Transmission system for waves with nonlinear weights and delay

Document Type : Research Paper

Authors

Applied Mathematical Laboratory (LaMa), Faculty of Sciences, University of Setif 1- SETIF, 19000, Algeria

Abstract

In this paper we consider a transmission problem for one dimensional waves with nonlinear weights on the frictional damping and time delay. We prove first, the existence and the uniqueness of the solution using the semigroup theory. Second, we chow the exponential stability of the solution by introducing a suitable Lyaponov functional.

Keywords


[1] K. Ammari, S. Nicaise and C. Pignotti. Feedback boundary stabilization of wave equations with interior delay,
Syst. Cont. Lett. 59 (2010), 23–62[2] E. Balm`es and S. Germ`es, Tools for viscoelastic treatmentdesign. Application to an automotive floor panel, In
ISMA Conf. Pro., 2002.
[3] A. Benseghir, Existence and exponential decay of solutions for transmission problems with delay, Electronic J.
Differ. Equ. 212 (2014), 1–11.
[4] H. Benseridi, Y. Letoufa and M. Dilmi, On the asymptotic behavior of an iInterface pProblem in a thin domain,
Proc. Nat. Acad. Sci. India Sect. A Phys. Sci. 90 (2020), no. 3, 547–556.
[5] R. Datko, Two questions concerning the boundary control of certain elastic systems, J. Differ. Equ. 92 (1991),
no. 1, 27–44.
[6] 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), no. 1, 152–156.
[7] T.F. Ma and H.P. Oquendo, A transmission problem for beams on nonlinear supports, Bound. Value Probl.
2006 (2006), Art. ID 75107.
[8] A. Guesmia, Well-posedness and exponential stability of an abstract evolution equation with infinity memory and
time delay, IAM J. Math. Control Inf. 30 (2013), no. 4, 505–526.
[9] S. Manaa, H. Benseridi and M. Dilmi, 3D–2D asymptotic analysis of an interface problem with a dissipative term
in a dynamic regime, Bol. Soc. Mat. Mexicana 27 (2021), 1–26.
[10] A. Marzocchi, J.E. Mu˜noz Rivera and M.G. Naso, Asymptotic behavior and exponential stability for a transmission
problem in thermoelasticity, Math. Meth. Appl. Sci. 25 (2002), 955–980.
[11] A. Marzocchi, J.E. Mu˜noz Rivera and M.G. Naso, Transmission problem in thermoelasticity with symmetry, IMA
J. Appl. Math. 63 (2002), no. 1, 23–46.
[12] S.A. Messaoudi and B. Said-Houari. Energy decay in a transmission problem in thermoelasticity of type iii, IMA.
J. Appl. Math. 74 (2009), 344–360.
[13] 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), no. 5, 1561–1585, .
[14] J.E. Mu˜noz Rivera and H.P. Oquendo, The transmission problem of viscoelastic waves, Acta Appl. Math. 62
(2000), no. 1, 1–21.
[15] S. Zitouni, A. Abdelouahab, K. Zennir and A. Rachida. Existence and exponential stability of solutions for
transmission system with varying delay in R, Math. Moravica. 20 (2016), 143–161.
[16] 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.
Volume 13, Issue 2
July 2022
Pages 971-982
  • Receive Date: 23 April 2021
  • Revise Date: 04 July 2021
  • Accept Date: 05 July 2021