In this paper, we consider the standard linear solid model in \(\mathbb {R}^N\) coupled with the Fourier law of heat conduction. First, we give the appropriate functional setting to prove the well-posedness of this model under certain assumptions on the parameters (that is, \(0<\tau \le \beta \)). Second, using the energy method in the Fourier space, we obtain the optimal decay rate of a norm related to the solution. In particular, we prove that when \(0<\tau <\beta \) the decay rate is the same as in the Cauchy problem without heat conduction (see Pellicer and Said-Houari in Appl Math Optim 80: 447–478, 2019), and that it does not exhibit the well-known regularity loss phenomenon which is present in some of these models. When \(\tau =\beta >0\) (that is, when the only dissipation comes through the heat conduction), we still have asymptotic stability, but with a slower decay rate. We also prove the optimality of the previous decay rate for the solution itself by using the eigenvalues expansion method. Finally, we complete the results in Pellicer and Said-Houari (Appl Math Optim 80: 447–478, 2019) by showing how the condition \(0<\tau < \beta \) is not only sufficient but also necessary for the asymptotic stability of the problem without heat conduction.

在本文中，我们考虑\（\ mathbb {R} ^ N \）中的标准线性实体模型，并结合了热传导的傅立叶定律。首先，我们给出适当的函数设置，以证明在某些假设下参数（即\（0 <\ tau \ le \ beta \））下该模型的适定性。其次，在傅立叶空间中使用能量方法，我们获得了与解有关的范数的最佳衰减率。特别是，我们证明了当\（0 <\ tau <\ beta \）时的衰减率与无导热的柯西问题相同（请参阅Appl Math Optim 80：447–478，2019中的Pellicer和Said-Houari ），并且它不表现出某些模型中存在的众所周知的规则损失现象。当\（\ tau = \ beta> 0 \）（也就是说，当唯一的耗散通过热传导时），我们仍然具有渐近稳定性，但衰减速率较慢。我们还使用特征值展开法证明了解本身对先前衰减率的最优性。最后，我们通过显示条件\（0 <\ tau <\ beta \）不仅对于渐近而言是足够的，而且对于渐近而言是必要的，我们在Pellicer和Said-Houari（Appl Math Optim 80：447–478，2019）中完成了结果。没有热传导的问题的稳定性。