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

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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

Ferhat Mohamed and Hakem Ali ^,

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.

Keywords: Energy decay, viscoelastic term, time-varying delay term.

Mathematics Subject Classification: Primary: 35L05, 35L15; Secondary: 93D1.

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.

show all references

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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