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Applicable Analysis
An International Journal
Volume 101, 2022 - Issue 15
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Research Article

Stability of Timoshenko system coupled with thermal law of Gurtin-Pipkin affecting on shear force

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Pages 5171-5192 | Received 01 Feb 2020, Accepted 24 Jan 2021, Published online: 05 Feb 2021

References

  • Timoshenko SP. LXVI. On the correction for shear of the differential equation for transverse vibrations of prismatic bars. Lond Edinb Dublin Philos Mag J Sci. 1921;41(245):744–746.
  • Soufyane A. Stabilisation de la poutre de timoshenko. Comptes Rendus De L'Académie Des Sciences-Series I-Math. 1999;328(8):731–734.
  • Alves MS, Raposo CA, Rivera JEM, et al. Uniform stabilization for the transmission problem of the Timoshenko system with memory. J Math Anal Appl. 2010;369(1):323–345.
  • Ammar-Khodja F, Benabdallah A, Munoz Rivera JE, et al. Energy decay for Timoshenko systems of memory type. J Differ Equ. 2003;194(1):82–115.
  • Grasselli M, Pata V, Prouse G, et al. Longtime behavior of a viscoelastic Timoshenko beam. Discrete Contin Dyn Syst. 2004;10(1/2):337–348.
  • Munoz Rivera JE, Fernández Sare HD, et al. Exponential decay of Timoshenko systems with indefinite memory dissipation. Adv Differ Equ. 2008;13(7-8):733–752.
  • Messaoudi SA, Said-Houari B. Energy decay in a Timoshenko-type system of thermoelasticity of type iii. J Math Anal Appl. 2008;348(1):298–307.
  • Messaoudi SA, Said-Houari B. Energy decay in a Timoshenko-type system with history in thermoelasticity of type iii. Adv Differ Equ. 2009;14(3/4):375–400.
  • Ammar-Khodja F, Kerbal S, Soufyane A. Stabilization of the nonuniform Timoshenko beam. J Math Anal Appl. 2007;327(1):525–538.
  • Messaoudi SA, Soufyane A. Boundary stabilization of solutions of a nonlinear system of Timoshenko type. Nonlinear Anal Theor Meth Appl. 2007;67(7):2107–2121.
  • Messaoudi SA, Mustafa MI. On the stabilization of the Timoshenko system by a weak nonlinear dissipation. Math Methods Appl Sci. 2009;32(4):454–469.
  • Soufyane A, Whebe A. Uniform stabilization for the Timoshenko beam by a locally distributed damping. Electron J Differ Equ. 2003;2003(29):1–14.
  • Alves MO, Gomes Tavares EH, Jorge Silva MA, et al. On modeling and uniform stability of a partially dissipative viscoelastic Timoshenko system. SIAM J Math Anal. 2019;51:4520–4543.
  • Feng B, Pelicer ML. Global existence and exponential stability for a nonlinear Timoshenko system with delay. Bound Value Probl. 2015;206:1–13.
  • Feng B, Yang XG. Long-time dynamics for a nonlinear Timoshenko system with delay. Appl Anal. 2017;96:606–625.
  • Kim JU, Renardy Y. Boundary control of the Timoshenko beam. SIAM J Control Optim. 1987;25(6):1417–1429.
  • Raposo CA, Ferreira J, Santos ML, et al. Exponential stability for the Timoshenko system with two weak dampings. Appl Math Lett. 2005;18(5):535–541.
  • Wehbe A, Youssef W. Stabilization of the uniform Timoshenko beam by one locally distributed feedback. Appl Anal. 2009;88(7):1067–1078.
  • Guesmia A, Messaoudi SA, Soufyane A. Stabilization of a linear Timoshenko system with infinite history and applications to the Timoshenko-heat systems. Electron J Differ Equ. 2012;193:1–45.
  • Muñoz Rivera JE, Racke R. Timoshenko systems with indefinite damping. J Math Anal Appl. 2008;341(2):1068–1083.
  • Munoz Rivera JE, Fernández Sare HD. Stability of Timoshenko systems with past history. J Math Anal Appl. 2008;339(1):482–502.
  • Junior DSA, Ramos AJA. On the nature of dissipative Timoshenko systems at light of the second spectrum of frequency. Z Angew Math Phys. 2017;68(145):264.
  • Muñoz Rivera JE, Racke R. Mildly dissipative nonlinear Timoshenko systemsglobal existence and exponential stability. J Math Anal Appl. 2002;276(1):248–278.
  • Júnior DSA, Santos ML, Rivera JEM. Stability to 1-D thermoelastic Timoshenko beam acting on shear force. Zeitschrift Für Angewandte Mathematik Und Physik. 2014;65(6):1233–1249.
  • Alves MS, Jorge Silva MA, Ma TF, et al. Invariance of decay rate with respect to boundary conditions in thermoelastic Timoshenko systems. Z Angew Math Phys. 2016;67:16.
  • Alves MO, Caixeta AH, Jorge Silva MA, et al. On a Timoshenko system with thermal coupling on both the bending moment and the shear force. J Evol Equ. 2020;20:295–320.
  • Alves MS, Jorge Silva MA, Ma TF, et al. Non-homogeneous thermoelastic Timoshenko systems. Bull Braz Math Soc. 2017;48:461–484.
  • Sabri Öncü T, Bryant Moodie T. On the constitutive relations for second sound in elastic solids. Arch Ration Mech Anal. 1992;121(1):87–99.
  • Straughan B. Heat waves, applied mathematical sciences. Vol. 177. Berlin: Springer; 2017.
  • Santos ML, Almeida Júnior DS, Muñoz Rivera JE. The stability number of the Timoshenko system with second sound. J Differ Equ. 2012;253(9):2715–2733.
  • Gurtin ME, Pipkin AC. A general theory of heat conduction with finite wave speeds. Arch Ration Mech Anal. 1968;31(2):113–126.
  • Dell'Oro F, Pata V. On the stability of Timoshenko systems with Gurtin–Pipkin thermal law. J Differ Equ. 2014;257(2):523–548.
  • Feng B. On a semilinear Timoshenko-Coleman-Gurtin system, quasi-stability and attractors. Discrete Contin Dyn Syst. 2017;37(9):4729–4751.
  • Pazy ASemigroups of linear operators and applications to partial differential equations. Springer-Verlag New York; 2012.
  • Dafermos CM. Asymptotic stability in viscoelasticity. Arch Ration Mech Anal. 1970;37(4):297–308.
  • Grasselli M, Pata V. Uniform attractors of nonautonomous dynamical systems with memory. In: Lorenzi A., Ruf B. (eds) Evolution Equations, Semigroups and Functional Analysis. Progress in Nonlinear Differential Equations and Their Applications, vol 50. Birkhäuser, Basel, 2002, p. 155–178.
  • Zelati MC, DellOro F, Pata V. Energy decay of type iii linear thermoelastic plates with memory. J Math Anal Appl. 2013;401(1):357–366.
  • Gearhart L. Spectral theory for contraction semigroups on Hilbert space. Trans Am Math Soc. 1978;236:385–394.
  • Huang F. Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces. Ann Diff Equ. 1985;1:43–56.
  • Prüss J. On the spectrum of-semigroups. Trans Am Math Soc. 1984;284(2):847–857.
  • Messaoudi SA, Said-Houari B. Uniform decay in a Timoshenko-type system with past history. J Math Anal Appl. 2009;360(2):459–475.
  • Muñoz-Rivera J, Quintanilla R. On the time polynomial decay in elastic solids with voids. J Math Anal Appl. 2008;338(2):1296–1309.
  • Ramos AJA, Aouadi M, Almeida Junior DD, et al. A new stabilization scenario for Timoshenko systems with thermo-diffusion effects in second spectrum perspective. Arch Math. 2020. https://doi.org/10.1007/s00013-020-01526-4.

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