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L p – L q Estimates To The Solution Of The Cauchy Problem For Partial Differential Equations Describing Non‐Simple Thermoelastic Materials
Author(s) -
Gawinecki Jerzy,
Łazuka Jarosław
Publication year - 2005
Publication title -
pamm
Language(s) - English
Resource type - Journals
ISSN - 1617-7061
DOI - 10.1002/pamm.200510286
Subject(s) - thermoelastic damping , mathematical analysis , displacement (psychology) , simple (philosophy) , mathematics , partial differential equation , scalar (mathematics) , fourier transform , differential equation , scalar field , initial value problem , cauchy problem , classical mechanics , physics , thermodynamics , geometry , psychology , philosophy , epistemology , thermal , psychotherapist
Theory of non‐simple materials is different from that of simple materials because in it the first strain gradient is taken into consideration as the constitutive variable. The consequence of this fact, from mathematical point of view, is that the equation of motion consists either of higher order derivatives of displacement (four order derivatives) and some material parameters can depend not only on the temperature and the gradient of displacement but also on the second derivative of displacement. We consider the system of partial differential equations describing non‐simple thermoelastic materials. This system consists of four scalar equations, three equations of motion and one of energy balance, describing the field of displacement and the temperature in an elastic body. Using the Fourier transform, we found the L p – L q time decay estimates of the solution of the Cauchy problem for the system of equation describing the non‐simple thermoelastic materials, being important for proving the global‐in‐time solution of this problem. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)