On the Cauchy problem for the standard linear solid model with heat conduction: Fourier versus Cattaneo

Abstract

In this paper, we consider the standard linear solid model in $\mathbb{R}^N$ coupled, first, with the Fourier law of heat conduction and, second, with the Cattaneo law. First, we give the appropriate functional setting to prove the well-posedness of both models under certain assumptions on the parameters (that is, $0<τ\leq β$). Second, using the energy method in the Fourier space, we obtain the optimal decay rate of a norm related to the solution both in the Fourier and the Cattaneo heat conduction models under the same assumptions on the parameters. More concretely, we prove that, when $0<τ<β$, the model with heat conduction has the same decay rate as the Cauchy problem without heat conduction (see [Pellicer_Said-Houari_AMO_2017]) both under the Fourier and Cattaneo heat laws. Also, we are able to see that the difference between using a Fourier or Cattaneo law in the heat conduction is not in the decay rate, but in the fact that the Cattaneo coupling exhibits the well-know regularity loss phenomenon, that is, Cattaneo model requires a higher regularity of the initial data for the solution to decay. When $0<τ=β$ (that is, when the dissipation comes through the heat conduction) we still have asymptotic stability in both heat coupling models, but with a slower decay rate. As we prove later, such stability is not possible in the absence of the heat conduction. We also prove the optimality of the previous decay rates for both models by using the eigenvalues expansion method. Finally, we complete the results in [Pellicer_Said-Houari_AMO_2017] by showing how the condition $0<τ< β$ is not only sufficient but also necessary for the asymptotic stability of the problem without heat conduction.