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Lp‐Lq time decay estimate to the solution of the Cauchy problem of the system of equations for nonlocal model of hyperbolic thermoelasticity theory
Author(s) -
Gawinecki Jerzy,
Lazuka Jaroslaw,
Rafa Jozef
Publication year - 2008
Publication title -
pamm
Language(s) - English
Resource type - Journals
ISSN - 1617-7061
DOI - 10.1002/pamm.200810497
Subject(s) - mathematics , sobolev space , cauchy problem , mathematical analysis , initial value problem , matrix (chemical analysis) , cauchy distribution , hyperbolic partial differential equation , space (punctuation) , spacetime , physics , partial differential equation , computer science , materials science , quantum mechanics , composite material , operating system
The aim of this paper is to present a new system of equations describing nonlocal model of hyperbolic thermoelasticity theory. We used the Papkin and Gurtin approach based on the constitutive relations for internal energy e ( x ), and heat flux q ( x ), with integral terms. Such system of equations describes the propagation of thermal perturbation with finite velocity. 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 the constructed matrix of fundamental solutions in the explicit formula we represent the solution of the Cauchy problem to this system of equations in the form of some kind of convolutions. Next, applying the method of Sobolev spaces, we obtain the L p − L q time decay estimate to the solution of the Cauchy problem. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)