General energy decay for nonlinear wave equation of $\phi$−Laplacian type with a delay term in the internal feedback

Khaled ZENNIR

doi:10.26637/mjm302/002

Abstract

Under conditions on the delay term, using the multiplier method and general weighted integral inequalities, we study the question of asymptotic behavior of solutions for a nonlinear wave equation with $\phi$-Laplacian operator and a delay term in the internal feedback.

Keywords:

Nonlinear wave equation, Time delay term, Decay rate, Multiplier method, $\phi$−Laplacian

Mathematics Subject Classification:

35B40, 35L70

Authors Imprint References Funding Related Articles Downloads

Khaled ZENNIR Laboratory of LAMAHIS, Department of Mathematics, Faculty of Sciences, University 20 Août 1955 of Skikda, ALGERIA. https://orcid.org/0000-0002-7895-4168

Pages: 143-152
Date Published: 01-04-2015
Vol. 3 No. 02 (2015): Malaya Journal of Matematik (MJM)

F. Alabau-Boussouira, Convexity and weighted integral inequalities for energy decay rates of nonlinear dissipative hyperbolic systems, Appl. Math. Optim. 51:1 (2005), 61-105. DOI: https://doi.org/10.1007/s00245

V. I. Arnold, Mathematical Methods of Classical Mechanics, Translated from the Russian by K. Vogtmann and A. Weinstein. Second edition. Graduate Texts in Mathematics, 60. Springer-Verlag, New York, 1989. DOI: https://doi.org/10.1007/978-1-4757-2063-1

A. Benaissa and A. Guesmia, Energy decay for wave equations of $phi$-Laplacian type with weakly nonlinear dissipation, Electron. J. Differential Equations 2008, No. 109, 22.

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, J. Differential Equations 236:2 (2007), $407-459$. DOI: https://doi.org/10.1016/j.jde.2007.02.004

G. Chen, Control and stabilization for the wave equation in a bounded domain, SIAM J. Control Optim. $17: 1(1979), 66-81$. DOI: https://doi.org/10.1137/0317007

G. Chen, Control and stabilization for the wave equation in a bounded domain. II, SIAM J. Control Optim. 19:1(1981), 114-122. DOI: https://doi.org/10.1137/0319009

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 $2: 1(2009), 67-94$ DOI: https://doi.org/10.3934/dcdss.2009.2.67

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. 24:1 (1986), 152-156. DOI: https://doi.org/10.1137/0324007

M. Eller, J. E. Lagnese, and S. Nicaise, Decay rates for solutions of a Maxwell system with nonlinear boundary damping, Comput. Appl. Math. 21:1 (2002), 135-165.

A. Guesmia, Inégalités intégrales et application à la stabilisation des systèmes distribués non dissipatifs, HDR thesis, Paul Verlaine-Metz Univeristy, 2006.

A. Haraux, Two remarks on hyperbolic dissipative problems, in: Nonlinear partial differential equations and their applications. College de France seminar, Vol. VII (Paris, 1983-1984), 6, pp. 161-179, Res. Notes in Math., 122, Pitman, Boston, MA, 1985.

V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, RAM: Research in Applied Mathematics. Masson, Paris; John Wiley & Sons, Ltd., Chichester, 1994.

I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping, Differential Integral Equations 6:3 (1993), 507-533. DOI: https://doi.org/10.57262/die/1370378427

I. Lasiecka, Mathematical Control Theory of Coupled PDEs, CBMS-NSF Regional Conference Series in Applied Mathematics, 75. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002. DOI: https://doi.org/10.1137/1.9780898717099

I. Lasiecka and D. Toundykov, Energy decay rates for the semilinear wave equation with nonlinear localized damping and source terms, Nonlinear Anal. 64:8 (2006), 1757-1797. DOI: https://doi.org/10.1016/j.na.2005.07.024

W. J. Liu and E. Zuazua, Decay rates for dissipative wave equations, Ricerche Mat. 48 (1999), suppl., 61-75.

Mama Abdelli and Salim A. Messaoudi, Energy decay for degenerate Kirchhoff equations with weakly nonlinear dissipation, Electron. J. Differ. Equ., 222 (2013), 1-7.

M. Nakao, Decay of solutions of some nonlinear evolution equations, J. Math. Anal. Appl. 60:2 (1977), $542-549$. DOI: https://doi.org/10.1016/0022-247X(77)90040-3

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. 45:5 (2006), 1561-1585 (electronic). DOI: https://doi.org/10.1137/060648891

S. Nicaise and C. Pignotti, Stabilization of the wave equation with boundary or internal distributed delay, Differential Integral Equations 21:9-10 (2008), 935-958. DOI: https://doi.org/10.57262/die/1356038593

C. Q. Xu, S. P. Yung, and L. K. Li, Stabilization of wave systems with input delay in the boundary control, ESAIM Control Optim. Calc. Var. 12:4 (2006), 770-785 (electronic). DOI: https://doi.org/10.1051/cocv:2006021