2) : 491-506. doi: 10.3934/dcdsb.2017024
##### References:
 [1] C. Abdallah, P. Dorato, J. Benitez-Read and R. Byrne, Delayed Positive Feedback Can Stabilize Oscillatory System, ACC, San Francisco, (1993), 3106-3107. [2] V. I. Arnold, Mathematical Methods of Classical Mecanics, Springer-Verlag, New York, 1989.  doi: 10.1007/978-1-4757-2063-1. [3] A. Benaissa and A. Guesmia, Energy decay for wave equations of ϕ-Laplacian type with weakly nonlinear dissipation, Electron. J. Differ. Equations, 2008 (2008), 1-22. [4] M. M. Cavalcanti, V. D. Cavalcanti and I. Lasiecka, Well-posedness and optimal decay rates for the wave equation with nonlinear boundary damping-source interaction, Jour. Diff. Equa., 236 (2007), 407-459.  doi: 10.1016/j.jde.2007.02.004. [5] G. Chen, Control and stabilization for the wave equation in a bounded domain, Part Ⅰ, SIAM J. Control Optim., 17 (1979), 66-81.  doi: 10.1137/0317007. [6] G. Chen, Control and stabilization for the wave equation in a bounded domain, Part Ⅱ, SIAM J. Control Optim., 19 (1981), 114-122.  doi: 10.1137/0319009. [7] R. Datko, J. Lagnese and M. P. Polis, An example on the effect of time delays in boundary feedback stabilization of wave equations, SIAM J. Control Optim., 24 (1986), 152-156.  doi: 10.1137/0324007. [8] M. Eller, J. E. Lagnese and S. Nicaise, Decay rates for solutions of a Maxwell system with nonlinear boundary damping, Computational. Appl. Math., 21 (2002), 135-165. [9] A. Haraux, Two remarks on dissipative hyperbolic problems, Research Notes in Mathematics, Pitman: Boston, MA, 122 (1985), 161-179. [10] T. Kato, Linear and quasilinear equations of evolution of hyperbolic type, Hyperbolicity, C. I. M. E. Summer Sch. , Springer, Heidelberg, 72 (2011), 125-191. [11] T. Kato, Abstract Differential Equations and Nonlinear Mixed Problems, Lezioni Fermiane, [Fermi Lectures]. Scuola Normale Superiore, Pisa 1985. [12] V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, Masson-John Wiley, Paris, 1994. [13] I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equations with nonlinear boundary dampin, Diff. Integr. Equa., 6 (1993), 507-533. [14] J. L. Lions, Quelques méthodes de résolution des problémes aux limites non linéaires, Dunod, Paris, 1969. [15] W. J. Liu and E. Zuazua, Decay rates for dissipative wave equations, Ricerche di Matematica, 48 (1999), 61-75. [16] P. Martinez and J. Vancostenoble, Optimality of energy estimates for the wave equation with nonlinear boundary velocity feedbacks, SIAM J. Control Optim, 39 (2000), 776-797.  doi: 10.1137/S0363012999354211. [17] M. Nakao, Decay of solutions of some nonlinear evolution equations, J. Math. Anal. Appl, 60 (1977), 542-549. [18] S. Nicaise and C. Pignotti, Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks, SIAM J. Control Optim, 45 (2006), 1561-1585.  doi: 10.1137/060648891. [19] S. Nicaise and C. Pignotti, Stabilization of the wave equation with boundary or internal distributed delay, Differ. Integr. Equat, 21 (2008), 935-958. [20] S. Nicaise and J. Valein, Stabilization of second order evolution equations with unbounded feedback with delay, ESAIM Control Optim. Calc. Var, 16 (2010), 420-456.  doi: 10.1051/cocv/2009007. [21] S. Nicaise, C. Pignotti and J. Valein, Exponential stability of the wave equation with boundary time-varying delay, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 693-722.  doi: 10.3934/dcdss.2011.4.693. [22] S. Nicaise, J. Valein and E. Fridman, Stability of the heat and of the wave equations with boundary time-varying delays, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 559-581.  doi: 10.3934/dcdss.2009.2.559. [23] J. Y. Park and S. H. Park, General decay for a nonlinear beam equation with weak dissipation, J. Math. Phys. , 51 (2010), 073508, 8pp. [24] W. Rudin, Real and Complex Analysis, second edition, McGraw-Hill, Inc, New York, 1974. [25] F. G. Shinskey, Process Control Systems, McGraw-Hill Book Company, 1967. [26] I. H. Suh and Z. Bien, Use of time delay action in the controller design, IEEE Trans. Autom. Control, 25 (1980), 600-603. [27] C. Q. Xu, S. P. Yung and L. K. Li, Stabilization of the wave system with input delay in the boundary control, ESAIM Control Optim. Calc. Var, 12 (2006), 770-785.  doi: 10.1051/cocv:2006021. [28] Q. C. Zhong, Robust Control of Time-delay Systems, Springer, London, 2006.

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##### References:
 [1] C. Abdallah, P. Dorato, J. Benitez-Read and R. Byrne, Delayed Positive Feedback Can Stabilize Oscillatory System, ACC, San Francisco, (1993), 3106-3107. [2] V. I. Arnold, Mathematical Methods of Classical Mecanics, Springer-Verlag, New York, 1989.  doi: 10.1007/978-1-4757-2063-1. [3] A. Benaissa and A. Guesmia, Energy decay for wave equations of ϕ-Laplacian type with weakly nonlinear dissipation, Electron. J. Differ. Equations, 2008 (2008), 1-22. [4] M. M. Cavalcanti, V. D. Cavalcanti and I. Lasiecka, Well-posedness and optimal decay rates for the wave equation with nonlinear boundary damping-source interaction, Jour. Diff. Equa., 236 (2007), 407-459.  doi: 10.1016/j.jde.2007.02.004. [5] G. Chen, Control and stabilization for the wave equation in a bounded domain, Part Ⅰ, SIAM J. Control Optim., 17 (1979), 66-81.  doi: 10.1137/0317007. [6] G. Chen, Control and stabilization for the wave equation in a bounded domain, Part Ⅱ, SIAM J. Control Optim., 19 (1981), 114-122.  doi: 10.1137/0319009. [7] R. Datko, J. Lagnese and M. P. Polis, An example on the effect of time delays in boundary feedback stabilization of wave equations, SIAM J. Control Optim., 24 (1986), 152-156.  doi: 10.1137/0324007. [8] M. Eller, J. E. Lagnese and S. Nicaise, Decay rates for solutions of a Maxwell system with nonlinear boundary damping, Computational. Appl. Math., 21 (2002), 135-165. [9] A. Haraux, Two remarks on dissipative hyperbolic problems, Research Notes in Mathematics, Pitman: Boston, MA, 122 (1985), 161-179. [10] T. Kato, Linear and quasilinear equations of evolution of hyperbolic type, Hyperbolicity, C. I. M. E. Summer Sch. , Springer, Heidelberg, 72 (2011), 125-191. [11] T. Kato, Abstract Differential Equations and Nonlinear Mixed Problems, Lezioni Fermiane, [Fermi Lectures]. Scuola Normale Superiore, Pisa 1985. [12] V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, Masson-John Wiley, Paris, 1994. [13] I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equations with nonlinear boundary dampin, Diff. Integr. Equa., 6 (1993), 507-533. [14] J. L. Lions, Quelques méthodes de résolution des problémes aux limites non linéaires, Dunod, Paris, 1969. [15] W. J. Liu and E. Zuazua, Decay rates for dissipative wave equations, Ricerche di Matematica, 48 (1999), 61-75. [16] P. Martinez and J. Vancostenoble, Optimality of energy estimates for the wave equation with nonlinear boundary velocity feedbacks, SIAM J. Control Optim, 39 (2000), 776-797.  doi: 10.1137/S0363012999354211. [17] M. Nakao, Decay of solutions of some nonlinear evolution equations, J. Math. Anal. Appl, 60 (1977), 542-549. [18] S. Nicaise and C. Pignotti, Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks, SIAM J. Control Optim, 45 (2006), 1561-1585.  doi: 10.1137/060648891. [19] S. Nicaise and C. Pignotti, Stabilization of the wave equation with boundary or internal distributed delay, Differ. Integr. Equat, 21 (2008), 935-958. [20] S. Nicaise and J. Valein, Stabilization of second order evolution equations with unbounded feedback with delay, ESAIM Control Optim. Calc. Var, 16 (2010), 420-456.  doi: 10.1051/cocv/2009007. [21] S. Nicaise, C. Pignotti and J. Valein, Exponential stability of the wave equation with boundary time-varying delay, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 693-722.  doi: 10.3934/dcdss.2011.4.693. [22] S. Nicaise, J. Valein and E. Fridman, Stability of the heat and of the wave equations with boundary time-varying delays, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 559-581.  doi: 10.3934/dcdss.2009.2.559. [23] J. Y. Park and S. H. Park, General decay for a nonlinear beam equation with weak dissipation, J. Math. Phys. , 51 (2010), 073508, 8pp. [24] W. Rudin, Real and Complex Analysis, second edition, McGraw-Hill, Inc, New York, 1974. [25] F. G. Shinskey, Process Control Systems, McGraw-Hill Book Company, 1967. [26] I. H. Suh and Z. Bien, Use of time delay action in the controller design, IEEE Trans. Autom. Control, 25 (1980), 600-603. [27] C. Q. Xu, S. P. Yung and L. K. Li, Stabilization of the wave system with input delay in the boundary control, ESAIM Control Optim. Calc. Var, 12 (2006), 770-785.  doi: 10.1051/cocv:2006021. [28] Q. C. Zhong, Robust Control of Time-delay Systems, Springer, London, 2006.
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