The effect of the wave speeds and the frictional damping terms on the decay rate of the Bresse system

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December 2014, 3(4): 713-738. doi: 10.3934/eect.2014.3.713

The effect of the wave speeds and the frictional damping terms on the decay rate of the Bresse system

Abdelaziz Soufyane ^1, and Belkacem Said-Houari ^1,

1.	ALHOSN University, Mathematics and Natural Sciences Department, PO Box 38772, Abu Dhabi, United Arab Emirates, United Arab Emirates

Received July 2014 Revised September 2014 Published October 2014

Abstract
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In this paper, we consider the Bresse system with frictional damping terms. We investigated the relationship between the frictional damping terms, the wave speeds of propagation and their influence on the decay rate of the solution. We proved that in many cases the solution enjoys the decay property of regularity-loss type. We introduced a new assumption on the wave speeds that controls the behavior of the solution of the Bresse system. In addition, when the coefficient $l $ goes to zero, we showed that the solution of the Bresse system decays faster than the one of the Timoshenko system. This result seems to be the first one to give the decay rate of the solution of the Bresse system in unbounded domain.

Keywords: wave speeds., Bresse system, Decay rate, Timoshenko system, regularity loos.

Mathematics Subject Classification: 35B35, 35L55, 74D05, 93D15, 93D2.

Citation: Abdelaziz Soufyane, Belkacem Said-Houari. The effect of the wave speeds and the frictional damping terms on the decay rate of the Bresse system. Evolution Equations & Control Theory, 2014, 3 (4) : 713-738. doi: 10.3934/eect.2014.3.713

References:

[1]	F. Alabau Boussouira, J. E. Muñoz Rivera and D. S. Almeida Júnior, Stability to weak dissipative Bresse system,, J. Math. Anal. Appl., 374 (2011), 481. doi: 10.1016/j.jmaa.2010.07.046.
[2]	M. M. Cavalcanti, V. N Domingos Cavalcanti, F. A Falcão Nascimento, I Lasiecka and J. H Rodrigues, Uniform decay rates for the energy of Timoshenko system with the arbitrary speeds of propagation and localized nonlinear damping,, Z. Angew. Math. Phys., (2013), 1. doi: 10.1007/s00033-013-0380-7.
[3]	L. H. Fatori and R. N. Monteiro, The optimal decay rate for a weak dissipative Bresse system,, Appl. Math. Lett., 25 (2012), 600. doi: 10.1016/j.aml.2011.09.067.
[4]	L. H. Fatori and J. E. Muñoz Rivera, Rates of decay to weak thermoelastic Bresse system,, IMA Journal of Applied Mathematics, 75 (2010), 881. doi: 10.1093/imamat/hxq038.
[5]	M. Grobbelaar-Van Dalsen, Polynomial decay rate of a thermoelastic mindlin-Timoshenko plate model with dirichlet boundary conditions,, Z. Angew. Math. Phys., (2013), 1. doi: 10.1007/s00033-013-0391-4.
[6]	K. Ide, K. Haramoto and S. Kawashima, Decay property of regularity-loss type for dissipative Timoshenko system,, Math. Mod. Meth. Appl. Sci., 18 (2008), 647. doi: 10.1142/S0218202508002802.
[7]	K. Ide and S. Kawashima, Decay property of regularity-loss type and nonlinear effects for dissipative Timoshenko system,, Math. Mod. Meth. Appl. Sci., 18 (2008), 1001. doi: 10.1142/S0218202508002930.
[8]	Z. Liu and B. Rao, Energy decay rate of the thermoelastic Bresse system,, Z. Angew. Math. Phys., 60 (2009), 54. doi: 10.1007/s00033-008-6122-6.
[9]	A. Matsumura, On the asymptotic behavior of solutions of semi-linear wave equations,, Publ. Res. Inst. Math. Sci. Kyoto. Univ, 12 (1976), 169. doi: 10.2977/prims/1195190962.
[10]	L. Nirenberg, On elliptic partial differential equations,, Ann. Scuola Norm. Sup. Pisa, 13 (1959), 115.
[11]	N. Noun and A. Wehbe, Stabilisation faible interne locale de système élastique de Bresse,, C. R. Math. Acad. Sci. Paris, 350 (2012), 493. doi: 10.1016/j.crma.2012.04.003.
[12]	R. Racke and B. Said-Houari, Decay rates and global existence for semilinear dissipative Timoshenko systems,, Quart. Appl. Math., 71 (2013), 229. doi: 10.1090/S0033-569X-2012-01280-8.
[13]	C. A. Raposo, J. Ferreira, M. L. Santos and N. N. O. Castro, Exponential stability for the Timoshenko system with two weak dampings,, Appl. Math. Letters, 18 (2005), 535. doi: 10.1016/j.aml.2004.03.017.
[14]	M. L. Santos, A. Soufyane and D. S. A. Júnior, Asymptotic behavior to bresse system with past history,, Quarterly of Applied Mathematics, (2013).
[15]	J. A. Soriano, J. E. Muñoz Rivera and L. H. Fatori, Bresse system with indefinite damping,, J. Math. Anal. Appl., 387 (2012), 284. doi: 10.1016/j.jmaa.2011.08.072.

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References:

[1]	F. Alabau Boussouira, J. E. Muñoz Rivera and D. S. Almeida Júnior, Stability to weak dissipative Bresse system,, J. Math. Anal. Appl., 374 (2011), 481. doi: 10.1016/j.jmaa.2010.07.046.
[2]	M. M. Cavalcanti, V. N Domingos Cavalcanti, F. A Falcão Nascimento, I Lasiecka and J. H Rodrigues, Uniform decay rates for the energy of Timoshenko system with the arbitrary speeds of propagation and localized nonlinear damping,, Z. Angew. Math. Phys., (2013), 1. doi: 10.1007/s00033-013-0380-7.
[3]	L. H. Fatori and R. N. Monteiro, The optimal decay rate for a weak dissipative Bresse system,, Appl. Math. Lett., 25 (2012), 600. doi: 10.1016/j.aml.2011.09.067.
[4]	L. H. Fatori and J. E. Muñoz Rivera, Rates of decay to weak thermoelastic Bresse system,, IMA Journal of Applied Mathematics, 75 (2010), 881. doi: 10.1093/imamat/hxq038.
[5]	M. Grobbelaar-Van Dalsen, Polynomial decay rate of a thermoelastic mindlin-Timoshenko plate model with dirichlet boundary conditions,, Z. Angew. Math. Phys., (2013), 1. doi: 10.1007/s00033-013-0391-4.
[6]	K. Ide, K. Haramoto and S. Kawashima, Decay property of regularity-loss type for dissipative Timoshenko system,, Math. Mod. Meth. Appl. Sci., 18 (2008), 647. doi: 10.1142/S0218202508002802.
[7]	K. Ide and S. Kawashima, Decay property of regularity-loss type and nonlinear effects for dissipative Timoshenko system,, Math. Mod. Meth. Appl. Sci., 18 (2008), 1001. doi: 10.1142/S0218202508002930.
[8]	Z. Liu and B. Rao, Energy decay rate of the thermoelastic Bresse system,, Z. Angew. Math. Phys., 60 (2009), 54. doi: 10.1007/s00033-008-6122-6.
[9]	A. Matsumura, On the asymptotic behavior of solutions of semi-linear wave equations,, Publ. Res. Inst. Math. Sci. Kyoto. Univ, 12 (1976), 169. doi: 10.2977/prims/1195190962.
[10]	L. Nirenberg, On elliptic partial differential equations,, Ann. Scuola Norm. Sup. Pisa, 13 (1959), 115.
[11]	N. Noun and A. Wehbe, Stabilisation faible interne locale de système élastique de Bresse,, C. R. Math. Acad. Sci. Paris, 350 (2012), 493. doi: 10.1016/j.crma.2012.04.003.
[12]	R. Racke and B. Said-Houari, Decay rates and global existence for semilinear dissipative Timoshenko systems,, Quart. Appl. Math., 71 (2013), 229. doi: 10.1090/S0033-569X-2012-01280-8.
[13]	C. A. Raposo, J. Ferreira, M. L. Santos and N. N. O. Castro, Exponential stability for the Timoshenko system with two weak dampings,, Appl. Math. Letters, 18 (2005), 535. doi: 10.1016/j.aml.2004.03.017.
[14]	M. L. Santos, A. Soufyane and D. S. A. Júnior, Asymptotic behavior to bresse system with past history,, Quarterly of Applied Mathematics, (2013).
[15]	J. A. Soriano, J. E. Muñoz Rivera and L. H. Fatori, Bresse system with indefinite damping,, J. Math. Anal. Appl., 387 (2012), 284. doi: 10.1016/j.jmaa.2011.08.072.

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