# American Institute of Mathematical Sciences

March  2015, 20(2): 599-612. doi: 10.3934/dcdsb.2015.20.599

## Exponential decay for linear damped porous thermoelastic systems with second sound

 1 King Fahd University of Petroleum and Minerals, Department of Mathematics and Statistics, Dhahran 31261 2 Laboratoire de thèorie des operateurs et EDP: fondements et applications, Faculté des Sciences et de technologie, Universit, El Oued 39000, Algeria

Received  March 2014 Revised  July 2014 Published  January 2015

In this paper, we investigate two problems in porous thermoelasticity where the heat conduction is given by Cattaneo's law and prove exponential decay results in the presence of both macro- and micro-dissipations.
Citation: Salim A. Messaoudi, Abdelfeteh Fareh. Exponential decay for linear damped porous thermoelastic systems with second sound. Discrete & Continuous Dynamical Systems - B, 2015, 20 (2) : 599-612. doi: 10.3934/dcdsb.2015.20.599
##### References:
 [1] P. S. Casas and R. Quintanilla, Exponential decay in one-dimensional porous thermoelasticity,, Mech. Res. Comm., 32 (2005), 652. doi: 10.1016/j.mechrescom.2005.02.015. Google Scholar [2] H. D. Fernàndez-Sare and R. Racke, On the stability of damped Timoshenko system Cattaneo versus Fourier law$^*$,, Arch. Rat. Mech. Anal., 194 (2009), 221. doi: 10.1007/s00205-009-0220-2. Google Scholar [3] A. Guesmia, S. A. Messaoudi and A. Wehbe, Uniform decay in mildly damped Timoshenko system with non-equal wave speed propagation,, Dynamic Systems and Applications, 21 (2012), 133. Google Scholar [4] S. W. Hansen, Exponential energy decay in a linear thermoelastic rod,, J. Math. Anal. appl., 167 (1992), 429. doi: 10.1016/0022-247X(92)90217-2. Google Scholar [5] Z. J. Han and G. Q. Xu, Exponential decay result in non-uniform porous-thermo-elasticity model of Lord-Shulman type,, Disc. Cont. Dyn. Sys. B, 17 (2012), 57. doi: 10.3934/dcdsb.2012.17.57. Google Scholar [6] H. W. Lord and Y. Shulman, A generalized dynamical theory of thermoelasticity,, J. Mech. Phys. Sol., 15 (1967), 299. doi: 10.1016/0022-5096(67)90024-5. Google Scholar [7] A. Magaña and R. Quintanilla, On the time decay of solutions in one-dimensional theories of porous materials,, Internat. J. Solids Struct., 43 (2006), 3414. doi: 10.1016/j.ijsolstr.2005.06.077. Google Scholar [8] S. A. Messaoudi and A. Fareh, General decay for a porous thermoelastic system with memory: The case of equal speeds,, Nonlinear analysis: TMA, 74 (2011), 6895. doi: 10.1016/j.na.2011.07.012. Google Scholar [9] S. A. Messaoudi and A. Fareh, General decay for a porous thermoelastic system with memory: The case of nonequal speeds,, Acta Mathimatica Scientia, 33 (2013), 23. doi: 10.1016/S0252-9602(12)60192-1. Google Scholar [10] S. A. Messaoudi, M. Pokojovy and B. Said-Houari, Nonlinear damped Timoshenko system with second sound - Global existence and exponential stability,, Math. Meth. Appl. Sci., 32 (2009), 505. doi: 10.1002/mma.1049. Google Scholar [11] J. E. Muñoz Rivera, Energy decay rate in linear thermoelasticity,, Funkcial Ekvac., 35 (1992), 19. Google Scholar [12] J. E. Muñoz Rivera and R. Racke, Mildly dissipative nonlinear Timoshenko systems - global existence and exponential stability,, J. Math. Anal. Appl., 276 (2002), 248. doi: 10.1016/S0022-247X(02)00436-5. Google Scholar [13] J. E. Muñoz Rivera and R. Quintanilla, On the time polynomial decay in elastic solids with voids,, J. Math. Anal. Appl., 338 (2008), 1296. doi: 10.1016/j.jmaa.2007.06.005. Google Scholar [14] R. Quintanilla, Slow decay in one-dimensional porous dissipation elasticity,, Applied Math. Letters, 16 (2003), 487. doi: 10.1016/S0893-9659(03)00025-9. Google Scholar [15] R. Racke, Thermoelasticity with second sound, exponential stability in linear and non linear 1-d,, Math. Meth. Appl. Sci., 25 (2002), 409. doi: 10.1002/mma.298. Google Scholar [16] R. Racke, Asymptotic behavior of solutions in linear 2- or 3-D thermoelasticity with second sound,, Quart. Appl. Math., 61 (2003), 315. Google Scholar [17] M. L. Santos, D. S. Almeida Júnior and J. E. Muñoz Rivera, The stability number of the Timoshenko system with second sound,, J. Diff. Eqns., 253 (2012), 2715. doi: 10.1016/j.jde.2012.07.012. Google Scholar [18] A. Soufyane, Energy decay for porous-thermo-elasticity systems of memory type,, Appl. Anal., 87 (2008), 451. doi: 10.1080/00036810802035634. Google Scholar [19] M. A. Tarabek, On the existence of smooth solutions in one-dimensional nonlinear thermoelasticity with second sound,, Quarterly of Applied Mathematics, 50 (1992), 727. Google Scholar

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##### References:
 [1] P. S. Casas and R. Quintanilla, Exponential decay in one-dimensional porous thermoelasticity,, Mech. Res. Comm., 32 (2005), 652. doi: 10.1016/j.mechrescom.2005.02.015. Google Scholar [2] H. D. Fernàndez-Sare and R. Racke, On the stability of damped Timoshenko system Cattaneo versus Fourier law$^*$,, Arch. Rat. Mech. Anal., 194 (2009), 221. doi: 10.1007/s00205-009-0220-2. Google Scholar [3] A. Guesmia, S. A. Messaoudi and A. Wehbe, Uniform decay in mildly damped Timoshenko system with non-equal wave speed propagation,, Dynamic Systems and Applications, 21 (2012), 133. Google Scholar [4] S. W. Hansen, Exponential energy decay in a linear thermoelastic rod,, J. Math. Anal. appl., 167 (1992), 429. doi: 10.1016/0022-247X(92)90217-2. Google Scholar [5] Z. J. Han and G. Q. Xu, Exponential decay result in non-uniform porous-thermo-elasticity model of Lord-Shulman type,, Disc. Cont. Dyn. Sys. B, 17 (2012), 57. doi: 10.3934/dcdsb.2012.17.57. Google Scholar [6] H. W. Lord and Y. Shulman, A generalized dynamical theory of thermoelasticity,, J. Mech. Phys. Sol., 15 (1967), 299. doi: 10.1016/0022-5096(67)90024-5. Google Scholar [7] A. Magaña and R. Quintanilla, On the time decay of solutions in one-dimensional theories of porous materials,, Internat. J. Solids Struct., 43 (2006), 3414. doi: 10.1016/j.ijsolstr.2005.06.077. Google Scholar [8] S. A. Messaoudi and A. Fareh, General decay for a porous thermoelastic system with memory: The case of equal speeds,, Nonlinear analysis: TMA, 74 (2011), 6895. doi: 10.1016/j.na.2011.07.012. Google Scholar [9] S. A. Messaoudi and A. Fareh, General decay for a porous thermoelastic system with memory: The case of nonequal speeds,, Acta Mathimatica Scientia, 33 (2013), 23. doi: 10.1016/S0252-9602(12)60192-1. Google Scholar [10] S. A. Messaoudi, M. Pokojovy and B. Said-Houari, Nonlinear damped Timoshenko system with second sound - Global existence and exponential stability,, Math. Meth. Appl. Sci., 32 (2009), 505. doi: 10.1002/mma.1049. Google Scholar [11] J. E. Muñoz Rivera, Energy decay rate in linear thermoelasticity,, Funkcial Ekvac., 35 (1992), 19. Google Scholar [12] J. E. Muñoz Rivera and R. Racke, Mildly dissipative nonlinear Timoshenko systems - global existence and exponential stability,, J. Math. Anal. Appl., 276 (2002), 248. doi: 10.1016/S0022-247X(02)00436-5. Google Scholar [13] J. E. Muñoz Rivera and R. Quintanilla, On the time polynomial decay in elastic solids with voids,, J. Math. Anal. Appl., 338 (2008), 1296. doi: 10.1016/j.jmaa.2007.06.005. Google Scholar [14] R. Quintanilla, Slow decay in one-dimensional porous dissipation elasticity,, Applied Math. Letters, 16 (2003), 487. doi: 10.1016/S0893-9659(03)00025-9. Google Scholar [15] R. Racke, Thermoelasticity with second sound, exponential stability in linear and non linear 1-d,, Math. Meth. Appl. Sci., 25 (2002), 409. doi: 10.1002/mma.298. Google Scholar [16] R. Racke, Asymptotic behavior of solutions in linear 2- or 3-D thermoelasticity with second sound,, Quart. Appl. Math., 61 (2003), 315. Google Scholar [17] M. L. Santos, D. S. Almeida Júnior and J. E. Muñoz Rivera, The stability number of the Timoshenko system with second sound,, J. Diff. Eqns., 253 (2012), 2715. doi: 10.1016/j.jde.2012.07.012. Google Scholar [18] A. Soufyane, Energy decay for porous-thermo-elasticity systems of memory type,, Appl. Anal., 87 (2008), 451. doi: 10.1080/00036810802035634. Google Scholar [19] M. A. Tarabek, On the existence of smooth solutions in one-dimensional nonlinear thermoelasticity with second sound,, Quarterly of Applied Mathematics, 50 (1992), 727. Google Scholar
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