American Institute of Mathematical Sciences

Advanced
August  2022, 15(8): 2189-2207. doi: 10.3934/dcdss.2022050

Exponential stability of Timoshenko-Gurtin-Pipkin systems with full thermal coupling

Filippo Dell'Oro 1,, , Marcio A. Jorge Silva 2,†, and  Sandro B. Pinheiro 3,‡,

1. 

Politecnico di Milano, Dipartimento di Matematica, Via Bonardi 9, 20133 Milano, Italy
2. 

Department of Mathematics, State University of Londrina, Londrina 86057-970, Paraná, Brazil
3. 

Department of Mathematics, State University of Maringá, Maringá 87020-900, Paraná, Brazil

* Corresponding author: Filippo Dell'Oro

† The second author has been supported by the CNPq, Grant #301116/2019-9
‡ The third author has been supported by the CAPES, Finance Code 001 (PICME Scholarship)
Dedicated to Professor Maurizio Grasselli on his 60th birthday

Received  November 2021 Published  August 2022 Early access  March 2022

We analyze the stability properties of a linear thermoelastic Timoshenko-Gurtin-Pipkin system with thermal coupling acting on both the shear force and the bending moment. Under either the mixed Dirichlet-Neumann or else the full Dirichlet boundary conditions, we show that the associated solution semigroup in the history space framework of Dafermos is exponentially stable independently of the values of the structural parameters of the model.

Keywords: Timoshenko system, Gurtin-Pipkin law, thermal coupling, exponential stability.
Mathematics Subject Classification: Primary: 35B40, 45K05; Secondary: 47D03, 74D05.
Citation: Filippo Dell'Oro, Marcio A. Jorge Silva, Sandro B. Pinheiro. Exponential stability of Timoshenko-Gurtin-Pipkin systems with full thermal coupling. Discrete and Continuous Dynamical Systems - S, 2022, 15 (8) : 2189-2207. doi: 10.3934/dcdss.2022050
References:
[1]

D. S. Almeida Júnior, M. L. Santos and J. E. Muñoz Rivera, Stability to 1-D thermoelastic Timoshenko beam acting on shear force, Z. Angew. Math. Phys., 65 (2014), 1233-1249.  doi: 10.1007/s00033-013-0387-0.

[2]

M. O. Alves, A. H. Caixeta, M. A. Jorge Silva, J. H. Rodrigues and D. S. Almeida Júnior, On a Timoshenko system with thermal coupling on both the bending moment and the shear force, J. Evol. Equ., 20 (2020), 295-320.  doi: 10.1007/s00028-019-00522-8.

[3]

M. S. Alves, M. A. Jorge Silva, T. F. Ma and J. E. Muñoz Rivera, Non-homogeneous thermoelastic Timoshenko systems, Bull. Braz. Math. Soc., 48 (2017), 461-484.  doi: 10.1007/s00574-017-0030-3.

[4]

W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander, Vector-Valued Laplace Transforms and Cauchy Problems, Birkhäuser, Basel, 2011. doi: 10.1007/978-3-0348-5075-9.

[5]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011. doi: 10.1007/978-0-387-70914-7.

[6]

C. L. Cardozo, M. A. Jorge Silva, T. F. Ma and J. E. Muñoz Rivera, Stability of Timoshenko systems with thermal coupling on the bending moment, Math. Nachr., 292 (2019), 2537-2555.  doi: 10.1002/mana.201800546.

[7]

C. Cattaneo, Sulla conduzione del calore, Atti Semin. Matemat. Univ. Modena, 3 (1949), 83-101.  doi: 10.1007/978-3-642-11051-1_5.

[8]

C. I. Christov and P. M. Jordan, Heat conduction paradox involving second-sound propagation in moving media, Phys. Rev. Lett., 94 (2005), 154301.  doi: 10.1103/PhysRevLett.94.154301.

[9]

B. D. Coleman and M. E. Gurtin, Equipresence and constitutive equations for rigid heat conductors, Z. Angew. Math. Phys., 18 (1967), 199-208.  doi: 10.1007/BF01596912.

[10]

M. Conti, V. Pata and M. Squassina, Singular limit of differential systems with memory, Indiana Univ. Math. J., 55 (2006), 169-215.  doi: 10.1512/iumj.2006.55.2661.

[11]

C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal., 37 (1970), 297-308.  doi: 10.1007/BF00251609.

[12]

V. Danese and F. Dell'Oro, The lack of exponential stability for a class of second-order systems with memory, Proc. Roy. Soc. Edinburgh Sect. A, 147 (2017), 683-702.  doi: 10.1017/S0308210516000330.

[13]

F. Dell'Oro, On the stability of Bresse and Timoshenko systems with hyperbolic heat conduction, J. Differential Equations, 281 (2021), 148-198.  doi: 10.1016/j.jde.2021.02.009.

[14]

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.

[15]

F. Djellali, S. Labidi and F. Taallah, Exponential stability of thermoelastic Timoshenko system with Cattaneo's law, Ann. Univ. Ferrara Sez. VII Sci. Mat., 67 (2021), 43-57.  doi: 10.1007/s11565-021-00360-y.

[16]

L. Gearhart, Spectral theory for contraction semigroups on Hilbert space, Trans. Amer. Math. Soc., 236 (1978), 385-394.  doi: 10.1090/S0002-9947-1978-0461206-1.

[17]

M. Grasselli and V. Pata, Uniform attractors of nonautonomous systems with memory in "Evolution Equations, Semigroups and Functional Analysis"(A. Lorenzi and B. Ruf, Eds.), Progr. Nonlinear Differential Equations Appl., Birkhäuser, Boston, 50 (2002), 155–178. doi: 10.1007/978-3-0348-8221-7_9.

[18]

M. E. Gurtin and A. C. Pipkin, A general theory of heat conduction with finite wave speeds, Arch. Rational Mech. Anal., 31 (1968), 113-126.  doi: 10.1007/BF00281373.

[19]

F. L. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Differential Equations, 1 (1985), 43-56. 

[20]

Z. Liu and S. Zheng, Semigroups Associated with Dissipative Systems, Chapman & Hall/CRC, Boca Raton, 1999.

[21]

J. E. Muñoz Rivera and A. I. Ávila, Rates of decay to non homogeneous Timoshenko model with tip body, J. Differential Equations, 258 (2015), 3468-3490.  doi: 10.1016/j.jde.2015.01.011.

[22]

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.

[23]

V. Pata, Stability and exponential stability in linear viscoelasticity, Milan J. Math., 77 (2009), 333-360.  doi: 10.1007/s00032-009-0098-3.

[24]

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

[25]

J. Prüss, On the spectrum of $C_0$-semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857.  doi: 10.2307/1999112.

[26]

M. L. Santos, D. 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.

[27]

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

[28]

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

show all references

References:
[1]

D. S. Almeida Júnior, M. L. Santos and J. E. Muñoz Rivera, Stability to 1-D thermoelastic Timoshenko beam acting on shear force, Z. Angew. Math. Phys., 65 (2014), 1233-1249.  doi: 10.1007/s00033-013-0387-0.

[2]

M. O. Alves, A. H. Caixeta, M. A. Jorge Silva, J. H. Rodrigues and D. S. Almeida Júnior, On a Timoshenko system with thermal coupling on both the bending moment and the shear force, J. Evol. Equ., 20 (2020), 295-320.  doi: 10.1007/s00028-019-00522-8.

[3]

M. S. Alves, M. A. Jorge Silva, T. F. Ma and J. E. Muñoz Rivera, Non-homogeneous thermoelastic Timoshenko systems, Bull. Braz. Math. Soc., 48 (2017), 461-484.  doi: 10.1007/s00574-017-0030-3.

[4]

W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander, Vector-Valued Laplace Transforms and Cauchy Problems, Birkhäuser, Basel, 2011. doi: 10.1007/978-3-0348-5075-9.

[5]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011. doi: 10.1007/978-0-387-70914-7.

[6]

C. L. Cardozo, M. A. Jorge Silva, T. F. Ma and J. E. Muñoz Rivera, Stability of Timoshenko systems with thermal coupling on the bending moment, Math. Nachr., 292 (2019), 2537-2555.  doi: 10.1002/mana.201800546.

[7]

C. Cattaneo, Sulla conduzione del calore, Atti Semin. Matemat. Univ. Modena, 3 (1949), 83-101.  doi: 10.1007/978-3-642-11051-1_5.

[8]

C. I. Christov and P. M. Jordan, Heat conduction paradox involving second-sound propagation in moving media, Phys. Rev. Lett., 94 (2005), 154301.  doi: 10.1103/PhysRevLett.94.154301.

[9]

B. D. Coleman and M. E. Gurtin, Equipresence and constitutive equations for rigid heat conductors, Z. Angew. Math. Phys., 18 (1967), 199-208.  doi: 10.1007/BF01596912.

[10]

M. Conti, V. Pata and M. Squassina, Singular limit of differential systems with memory, Indiana Univ. Math. J., 55 (2006), 169-215.  doi: 10.1512/iumj.2006.55.2661.

[11]

C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal., 37 (1970), 297-308.  doi: 10.1007/BF00251609.

[12]

V. Danese and F. Dell'Oro, The lack of exponential stability for a class of second-order systems with memory, Proc. Roy. Soc. Edinburgh Sect. A, 147 (2017), 683-702.  doi: 10.1017/S0308210516000330.

[13]

F. Dell'Oro, On the stability of Bresse and Timoshenko systems with hyperbolic heat conduction, J. Differential Equations, 281 (2021), 148-198.  doi: 10.1016/j.jde.2021.02.009.

[14]

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.

[15]

F. Djellali, S. Labidi and F. Taallah, Exponential stability of thermoelastic Timoshenko system with Cattaneo's law, Ann. Univ. Ferrara Sez. VII Sci. Mat., 67 (2021), 43-57.  doi: 10.1007/s11565-021-00360-y.

[16]

L. Gearhart, Spectral theory for contraction semigroups on Hilbert space, Trans. Amer. Math. Soc., 236 (1978), 385-394.  doi: 10.1090/S0002-9947-1978-0461206-1.

[17]

M. Grasselli and V. Pata, Uniform attractors of nonautonomous systems with memory in "Evolution Equations, Semigroups and Functional Analysis"(A. Lorenzi and B. Ruf, Eds.), Progr. Nonlinear Differential Equations Appl., Birkhäuser, Boston, 50 (2002), 155–178. doi: 10.1007/978-3-0348-8221-7_9.

[18]

M. E. Gurtin and A. C. Pipkin, A general theory of heat conduction with finite wave speeds, Arch. Rational Mech. Anal., 31 (1968), 113-126.  doi: 10.1007/BF00281373.

[19]

F. L. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Differential Equations, 1 (1985), 43-56. 

[20]

Z. Liu and S. Zheng, Semigroups Associated with Dissipative Systems, Chapman & Hall/CRC, Boca Raton, 1999.

[21]

J. E. Muñoz Rivera and A. I. Ávila, Rates of decay to non homogeneous Timoshenko model with tip body, J. Differential Equations, 258 (2015), 3468-3490.  doi: 10.1016/j.jde.2015.01.011.

[22]

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.

[23]

V. Pata, Stability and exponential stability in linear viscoelasticity, Milan J. Math., 77 (2009), 333-360.  doi: 10.1007/s00032-009-0098-3.

[24]

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

[25]

J. Prüss, On the spectrum of $C_0$-semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857.  doi: 10.2307/1999112.

[26]

M. L. Santos, D. 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.

[27]

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

[28]

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

[1]

Xiuxiang Zhou, Shu Luan. Controllability to rest of the Gurtin-Pipkin model. Mathematical Control and Related Fields, 2021  doi: 10.3934/mcrf.2021051
[2]

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
[3]

Luciano Pandolfi. Joint identification via deconvolution of the flux and energy relaxation kernels of the Gurtin-Pipkin model in thermodynamics with memory. Discrete and Continuous Dynamical Systems - S, 2020, 13 (5) : 1589-1599. doi: 10.3934/dcdss.2020090
[4]

Jaime E. Muñoz Rivera, Maria Grazia Naso. About the stability to Timoshenko system with pointwise dissipation. Discrete and Continuous Dynamical Systems - S, 2022, 15 (8) : 2289-2303. doi: 10.3934/dcdss.2022078
[5]

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
[6]

Salim A. Messaoudi, Muhammad I. Mustafa. A general stability result in a memory-type Timoshenko system. Communications on Pure and Applied Analysis, 2013, 12 (2) : 957-972. doi: 10.3934/cpaa.2013.12.957
[7]

Pedro Roberto de Lima, Hugo D. Fernández Sare. General condition for exponential stability of thermoelastic Bresse systems with Cattaneo's law. Communications on Pure and Applied Analysis, 2020, 19 (7) : 3575-3596. doi: 10.3934/cpaa.2020156
[8]

Gilbert Peralta. Uniform exponential stability of a fluid-plate interaction model due to thermal effects. Evolution Equations and Control Theory, 2020, 9 (1) : 39-60. doi: 10.3934/eect.2020016
[9]

Yinsong Bai, Lin He, Huijiang Zhao. Nonlinear stability of rarefaction waves for a hyperbolic system with Cattaneo's law. Communications on Pure and Applied Analysis, 2021, 20 (7&8) : 2441-2474. doi: 10.3934/cpaa.2021049
[10]

Mokhtar Kirane, Belkacem Said-Houari, Mohamed Naim Anwar. Stability result for the Timoshenko system with a time-varying delay term in the internal feedbacks. Communications on Pure and Applied Analysis, 2011, 10 (2) : 667-686. doi: 10.3934/cpaa.2011.10.667
[11]

J.E. Muñoz Rivera, Reinhard Racke. Global stability for damped Timoshenko systems. Discrete and Continuous Dynamical Systems, 2003, 9 (6) : 1625-1639. doi: 10.3934/dcds.2003.9.1625
[12]

Jing Zhang. The analyticity and exponential decay of a Stokes-wave coupling system with viscoelastic damping in the variational framework. Evolution Equations and Control Theory, 2017, 6 (1) : 135-154. doi: 10.3934/eect.2017008
[13]

Xiu-Fang Liu, Gen-Qi Xu. Exponential stabilization of Timoshenko beam with input and output delays. Mathematical Control and Related Fields, 2016, 6 (2) : 271-292. doi: 10.3934/mcrf.2016004
[14]

Hichem Kasri, Amar Heminna. Exponential stability of a coupled system with Wentzell conditions. Evolution Equations and Control Theory, 2016, 5 (2) : 235-250. doi: 10.3934/eect.2016003
[15]

István Györi, Ferenc Hartung. Exponential stability of a state-dependent delay system. Discrete and Continuous Dynamical Systems, 2007, 18 (4) : 773-791. doi: 10.3934/dcds.2007.18.773
[16]

Alain Miranville, Giulio Schimperna. On a doubly nonlinear Cahn-Hilliard-Gurtin system. Discrete and Continuous Dynamical Systems - B, 2010, 14 (2) : 675-697. doi: 10.3934/dcdsb.2010.14.675
[17]

J. J. Morgan, Hong-Ming Yin. On Maxwell's system with a thermal effect. Discrete and Continuous Dynamical Systems - B, 2001, 1 (4) : 485-494. doi: 10.3934/dcdsb.2001.1.485
[18]

Denis Mercier, Virginie Régnier. Decay rate of the Timoshenko system with one boundary damping. Evolution Equations and Control Theory, 2019, 8 (2) : 423-445. doi: 10.3934/eect.2019021
[19]

Ebenezer Bonyah, Fatmawati. An analysis of tuberculosis model with exponential decay law operator. Discrete and Continuous Dynamical Systems - S, 2021, 14 (7) : 2101-2117. doi: 10.3934/dcdss.2021057
[20]

Qiong Zhang. Exponential stability of a joint-leg-beam system with memory damping. Mathematical Control and Related Fields, 2015, 5 (2) : 321-333. doi: 10.3934/mcrf.2015.5.321

2021 Impact Factor: 1.865

Tools

Metrics

  • PDF downloads (163)
  • HTML views (103)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy