Abstract: In the present work, we consider a one-dimensional swelling porous-heat system with single time-delay in a bounded domain under Dirichlet-Neumann boundary conditions subject to thermodiffusion effects and frictional damping to control the delay term. The coupling gives new contributions to the theory associated with asymptotic behaviors of swelling porous-heat. At first, we state and prove the well-posedness of the solution of the system by the semigroup approach using Lumer-Philips theorem under suitable assumption on the weight of the delay. Then, we show that the considered dissipation in which we depended on are strong enough to guarantee an exponential decay result by using the energy method that consists to construct an appropriate Lyapunov functional based on the multiplier technique, this result is obtained without the equal-speed requirement. Our result is new and an extension of many other works in this area.
For citation: Douib, M. and Zitouni, S. Exponential Stability for a Swelling Porous-Heat System with Thermodiffusion Effects and Delay,
Vladikavkaz Math. J., 2023, vol. 25, no. 2, pp. 65-77.
DOI 10.46698/y2253-0872-2762-l
1. Iesan, D. and Quintanilla, R. Existence and Continuous
Dependence Results in the Theory of Interacting Continua,
Journal of Elasticity, 1994, vol. 36, no. 1, pp. 85-98.
DOI: 10.1007/BF00042493.
2. Apalara, T. A. Well-Posedness and Exponential
Stability for a Linear Damped Timoshenko System with Second Sound and
Internal Distributed Delay, Electronic Journal of Differential Equations,
2014, vol. 2014, no. 254, pp. 1-15.
3. Choucha, A., Boulaaras, S. M., Ouchenane, D., Cherif, B. B. and Abdalla, M.
Exponential Stability of Swelling Porous Elastic with a
Viscoelastic Damping and Distributed Delay Term,
Journal of Function Spaces, 2021, vol. 2021, pp. 1-8.
DOI: 10.1155/2021/5581634.
4. Kafini, M., Messaoudi, S. A., Mustafa, M. I. and Apalara, T.
Well-Posedness and Stability Results in a Timoshenko-Type System of
Thermoelasticity of Type III with Delay, Zeitschrift fur Angewandte Mathematik
und Physik, 2015, vol. 66, no. 4, pp. 1499-1517. DOI: 10.1007/s00033-014-0475-9.
5. Mustafa, M. I. A Uniform Stability Result for Thermoelasticity of Type III with Boundary
Distributed Delay, Journal of Mathematical Analysis and Applications,
2014, vol. 415, no. 1, pp. 148-158. DOI: 10.1016/j.jmaa.2014.01.080.
6. Nicaise, S. and Pignotti, C. Stability and Instability Results of the Wave Equation
with a Delay Term in the Boundary or Internal Feedbacks, SIAM Journal on Control and Optimization,
2006, vol. 45, no. 5, pp. 1561-1585. DOI: 10.1137/060648891.
7. Nicaise, S. and Pignotti, C. Stabilization of the Wave Equation with Boundary or Internal
Distributed Delay, Differential and Integral Equations,
2008, vol. 21, no. 9-10, pp. 935-958. DOI: 10.57262/die/1356038593.
8. Ramos, A. J. A., Almeida Junior, D. S., Freitas, M. M., Noe, A. S. and Dos Santos, M. J.
Stabilization of Swelling Porous Elastic Soils with Fluid Saturation and Delay Time Terms,
Journal of Mathematical Physics, 2021, vol. 62, no. 2, pp. 1-10. DOI: 10.1063/5.0018795.
9. Datko, R. Not All Feedback Stabilized Hyperbolic
Systems are Robust with Respect to Small Time Delays in their Feedbacks,
SIAM Journal on Control and Optimization, 1988, vol. 26, no. 3, pp. 697-713. DOI: 10.1137/0326040.
10. Datko, R., Lagnese, J. and Polis, M. P. An Example on the Effect of Time Delays in Boundary
Feedback Stabilization of Wave Equations, SIAM Journal on Control and Optimization,
1986, vol. 24, no. 1, pp. 152-156. DOI: 10.1137/0324007.
11. Racke, R. Instability of Coupled Systems with Delay,
Communications on Pure and Applied Analysis,
2012, vol. 11, no. 5, pp. 1753-1773. DOI: 10.3934/cpaa.2012.11.1753.
12. Apalara, T. A. General Stability Result of Swelling Porous Elastic
Soils with a Viscoelastic Damping, Zeitschrift fur Angewandte Mathematik und Physik,
2020, vol. 71, no. 6, pp. 1-10. DOI: 10.1007/s00033-020-01427-0.
13. Bofill, F. and Quintanilla, R. Anti-Plane Shear Deformations of Swelling Porous Elastic Soils,
International Journal of Engineering Science, 2003, vol. 41, no. 8, pp. 801-816.
DOI: 10.1016/S0020-7225(02)00281-1.
14. Choucha, A., Boulaaras, S. M., Ouchenane, D., Cherif, B. B.,
Hidan, M. and Abdalla, M. Exponential Stabilization of a Swelling
Porous-Elastic System with Microtemperature Effect and Distributed Delay,
Journal of Function Spaces, 2021, vol. 2021, art. ID 5513981, pp. 1-11.
DOI: 10.1155/2021/5513981.
15. Keddi, A., Messaoudi, S. A. and Alahyane, M. Well-posedness and Stability Results
for a Swelling Porous-Heat System of Second Sound, Journal of Thermal Stresses,
2021, vol. 44, no. 12, pp. 1427-1440. DOI: 10.1080/01495739.2021.2003274.
16. Murad, M. A. and Cushman, J. H. Thermomechanical Theories for Swelling
Porous Media with Microstructure, International Journal of Engineering Science,
2000, vol. 38, no. 5, pp. 517-564. DOI: 10.1016/S0020-7225(99)00054-3.
17. Quintanilla, R. Exponential Stability for One-Dimensional Problem
of Swelling Porous Elastic Soils with Fluid Saturation,
Journal of Computational and Applied Mathematics,
2002, vol. 145, no. 2, pp. 525-533. DOI: 10.1016/S0377-0427(02)00442-9.
18. Quintanilla, R. Exponential Stability of Solutions of Swelling
Porous Elastic Soils, Meccanica, 2004, vol. 39, no. 2, pp. 139-145.
DOI: 10.1023/B:MECC.0000005105.45175.61.
19. Quintanilla, R. On the Linear Problem of Swelling Porous Elastic Soils
with Incompressible Fluid, International Journal of Engineering Science,
2002, vol. 40, no. 13, pp. 1485-1494. DOI: 10.1016/S0020-7225(02)00021-6.
20. Wang, J. M. and Guo, B. Z. On the Stability of Swelling
Porous Elastic Soils with Fluid Saturation by one Internal Damping,
IMA Journal of Applied Mathematics, 2006, vol. 71, no. 4, pp. 565-582.
DOI: 10.1093/imamat/hxl009.
21. Goldstein, J. A. Semigroups of Linear Operators and Applications,
Oxford Mathematical Monographs, New York,
The Clarendon Press, Oxford University Press, 1985.
22. Pazy, A. Semigroups of Linear Operators and
Applications to Partial Differential Equations,
New York, Springer, 1983.
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