In this paper, we provide necessary and sufficient conditions for obtaining the stabilization properties for the one-dimensional Lord–Shulman thermoelastic theory with porosity subject to microtemperature but without temperature where the microtemperature conduction equations are governed by Cattaneo–Maxwell’s law. Based on recent results due to Bazarra et al., we introduce a stability number involving all coefficients of the system, and we prove that the exponential decay of the corresponding semigroup holds if and only if . Otherwise, we show that the system loses exponential stability and its solution decays polynomially with a rate equal to .
MagañaAQuintanillaR.On the time decay of solutions in one-dimensional theories of porous materials. Int J Solids Struct2006; 43(11–12): 3414–3427.
13.
SantosMLCampeloADSAlmeida JúniorDS.On the decay rates of porous elastic systems. J Elasticity2016; 127(1): 79–101.
14.
AouadiMGuerineS.Decay in full Von Kármán beam with temperature and microtemperatures effects. Math Model Nat Phenom2023; 18: 3.
15.
ApalaraTA.On the stability of porous-elastic system with microtemperatures. J Therm Stresses2019; 42: 265–278.
16.
ChiritaSCiarlettaMD’ApiceC.On the theory of thermoelasticity with microtemperatures. J Math Anal Appl2013; 397(1): 349–361.
17.
MagañaQuintanillaR. Exponential stability in type III thermoelasticity with microtemperatures. Z Angew Math Phys2018; 69(5): 129.
18.
MagañaAQuintanillaR.Exponential stability in three-dimensional type III thermo-porouselasticity with microtemperatures. J Elasticity2020; 139(1): 153–161.
19.
QuintanillaR.On growth and continuous dependence in thermoelasticity with microtemperatures. J Therm Stresses2011; 34(9): 911–922.
20.
QuintanillaR.On the logarithmic convexity in thermoelasticity with microtemperatures. J Therm Stresses2013; 34(6): 378–386.
21.
Almeida JúniorDSRamosAJAFreitasMM, et al. Polynomial stability for the equations of porous elasticity in one-dimensional bounded domains. Math Mech Solids2022; 27(2): 308–318.
22.
ApalaraTA.General decay of solutions in one-dimensional porous-elastic system with memory. J Math Anal Appl2019; 469(2): 457–471.
23.
FengBYanLAlmeida JúniorDS.Stabilization for an inhomogeneous porous-elastic system with temperature and microtemperature. ZAMM – Z Angew Math Me2021; 101(6): e202000058.
24.
FengBApalaraTA.Optimal decay for a porous elasticity system with memory. J Math Anal Appl2019; 470(2): 1108–1128.
25.
MessaoudiSAApalaraTA.General stability result in a memory-type porous thermoelasticity system of type III. Arab J Math Sci2014; 20(2): 213–232.
26.
RamosAJAAlmeida JúniorDSFreitasMM. About polynomial stability for the porous-elastic system with Fourier’s law. Z Angew Math Phys2022; 73: 57.
27.
SantosMLCampeloADSOliveiraMLS. On porous-elastic systems with Fourier law. Math Mech Solids2019; 98(6): 1181–1197.
28.
BazarraNFernandezJRQuintanillaR.Lord-Shulman thermoelasticity with microtemperatures. Appl Math Opt2021; 84(2): 1667–1685.
29.
AouadiMRamosAJACastejónA.Stability conditions for thermodiffusion Timoshenko system with second sound. Z Angew Math Phys2021; 72: 1–32.
30.
SantosMLAlmeida JúniorDSMuñoz RiveraJE.The stability number of the Timoshenko system with second sound. J Differ Equations2012; 253(9): 2715–2733.
31.
HuangF.Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces. Ann Differ Equations1985; 1: 43–56.
32.
PrüssJ.On the spectrum of semigroups. Trans Amer Math Soc1983; 284: 847–857.
33.
BorichevATomilovY.Optimal polynomial decay of functions and operator semigroups. Math Ann2010; 347: 455–478.
34.
PazyA.Semigroups of linear operators and applications to partial differential equations. New York: Springer-Verlag, 1983.
35.
KatoT.Perturbation theory for linear operators. New York: Springer-Verlag, 1980.
36.
Dell’OroF.On the stability of Bresse and Timoshenko systems with hyperbolic heat conduction. J Differ Equations2021; 281: 148–198.
