TY - JOUR T1 - Exponential stability of a Timoshenko type thermoelastic system with Gurtin-Pipkin thermal law and frictional damping AU - Fareh, Abdelfeteh PY - 2022 DA - March Y2 - 2021 DO - 10.31801/cfsuasmas.847038 JF - Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics JO - Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. PB - Ankara University WT - DergiPark SN - 1303-5991 SP - 95 EP - 115 VL - 71 IS - 1 LA - en AB - In this paper we consider a linear thermoelastic system of Timoshenko typewhere the heat conduction is given by the linearized law of Gurtin-Pipkin. Anexistence 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. KW - Timeshenko system KW - well-posedness KW - exponential stability CR - 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 CR - 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 CR - 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 CR - 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 CR - 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 CR - 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 CR - 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 CR - 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 CR - 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. CR - 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 CR - 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. CR - 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 CR - 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 CR - Green, A. E., Naghdi, P. M., Thermoelasticity without energy-dissipation. J. Elast., 31(3) (1993), 189-208. https://doi.org/10.1007/BF00044969 CR - 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 CR - 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 CR - 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 CR - 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 CR - 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 CR - 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 CR - 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 CR - 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. CR - 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 CR - 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 CR - 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 CR - 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 CR - Pazy, A., Semigroups of Linear Pperators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. CR - Quintanilla, R., Racke, R., Qualitative aspects of solutions in resonators, Arch. Mech., 60(4) (2008), 345–360. CR - 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 CR - 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 CR - 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. CR - 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 CR - Vrabie, I. I., C0-Semigroups and Applications, Elsevier Science B.V., Amesterdam, 2003 UR - https://doi.org/10.31801/cfsuasmas.847038 L1 - https://dergipark.org.tr/en/download/article-file/1468775 ER -