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

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

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.

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 JúniorM. 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. AlvesA. H. CaixetaM. A. Jorge SilvaJ. 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. AlvesM. A. Jorge SilvaT. 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. CardozoM. A. Jorge SilvaT. 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. ContiV. 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. DjellaliS. 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. 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.

[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 JúniorM. 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. AlvesA. H. CaixetaM. A. Jorge SilvaJ. 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. AlvesM. A. Jorge SilvaT. 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. CardozoM. A. Jorge SilvaT. 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. ContiV. 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. DjellaliS. 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. 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.

[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.

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