Research Article
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Exponential stability of a Timoshenko type thermoelastic system with Gurtin-Pipkin thermal law and frictional damping

Year 2022, Volume: 71 Issue: 1, 95 - 115, 30.03.2022

Abdelfeteh Fareh

https://doi.org/10.31801/cfsuasmas.847038
Cited By: 6

Abstract

In this paper we consider a linear thermoelastic system of Timoshenko type
where the heat conduction is given by the linearized law of Gurtin-Pipkin. An
existence and uniqueness result is proved by the use of a semigroup approach. We establish an exponential stability result without any assumption on the wave speeds once here we have a fully damped system.

Keywords

Timeshenko system well-posedness exponential stability

Supporting Institution

Algerian ministry of higher education and scientific researches

Project Number

PRFU C00L03UN390120180002

References

  • Almeida Junior, D. S., Santos, M. L., Munoz Rivera, J. E., Stability to 1-D thermoelastic Timoshenko beam acting on shear force, Z. Angew. Math. Phys., 65 (2014), 1233–1249. https://doi.org/10.1007/s00033-013-0387-0
  • Alves, M. S., Jorge Silva, M. A., Ma, T. F., Munoz Rivera, J. E., Invariance of decay rate with respect to boundary conditions in thermoelastic Timoshenko systems, Z. Angew. Math. Phys., 67 (2016), 70. https://doi.org/ 10.1007/s00033-016-0662-y
  • Alves, M. S., Jorge Silva, M. A., Ma, T. F., Munoz Rivera, J. E., Non-homogeneous thermoelastic Timoshenko systems, Bull. Braz. Math. Soc. (N. S.), 48 (2017), 461–484. https://doi.org/10.1007/s00574-017-0030-3
  • Apalara, T. A., Uniform stability of a laminated beam with structural damping and second sound, Z. Angew. Math. Phys., 68(2) (2017). https://doi.org/10.1007/s00033-017-0784-x
  • Choucha, A., Ouchenane, D., Boulaaras, S., Well posedness and stability result for a thermoelastic laminated Timoshenko beam with distributed delay term, Math. Meth. Appl. Sci., 43(17) (2020), 1–22.https://doi.org/10.1002/mma.6673
  • Choucha, A., Boulaaras, S., Ouchenane, D., Alkhalaf, S., Stability result and well posedness for Timoshenko’s beam laminated with termoelastic and past history, Fractals, 29 (2021), 1–26. https://doi.org/10.1142/S0218348X21400259
  • Dell’Oro, F., Pata, V., On the stability of Timoshenko systems with Gurtin-Pipkin thermal law, J. Diff. Equa., 257 (2014), 523–548. https://doi.org/10.1016/j.jde.2014.04.009
  • Fatori, L. H., Munoz Rivera, J. E., Energy decay for hyperbolic thermoelastic systems of memory type, Quart. Appl. Math., 59 (2001), 441–458. https://doi.org/10.1090/qam/1848527
  • Fatori, L. H., Munoz Rivera, J. E., Monteiro, R. N., Energy decay to Timoshenko’s system with thermoelasticity of type III, Asymptotic Analysis, 86 (2014), 227-247.
  • Fernandez-Sare H. D., Racke, R., On the stability of damped Timoshenko system Cattaneo versus Fourier law, Arch. Rat. Mech. Anal., 194 (2009), 221–251. https://doi.org/10.1007/s00205-009-0220-2
  • Giorgi, C., Naso, M. G., Pata, V., Exponential stability in linear heat conduction with memory: a semigroup approach, Commun. Appl. Anal., 5 (2001), 121–134.
  • Green, A. E., Naghdi, P. M., A re-examination of the basic postulates of thermomechanics, Proc. Roy. Soc. London Ser., A432 (1991), 171–194. https://doi.org/10.1098/rspa.1991.0012
  • Green, A. E., Naghdi, P. M., On undamped heat waves in an elastic solid, J. Thermal Stresses, 15 (1992), 253–264. https://doi.org/10.1080/01495739208946136
  • Green, A. E., Naghdi, P. M., Thermoelasticity without energy-dissipation. J. Elast., 31(3) (1993), 189-208. https://doi.org/10.1007/BF00044969
  • Grobbelaar-Van Dalsen, M., Strong stabilization of models incorporating the thermoelastic Reissner–Mindlin plate equations with second sound, Applicable Analysis, 90(9) (2011), 1419– 1449. https://doi.org/10.1080/00036811.2010.530259
  • Gurtin, M. E., Pipkin, A. C., A general theory of heat conduction with finite wave speeds, Arch. Ration. Mech. Anal., 31 (1968),113–126. https://doi.org/10.1007/BF00281373
  • Guesmia, A., Messaoudi, S. A., Wahbe, A., Uniform decay in mildly damped Timoshenko systems with non-equal wave speed propagation, Dynamic Systems and Applications, 21 (2012), 133–146. https://hal.inria.fr/hal-01281866
  • Jorge Silva, M. A., Pinheiro, S. B., Improvement on the polynomial stability for a Timoshenko system with type III thermoelasticity, Applied Mathematics Letters, 96 (2019), 95– 100. https://doi.org/10.1016/j.aml.2019.04.014
  • Jorge Silva, M. A., Racke, R., Effects of history and heat models on the stability of thermoelastic Timoshenko systems, J. Diff. Equ., 275 (2021), 167-203. https://doi.org/10.1016/j.jde.2020.11.041
  • Liu, W. J., Zhao, W., Exponential and polynomial decay for a laminated beam with Fourier’s type heat conduction, Preprints, (2017) https://doi.org/10.20944/preprints201702.0058.v2
  • Lord H. W., Shulman, Y., A generalized dynamical theory of thermoelasticity, J. Mech. Phys. Sol., 15 (1967), 299–309. https://doi.org/10.1016/0022-5096(67)90024-5
  • Messaoudi, S. A., Pokojovy, M., Said-Houari, B., Nonlinear damped Timoshenko systems with second: global existence and exponential stability, Math. Method. Appl. Sci., 32 (2009), 505-534. https://doi.org/10.1002/mma.1049.
  • Messaoudi, S. A., Fareh, A., Energy decay in a Timoshenko-type system of thermoelasticity of type III with different wave-propagation speeds, Arab J Math., 2 (2013), 199–207. https://doi.org/10.1007/s40065-012-0061-y
  • Messaoudi, S. A., Said-Houari, B., Energy decay in a Timoshenko-type system of thermoelasticity of type III. J. Math. Anal. Appl., 348 (2008), 298–307. https://doi.org/10.1016/j.jmaa.2008.07.036
  • Munoz Rivera, J. E., Racke, R., Mildly dissipative nonlinear Timoshenko systemsglobal existence and exponential stability, J. Math. Anal. Appl., 276 (2002), 248–278. https://doi.org/10.1016/S0022-247X(02)00436-5
  • Pata, V., Vuk, E., On the exponential stability of linear thermoelasticity, Contin. Mech. Thermodyn., 12 (2000), 121–130. https://doi.org/doi:10.1007/s001610050131
  • Pazy, A., Semigroups of Linear Pperators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983.
  • Quintanilla, R., Racke, R., Qualitative aspects of solutions in resonators, Arch. Mech., 60(4) (2008), 345–360.
  • Raposo, C. A., Exponential stability for a structure with interfacial slip and frictional damping, Applied Mathematics Letters, 53 (2016), 85–91. https://doi.org/10.1016/j.aml.2015.10.005
  • Santos, M. L., Almeida Junior, D. S., On Timoshenko-type systems with type III thermoelasticity: asymptotic behavior, J. Math. Appl., 448 (2017), 650–671. https://doi.org/10.1016/j.jmaa.2016.10.074
  • Santos, M. L., Almeida Junior, D. S., Munoz Rivera, J. E., The stability number of the Timoshenko system with second sound, J. Differential Equations, 253 (2012), 2715–2733. https://doi.org/10.1016/j.jde.2012.07.012.
  • Timoshenko, S. P., On the correction for shear of the differential equation for transverse vibrations of bars of Prismatic bars, Dubl. Philos. Mag., 41 (1921), 744–746. https://doi.org/10.1080/14786442108636264
  • Vrabie, I. I., C0-Semigroups and Applications, Elsevier Science B.V., Amesterdam, 2003

Year 2022, Volume: 71 Issue: 1, 95 - 115, 30.03.2022

Abdelfeteh Fareh

https://doi.org/10.31801/cfsuasmas.847038
Cited By: 6

Abstract

Project Number

PRFU C00L03UN390120180002

References

  • Almeida Junior, D. S., Santos, M. L., Munoz Rivera, J. E., Stability to 1-D thermoelastic Timoshenko beam acting on shear force, Z. Angew. Math. Phys., 65 (2014), 1233–1249. https://doi.org/10.1007/s00033-013-0387-0
  • Alves, M. S., Jorge Silva, M. A., Ma, T. F., Munoz Rivera, J. E., Invariance of decay rate with respect to boundary conditions in thermoelastic Timoshenko systems, Z. Angew. Math. Phys., 67 (2016), 70. https://doi.org/ 10.1007/s00033-016-0662-y
  • Alves, M. S., Jorge Silva, M. A., Ma, T. F., Munoz Rivera, J. E., Non-homogeneous thermoelastic Timoshenko systems, Bull. Braz. Math. Soc. (N. S.), 48 (2017), 461–484. https://doi.org/10.1007/s00574-017-0030-3
  • Apalara, T. A., Uniform stability of a laminated beam with structural damping and second sound, Z. Angew. Math. Phys., 68(2) (2017). https://doi.org/10.1007/s00033-017-0784-x
  • Choucha, A., Ouchenane, D., Boulaaras, S., Well posedness and stability result for a thermoelastic laminated Timoshenko beam with distributed delay term, Math. Meth. Appl. Sci., 43(17) (2020), 1–22.https://doi.org/10.1002/mma.6673
  • Choucha, A., Boulaaras, S., Ouchenane, D., Alkhalaf, S., Stability result and well posedness for Timoshenko’s beam laminated with termoelastic and past history, Fractals, 29 (2021), 1–26. https://doi.org/10.1142/S0218348X21400259
  • Dell’Oro, F., Pata, V., On the stability of Timoshenko systems with Gurtin-Pipkin thermal law, J. Diff. Equa., 257 (2014), 523–548. https://doi.org/10.1016/j.jde.2014.04.009
  • Fatori, L. H., Munoz Rivera, J. E., Energy decay for hyperbolic thermoelastic systems of memory type, Quart. Appl. Math., 59 (2001), 441–458. https://doi.org/10.1090/qam/1848527
  • Fatori, L. H., Munoz Rivera, J. E., Monteiro, R. N., Energy decay to Timoshenko’s system with thermoelasticity of type III, Asymptotic Analysis, 86 (2014), 227-247.
  • Fernandez-Sare H. D., Racke, R., On the stability of damped Timoshenko system Cattaneo versus Fourier law, Arch. Rat. Mech. Anal., 194 (2009), 221–251. https://doi.org/10.1007/s00205-009-0220-2
  • Giorgi, C., Naso, M. G., Pata, V., Exponential stability in linear heat conduction with memory: a semigroup approach, Commun. Appl. Anal., 5 (2001), 121–134.
  • Green, A. E., Naghdi, P. M., A re-examination of the basic postulates of thermomechanics, Proc. Roy. Soc. London Ser., A432 (1991), 171–194. https://doi.org/10.1098/rspa.1991.0012
  • Green, A. E., Naghdi, P. M., On undamped heat waves in an elastic solid, J. Thermal Stresses, 15 (1992), 253–264. https://doi.org/10.1080/01495739208946136
  • Green, A. E., Naghdi, P. M., Thermoelasticity without energy-dissipation. J. Elast., 31(3) (1993), 189-208. https://doi.org/10.1007/BF00044969
  • Grobbelaar-Van Dalsen, M., Strong stabilization of models incorporating the thermoelastic Reissner–Mindlin plate equations with second sound, Applicable Analysis, 90(9) (2011), 1419– 1449. https://doi.org/10.1080/00036811.2010.530259
  • Gurtin, M. E., Pipkin, A. C., A general theory of heat conduction with finite wave speeds, Arch. Ration. Mech. Anal., 31 (1968),113–126. https://doi.org/10.1007/BF00281373
  • Guesmia, A., Messaoudi, S. A., Wahbe, A., Uniform decay in mildly damped Timoshenko systems with non-equal wave speed propagation, Dynamic Systems and Applications, 21 (2012), 133–146. https://hal.inria.fr/hal-01281866
  • Jorge Silva, M. A., Pinheiro, S. B., Improvement on the polynomial stability for a Timoshenko system with type III thermoelasticity, Applied Mathematics Letters, 96 (2019), 95– 100. https://doi.org/10.1016/j.aml.2019.04.014
  • Jorge Silva, M. A., Racke, R., Effects of history and heat models on the stability of thermoelastic Timoshenko systems, J. Diff. Equ., 275 (2021), 167-203. https://doi.org/10.1016/j.jde.2020.11.041
  • Liu, W. J., Zhao, W., Exponential and polynomial decay for a laminated beam with Fourier’s type heat conduction, Preprints, (2017) https://doi.org/10.20944/preprints201702.0058.v2
  • Lord H. W., Shulman, Y., A generalized dynamical theory of thermoelasticity, J. Mech. Phys. Sol., 15 (1967), 299–309. https://doi.org/10.1016/0022-5096(67)90024-5
  • Messaoudi, S. A., Pokojovy, M., Said-Houari, B., Nonlinear damped Timoshenko systems with second: global existence and exponential stability, Math. Method. Appl. Sci., 32 (2009), 505-534. https://doi.org/10.1002/mma.1049.
  • Messaoudi, S. A., Fareh, A., Energy decay in a Timoshenko-type system of thermoelasticity of type III with different wave-propagation speeds, Arab J Math., 2 (2013), 199–207. https://doi.org/10.1007/s40065-012-0061-y
  • Messaoudi, S. A., Said-Houari, B., Energy decay in a Timoshenko-type system of thermoelasticity of type III. J. Math. Anal. Appl., 348 (2008), 298–307. https://doi.org/10.1016/j.jmaa.2008.07.036
  • Munoz Rivera, J. E., Racke, R., Mildly dissipative nonlinear Timoshenko systemsglobal existence and exponential stability, J. Math. Anal. Appl., 276 (2002), 248–278. https://doi.org/10.1016/S0022-247X(02)00436-5
  • Pata, V., Vuk, E., On the exponential stability of linear thermoelasticity, Contin. Mech. Thermodyn., 12 (2000), 121–130. https://doi.org/doi:10.1007/s001610050131
  • Pazy, A., Semigroups of Linear Pperators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983.
  • Quintanilla, R., Racke, R., Qualitative aspects of solutions in resonators, Arch. Mech., 60(4) (2008), 345–360.
  • Raposo, C. A., Exponential stability for a structure with interfacial slip and frictional damping, Applied Mathematics Letters, 53 (2016), 85–91. https://doi.org/10.1016/j.aml.2015.10.005
  • Santos, M. L., Almeida Junior, D. S., On Timoshenko-type systems with type III thermoelasticity: asymptotic behavior, J. Math. Appl., 448 (2017), 650–671. https://doi.org/10.1016/j.jmaa.2016.10.074
  • Santos, M. L., Almeida Junior, D. S., Munoz Rivera, J. E., The stability number of the Timoshenko system with second sound, J. Differential Equations, 253 (2012), 2715–2733. https://doi.org/10.1016/j.jde.2012.07.012.
  • Timoshenko, S. P., On the correction for shear of the differential equation for transverse vibrations of bars of Prismatic bars, Dubl. Philos. Mag., 41 (1921), 744–746. https://doi.org/10.1080/14786442108636264
  • Vrabie, I. I., C0-Semigroups and Applications, Elsevier Science B.V., Amesterdam, 2003
There are 33 citations in total.

Details

Primary Language English
Subjects Applied Mathematics
Journal Section Research Articles
Authors

Abdelfeteh Fareh 0000-0003-3522-324X

Project Number PRFU C00L03UN390120180002
Publication Date March 30, 2022
Submission Date December 25, 2020
Acceptance Date December 14, 2021
Published in Issue Year 2022 Volume: 71 Issue: 1

Cite

APA Fareh, A. (2022). Exponential stability of a Timoshenko type thermoelastic system with Gurtin-Pipkin thermal law and frictional damping. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 71(1), 95-115. https://doi.org/10.31801/cfsuasmas.847038
AMA Fareh A. Exponential stability of a Timoshenko type thermoelastic system with Gurtin-Pipkin thermal law and frictional damping. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. March 2022;71(1):95-115. doi:10.31801/cfsuasmas.847038
Chicago Fareh, Abdelfeteh. “Exponential Stability of a Timoshenko Type Thermoelastic System With Gurtin-Pipkin Thermal Law and Frictional Damping”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 71, no. 1 (March 2022): 95-115. https://doi.org/10.31801/cfsuasmas.847038.
EndNote Fareh A (March 1, 2022) Exponential stability of a Timoshenko type thermoelastic system with Gurtin-Pipkin thermal law and frictional damping. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 71 1 95–115.
IEEE A. Fareh, “Exponential stability of a Timoshenko type thermoelastic system with Gurtin-Pipkin thermal law and frictional damping”, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., vol. 71, no. 1, pp. 95–115, 2022, doi: 10.31801/cfsuasmas.847038.
ISNAD Fareh, Abdelfeteh. “Exponential Stability of a Timoshenko Type Thermoelastic System With Gurtin-Pipkin Thermal Law and Frictional Damping”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 71/1 (March 2022), 95-115. https://doi.org/10.31801/cfsuasmas.847038.
JAMA Fareh A. Exponential stability of a Timoshenko type thermoelastic system with Gurtin-Pipkin thermal law and frictional damping. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2022;71:95–115.
MLA Fareh, Abdelfeteh. “Exponential Stability of a Timoshenko Type Thermoelastic System With Gurtin-Pipkin Thermal Law and Frictional Damping”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, vol. 71, no. 1, 2022, pp. 95-115, doi:10.31801/cfsuasmas.847038.
Vancouver Fareh A. Exponential stability of a Timoshenko type thermoelastic system with Gurtin-Pipkin thermal law and frictional damping. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2022;71(1):95-115.

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Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics.

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