Decay of solutions for a mixture of thermoelastic solids with different temperatures

Abstract

Abstract. We study a system modeling thermomechanical deformations for mixtures of thermoelastic solids with two different temperatures, that is, when each component of the mixture has its own temperature. In particular, we investigate the asymptotic behavior of the related solutions. We prove the exponential stability of solutions for a generic class of materials. In case of the coupling matrix B being singular, we find that in general the corresponding semigroup is not exponentially stable. In this case we obtain that the corresponding solution decays polynomially as t −1/2 in case of Neumann boundary condition. Additionally, we show that the rate of decay is optimal. For Dirichlet boundary condition, we prove that the rate of decay is t −1/6 . Finally, we demonstrate the impossibility of time-localization of solutions in case that two coefficients (related with the thermal conductivity constants) agree.

Energy decay of a mixture with local viscoelasticity

“…A complete description of the decay of the solutions for the one‐dimensional thermoelastic mixtures was obtained in Rivera et al [27]. Several extensions to the other thermal theories can be found in previous works [28, 29]. It is worth recalling the recent papers concerning the analyticity of solutions in the case of isothermal viscoelasticity [30] and some numerical aspects concerning the same problem [31].…”

A type III thermoelastic problem with mixtures

“…Thermoelastic mixtures is the objective of study of different papers in the last decades (see, for instance, [2,3,4,5,6,7,8,9,10,11,11,12,13,14,15,16]).…”

Stability of a thermoelastic mixture with second sound

“…On the other hand, the theory of the thermoelastic mixture of solids was explored by several authors, such as Truesdell, Toupin, Green, Naghdi, Bowen, Wiese, Atkin, Craine, Bedford, and Drumheller (see [9–15] and the references therein). The asymptotic behavior to this model of mixtures with the heat flux defined by the Fourier law has been studied by a number of authors [1, 16–24]. Alves et al [1] proved that equation (1) with the Fourier law ( τ 0 = 0 ) is exponentially stable if and only if…”