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L p – L q time decay estimate to the solution of the Cauchy problem for the system of equations describing nonlocal model of the thermoviscoelastic body
Author(s) -
Gawinecki Jerzy,
Lazuka Jaroslaw,
Rafa Jozef
Publication year - 2009
Publication title -
pamm
Language(s) - English
Resource type - Journals
ISSN - 1617-7061
DOI - 10.1002/pamm.200910228
Subject(s) - sobolev space , cauchy problem , initial value problem , matrix (chemical analysis) , mathematical analysis , mathematics , cauchy distribution , tensor (intrinsic definition) , mathematical physics , space (punctuation) , cauchy stress tensor , physics , pure mathematics , chemistry , chromatography , linguistics , philosophy
The 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 L p – L q time decay estimate to the solution of the Cauchy problem. (© 2009 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)