Well-posedness and decay property for the Cauchy problem of the standard linear solid model with thermoelasticity of type III

In this paper, we investigate the Cauchy problem related to the standard linear solid model with thermoelasticity of type III in the whole space. By using semigroup theory, we first establish the well-posedness result under certain assumptions on the parameters. By applying the energy method in the Fourier space, we then prove the optimal decay estimate results for both the non-critical and critical cases. Specifically, the decay property of the system is not of the regularity-loss type. In addition, we study the asymptotic expansion of the eigenvalues to analyze the optimality of the decay results.