We study the large time behavior of solutions to a linear transmission problem in one space dimension. The problem at hand models a thermoelastic material with second sound confined by a purely elastic one. We shall characterize all equilibrium states of the considered system and prove that every solution approaches one designated equilibrium state with an exponential rate as time goes to infinity. Hereto, we apply methods from the theory of strongly continuous semigroups. In particular, we obtain uniform resolvent bounds for the underlying generator. This removes the largeness assumption of elastic wave speeds imposed in Meng and Wang (Anal Appl (Singap) 13, 2015) for having an exponential energy decay rate when the problem only has the trivial equilibrium. In an appendix, we provide a similar exponential stability result for the case where heat conduction is modeled using Fourier’s law.

我们研究在一维空间中线性传输问题解的较大时间行为。当前的问题是模拟具有第二声音的热弹性材料，第二声音受纯弹性声音限制。我们将刻画所考虑系统的所有平衡状态，并证明随着时间到无穷大，每个解都以指数速率接近一个指定的平衡状态。到目前为止，我们采用强连续半群理论的方法。特别是，我们为基础生成器获得统一的可分辨边界。这消除了在Meng和Wang（Anal Appl（Singap）2015年13月）中强加的弹性波速度假设，即当问题仅具有微不足道的平衡时，弹性波速度具有指数能量衰减率。在附录中