American Institute of Mathematical Sciences

Advanced
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

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

L. Nirenberg, On elliptic partial differential equations,, Ann. Scuola Norm. Sup. Pisa, 13 (1959), 115.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

M. L. Santos, A. Soufyane and D. S. A. Júnior, Asymptotic behavior to bresse system with past history,, Quarterly of Applied Mathematics, (2013).   Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

L. Nirenberg, On elliptic partial differential equations,, Ann. Scuola Norm. Sup. Pisa, 13 (1959), 115.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

M. L. Santos, A. Soufyane and D. S. A. Júnior, Asymptotic behavior to bresse system with past history,, Quarterly of Applied Mathematics, (2013).   Google Scholar

[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.  Google Scholar

[1]

Adel M. Al-Mahdi, Mohammad M. Al-Gharabli, Salim A. Messaoudi. New general decay result for a system of viscoelastic wave equations with past history. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020273
[2]

Abdelghafour Atlas, Mostafa Bendahmane, Fahd Karami, Driss Meskine, Omar Oubbih. A nonlinear fractional reaction-diffusion system applied to image denoising and decomposition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020321
[3]

Manil T. Mohan. First order necessary conditions of optimality for the two dimensional tidal dynamics system. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020045
[4]

Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of a Sobolev type impulsive functional evolution system in Banach spaces. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020049
[5]

Helmut Abels, Andreas Marquardt. On a linearized Mullins-Sekerka/Stokes system for two-phase flows. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020467
[6]

Yichen Zhang, Meiqiang Feng. A coupled $ p $-Laplacian elliptic system: Existence, uniqueness and asymptotic behavior. Electronic Research Archive, 2020, 28 (4) : 1419-1438. doi: 10.3934/era.2020075
[7]

Youshan Tao, Michael Winkler. Critical mass for infinite-time blow-up in a haptotaxis system with nonlinear zero-order interaction. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 439-454. doi: 10.3934/dcds.2020216
[8]

Jianquan Li, Xin Xie, Dian Zhang, Jia Li, Xiaolin Lin. Qualitative analysis of a simple tumor-immune system with time delay of tumor action. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020341
[9]

Denis Bonheure, Silvia Cingolani, Simone Secchi. Concentration phenomena for the Schrödinger-Poisson system in $ \mathbb{R}^2 $. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020447
[10]

Xavier Carvajal, Liliana Esquivel, Raphael Santos. On local well-posedness and ill-posedness results for a coupled system of mkdv type equations. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020382
[11]

Fathalla A. Rihan, Hebatallah J. Alsakaji. Stochastic delay differential equations of three-species prey-predator system with cooperation among prey species. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020468
[12]

Mathew Gluck. Classification of solutions to a system of $ n^{\rm th} $ order equations on $ \mathbb R^n $. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5413-5436. doi: 10.3934/cpaa.2020246
[13]

Kihoon Seong. Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5437-5473. doi: 10.3934/cpaa.2020247
[14]

Mengni Li. Global regularity for a class of Monge-Ampère type equations with nonzero boundary conditions. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020267
[15]

Serge Dumont, Olivier Goubet, Youcef Mammeri. Decay of solutions to one dimensional nonlinear Schrödinger equations with white noise dispersion. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020456
[16]

Ahmad Z. Fino, Wenhui Chen. A global existence result for two-dimensional semilinear strongly damped wave equation with mixed nonlinearity in an exterior domain. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5387-5411. doi: 10.3934/cpaa.2020243
[17]

Jerry L. Bona, Angel Durán, Dimitrios Mitsotakis. Solitary-wave solutions of Benjamin-Ono and other systems for internal waves. I. approximations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 87-111. doi: 10.3934/dcds.2020215
[18]

Hai-Feng Huo, Shi-Ke Hu, Hong Xiang. Traveling wave solution for a diffusion SEIR epidemic model with self-protection and treatment. Electronic Research Archive, , () : -. doi: 10.3934/era.2020118
[19]

Omid Nikan, Seyedeh Mahboubeh Molavi-Arabshai, Hossein Jafari. Numerical simulation of the nonlinear fractional regularized long-wave model arising in ion acoustic plasma waves. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020466

2019 Impact Factor: 0.953

Tools

Metrics

  • PDF downloads (61)
  • HTML views (0)
  • Cited by (19)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences