September  2017, 37(9): 4729-4751. doi: 10.3934/dcds.2017203

On a semilinear Timoshenko-Coleman-Gurtin system: Quasi-stability and attractors

Department of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu 611130, China

Received  December 2016 Revised  April 2017 Published  June 2017

A semilinear Timoshenko-Coleman-Gurtin system is studied. The system describes a Timoshenko beam coupled with a temperature with Coleman-Gurtin law. Under some assumptions on nonlinear damping terms and nonlinear source terms, we establish the global well-posedness of the system. The main result is the long-time dynamics of the system. By using the methods developed by Chueshov and Lasiecka, we get the quasi-stability property of the system and obtain the existence of a global attractor which has finite fractal dimension. Result on exponential attractors of the system is also proved.

Citation: Baowei Feng. On a semilinear Timoshenko-Coleman-Gurtin system: Quasi-stability and attractors. Discrete and Continuous Dynamical Systems, 2017, 37 (9) : 4729-4751. doi: 10.3934/dcds.2017203
References:
[1]

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

[2]

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

[3]

F. Ammar-KhodjaS. 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.

[4]

F. Ammar-KhodjaA. BenabdallahJ. 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.

[5]

A. R. A. Barbosa and T. F. Ma, long-time dynamics of an extensible plate equation with thermal memory, J. Math. Anal. Appl., 416 (2014), 143-165.  doi: 10.1016/j.jmaa.2014.02.042.

[6]

V. Barbu, Nonlinear Differential Equations of Monotone Types in Banach Spaces, Springer, New York, 2010. doi: 10.1007/978-1-4419-5542-5.

[7]

M. M. CavalcantiV. N. Domingos CavalcantiF. A. Falcão NascimentoI. 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]

W. CharlesJ. A. SorianoF. A. Falcão Nascimento and J. H. Rodrigues, Decay rates for Bresse system with arbitrary nonlinear localized damping, J. Differential Equations, 255 (2013), 2267-2290.  doi: 10.1016/j.jde.2013.06.014.

[9]

I. D. Chueshov, Introduction to the Theory of Infinite Dimensional Dissipative Systems, AKTA, Kharkiv, 1999.

[10]

I. D. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping in Mem. Amer. Math. Soc. , 195 (2008), ⅷ+183 pp. doi: 10.1007/978-0-387-87712-9.

[11]

I. D. Chueshov and I. Lasiecka, Von Karman Evolution Equations Springer Verlag, 2010. doi: 10.1016/j.jde.2014.04.009.

[12]

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.

[13]

L. H. FatoriM. A. Jorge Silva and V. Narciso, Quasi-stability property and attractors for a semilinear Timoshenko system, Discrete Conti. Dyn. Sys., 36 (2016), 6117-6132.  doi: 10.3934/dcds.2016067.

[14]

L. H. FatoriR. 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.

[15]

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

[16]

B. Feng and X. Yang, Long-time dynamics for a nonlinear Timoshenko system with delay, Appl. Anal., 96 (2017), 606-625.  doi: 10.1080/00036811.2016.1148139.

[17]

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

[18]

C. GiorgiA. Marzocchi and V. Pata, Asymptotic behavior of a semilinear problem in heat conduction with memory, NoDEA Nonlinear Differ. Equ. Appl., 5 (1998), 333-354.  doi: 10.1007/s000300050049.

[19]

C. Giorgi and V. Pata, Stability of abstract linear thermoelastic systems with memory, Math. Models Methods Appl. Sci., 11 (2001), 627-644.  doi: 10.1142/S0218202501001021.

[20]

M. Grasselli and V. Pata, Uniform attractors ofnonautonomous dynamical systems with memory, in Evolution Equations, Semigroups and Functional Analysis (eds. A. Lorenzi and B. Rus), (2000), 155–178, Progr. Nonlinear Differential Equations Appl., 50, Birkhäuser, Basel, 2002.

[21]

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

[22]

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

[23]

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

[24]

T. F. Ma and R. N. Monteiro, Singular limit and long-time dynamics of Bresse systems, preprint, arXiv: 1511.06786.

[25]

S. A. Messaoudi and M. I. Mustafa, On the Internal and Boundary Stabilization of Timoshenko Beams, NoDEA Nonlinear Differential Equations Appl., 15 (2008), 655-671.  doi: 10.1007/s00030-008-7075-3.

[26]

S. A. Messaoudi and A. Soufyane, Boundary stabilization of a nonlinear system of Timoshenko type, Nonlinear Anal., 67 (2007), 2107-2121.  doi: 10.1016/j.na.2006.08.039.

[27]

S. A. Messaoudi and B. Said-Houari, Uniform decay in a Timoshenko-type system with past history, J. Math. Anal. Appl., 360 (2009), 459-475.  doi: 10.1016/j.jmaa.2009.06.064.

[28]

S. A. Messaoudi and B. Said-Houari, Energy decay in a Timoshenko-type system of thermoelasticity of type Ⅲ, J. Math. Anal. Appl., 348 (2008), 298-307.  doi: 10.1016/j.jmaa.2008.07.036.

[29]

S. A. Messaoudi and B. Said-Houari, Energy decay in a Timoshenko-type system with history in thermoelasticity of type Ⅲ, Adv. Differ. Equ., 14 (2009), 375-400. 

[30]

M. I. Mustafa and S. A. Messaoudi, General energy decay rates for a weakly damped Timoshenko system, J. Dyna. Control Sys., 16 (2010), 211-226.  doi: 10.1007/s10883-010-9090-z.

[31]

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

[32]

C. A. RaposoJ. FerreiraM. L. Santos and N. N. O. Castro, Exponential stability for the Timoshenko system with two weak dampings, Appl. Math. Let., 18 (2005), 535-541.  doi: 10.1016/j.aml.2004.03.017.

[33]

J. C. Robinson, Infinite-Dimensional Dynamical Systems, An introduction to dissipative parabolic PDEs and the theory of global attractor, Cambridge University Press, 2001. doi: 10.1007/978-94-010-0732-0.

[34]

M. L. SantosD. S. Almeida Júnior and J. E. Muñoz Rivera, 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.

[35]

M. L. Santos and D. S. Almeida Júnior, On Timoshenko-type systems with type Ⅲ thermoelasticity: Asymptotic behavior, J. Math. Anal. Appl., 448 (2017), 650-671.  doi: 10.1016/j.jmaa.2016.10.074.

[36]

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.

[37]

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.

[38]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Appl. Math. Sci., 68 Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-0645-3.

[39]

S. Timoshenko, On the correction for shear of the differential equation for transverse vibrations of prismaticbars, Philos. Mag., 41 (1921), 744-746. 

[40]

G. Q. Xu, Feedback exponential stabilization of a Timoshenko beam with both ends free, Int. J. Control, 78 (2005), 286-297.  doi: 10.1080/00207170500095148.

[41]

G. Q. Xu and S. P. Yung, Exponential decay rate for a Timoshenko beam with boundary damping, J. Optim. Theory Appl., 123 (2004), 669-693.  doi: 10.1007/s10957-004-5728-x.

show all references

References:
[1]

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

[2]

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

[3]

F. Ammar-KhodjaS. 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.

[4]

F. Ammar-KhodjaA. BenabdallahJ. 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.

[5]

A. R. A. Barbosa and T. F. Ma, long-time dynamics of an extensible plate equation with thermal memory, J. Math. Anal. Appl., 416 (2014), 143-165.  doi: 10.1016/j.jmaa.2014.02.042.

[6]

V. Barbu, Nonlinear Differential Equations of Monotone Types in Banach Spaces, Springer, New York, 2010. doi: 10.1007/978-1-4419-5542-5.

[7]

M. M. CavalcantiV. N. Domingos CavalcantiF. A. Falcão NascimentoI. 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]

W. CharlesJ. A. SorianoF. A. Falcão Nascimento and J. H. Rodrigues, Decay rates for Bresse system with arbitrary nonlinear localized damping, J. Differential Equations, 255 (2013), 2267-2290.  doi: 10.1016/j.jde.2013.06.014.

[9]

I. D. Chueshov, Introduction to the Theory of Infinite Dimensional Dissipative Systems, AKTA, Kharkiv, 1999.

[10]

I. D. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping in Mem. Amer. Math. Soc. , 195 (2008), ⅷ+183 pp. doi: 10.1007/978-0-387-87712-9.

[11]

I. D. Chueshov and I. Lasiecka, Von Karman Evolution Equations Springer Verlag, 2010. doi: 10.1016/j.jde.2014.04.009.

[12]

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.

[13]

L. H. FatoriM. A. Jorge Silva and V. Narciso, Quasi-stability property and attractors for a semilinear Timoshenko system, Discrete Conti. Dyn. Sys., 36 (2016), 6117-6132.  doi: 10.3934/dcds.2016067.

[14]

L. H. FatoriR. 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.

[15]

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

[16]

B. Feng and X. Yang, Long-time dynamics for a nonlinear Timoshenko system with delay, Appl. Anal., 96 (2017), 606-625.  doi: 10.1080/00036811.2016.1148139.

[17]

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

[18]

C. GiorgiA. Marzocchi and V. Pata, Asymptotic behavior of a semilinear problem in heat conduction with memory, NoDEA Nonlinear Differ. Equ. Appl., 5 (1998), 333-354.  doi: 10.1007/s000300050049.

[19]

C. Giorgi and V. Pata, Stability of abstract linear thermoelastic systems with memory, Math. Models Methods Appl. Sci., 11 (2001), 627-644.  doi: 10.1142/S0218202501001021.

[20]

M. Grasselli and V. Pata, Uniform attractors ofnonautonomous dynamical systems with memory, in Evolution Equations, Semigroups and Functional Analysis (eds. A. Lorenzi and B. Rus), (2000), 155–178, Progr. Nonlinear Differential Equations Appl., 50, Birkhäuser, Basel, 2002.

[21]

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

[22]

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

[23]

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

[24]

T. F. Ma and R. N. Monteiro, Singular limit and long-time dynamics of Bresse systems, preprint, arXiv: 1511.06786.

[25]

S. A. Messaoudi and M. I. Mustafa, On the Internal and Boundary Stabilization of Timoshenko Beams, NoDEA Nonlinear Differential Equations Appl., 15 (2008), 655-671.  doi: 10.1007/s00030-008-7075-3.

[26]

S. A. Messaoudi and A. Soufyane, Boundary stabilization of a nonlinear system of Timoshenko type, Nonlinear Anal., 67 (2007), 2107-2121.  doi: 10.1016/j.na.2006.08.039.

[27]

S. A. Messaoudi and B. Said-Houari, Uniform decay in a Timoshenko-type system with past history, J. Math. Anal. Appl., 360 (2009), 459-475.  doi: 10.1016/j.jmaa.2009.06.064.

[28]

S. A. Messaoudi and B. Said-Houari, Energy decay in a Timoshenko-type system of thermoelasticity of type Ⅲ, J. Math. Anal. Appl., 348 (2008), 298-307.  doi: 10.1016/j.jmaa.2008.07.036.

[29]

S. A. Messaoudi and B. Said-Houari, Energy decay in a Timoshenko-type system with history in thermoelasticity of type Ⅲ, Adv. Differ. Equ., 14 (2009), 375-400. 

[30]

M. I. Mustafa and S. A. Messaoudi, General energy decay rates for a weakly damped Timoshenko system, J. Dyna. Control Sys., 16 (2010), 211-226.  doi: 10.1007/s10883-010-9090-z.

[31]

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

[32]

C. A. RaposoJ. FerreiraM. L. Santos and N. N. O. Castro, Exponential stability for the Timoshenko system with two weak dampings, Appl. Math. Let., 18 (2005), 535-541.  doi: 10.1016/j.aml.2004.03.017.

[33]

J. C. Robinson, Infinite-Dimensional Dynamical Systems, An introduction to dissipative parabolic PDEs and the theory of global attractor, Cambridge University Press, 2001. doi: 10.1007/978-94-010-0732-0.

[34]

M. L. SantosD. S. Almeida Júnior and J. E. Muñoz Rivera, 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.

[35]

M. L. Santos and D. S. Almeida Júnior, On Timoshenko-type systems with type Ⅲ thermoelasticity: Asymptotic behavior, J. Math. Anal. Appl., 448 (2017), 650-671.  doi: 10.1016/j.jmaa.2016.10.074.

[36]

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.

[37]

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.

[38]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Appl. Math. Sci., 68 Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-0645-3.

[39]

S. Timoshenko, On the correction for shear of the differential equation for transverse vibrations of prismaticbars, Philos. Mag., 41 (1921), 744-746. 

[40]

G. Q. Xu, Feedback exponential stabilization of a Timoshenko beam with both ends free, Int. J. Control, 78 (2005), 286-297.  doi: 10.1080/00207170500095148.

[41]

G. Q. Xu and S. P. Yung, Exponential decay rate for a Timoshenko beam with boundary damping, J. Optim. Theory Appl., 123 (2004), 669-693.  doi: 10.1007/s10957-004-5728-x.

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