On the Cauchy problem of the standard linear solid model with Fourier heat conduction

Abstract

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.