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), 481-498. 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), 1-18. 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), 600-604. 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), 881-904. 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), 1-16. 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), 647-667. 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), 1001-1025. 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), 54-69. 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), 169-189. doi: 10.2977/prims/1195190962.  Google Scholar

[10]

L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa, 13 (1959), 115-162.  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), 493-498. 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), 229-266. 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), 535-541. 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, Accepted, 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), 284-290. 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), 481-498. 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), 1-18. 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), 600-604. 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), 881-904. 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), 1-16. 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), 647-667. 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), 1001-1025. 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), 54-69. 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), 169-189. doi: 10.2977/prims/1195190962.  Google Scholar

[10]

L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa, 13 (1959), 115-162.  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), 493-498. 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), 229-266. 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), 535-541. 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, Accepted, 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), 284-290. doi: 10.1016/j.jmaa.2011.08.072.  Google Scholar

[1]

Denis Mercier, Virginie Régnier. Decay rate of the Timoshenko system with one boundary damping. Evolution Equations & Control Theory, 2019, 8 (2) : 423-445. doi: 10.3934/eect.2019021
[2]

Ammar Khemmoudj, Taklit Hamadouche. General decay of solutions of a Bresse system with viscoelastic boundary conditions. Discrete & Continuous Dynamical Systems, 2017, 37 (9) : 4857-4876. doi: 10.3934/dcds.2017209
[3]

Salim A. Messaoudi, Jamilu Hashim Hassan. New general decay results in a finite-memory bresse system. Communications on Pure & Applied Analysis, 2019, 18 (4) : 1637-1662. doi: 10.3934/cpaa.2019078
[4]

Abbes Benaissa, Abderrahmane Kasmi. Well-posedeness and energy decay of solutions to a bresse system with a boundary dissipation of fractional derivative type. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4361-4395. doi: 10.3934/dcdsb.2018168
[5]

Yuming Qin, Xinguang Yang, Zhiyong Ma. Global existence of solutions for the thermoelastic Bresse system. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1395-1406. doi: 10.3934/cpaa.2014.13.1395
[6]

Zhuangyi Liu, Ramón Quintanilla. Energy decay rate of a mixed type II and type III thermoelastic system. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1433-1444. doi: 10.3934/dcdsb.2010.14.1433
[7]

Yanxia Niu, Yinxia Wang, Qingnian Zhang. Decay rate of global solutions to three dimensional generalized MHD system. Evolution Equations & Control Theory, 2021, 10 (2) : 249-258. doi: 10.3934/eect.2020064
[8]

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, 2021, 20 (1) : 389-404. doi: 10.3934/cpaa.2020273
[9]

Tong Li, Hailiang Liu. Critical thresholds in a relaxation system with resonance of characteristic speeds. Discrete & Continuous Dynamical Systems, 2009, 24 (2) : 511-521. doi: 10.3934/dcds.2009.24.511
[10]

Xing Wu, Keqin Su. Global existence and optimal decay rate of solutions to hyperbolic chemotaxis system in Besov spaces. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021002
[11]

Hai-Liang Li, Hongjun Yu, Mingying Zhong. Spectrum structure and optimal decay rate of the relativistic Vlasov-Poisson-Landau system. Kinetic & Related Models, 2017, 10 (4) : 1089-1125. doi: 10.3934/krm.2017043
[12]

Belkacem Said-Houari, Radouane Rahali. Asymptotic behavior of the solution to the Cauchy problem for the Timoshenko system in thermoelasticity of type III. Evolution Equations & Control Theory, 2013, 2 (2) : 423-440. doi: 10.3934/eect.2013.2.423
[13]

Luci H. Fatori, Marcio A. Jorge Silva, Vando Narciso. Quasi-stability property and attractors for a semilinear Timoshenko system. Discrete & Continuous Dynamical Systems, 2016, 36 (11) : 6117-6132. doi: 10.3934/dcds.2016067
[14]

Salim A. Messaoudi, Muhammad I. Mustafa. A general stability result in a memory-type Timoshenko system. Communications on Pure & Applied Analysis, 2013, 12 (2) : 957-972. doi: 10.3934/cpaa.2013.12.957
[15]

Baowei Feng. On a semilinear Timoshenko-Coleman-Gurtin system: Quasi-stability and attractors. Discrete & Continuous Dynamical Systems, 2017, 37 (9) : 4729-4751. doi: 10.3934/dcds.2017203
[16]

Makram Hamouda, Ahmed Bchatnia, Mohamed Ali Ayadi. Numerical solutions for a Timoshenko-type system with thermoelasticity with second sound. Discrete & Continuous Dynamical Systems - S, 2021, 14 (8) : 2975-2992. doi: 10.3934/dcdss.2021001
[17]

Matt Holzer. A proof of anomalous invasion speeds in a system of coupled Fisher-KPP equations. Discrete & Continuous Dynamical Systems, 2016, 36 (4) : 2069-2084. doi: 10.3934/dcds.2016.36.2069
[18]

Juan Dávila, Olivier Goubet. Partial regularity for a Liouville system. Discrete & Continuous Dynamical Systems, 2014, 34 (6) : 2495-2503. doi: 10.3934/dcds.2014.34.2495
[19]

Jing Zhang. The analyticity and exponential decay of a Stokes-wave coupling system with viscoelastic damping in the variational framework. Evolution Equations & Control Theory, 2017, 6 (1) : 135-154. doi: 10.3934/eect.2017008
[20]

Nguyen Thanh Long, Hoang Hai Ha, Le Thi Phuong Ngoc, Nguyen Anh Triet. Existence, blow-up and exponential decay estimates for a system of nonlinear viscoelastic wave equations with nonlinear boundary conditions. Communications on Pure & Applied Analysis, 2020, 19 (1) : 455-492. doi: 10.3934/cpaa.2020023

2020 Impact Factor: 1.081

Tools

Metrics

  • PDF downloads (135)
  • HTML views (0)
  • Cited by (21)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences