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Local existence of the solution to the initial‐boundary value problem in nonlinear thermomicroelasticity theory
Author(s) -
Gawinecki Jerzy,
Łazuka Jarosław
Publication year - 2004
Publication title -
pamm
Language(s) - English
Resource type - Journals
ISSN - 1617-7061
DOI - 10.1002/pamm.200410265
Subject(s) - mathematics , uniqueness , sobolev space , fixed point theorem , picard–lindelöf theorem , banach space , banach fixed point theorem , nonlinear system , boundary value problem , mathematical analysis , initial value problem , space (punctuation) , fixed point , physics , computer science , quantum mechanics , operating system
We prove a theorem about local existence (in time) of the solution to the first initial‐boundary value problem for a nonlinear system of equation of the thermomicroelasticity theory. At first, we prove existence, uniqueness and regularity of the solution to this problem for the associated linearized system by using the method of semi‐group theory. Next, basing on this theorem, we prove an energy estimate for the solution to the linearized system by applying the method of Sobolev space. At the end, using the Banach fixed point theorem, we prove that the solution of our nonlinear problem exists and is unique. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)