General Decay of Solutions in One-Dimensional Porous-Elastic with Memory and Distributed Delay Term


  • Abdelbaki Choucha Department of Mathematics, University of El Oued, B.P. 789, El Oued 39000, Algerie
  • Djamel Ouchenane Laboratory of pure Mathematics and Applications, Laghouat, Algeria
  • Khaled Zennir Department of Mathematics, College of Sciences and Arts, Al-Ras, Qassim University, Kingdom of Saudi Arabia



Porous system; General decay; Exponential Decay; Memory term; delay term; Relaxation function.


As a continuity to the study by T. A. Apalarain[3], we consider a one-dimensional porous-elastic system with the presence of both memory and distributed delay terms in the second equation. Using the well known energy method combined with Lyapunov functionals approach, we prove a general decay result given in Theorem 2.1.


A. Soufyane, M. Afilal, T. Aouam and M. Chacha. General decay of solutions of a linear one-dimensional porous-thermoelasticity system with a boundary control of memory type. Nonl. Anal., 72(11):3903-3910, 2010.

T. A. Apalara, A general decay for a weakly nonlinearly damped porous system. J. Dyn. Control Syst., 25(3):311-322, 2019.

T. A. Apalara. General decay of solution in one-dimensional porous-elastic system with memory. J. Math. Anal. Appl., 469(2):457-471, 2017.

T. A. Apalara. On the stabilization of a memory-type porous thermoelastic system. Bull. Malays. Math. Sci. Soc., 43:1433–1448(2020),, 2019.

T. A. Apalara, C. A. Raposo and J. O. Ribeiro. Analyticity to transmission problem with delay in porous-elasticity. J. Math. Anal. Appl., 466(1):819-834, 2018.

S. P. Yung, C. Q. Xu and L. K. Li. Stabilization of wave systems with input delay in the boundary control. ESAIM Control Optim. Calc. Var., 12(4):770-785, 2006.

P. S. Casas and R. Quintanilla. Exponential decay in one-dimensional porous- thermoelasticity. Mech. Res. Commun., 32(6):652-658, 2005.

P. S. Casas and R. Quintanilla. Exponential stability in thermoelasticity with microtemper- atures. Int. J. Eng. Sci., 43(1-2):33-47, 2005.

S. C. Cowin and J. W. Nunziato. Linear elastic materials with voids. J. Elast., 13(2):125-147, 1983.

S. C. Cowin and J. W. Nunziato. The viscoelastic behavior of linear elastic materials with voids. J. Elast., 15(2):185-191, 1985.

R. Datko. Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks. SIAM J. Control Optim., 26(3):697-713, 1988.

E. H. Dill. Continum mechanics: elasticity, plasticity, viscoelasticity. CRC Press,Taylor- Francis Group, New York, page pp 368, 2006.

A Fareh and S. A. Messaoud. Energy decay for a porous thermoelastic system with thermoelasticity of second sound and with a non-necessary positive definite energy. Appl. Math. Comput., 293(15):493-507, 2017.

M. A. Goodman and S. C. Cowin. A continuum theory for granular materials. Arch. Ration. Mech. Anal., 44(4):249-266, 1972.

A. Magana and R. Quintanilla. On the time decay of solutions in one-dimensional theories of porous materials. Int.J.Solids Struct., 43(11-12):3414-3427, 2006.

A. S. Nicaise and C. Pignotti. Stabilization of the wave equation with boundary or internal distributed delay. Diff. Int. Equs., 21(9-10):935-958, 2008.

J. W. Nunziato and S. C. Cowin. A nonlinear theory of elastic materials with voids. Arch. Ration. Mech. Anal., 72(2):175-201, 1979.

D. Ouchenane. A stability result of a timoshenko system in thermoelasticity of second sound with a delay term in the internal feedback. Georgian Math. J., 21(4):475-489, 2014.

J. E. M. Rivera, P. X. Pamplona and R. Quintanilla. Stabilization in elastic solids with voids. J. Math. Anal. Appl., 350(1):37-49, 2009.

J. E. M. Rivera, P. X. Pamplona and R. Quintanilla. On the decay of solutions for porous- elastic system with history. J. Math. Anal. Appl., 379(2):682-705, 2011.

R. Quintanilla. Slow decay in one-dimensional porous dissipation elasticity. Appl. Math. Lett., 16(4):487-491, 2003.

J. Lagnese, R. Datko and M. P. Polis. An example on the effect of time delays in boundary feedback stabilization of wave equations. SIAM J. Control Optim., 24(1):152-156, 1986.

J. E. M. Rivera and R. Quintanilla. On the time polynomial decay in elastic solids with voids. J. Math. Anal. Appl., 338(2):1296-1309, 2008.

M. L. Santos and D. A. Junior. On porous-elastic system with localized damping. Z. Angew. Math. Phys., 67(63):1-18, 2016.

A. Soufyane. Energy decay for porous-thermo-elasticity systems of memory type. Appl. Anal., 87(4):451-464, 2008.

I. H. Suh and Z. Bien. Use of time delay action in the controller design. IEEE Trans. Automat. Control., 25(3):600-603, 1980.

S. Zheng and Z. Liu. Semigroups associated with dissipative systems. ChapmanHall/CRC:Boca, Raton, page pp 224, 1999.




How to Cite

Choucha, A., Ouchenane, D., & Zennir, K. (2021). General Decay of Solutions in One-Dimensional Porous-Elastic with Memory and Distributed Delay Term. Tamkang Journal of Mathematics, 52(4), 479–495.