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July  2020, 19(7): 3575-3596. doi: 10.3934/cpaa.2020156

General condition for exponential stability of thermoelastic Bresse systems with Cattaneo's law

Pedro Roberto de Lima 1,, and  Hugo D. Fernández Sare 2,

1. 

Departamento de Matemática, Universidade Estadual do Centro-Oeste, Guarapuava, PR, CEP 85040-167, Brazil
2. 

Departamento de Matemática, Universidade Federal de Juiz de Fora, Juiz de Fora, MG, CEP 36036-900, Brazil

*Corresponding author

Received  May 2019 Revised  January 2020 Published  April 2020

Fund Project: The first author was supported by CAPES at the Instituto de Matemática of the Universidade Federal do Rio de Janeiro when this work was submitted

In this paper, we give a new and more general sufficient condition for exponential stability of thermoelastic Bresse systems with heat flux given by Cattaneo's law acting in shear and longitudinal motion equations. This condition, which we also prove to be necessary in some special cases, is given by a relation between the constants of the system and generalizes the well-known equal wave speed condition.

Keywords: Bresse system, exponential Stability, Cattaneo's law.
Mathematics Subject Classification: Primary: 35B40, 35E15, 35Q74; Secondary: 74B05, 74F05.
Citation: Pedro Roberto de Lima, Hugo D. Fernández Sare. General condition for exponential stability of thermoelastic Bresse systems with Cattaneo's law. Communications on Pure & Applied Analysis, 2020, 19 (7) : 3575-3596. doi: 10.3934/cpaa.2020156
References:
[1]

M. Afilal, A. Guesmia and A. Soufyane, New stability results for a linear thermoelastic Bresse system with second sound, Appl. Math. Optim., (2019), 40 pp. doi: 10.1007/s00245-019-09560-7.  Google Scholar

[2]

P. R. de Lima and H. D. Fernández Sare, Stability of thermoelastic Bresse systems, Z. Angew. Math. Phys., 70 (2019), 30 pp. doi: 10.1007/s00033-018-1057-z.  Google Scholar

[3]

F. Dell'Oro, Asymptotic stability of thermoelastic systems of Bresse type, J. Differ. Equ., 258 (2015), 3902-3927.  doi: 10.1016/j.jde.2015.01.025.  Google Scholar

[4]

K. J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer-Verlag, New York, 2000. doi: 10.1007/b97696.  Google Scholar

[5]

G. Garbugio, Modelagem e Estabilidade Uniforme de Vigas Curvas Termoelásticas, (Portuguese) [Modeling and Uniform Stability of Thermoelastic Curved Beams], Ph.D thesis, Laboratório Nacional de Computação Científica, Petrópolis, Brazil, 2015. Google Scholar

[6]

A. Guesmia, The effect of the heat conduction of types I and III on the decay rate of the Bresse system via the longitudinal displacement, Arab. J. Math., 8 (2019), 15-41.  doi: 10.1007/s40065-018-0210-z.  Google Scholar

[7]

S. Kesavan, Functional Analysis, Hindustan Book Agency, New Delhi, 2009. doi: 10.1007/978-93-86279-42-2.  Google Scholar

[8]

J. E. Lagnese, G. Leugering and E. J. P. G. Schmidt, Modeling, Analysis and Control of Dynamic Elastic Multi-Link Structures, Birkhäuser, Boston, 1994. doi: 10.1007/978-1-4612-0273-8.  Google Scholar

[9]

J. E. Lagnese, G. Leugering and E. J. P. G. Schmidt, Modelling of dynamic networks of thin thermoelastic beams, Math. Meth. Appl. Sci., 16 (1993), 327-358.  doi: 10.1002/mma.1670160503.  Google Scholar

[10]

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

[11]

Z. Liu and S. Zheng, Semigroups Associated with Dissipative Systems, Chapman & Hall, London, 1999.  Google Scholar

[12]

M. L. Santos, D. S. Almeida Júnior and J. E. Muñoz Rivera, The stability number of the Timoshenko system with second sound, J. Differ. Equ., 253 (2012), 2715-2733.  doi: 10.1016/j.jde.2012.07.012.  Google Scholar

show all references

References:
[1]

M. Afilal, A. Guesmia and A. Soufyane, New stability results for a linear thermoelastic Bresse system with second sound, Appl. Math. Optim., (2019), 40 pp. doi: 10.1007/s00245-019-09560-7.  Google Scholar

[2]

P. R. de Lima and H. D. Fernández Sare, Stability of thermoelastic Bresse systems, Z. Angew. Math. Phys., 70 (2019), 30 pp. doi: 10.1007/s00033-018-1057-z.  Google Scholar

[3]

F. Dell'Oro, Asymptotic stability of thermoelastic systems of Bresse type, J. Differ. Equ., 258 (2015), 3902-3927.  doi: 10.1016/j.jde.2015.01.025.  Google Scholar

[4]

K. J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer-Verlag, New York, 2000. doi: 10.1007/b97696.  Google Scholar

[5]

G. Garbugio, Modelagem e Estabilidade Uniforme de Vigas Curvas Termoelásticas, (Portuguese) [Modeling and Uniform Stability of Thermoelastic Curved Beams], Ph.D thesis, Laboratório Nacional de Computação Científica, Petrópolis, Brazil, 2015. Google Scholar

[6]

A. Guesmia, The effect of the heat conduction of types I and III on the decay rate of the Bresse system via the longitudinal displacement, Arab. J. Math., 8 (2019), 15-41.  doi: 10.1007/s40065-018-0210-z.  Google Scholar

[7]

S. Kesavan, Functional Analysis, Hindustan Book Agency, New Delhi, 2009. doi: 10.1007/978-93-86279-42-2.  Google Scholar

[8]

J. E. Lagnese, G. Leugering and E. J. P. G. Schmidt, Modeling, Analysis and Control of Dynamic Elastic Multi-Link Structures, Birkhäuser, Boston, 1994. doi: 10.1007/978-1-4612-0273-8.  Google Scholar

[9]

J. E. Lagnese, G. Leugering and E. J. P. G. Schmidt, Modelling of dynamic networks of thin thermoelastic beams, Math. Meth. Appl. Sci., 16 (1993), 327-358.  doi: 10.1002/mma.1670160503.  Google Scholar

[10]

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

[11]

Z. Liu and S. Zheng, Semigroups Associated with Dissipative Systems, Chapman & Hall, London, 1999.  Google Scholar

[12]

M. L. Santos, D. S. Almeida Júnior and J. E. Muñoz Rivera, The stability number of the Timoshenko system with second sound, J. Differ. Equ., 253 (2012), 2715-2733.  doi: 10.1016/j.jde.2012.07.012.  Google Scholar

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