In this paper, we are interested in studying the well-posedness, optimal polynomial stability, and the lack of exponential stability for a class of thermoelastic system of Reissner–Mindlin–Timoshenko plates with structural damping, that is, with the dissipation of Kelvin–Voigt type on the equations for the rotation angles. We also consider the thermal effect with thermal variables described by Fourier’s law of heat conduction.
F.D.Araruna, P.Braz e Silva and P.Queiroz-Souza, Asymptotic limits and stabilization for the 2D nonlinear Mindlin–Timoshenko system, Analysis & PDE11(2) (2017), 351–382. doi:10.2140/apde.2018.11.351.
2.
A.Borichev and Y.Tomilov, Optimal polynomial decay of functions and operator semigroups, Math Ann.347(2) (2009), 455–478. doi:10.1007/s00208-009-0439-0.
3.
H.Brezis, Analyse Fonctionelle, Théorie et Applications, Masson, Paris, 1992.
4.
A.D.S.Campelo, D.S.AlmeidaJúnior and M.L.Santos, Stability to the dissipative Reissner–Mindlin–Timoshenko acting on displacement equation, Eur. J. Appl. Math.27(2) (2016), 157–193. doi:10.1017/S0956792515000467.
5.
A.D.S.Campelo, D.S.AlmeidaJúnior and M.L.Santos, Stability of weakly dissipative Reissner–Mindlin–Timoshenko plates: A sharp result, Eur. J. Appl. Math.29(2) (2018), 226–252. doi:10.1017/S0956792517000092.
6.
J.F.Doyle, Wave Propagation in Structures: Spectral Analysis Using Fast Discrete Fourier Transforms, Springer, 1997.
7.
H.D.Fernándes Sare, On the stability of Mindlin–Timoshenko plates, Q. Appl. Math.67 (2009), 249–263. doi:10.1090/S0033-569X-09-01110-2.
8.
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.
9.
M.Grobbelaar-Van Dalsen, Strong stabilization of models incorporating the thermoelastic Reissner–Mindlin plate equations with second sound, Appl. Anal.90 (2011), 1419–1449. doi:10.1080/00036811.2010.530259.
10.
M.Grobbelaar-Van Dalsen, On the dissipative effect of a magnetic field in a Mindlin–Timoshenko plate model, Z. Angew. Math. Phys.63(6) (2012), 1047–1065. doi:10.1007/s00033-012-0206-z.
11.
M.Grobbelaar-Van Dalsen, Stabilization of a thermoelastic Mindlin–Timoshenko plate model revisited, Z. Angew. Math. Phys.64 (2013), 1305–1325. doi:10.1007/s00033-012-0289-6.
12.
M.Grobbelaar-Van Dalsen, Polynomial decay rate of a thermoelastic Mindlin–Timoshenko plate model with Dirichlet boundary conditions, Z. Angew. Math. Phys.66(1) (2015), 113–128. doi:10.1007/s00033-013-0391-4.
13.
F.L.Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Differential Equations.1 (1985), 43–56.
14.
J.Lagnese, Boundary Stabilization of Thin Plates, SIAM, Philadelphia, 1989.
15.
J.Lagnese and J.Lions, Modelling, Analysis and Control of Thin Plates, Collection RMA, Masson, Paris, 1988.
16.
M.Pokojovy, On stability of hyperbolic thermoelastic Reissner–Mindlin–Timoshenko plates, Mathematical Methods in Applied Science.38(7) (2015), 1225–1246. doi:10.1002/mma.3140.
17.
J.Prüss, On the spectrum of -semigroups, Trans Am Math Soc.284 (1984), 847–857.
18.
M.J.Silva, T.F.Ma and J.E.Muñoz Rivera, Mindlin–Timoshenko systems with Kelvin–Voigt: Analyticity and optimal decay rates, J. Math. Anal. Appl.417(1) (2014), 164–179. doi:10.1016/j.jmaa.2014.02.066.
