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Mathematical and physical aspects of the solution to the Cauchy problem for a time‐nonlocal model of thermoelasticity theory
Author(s) -
Gawinecki Jerzy August,
Rafa Józef,
Łazuka Jarosław
Publication year - 2020
Publication title -
zamm ‐ journal of applied mathematics and mechanics / zeitschrift für angewandte mathematik und mechanik
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.449
H-Index - 51
eISSN - 1521-4001
pISSN - 0044-2267
DOI - 10.1002/zamm.201800326
Subject(s) - mathematics , sobolev space , mathematical analysis , cauchy problem , cauchy distribution , initial value problem , matrix (chemical analysis) , type (biology) , wave equation , heat equation , heat flux , physics , heat transfer , mechanics , ecology , materials science , composite material , biology
Abstract The thermoelasticity theories describing a finite speed for the propagation of thermal perturbation have been investigated over the last 50 years. In contrast to the classical thermoelasticity theory, these nonclassical theories involve a hyperbolic type heat transport equation and they are motivated by experiments exhibiting the actual occurence of wave‐type heat transport, so‐called second sound (cf. [6–8,11,26]). The aim of this paper is to present a new system of equations describing nonlocal model of thermoelasticity theory. In order to obtain it we used the Pipkin and Gurtin [26] approach to the constitutive relations for the internal energy and the heat flux. Using the modified Cagniard‐de Hoop method we constructed the matrix of fundamental solutions to this system of equations in the three‐dimensional spaces. Basing on the constructed matrix of fundamental solutions we obtained the explicit formulae for the solution to the Cauchy problem to this system of equations in the form of some kind of convolutions. Applying the method of Sobolev spaces we proved theL p − L qtime decay estimate for the solution to the considered Cauchy problem to this new system of equations. We are interested in physical and mathematical aspects of this new system of equations. We discovered the new type of thermal wave appearing in the medium described by the nonlocal model of thermoelasticity theory.