In this paper, we consider the long time behavior of the coupling problem between the fluid and thermoelastic plate under external forces, while the heat effects are modeled by the Cattaneo law giving rise to a “second sound” effect. By virtue of the semigroup method combined with the boundedness of natural energy, we prove the existence of global solution, through using the polishing technique to realize the zero-average multipliers and the Stokes operator to solve the matching problem of multipliers between fluid and plate equations. We then construct the appropriate Lyapunov functional to obtain asymptotic compactness of the dynamical system and prove the existence of a smooth global attractor.
CattaneoC. Sulla conduzione del calore. Atti Sem Mat Fis Univ Modena1949; 3: 83–101.
2.
ChueshovIRyzhkovaI. A global attractor for a fluid-plate interaction model. Commun Pure Appl Anal2013; 12(4): 1635–1656.
3.
KimJU. On the energy decay of a linear thermoelastic bar and plate. SIAM J Math Anal1992; 23(4): 889–899.
4.
LiuZRenardyM. A note on the equations of a thermoelastic plate. Appl Math Lett1995; 8(3): 1–6.
5.
LiuZZhengS. Exponential stability of the Kirchhoff plate with thermal or viscoelastic damping. Quart Appl Math1997; 55(3): 551–564.
6.
Fernández SareHDMuñoz RiveraJE. Optimal rates of decay in 2-d thermoelasticity with second sound. J Math Phys2012; 53: 073509.
7.
Muñoz RiveraJERackeRSepúlvedaM, et al. On exponential stability for the thermoelastic plate: comparison and singular limits. Appl Math Optim2021; 84(1): 1045–1081.
8.
FengBYangXGQinY. Uniform attractors for a nonautonomous extensible plate equation with a strong damping. Math Methods Appl Sci2017; 40(10): 3479–3492.
9.
MaTFNarcisoV. Global attractor for a model of extensible beam with nonlinear damping and source terms. Nonlinear Anal2010; 73(10): 3402–3412.
10.
YangZ. On an extensible beam equation with nonlinear damping and source terms. J Differential Equations2013; 254(9): 3903–3927.
11.
LasieckaIPokojovyMWanX. Long-time behavior of quasilinear thermoelastic Kirchhoff-Love plates with second sound. Nonlinear Anal2019; 186: 219–258.
12.
AouadiM. Global and exponential attractors for extensible thermoelastic plate with time-varying delay. J Differential Equations2020; 269(5): 4079–4115.
13.
LionsJLZuazuaE. Approximate controllability of a hydro-elastic coupled system. ESAIM Contrôle Optim Calc Var1995/1996; 1: 1–15.
14.
AvalosGBucciF. Rational rates of uniform decay for strong solutions to a fluid-structure PDE system. J Differential Equations2015; 258(12): 4398–4423.
15.
AvalosGClarkTJ. A mixed variational formulation for the wellposedness and numerical approximation of a PDE model arising in a 3-D fluid-structure interaction. Evol Equ Control Theory2014; 3(4): 557–578.
16.
QinYGuoYYaoPF. Energy decay and global smooth solutions for a free boundary fluid-nonlinear elastic structure interface model with boundary dissipation. Discrete Contin Dyn Syst2020; 40(3): 1555–1593.
17.
QinYYaoPF. Energy decay and global solutions for a damped free boundary fluid–elastic structure interface model with variable coefficients in elasticity. Appl Anal2020; 99(11): 1953–1971.
18.
ShenLWangSFengY. Existence of global weak solutions for the high frequency and small displacement oscillation fluid-structure interaction systems. Math Methods Appl Sci2021; 44(5): 3249–3259.
19.
PeraltaG. Uniform exponential stability of a fluid-plate interaction model due to thermal effects. Evol Equ Control Theory2020; 9(1): 39–60.
20.
JiangJLiuW. Well-posedness and exponential stability of a coupled fluid-thermoelastic plate interaction model with second sound. Z Angew Math Phys2023; 74(4): 132.
21.
ChueshovI. A global attractor for a fluid-plate interaction model accounting only for longitudinal deformations of the plate. Math Methods Appl Sci2011; 34(14): 1801–1812.
22.
ChueshovIEllerMLasieckaI. On the attractor for a semilinear wave equation with critical exponent and nonlinear boundary dissipation. Commun Partial Differ Equ2002; 27(9–10): 1901–1951.
23.
LiYTangJYuS. Asymptotic stability of rarefaction wave for the compressible Navier-Stokes-Korteweg equations in the half space. Proc Roy Soc Edinburgh Sect A2022; 152(3): 756–779.
24.
Ryzhkova-GerasymovaI. Long time behaviour of strong solutions to interactive fluid-plate system without rotational inertia. Discrete Contin Dyn Syst Ser B2018; 23(3): 1243–1265.
25.
YangXFengBde SouzaTM, et al. Long-time dynamics for a non-autonomous Navier-Stokes-Voigt equation in Lipschitz domains. Discrete Contin Dyn Syst Ser B2019; 24(1): 363–386.
26.
FreitasMMRamosAJAÖzerAO, et al. Long-time dynamics for a fractional piezoelectric system with magnetic effects and fourier’s law. J Diff Equa2021; 280: 891–927.
27.
HarauxA. Decay rate of the range component of solutions to some semilinear evolution equations. Nonlinear Diff Equ Appl2006; 13(4): 435–445.
28.
ApalaraTAMessaoudiSA. An exponential stability result of a Timoshenko system with thermoelasticity with second sound and in the presence of delay. Appl Math Optim2015; 71(3): 449–472.
29.
BoulanouarFDrablaS. General boundary stabilization result of memory-type thermoelasticity with second sound. Electron J Diff Equ2014; 2014(202): 1–18.
30.
LiuWChenM. Well-posedness and exponential decay for a porous thermoelastic system with second sound and a time-varying delay term in the internal feedback. Contin Mech Thermodyn2017; 29(3): 731–746.
31.
LiuWWangDChenD. General decay of solution for a transmission problem in infinite memory-type thermoelasticity with second sound. J Therm Stresses2018; 41(6): 758–775.
32.
RisselMWangYG. Remarks on exponential stability for a coupled system of elasticity and thermoelasticity with second sound. J Evol Equ2021; 21(2): 1573–1599.
33.
LionsJLMagenesE. Non-homogeneous boundary value problems and applications, vol. 1. New York: Springer, 1972.
34.
TucsnakMWeissG. Observation and control for operator semigroups. Birkhäuser Advanced Texts: Basler Lehrbücher. Basel: Birkhäuser Verlag, 2009.
35.
MessaoudiSAMadaniB. A general decay result for a memory-type thermoelasticity with second sound. Appl Anal2014; 93(8): 1663–1673.
36.
TemamR. Navier-stokes equations, Studies in Mathematics and its Applications, vol. 2. Revised ed. Amsterdam: North-Holland Publishing Co., 1979.
37.
ChueshovILasieckaI. Von Karman evolution equations. Springer Monographs in Mathematics. New York: Springer, 2010.
38.
AvalosGTriggianiR. The coupled PDE system arising in fluid/structure interaction. I. Explicit semigroup generator and its spectral properties. In: Fluids and waves, Contemporary Mathematics, vol. 440. Providence, RI: American Mathematical Society, 2007, pp. 15–54.
39.
AvalosGDvorakM. A new maximality argument for a coupled fluid-structure interaction, with implications for a divergence-free finite element method. Appl Math2008; 35: 259–280.
40.
AvalosGTriggianiR. Fluid-structure interaction with and without internal dissipation of the structure: a contrast study in stability. Evol Equ Control Theory2013; 2(4): 563–598.
41.
LasieckaILuY. Interface feedback control stabilization of a nonlinear fluid-structure interaction. Nonlinear Anal2012; 75(3): 1449–1460.
