Stability of the Kawahara equation with time-varying delay

Document Type : Research Article

Authors

1 Universit\'e Oran1 Ahmed Ben Bella, Laboratoire de Mathématique et ses Applications, Oran, Algeria

2 Université Oran1 Ahmed Ben Bella, Laboratoire de Mathématique et ses Applications, Oran, Algeria

Abstract

In this work, we consider the nonlinear Kawahara equation with internal time-dependent delay in a bounded domain. We prove that this equation has a unique solution. Moreover, we use a Lyapunov functional approach to prove the exponential stability of the nonlinear system, under some assumptions on the weights of the feedbacks and on the time-dependent delay. We present some numerical simulations to illustrate the obtained results.

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References

[1] F.D. Araruna, R.A. Capistrano-Filho, G.G Doronin, Energy decay for the modified Kawahara equa-
tion posed in a bounded domain, J. Math. Anal. Appl. 385 (2012) 743–756.
[2] B. Chentouf, Well-posedness and exponential stability of the Kawahara equation with a time-
delayed localized damping, Math. Methods Appl. Sci. 45 (2022) 10312–10330.
[3] T. Kato, Linear evolution equations of “hyperbolic” type, J. Fac. Sci. Univ. Tokyo Sect. I 17 (1970)
241–258.
[4] S. Nicaise, J. Valein, E. Fridman, Stability of the heat and of the wave equations with boundary
time-varying delays, Discrete Contin. Dyn. Syst. S 2 (2009) 559–581.
[5] S. Nicaise, C. Pignotti, J. Valein, Exponential stability of the wave equation with boundary time-
varying delay, Discrete Contin. Dyn. Syst. S 4 (2011) 693–722.
[6] H. Parada, C. Timimoun and J. Valein, Stability results for the KdV equation with time-varying
delay, Syst. Control Lett. 177 (2023) 105547.
[7] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations (Ap-
plied Mathematical Sciences), Springer-Verlag, New York, 1983.
[8] J. Valein, On the asymptotic stability of the Korteweg-de Vries equation with time-delayed internal
feedback, Math. Control Relat. Fields 12 (2022) 667–694.
[9] C.F. Vasconcellos, P.N. Da Silva, Stabilisation of the Kawahara equation with localized damping,
ESAIM Control Optim. Ca. 17 (2011) 102–116.
 
Journal of Mathematical Modeling
Volume 13, Issue 2
May 2025
Pages 429-443

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