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November  2016, 36(11): 6117-6132. doi: 10.3934/dcds.2016067

Quasi-stability property and attractors for a semilinear Timoshenko system

1. 

Department of Mathematics, State University of Londrina, Londrina PR, 86057-970, Brazil, Brazil

2. 

Center of exact sciences, State University of Mato Grosso do Sul, Dourados, 79804-970

Received  December 2015 Revised  February 2016 Published  August 2016

This paper is concerned with the classical Timoshenko system for vibrations of thin rods. It has been studied by many authors and most of known results are concerned with decay rates of the energy, controllability and numerical approximations. There are just a few references on the long-time dynamics of such systems. Motivated by this scenario we establish the existence of global and exponential attractors for a class of semilinear Timoshenko systems with linear frictional damping acting on the whole system and without assuming the well-known equal wave speeds condition.
Citation: Luci H. Fatori, Marcio A. Jorge Silva, Vando Narciso. Quasi-stability property and attractors for a semilinear Timoshenko system. Discrete and Continuous Dynamical Systems, 2016, 36 (11) : 6117-6132. doi: 10.3934/dcds.2016067
References:
[1]

F. Alabau-Boussouira, Asymptotic behavior for Timoshenko beams subject to a single nonlinear feedback control, NoDEA Nonlinear Differential Equations Appl., 14 (2007), 643-669. doi: 10.1007/s00030-007-5033-0.

[2]

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

[3]

D. S. Almeida Júnior, J. E. Muñoz Rivera and M. L. Santos, Stability to weakly dissipative Timoshenko systems, Math. Meth. Appl. Sci., 36 (2013), 1965-1976. doi: 10.1002/mma.2741.

[4]

F. Ammar-Khodja, S. Kerbal and A. Soufyane, Stabilization of the nonuniform Timoshenko beam, J. Math. Anal. Appl., 327 (2007), 525-538. doi: 10.1016/j.jmaa.2006.04.016.

[5]

F. Ammar-Khodja, A. Benabdallah, J. E. Muñoz Rivera and R. Racke, Energy decay for Timoshenko systems of memory type, J. Differential Equations, 194 (2003), 82-115. doi: 10.1016/S0022-0396(03)00185-2.

[6]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, Studies in Mathematics and its Application 25, North-Holland, Amsterdam, 1992.

[7]

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., 65 (2014), 1189-1206. doi: 10.1007/s00033-013-0380-7.

[8]

I. Chueshov and I. Lasiecka, Long-Time Behavior of Second Order Evolution Equations with Nonlinear Damping, Mem. Amer. Math. Soc. 195, no. 912, Providence, 2008. doi: 10.1090/memo/0912.

[9]

I. Chueshov and I. Lasiecka, Von Karman Evolution Equations. Well-Posedness and Long-Time Dynamics, Springer Monographs in Mathematics, Springer, New York, 2010. doi: 10.1007/978-0-387-87712-9.

[10]

F. Dell'Oro and V. Pata, On the stability of Timoshenko systems with Gurtin-Pipkin thermal law, J. Differential Equations, 257 (2014), 523-548. doi: 10.1016/j.jde.2014.04.009.

[11]

A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative Evolution Equations, RAM: Research in Applied Mathematics, 37. Masson, Paris; John Wiley & Sons, Ltd., Chichester, 1994.

[12]

D-X Feng and D-H Shi, Exponential decay of Timoshenko beam with locally distributed feedback, IMA J. Math. Control Inform., 18 (2001), 395-403. doi: 10.1093/imamci/18.3.395.

[13]

L. H. Fatori, R. N. Monteiro and H. D. Fernández Sare, The Timoshenko system with history and Cattaneo law, Appl. Math. Comput., 228 (2014), 128-140. doi: 10.1016/j.amc.2013.11.054.

[14]

L. H. Fatori, R. N. Monteiro and J. E. Muñoz Rivera, Energy decay to Timoshenko's system with thermoelasticity of type III, Asymptot. Anal., 86 (2014), 227-247.

[15]

H. D. Fernández Sare and R. Racke, On the stability of damped Timoshenko systems: Cattaneo versus Fourier law, Arch. Ration. Mech. Anal., 194 (2009), 221-251. doi: 10.1007/s00205-009-0220-2.

[16]

M. Grasselli, V. Pata and G. Prouse, Longtime behavior of a viscoelastic Timoshenko beam, Discrete Contin. Dyn. Syst., 10 (2004), 337-348. doi: 10.3934/dcds.2004.10.337.

[17]

A. Guesmia and S. A. Messaoudi, A general stability result in a Timoshenko system with infinite memory: a new approach, Math. Methods Appl. Sci., 37 (2014), 384-392. doi: 10.1002/mma.2797.

[18]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs, 25. American Mathematical Society, Providence, RI, 1988.

[19]

M. A. Jorge Silva and V. Narciso, Attractors and their properties for a class of nonlocal extensible beams, Discrete Contin. Dyn. Syst., 35 (2015), 985-1008. doi: 10.3934/dcds.2015.35.985.

[20]

J. U. Kim and Y. Renardy, Boundary control of the Timoshenko beam, SIAM J. Control Optim., 25 (1987), 1417-1429. doi: 10.1137/0325078.

[21]

J. E. Muñoz Rivera and R. Racke, Mildly dissipative nonlinear Timoshenko systems-global existence and exponential stability, J. Math. Anal. Appl., 276 (2002), 248-278. doi: 10.1016/S0022-247X(02)00436-5.

[22]

J. E. Muñoz Rivera and R. Racke, Global stability for damped Timoshenko systems, Discrete Contin. Dyn. Syst., 9 (2003), 1625-1639. doi: 10.3934/dcds.2003.9.1625.

[23]

J. E. Muñoz Rivera and R. Racke, Timoshenko systems with indefinite damping, J. Math. Anal. Appl., 341 (2008), 1068-1083. doi: 10.1016/j.jmaa.2007.11.012.

[24]

P. Olsson and G. Kristensson, Wave splitting of the Timoshenko beam equation in the time domain, Z. Angew. Math. Phys., 45 (1994), 866-881. doi: 10.1007/BF00952082.

[25]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[26]

M. Potomkin, Asymptotic behavior of thermoviscoelastic Berger plate, Commun. Pure Appl. Anal., 9 (2010), 161-192. doi: 10.3934/cpaa.2010.9.161.

[27]

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. Lett., 18 (2005), 535-541. doi: 10.1016/j.aml.2004.03.017.

[28]

D. H. Shi and D. X. Feng, Exponential decay of Timoshenko beam with locally distributed feedback, in Proceeding of the IFAC World Congress, Beijing, vol. F, 1999. doi: 10.1093/imamci/18.3.395.

[29]

A. Soufyane, Stabilisation de la poutre de Timoshenko, C. R. Acad. Sci. Paris, Sér. I Math., 328 (1999), 731-734. doi: 10.1016/S0764-4442(99)80244-4.

[30]

A. Soufyane, Exponential stability of the linearized nonuniform Timoshenko beam, Nonlinear Anal. Real World Appl., 10 (2009), 1016-1020. doi: 10.1016/j.nonrwa.2007.11.019.

[31]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences 68. Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4684-0313-8.

[32]

S. P. Timoshenko, On the correction for shear of the differential equation for transverse vibrations of prismatic bars, Philosophical Magazine, 41 (1921), 744-746. doi: 10.1080/14786442108636264.

[33]

S. P. Timoshenko, Vibration Problems in Engineering, Van Nostrand, New York, 1955.

show all references

References:
[1]

F. Alabau-Boussouira, Asymptotic behavior for Timoshenko beams subject to a single nonlinear feedback control, NoDEA Nonlinear Differential Equations Appl., 14 (2007), 643-669. doi: 10.1007/s00030-007-5033-0.

[2]

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

[3]

D. S. Almeida Júnior, J. E. Muñoz Rivera and M. L. Santos, Stability to weakly dissipative Timoshenko systems, Math. Meth. Appl. Sci., 36 (2013), 1965-1976. doi: 10.1002/mma.2741.

[4]

F. Ammar-Khodja, S. Kerbal and A. Soufyane, Stabilization of the nonuniform Timoshenko beam, J. Math. Anal. Appl., 327 (2007), 525-538. doi: 10.1016/j.jmaa.2006.04.016.

[5]

F. Ammar-Khodja, A. Benabdallah, J. E. Muñoz Rivera and R. Racke, Energy decay for Timoshenko systems of memory type, J. Differential Equations, 194 (2003), 82-115. doi: 10.1016/S0022-0396(03)00185-2.

[6]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, Studies in Mathematics and its Application 25, North-Holland, Amsterdam, 1992.

[7]

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., 65 (2014), 1189-1206. doi: 10.1007/s00033-013-0380-7.

[8]

I. Chueshov and I. Lasiecka, Long-Time Behavior of Second Order Evolution Equations with Nonlinear Damping, Mem. Amer. Math. Soc. 195, no. 912, Providence, 2008. doi: 10.1090/memo/0912.

[9]

I. Chueshov and I. Lasiecka, Von Karman Evolution Equations. Well-Posedness and Long-Time Dynamics, Springer Monographs in Mathematics, Springer, New York, 2010. doi: 10.1007/978-0-387-87712-9.

[10]

F. Dell'Oro and V. Pata, On the stability of Timoshenko systems with Gurtin-Pipkin thermal law, J. Differential Equations, 257 (2014), 523-548. doi: 10.1016/j.jde.2014.04.009.

[11]

A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative Evolution Equations, RAM: Research in Applied Mathematics, 37. Masson, Paris; John Wiley & Sons, Ltd., Chichester, 1994.

[12]

D-X Feng and D-H Shi, Exponential decay of Timoshenko beam with locally distributed feedback, IMA J. Math. Control Inform., 18 (2001), 395-403. doi: 10.1093/imamci/18.3.395.

[13]

L. H. Fatori, R. N. Monteiro and H. D. Fernández Sare, The Timoshenko system with history and Cattaneo law, Appl. Math. Comput., 228 (2014), 128-140. doi: 10.1016/j.amc.2013.11.054.

[14]

L. H. Fatori, R. N. Monteiro and J. E. Muñoz Rivera, Energy decay to Timoshenko's system with thermoelasticity of type III, Asymptot. Anal., 86 (2014), 227-247.

[15]

H. D. Fernández Sare and R. Racke, On the stability of damped Timoshenko systems: Cattaneo versus Fourier law, Arch. Ration. Mech. Anal., 194 (2009), 221-251. doi: 10.1007/s00205-009-0220-2.

[16]

M. Grasselli, V. Pata and G. Prouse, Longtime behavior of a viscoelastic Timoshenko beam, Discrete Contin. Dyn. Syst., 10 (2004), 337-348. doi: 10.3934/dcds.2004.10.337.

[17]

A. Guesmia and S. A. Messaoudi, A general stability result in a Timoshenko system with infinite memory: a new approach, Math. Methods Appl. Sci., 37 (2014), 384-392. doi: 10.1002/mma.2797.

[18]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs, 25. American Mathematical Society, Providence, RI, 1988.

[19]

M. A. Jorge Silva and V. Narciso, Attractors and their properties for a class of nonlocal extensible beams, Discrete Contin. Dyn. Syst., 35 (2015), 985-1008. doi: 10.3934/dcds.2015.35.985.

[20]

J. U. Kim and Y. Renardy, Boundary control of the Timoshenko beam, SIAM J. Control Optim., 25 (1987), 1417-1429. doi: 10.1137/0325078.

[21]

J. E. Muñoz Rivera and R. Racke, Mildly dissipative nonlinear Timoshenko systems-global existence and exponential stability, J. Math. Anal. Appl., 276 (2002), 248-278. doi: 10.1016/S0022-247X(02)00436-5.

[22]

J. E. Muñoz Rivera and R. Racke, Global stability for damped Timoshenko systems, Discrete Contin. Dyn. Syst., 9 (2003), 1625-1639. doi: 10.3934/dcds.2003.9.1625.

[23]

J. E. Muñoz Rivera and R. Racke, Timoshenko systems with indefinite damping, J. Math. Anal. Appl., 341 (2008), 1068-1083. doi: 10.1016/j.jmaa.2007.11.012.

[24]

P. Olsson and G. Kristensson, Wave splitting of the Timoshenko beam equation in the time domain, Z. Angew. Math. Phys., 45 (1994), 866-881. doi: 10.1007/BF00952082.

[25]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[26]

M. Potomkin, Asymptotic behavior of thermoviscoelastic Berger plate, Commun. Pure Appl. Anal., 9 (2010), 161-192. doi: 10.3934/cpaa.2010.9.161.

[27]

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. Lett., 18 (2005), 535-541. doi: 10.1016/j.aml.2004.03.017.

[28]

D. H. Shi and D. X. Feng, Exponential decay of Timoshenko beam with locally distributed feedback, in Proceeding of the IFAC World Congress, Beijing, vol. F, 1999. doi: 10.1093/imamci/18.3.395.

[29]

A. Soufyane, Stabilisation de la poutre de Timoshenko, C. R. Acad. Sci. Paris, Sér. I Math., 328 (1999), 731-734. doi: 10.1016/S0764-4442(99)80244-4.

[30]

A. Soufyane, Exponential stability of the linearized nonuniform Timoshenko beam, Nonlinear Anal. Real World Appl., 10 (2009), 1016-1020. doi: 10.1016/j.nonrwa.2007.11.019.

[31]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences 68. Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4684-0313-8.

[32]

S. P. Timoshenko, On the correction for shear of the differential equation for transverse vibrations of prismatic bars, Philosophical Magazine, 41 (1921), 744-746. doi: 10.1080/14786442108636264.

[33]

S. P. Timoshenko, Vibration Problems in Engineering, Van Nostrand, New York, 1955.

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