Exponential Stability of a Transmission Problem with History and Delay

  • Beniani Abderrahmane Center University of Belhadj Bouchaib, Ain Temouchent, Algeria.
  • Noureddine Bahri
Keywords: Wave equation, transmission problem, past history, delay term.


In this paper, we consider a transmission problem in the presence of history and delay terms.Under appropriate assumptions, we prove well-posedness by using the semigroup theory. Our stability estimate proves that the unique dissipation given by the history term is strong enough to stabilize exponentially the system in presence of delay by introducing a suitable Lyaponov functional.


A. Benseghir, Existence and exponential decay of solutions for transmission problems with delay. Electronic Journal of Differential Equations, 212 (2014) , 1–11.

S. Berrimi, S. A. Messaoudi; Existence and decay of solutions of a viscoelastic equation with a nonlinear source, Nonlinear Anal.,64 (2006), no. 10, 2314–2331.

M. M. Cavalcanti, V. N. Domingos Cavalcanti, J. A. Soriano; Exponential decay for the solution of semilinear viscoelastic wave equations with localized damping, Electron. J. Differential Equations (2002), No. 44, 14 pp.

M. M. Cavalcanti et al., Existence and uniform decay rates for viscoelastic problems with nonlinear boundary damping, Differential Integral Equations, 14 (2001), no. 1, 85–116.

C. M. Dafermos, Asymptotic stability in viscoelasticity. Archive for rational mechanics and analysis, 37(4) (1970), 297-308.

A. Guesmia, Well-posedness and exponential stability of an abstract evolution equation with infinite memory and time delay. IMA Journal of Mathematical Control and Information, 30(4) (2013), 507-526.

G.Li,D. Wang, , B. Zhu,Well-posedness and decay of solutions for a transmission problem with history and delay. Electronic Journal of Differential Equations. Vol. 2016 (2016), No. 23, pp. 1–21.

W. J. Liu, K. W. Chen, Existence and general decay for nondissipative distributed systems with boundary frictional and memory dampings and acoustic boundary conditions, Z. Angew. Math. Phys., 66 (2015), no. 4, 1595–1614.

W. J. Liu, Arbitrary rate of decay for a viscoelastic equation with acoustic boundary conditions, Appl. Math. Lett., 38 (2014),155–161.

W. J. Liu, Y. Sun, General decay of solutions for a weak viscoelastic equation with acoustic boundary conditions, Z. Angew. Math.Phys., 65 (2014), no. 1, 125–134.

S. A.Messaoudi, A.Fareh and N. Doudi, Well posedness and exponential stability in a wave equation with a strong damping and a strong delay, Journal of Mathematical Physics, vol. 57, no 11, p. 111501(2016).

S. A. Messaoudi, General decay of solutions of a viscoelastic equation, J. Math. Anal. Appl., 341 (2008), 1457-1467.

A. Marzocchi, J. E. Mu˜ noz Rivera, M. G. Naso, Asymptotic behaviour and exponential stability for a transmission problem in thermoelasticity, Math. Methods Appl. Sci., (2002), no. 11, 955–980.

S. Nicaise, C. Pignotti, Stability and instability results of the wave equation with a delay, 45 (2006), no. 5, 1561–1585.

A. Pazy, Semigroups of linear operators and applications to partial differential equations, Springer New York 198 (1983).

S. B. Gazi Karakoc A Quartic Subdomain Finite Element Method for the Modified KdV Equation, Stat., Optim. Inf. Comput., Vol.6, December 2018, pp 609-618.

S. Battal Gazi Karakoc and Halil Zeybek, A cubic B-spline Galerkin approach for the numerical simulation of the GEW equation,Stat., Optim. Inf. Comput., Vol. 4, March 2016, pp 301.

F. Tahamtani, A. Peyravi, Asymptotic behavior and blow-up of solutions for a nonlinear viscoelastic wave equation with boundary dissipation, Taiwanese J. Math., 17 (2013), no. 6, 1921–1943.

O. P. Yadav and R. Jiwari, Finite element analysis and approximation of Burgers-Fisher equation, Numerical Methods for Partial Differential Equations, 33 (5), 1652-1677 (2017).

How to Cite
Abderrahmane, B., & Bahri, N. (2019). Exponential Stability of a Transmission Problem with History and Delay. Statistics, Optimization & Information Computing, 7(4), 731-747. https://doi.org/10.19139/soic-2310-5070-728
Research Articles