EXPONENTIAL DECAY FOR A NEUTRAL WAVE EQUATION

Nasser-eddine Tatar

doi:10.11948/2017077

2017 Volume 7 Issue 4

Article Contents

Article navigation > Journal of Applied Analysis & Computation > 2017 > 7(4): 1267-1274

Next Article Previous Article

Nasser-eddine Tatar. EXPONENTIAL DECAY FOR A NEUTRAL WAVE EQUATION[J]. Journal of Applied Analysis & Computation, 2017, 7(4): 1267-1274. doi: 10.11948/2017077

Citation:

Nasser-eddine Tatar. EXPONENTIAL DECAY FOR A NEUTRAL WAVE EQUATION[J]. Journal of Applied Analysis & Computation, 2017, 7(4): 1267-1274. doi: 10.11948/2017077

EXPONENTIAL DECAY FOR A NEUTRAL WAVE EQUATION

Nasser-eddine Tatar

Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran, 31261, Saudi Arabia

Fund Project:

Abstract

Abstract

A strongly damped wave equation involving a delay of neutral type in its second order derivative is considered. It is proved that solutions decay to zero exponentially despite the fact that delays are, in general, sources of instability
- Exponential decay /
- modified energy /
- multiplier technique /
- neutral delay /
- stability
MSC: 35L05;34K40

FullText(HTML)

References

[1]	J. Bontsema and S. A. De Vries, Robustness of flexible systems against small time delays, in Proc. 27th Conference on Decision and Control, Austin, Texas, Dec. 1988. Google Scholar
[2]	T. Candan and R. S. Dahiya, Positive solutions of first-order neutral differential equations, Appl. Math. Letters, 2009, 22, 1266-1270. Google Scholar
[3]	H. Chen, Some improved criteria on exponential stability of neutral differential equation, Adv. Difference Eqs., 2012, 2012, 170. DOI:10.1186/1687-1847-2012-170. Google Scholar
[4]	Q. Chuanxi and G. Ladas, Existence of positive solutions for neutral differential equations, J. Appl. Math. Simul., 1989, 2, 267-276. Google Scholar
[5]	R. Datko, J. Lagnese and M. P. Polis, An example of the effect of time delays in boundary feedback stabilization of wave equations, SIAM J. Control Optim., 1986, 24, 152-156. Google Scholar
[6]	R. Datko, Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks, SIAM J. Control Optim., 1988, 26, 697-713. Google Scholar
[7]	W. Desch, K. B. Hannsgen, Y. Renardy and R. L. Wheeler, Boundary stabilization of an Euler-Bernoulli beam with viscoelastic damping, in Proc. 26th Conference on Decision and Control, Los Angeles, CA, Dec., 1987. Google Scholar
[8]	R. D. Driver, A mixed neutral system, Nonl. Anal. T. M. A., 1984, 8(2), 155-158. Google Scholar
[9]	M. Eduardo Hernández and M. L. Pelicier, Existence results for a second-order abstract Cauchy problem with nonlocal conditions, Electron. J. Diff. Eqs., 2005, 2005(73), 1-17. Google Scholar
[10]	R. Grimmer, R. Lenczewski and W. Schappacher, Well-posedness of hyperbolic equations with delay in the boundary conditions, in Semigroup Theory and Applications, P. Clement, S. Invernizzi, E. Mitidieri, and I. Vrabie, eds., Lecture Notes in Pure and Appl. Math. 116, Marcel Dekker, New York and Basel, 1989. Google Scholar
[11]	I. Gyori and G. Ladas, Oscillation Theory of Delay Differential Equations with Applications, Clarendon Press-Oxford, 1991. Google Scholar
[12]	J. K. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York, 1997. Google Scholar
[13]	J. K. Hale and S.M. Verduyn Lunel, Introduction to Functional Differential Equations, in:Applied Mathematical Sciences, vol. 99, Springer Verlag, New York, 2003. Google Scholar
[14]	K. B. Hannsgen, Y. Renardy and R. L. Wheeler, Effectiveness and robustness with respect to time delays of boundary feedback stabilization in onedimensional viscoelasticity, SIAM J. Control Optim., 1988, 26, 1200-1234. Google Scholar
[15]	V. Kolmanovskii and A. Myshkis, Introduction to the Theory and Applications of Functional Differential Equations, Kluwer Academic Publishers, Dordrecht, 1999. Google Scholar
[16]	G. S. Ladde, V. Lakshmikantham and B. G. Zhang, Oscillation Theory of Differential Equations with Deviating Arguments, Marcel Dekker, New York, 1987. Google Scholar
[17]	G. Liu and J. Yan, Global asymptotic stability of nonlinear neutral differential equation, Commun. Nonlinear Sci. Numer. Simulat., 2014, 19, 1035-1041. Google Scholar
[18]	S. K. Ntouyas, Y. G. Sficas and P. Ch. Tsamatos, Existence results for initial value problems for neutral functional-differential equations, J. Diff. Eqs., 1994, 114(2), 527-537. Google Scholar
[19]	S. K. Ntouyas and D. O'Regan, Existence results for semilinear neutral functional differential inclusions with nonlocal conditions, Diff. Eqs. Appl., 2009, 1(1), 41-65. Google Scholar
[20]	T. Rojsiraphisal and P. Niamsup, Exponential stability of certain neutral differential equations, Appl. Math. Comput., 2010, 217, 3875-3880. Google Scholar
[21]	J. Sánchez and V. Vergara, Long-time behavior of nonlinear integro-differential evolution equations, Nonlin. Anal. T. M. A., 2013, 91, 20-31. Google Scholar
[22]	J. Wang, Existence and stability of solutions for neutral differential equations with delay, International Conference on Multimedia Technology (ICMT), 2011, 2462-2465, 10.1109/ICMT.2011.6002527. Google Scholar
[23]	W. Wang, Q. Fan, Y. Zhang and S. Li, Asymptotic stability of solution to nonlinear neutral and Volterra functional differential equations in Banach spaces, Appl. Math. Comput., 2014, 237, 217-226. Google Scholar
[24]	W.-S. Wang, S.-F. Li and R.-S. Yang, Contractivity and exponential stability of solutions to nonlinear neutral functional differential equations in Banach spaces, Acta Math. Appl. Sinica, English Series, 2012, 28(2), 289-304. Google Scholar
[25]	L. Wen, W. Wang and Y. Yu, Dissipativity and asymptotic stability of nonlinear neutral delay integro-differential equations, Nonlin. Anal. T. M. A., 2010, 72, 1746-1754. Google Scholar
[26]	J. Wu and H. Xia, Self-sustained oscillations in a ring array of coupled lossless transmission lines, J. Diff. Eqs., 1996, 124(1), 247-278. Google Scholar
[27]	S. Xie, Solvability of impulsive partial neutral second-order functional integrodifferential equations with infinite delay, Boundary Value Prob., 2013, 2013, 203. Google Scholar
[28]	R. Ye and G. Zhang, Neutral functional differential equations of second-order with infinite delays, Electron. J. Diff. Eqs., 2010, 2010(36), 1-12. Google Scholar