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 & 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. Google Scholar

[2] V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer-Verlag, New York, 1989. Google Scholar
[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. Google Scholar

[4]

C. Cattaneo, Sulla condizione del calore, Atti Del Semin. Matem. E Fis. Della Univ. Modena, 3 (1949), 83-101. Google Scholar

[5]

C. CavalcantiM. M. DomingosV. 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. Google Scholar

[6]

B. D. ColemanW. 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. Google Scholar

[7]

I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equation with nonlinear boundary damping, Diff. Integral Eq., 6 (1993), 507-533. Google Scholar

[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. Google Scholar

[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. Google Scholar

[10]

W. J. Liu and E. Zuazua, Decay rates for dissipative wave equations, Ricerche Mat., 48 (1999), 61-75. Google Scholar

[11]

J. C. Maxwell, On the Dynamical Theory of Gases, The Philosophical Transactions of the Royal Society, 157 (1867), 49-88. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[17]

, Blow up of solutions with positive energy in nonlinear thermoelasticity with second sound, J. Appl. Math., 2004 (2004), 201-211. Google Scholar

[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. Google Scholar

[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. Google Scholar

[20]

Y. QinZ. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

show all references

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. Google Scholar

[2] V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer-Verlag, New York, 1989. Google Scholar
[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. Google Scholar

[4]

C. Cattaneo, Sulla condizione del calore, Atti Del Semin. Matem. E Fis. Della Univ. Modena, 3 (1949), 83-101. Google Scholar

[5]

C. CavalcantiM. M. DomingosV. 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. Google Scholar

[6]

B. D. ColemanW. 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. Google Scholar

[7]

I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equation with nonlinear boundary damping, Diff. Integral Eq., 6 (1993), 507-533. Google Scholar

[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. Google Scholar

[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. Google Scholar

[10]

W. J. Liu and E. Zuazua, Decay rates for dissipative wave equations, Ricerche Mat., 48 (1999), 61-75. Google Scholar

[11]

J. C. Maxwell, On the Dynamical Theory of Gases, The Philosophical Transactions of the Royal Society, 157 (1867), 49-88. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[17]

, Blow up of solutions with positive energy in nonlinear thermoelasticity with second sound, J. Appl. Math., 2004 (2004), 201-211. Google Scholar

[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. Google Scholar

[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. Google Scholar

[20]

Y. QinZ. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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