# American Institute of Mathematical Sciences

June  2017, 22(4): 1329-1339. doi: 10.3934/dcdsb.2017064

## A general decay result for a multi-dimensional weakly damped thermoelastic system with second sound

 1 Applied Mathematics Laboratory, Setif 1 University, 19000, Algeria 2 Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia

* Corresponding author: Salah Drabla

Received  April 2015 Revised  December 2016 Published  February 2017

In this article we consider an n-dimensional system of thermoelasticity with second sound in the presence of a weak frictional damping. We establish an explicit and general decay rate result, using some properties of convex functions. Our result is obtained without imposing any restrictive growth assumption on the frictional damping term.

Citation: Salah Drabla, Salim A. Messaoudi, Fairouz Boulanouar. A general decay result for a multi-dimensional weakly damped thermoelastic system with second sound. Discrete and Continuous Dynamical Systems - B, 2017, 22 (4) : 1329-1339. doi: 10.3934/dcdsb.2017064
##### References:
 [1] F. Alabau-Boussouira, Convexity and weighted integral inequalities for energy decay rates of nonlinear dissipative hyperbolic systems, Appl. Math. Optim., 51 (2005), 61-105.  doi: 10.1007/s00245. [2] V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer-Verlag, New York, 1989. [3] F. Boulanouar and S. Drabla, General boundary stabilization result of memory-type thermoelasticity with second sound, Electron. J. Diff. Equ., 2014 (2014), 1-18. [4] C. Cattaneo, Sulla condizione del calore, Atti Del Semin. Matem. E Fis. Della Univ. Modena, 3 (1949), 83-101. [5] C. Cavalcanti, M. M. Domingos, V. N. Cavalcanti and I. Lasiecka, Well-posedness and optimal decay rates for the wave equation with nonlinear boundary damping-source interaction, J. Differential Equations, 236 (2007), 407-459.  doi: 10.1016/j.jde.2007.02.004. [6] B. D. Coleman, W. J. Hrusa and D. R. Owen, Stability of equilibrium for a nonlinear hyperbolic system describing heat propagation by second sound in solids, Arch. Ration. Mech. Anal., 94 (1986), 267-289.  doi: 10.1007/BF00279867. [7] I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equation with nonlinear boundary damping, Diff. Integral Eq., 6 (1993), 507-533. [8] I. Lasiecka and D. Toundykov, Energy decay rates for the semilinear wave equation with nonlinear localized damping and source terms, Nonlinear Anal., 64 (2006), 1757-1797.  doi: 10.1016/j.na.2005.07.024. [9] , Regularity of higher energies of wave equation with nonlinear localized damping and a nonlinear source, Nonlinear Anal., 69 (2008), 898-910.  doi: 10.1016/j.na.2008.02.069. [10] W. J. Liu and E. Zuazua, Decay rates for dissipative wave equations, Ricerche Mat., 48 (1999), 61-75. [11] J. C. Maxwell, On the Dynamical Theory of Gases, The Philosophical Transactions of the Royal Society, 157 (1867), 49-88. [12] S. A. Messaoudi, Local Existence and blow up in thermoelasticity with second sound, Comm. Partial Diff. Eqns., 27 (2002), 1681-1693.  doi: 10.1081/PDE-120005852. [13] S. Messaoudi and A. Al-Shehri, General boundary stabilization of memory-type thermoelasticity with second sound, Z. Anal. Anwend., 31 (2012), 441-461.  doi: 10.4171/ZAA/1468. [14] S. A. Messaoudi and B. Madani, A general decay result for a memory-type thermoelasticity with second sound, Applicable Analysis., 93 (2014), 1663-1673.  doi: 10.1080/00036811.2013.842230. [15] S. A. Messaoudi and M. I. Mustafa, General energy decay rates for a weakly damped wave equation, Commun. Math. Anal., 9 (2010), 67-76. [16] S. A. Messaoudi and B. Said-Houari, Exponential Stability in one-dimensional nonlinear thermoelasticity with second sound, Math. Meth. Appl. Sci., 28 (2005), 205-232.  doi: 10.1002/mma.556. [17] , Blow up of solutions with positive energy in nonlinear thermoelasticity with second sound, J. Appl. Math., 2004 (2004), 201-211. [18] M. I. Mustafa, Boundary stabilization of memory-type thermoelasticity with second sound, Z. Angew. Math. Phys., 63 (2012), 777-792.  doi: 10.1007/s00033-011-0190-8. [19] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied mathematical Sciences 44, Springer-Verlag, 1983. doi: 10.1007/978-1-4612-5561-1. [20] Y. Qin, Z. Ma and X. Yang, Exponential stability for nonlinear thermoelastic equations with second sound, Nonlinear Anal. Real World Appl., 11 (2010), 2502-2513.  doi: 10.1016/j.nonrwa.2009.08.006. [21] R. Racke, Thermoelasticity with second sound-exponential stability in linear and nonlinear 1-d, Math. Meth. Appl. Sci., 25 (2002), 409-441.  doi: 10.1002/mma.298. [22] , Asymptotic behavior of solutions in linear 2-or 3-d thermoelasticity with second sound, Quart. Appl. Math., 61 (2003), 315-328.  doi: 10.1090/qam/1976372. [23] R. Racke and Y. Wang, Nonlinear well-posedness and rates of decay in thermoelasticity with second sound, J. Hyperbolic Differ. Equ., 5 (2008), 25-43.  doi: 10.1142/S021989160800143X. [24] M. A. Tarabek, On the existence of smooth solutions in one-dimensional thermoelasticity with second sound, Quart. Appl. Math., 50 (1992), 727-742.  doi: 10.1090/qam/1193663.

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##### References:
 [1] F. Alabau-Boussouira, Convexity and weighted integral inequalities for energy decay rates of nonlinear dissipative hyperbolic systems, Appl. Math. Optim., 51 (2005), 61-105.  doi: 10.1007/s00245. [2] V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer-Verlag, New York, 1989. [3] F. Boulanouar and S. Drabla, General boundary stabilization result of memory-type thermoelasticity with second sound, Electron. J. Diff. Equ., 2014 (2014), 1-18. [4] C. Cattaneo, Sulla condizione del calore, Atti Del Semin. Matem. E Fis. Della Univ. Modena, 3 (1949), 83-101. [5] C. Cavalcanti, M. M. Domingos, V. N. Cavalcanti and I. Lasiecka, Well-posedness and optimal decay rates for the wave equation with nonlinear boundary damping-source interaction, J. Differential Equations, 236 (2007), 407-459.  doi: 10.1016/j.jde.2007.02.004. [6] B. D. Coleman, W. J. Hrusa and D. R. Owen, Stability of equilibrium for a nonlinear hyperbolic system describing heat propagation by second sound in solids, Arch. Ration. Mech. Anal., 94 (1986), 267-289.  doi: 10.1007/BF00279867. [7] I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equation with nonlinear boundary damping, Diff. Integral Eq., 6 (1993), 507-533. [8] I. Lasiecka and D. Toundykov, Energy decay rates for the semilinear wave equation with nonlinear localized damping and source terms, Nonlinear Anal., 64 (2006), 1757-1797.  doi: 10.1016/j.na.2005.07.024. [9] , Regularity of higher energies of wave equation with nonlinear localized damping and a nonlinear source, Nonlinear Anal., 69 (2008), 898-910.  doi: 10.1016/j.na.2008.02.069. [10] W. J. Liu and E. Zuazua, Decay rates for dissipative wave equations, Ricerche Mat., 48 (1999), 61-75. [11] J. C. Maxwell, On the Dynamical Theory of Gases, The Philosophical Transactions of the Royal Society, 157 (1867), 49-88. [12] S. A. Messaoudi, Local Existence and blow up in thermoelasticity with second sound, Comm. Partial Diff. Eqns., 27 (2002), 1681-1693.  doi: 10.1081/PDE-120005852. [13] S. Messaoudi and A. Al-Shehri, General boundary stabilization of memory-type thermoelasticity with second sound, Z. Anal. Anwend., 31 (2012), 441-461.  doi: 10.4171/ZAA/1468. [14] S. A. Messaoudi and B. Madani, A general decay result for a memory-type thermoelasticity with second sound, Applicable Analysis., 93 (2014), 1663-1673.  doi: 10.1080/00036811.2013.842230. [15] S. A. Messaoudi and M. I. Mustafa, General energy decay rates for a weakly damped wave equation, Commun. Math. Anal., 9 (2010), 67-76. [16] S. A. Messaoudi and B. Said-Houari, Exponential Stability in one-dimensional nonlinear thermoelasticity with second sound, Math. Meth. Appl. Sci., 28 (2005), 205-232.  doi: 10.1002/mma.556. [17] , Blow up of solutions with positive energy in nonlinear thermoelasticity with second sound, J. Appl. Math., 2004 (2004), 201-211. [18] M. I. Mustafa, Boundary stabilization of memory-type thermoelasticity with second sound, Z. Angew. Math. Phys., 63 (2012), 777-792.  doi: 10.1007/s00033-011-0190-8. [19] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied mathematical Sciences 44, Springer-Verlag, 1983. doi: 10.1007/978-1-4612-5561-1. [20] Y. Qin, Z. Ma and X. Yang, Exponential stability for nonlinear thermoelastic equations with second sound, Nonlinear Anal. Real World Appl., 11 (2010), 2502-2513.  doi: 10.1016/j.nonrwa.2009.08.006. [21] R. Racke, Thermoelasticity with second sound-exponential stability in linear and nonlinear 1-d, Math. Meth. Appl. Sci., 25 (2002), 409-441.  doi: 10.1002/mma.298. [22] , Asymptotic behavior of solutions in linear 2-or 3-d thermoelasticity with second sound, Quart. Appl. Math., 61 (2003), 315-328.  doi: 10.1090/qam/1976372. [23] R. Racke and Y. Wang, Nonlinear well-posedness and rates of decay in thermoelasticity with second sound, J. Hyperbolic Differ. Equ., 5 (2008), 25-43.  doi: 10.1142/S021989160800143X. [24] M. A. Tarabek, On the existence of smooth solutions in one-dimensional thermoelasticity with second sound, Quart. Appl. Math., 50 (1992), 727-742.  doi: 10.1090/qam/1193663.
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