Optimal decay rate for the Cauchy problem of the standard linear solid model with Gurtin–Pipkin thermal law

Abstract

In this paper, we consider the Cauchy problem for the standard linear solid model in the whole space Rn, where the heat conduction is given by the Gurtin–Pipkin thermal law. The energy method in the Fourier space is used to obtain the optimal decay rate of the L2–norm of the solution. More precisely, we prove that the decay rate of the solution is of the form (1+t)−n/4 and of regularity–loss type. Hence, we improve and extend the result obtained in [20].