Lp – Lq time decay estimate to the solution of the Cauchy problem for the system of equations describing nonlocal model of the thermoviscoelastic body

Abstract

AbstractThe aim of this paper is to present a new system of equations describing nonlocal model of thermoviscoelastic theory. We used the Papkin and Gurtin approach based on the constitutive relations for stress tensor σ(x), internal energy e(x) and heat flux q(x), with integral terms. Using the modified Cagniard‐de Hoop's method we constructed the matrix of fundamental solutions for this system of equations in three‐dimensional space. Basing on this matrix we represent in the explicit formula the solution of the Cauchy problem to this system of equations. Next, applying the method of Sobolev spaces, we proved the Lp–Lq time decay estimate to the solution of the Cauchy problem. (© 2009 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)