Journal of Applied Nonlinear Dynamics
WellPosedness and General Decay for Nonlinear Damped Porous Thermoelastic System with Second Sound and Distributed Delay Terms
Journal of Applied Nonlinear Dynamics 11(1) (2022) 153170  DOI:10.5890/JAND.2022.03.009
Djamel
Ouchenane$^1$ , Khaled Zennir$^{2, 3}$, Derradji Guidad$^4$
$^1$ Laboratory of pure and applied Mathematics , Amar Teledji Laghouat
University, Algeria
$^2$ Department of Mathematics, College of Sciences and Arts, AlRass,
Qassim University, Kingdom of Saudi Arabia
$^3$ Laboratoire de Mathematiques Appliquees et de Modelisation,
Universite 8 Mai 1945 Guelma. B.P. 401 Guelma 24000 Algerie
$^4$ Department of Mathematics, College of Sciences, Mohamed Khider Biskra University, Algeria
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Abstract
As a continuity to the study by M. M. AlGharabli et al. in [1], we consider a onedimensional porous thermoelastic system with second sound, distributed delay term and nonlinear feedback. We show the wellposedness,
using the semigroup theory, and establish an explicit and general decay rate
result, using some properties of convex functions and the multiplier method.
Our result is obtained under suitable assumption on delay without
imposing any restrictive growth assumption on the damping term.
Acknowledgments
The authors are highly grateful to the anonymous referee for his/her valuable comments and suggestions for the improvement of the paper. This research work is supported by the General Direction of Scientific Research and Technological
Development (DGRSDT), Algeria.
References

[1]  AlGharabli, M.M., Messaoudi, A.S. (2019), Wellposedness and a general decay for a nonlinear damped porous thermoelastic system with second sound,
Georgian Math. J., 26(1), 111.


[2]  Goodman, M.A. and Cowin, S.C. (1972), A continuum theory for
granular materials, Arch. Ration. Mech. Anal., 44(4), 249266.


[3]  Nunziato, J.W. and Cowin, S.C. (1979/80), A nonlinear theory of
elastic materials with voids, Arch. Ration. Mech. Anal., 72(2), 175201.


[4]  Magana, A. and Quintanilla, R. (2006), On the time decay of
solutions in onedimensional theories of porous materials, Internat. J.
Solids Structures, 43(111), 34143427.


[5]  Casas, P.S. and Quintanilla, R. (2005), Exponential decay in
onedimensional porousthermoelasticity, Mech. Res. Comm.,
32(6), 652658.


[6]  Quintanilla, R. (2003), Slow decay for onedimensional porous
dissipation elasticity, Appl. Math. Lett., 16(4), 487491.


[7]  Pamplona, P.X., Munoz Rivera, J.E., and Quintanilla, R. (2009),
Stabilization in elastic solids with voids, J. Math. Anal. Appl., 350(1), 3749.


[8]  Cowin, S.C. (1985), The viscoelastic behavior of linear elastic
materials with voids, J. Elasticity, 15(2), 185191.


[9]  Cowin, S.C. and Nunziato, J.W. (1983), Linear elastic materials
with voids, J. Elasticity, 13(2), 125147.


[10]  Munoz Rivera, J.E. and Quintanilla, R. (2008), On the time polynomial
decay in elastic solids with voids, J. Math. Anal. Appl., 338(2), 12961309.


[11]  Messaoudi, S.A. and Fareh, A. (2015), Exponential decay for
linear damped porous thermoelastic systems with second sound, Discrete
Contin. Dyn. Syst. Ser. B, 20(2), 599612.


[12]  Casas, P.S. and Quintanilla, R. (2005), Exponential stability in
thermoelasticity with microtemperatures, Internat. J. Engrg. Sci., 43(12), 3347.


[13]  Choucha, A., Ouchenane, D., and Zennir, Kh. (2021), General decay of
solutions in onedimensional porouselastic with memory and distributed
delay term, Tamkang J. Math., 52, http://dx.doi.org/10.5556/j.tkjm.52.2021.3519.


[14]  Choucha, A., Ouchenane, D., Zennir, Kh., and Feng, B. (2020), Global
wellposedness and exponential stability results of a class of
BresseTimoshenkotype systems with distributed delay term,
Math. Meth. Appl.
Sci., 126. DOI: 10.1002/mma.6437.


[15]  Choucha, A., Ouchenane, D., and Boulaars, S. (2020), Well Posedness
and Stability result for a Thermoelastic Laminated Timoshenko Beam with
distributed delay term, Math Meth Appl Sci., 43(17), 998310004.


[16]  Feng, B. (2020), On a Thermoelastic Laminated Timoshenko Beam: Well
Posedness and Stability, Hindawi, Complexity, Volume (2020), Article ID
5139419, 13 pages.


[17]  Nicaise, A.S. and Pignotti, C. (2008), Stabilization of the wave
equation with boundary or internal distributed delay, Diff. Int. Equs.,
21(910), 935958.


[18]  Lasiecka, I. and Tataru, D. (1993), Uniform boundary stabilization
of semilinear wave equations with nonlinear boundary damping, Diff.
Integ. Equ., 6(3), 507533.


[19]  Pazy, A. (1983), Semigroups of linear operateus and applications to
partial differential equations, Vol 44 of applied Math. Sciences,
SpringerVerlag, New York.


[20]  Komornik, V. (1994), Exact Controllability and Stabilization. The
Multiplier Method, Res. Appl. Math., 36, John Wiley and Sons, Chichester.


[21]  Arnold, L.V.I. (1989), Mathematical Methods of Classical
Mechanics, 2nd ed., Grad. Texts in Math. 60, Springer, New York.
