Skip Navigation Links
Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA


Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania


Well-posedness and Stability for a Moore-Gibson-Thompson Equation with Internal Distributed Delay

Discontinuity, Nonlinearity, and Complexity 10(4) (2021) 693--703 | DOI:10.5890/DNC.2021.12.009

Abdelkader Braik$^1$, Abderrahmane Beniani$^2$, Khaled Zennir$^3$

$^1$ Department of Sciences and Technology, University of Hassiba Ben Bouali, Chlef, Algeria

$^2$ Department of Mathematics, BP 284, University Centre BELHADJ Bouchaib Ain Tmouchent 46000, Algeri

$^3$ Department of Mathematics, College of Sciences and Arts, Qassim University, Ar-Rass, Saudi Arabia

Download Full Text PDF



In this work, we consider the Moore-Gibson-Thompson equation with distributed delay. We prove, under an appropriate assumptions and a smallness conditions on the parameters $\alpha $, $\beta$, $\gamma$ and $\mu$, that this problem is well-posed and then by introducing suitable energy and Lyapunov functionals, the solution of \eqref{p1} and \eqref{p2} decays to zero as $t$ tends to infinity.


  1. [1]  Kaltenbacher, B., Lasiecka, I., and Marchand, R. (2011), Wellposedness and exponential decay rates for the Moore-Gibson-Thompson equation arising in high intensity ultrasound, Control Cybernet., {\bf 40}(4), 971-988.
  2. [2] Caixeta, A.H., Lasiecka, I., and Cavalcanti, C.V.N. (2016), Global attractors for a third order in time nonlinear dynamics, J. Diff. Equ., {261}(1), 113-147.
  3. [3]  Lasiecka, I. and Wang, X. (2016), Moore-Gibson-Thompson equation with memory, part I: exponential decay of energy, Z. Angew. Math. Phys., {\bf 67}(17),
  4. [4]  Lasiecka, I. and Wang, X. (2015), Moore-Gibson-Thompson equation with memory part II: general decay of energy, J. Diff. Equ., {\bf 259}(12), 7610-7635.
  5. [5]  DellOro, F., Lasiecka, I., and Pata V. (2016), The Moore-Gibson-Thompson equation with memory in the critical case, J. Diff. Equ., {\bf 261}(7), 4188-4222.
  6. [6]  Nicaise, S. and Pignotti, C. (2008), Stabilization of the wave equation with boundary or internal distributed delay, Diff. Int. Equa., {\bf 9}(10), 935-958.
  7. [7]  Nicaise, S. and Pignotti, C. (2006), Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks, SIAM J. Control Optim., {\bf 5}, 1561-1585.
  8. [8]  Nicaise, S. and Pignotti, C. (2011), Interior feedback stabilization of wave equations with time dependent delay, Electron. J. Diff. Equ., {\bf 2011}(41), 1-20.
  9. [9]  Nicaise, S. and Pignotti, C. (2011), Exponential stability of the wave equation with boundary time-varying delay, Disc. Cont. Dynam. Syst., {\bf 3}, 693-722.
  10. [10]  Nicaise, S. and Valein, J. (2010), Stabilization of second order evolution equations with unbounded feedback with delay, ESAIM Control Optim., {\bf 2}, 420-456.
  11. [11]  Mustafa, M.I. (2014), A uniform stability result for thermoelasticity of type III with boundary distributed delay, J. Math. Anal. Appl., {\bf 415}, 148-158.
  12. [12]  Apalara, T.A. (2014), Well-posedness and exponential stability for a linear damped Timoshenko system with second sound and internal distributed delay, Electron. J. Diff. Equ., {\bf 254}, 1-15.
  13. [13]  Mustafa, M.I. and Kafini, M. (2013), Exponential decay in thermoelastic systems with internal distributed delay, Palestine J. Math., {\bf 2}(2), 287-299.
  14. [14]  Messaoudi, S.A., Fareh, A., and Doudi, N. (2016), Well posedness and exponential satbility in a wave equation with a strong damping and a strong delay, J. Math. Phys., {\bf 57}, 111501.
  15. [15]  Boulaaras, S., Draifia, A., and Zarai, A. (2019), Galerkin method for nonlocal mixed boundary value problem for the Moore-Gibson-Thompson equation with integral condition, Math. Meth. App. Sci., {\bf 42}, 2664-2679.
  16. [16]  Caixeta, A.H., Lasiecka, I., and Domingos, C.V.N. (2016), On long time behavior of Moore-Gibson-Thompson equation with molecular relaxation, Evol. Equ. Control Theory., 5(4), 661-676.
  17. [17]  Conejero, J.A., Lizama, C., and Rodenas, F. (2015), Chaotic behaviour of the solutions of the Moore-Gibson-Thompson equation, Appl. Math. Inf. Sci., {\bf 9}(5), 2233-2238.
  18. [18]  DellOro, F. and Pata, V. (2017), On the Moore-Gibson-Thompson equation and its relation to linear viscoelasticity, Appl. Math. Optim., {\bf 76}(3), 641-655.
  19. [19]  DellOro, F. and Pata, V. (2017), On a fourth-order equation of Moore-Gibson-Thompson type, Milan J. Math., {\bf 85}(2), 215-234.
  20. [20]  Kaltenbacher, B. and Lasiecka, I. (2012), Exponential decay for low and higher energies in the third order linear Moore-Gibson-Thompson equation with variable viscosity, Palest. J. Math., {\bf 1}(1), 1-10.
  21. [21]  Pazy, A. (1983), Semi-groups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York.
  22. [22]  Komornik, V. (1994), Exact Controllability and Stabilization. The Multiplier Method, Paris-Masson-John Wiley.
  23. [23]  Guesmia, A. (2013), Well-posedness and exponential stability of an abstract evolution equation with infinite memory and time delay, IMA J. Math. Control Inform., {\bf 30}(4), 507-526.