Global regular solutions to two-dimensional thermoviscoelasticity

Jerzy  Gawinecki; Wojciech M. Zajączkowski

doi:10.3934/cpaa.2016.15.1009

Article Contents

2016, Volume 15, Issue 3: 1009-1028. Doi: 10.3934/cpaa.2016.15.1009

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Global regular solutions to two-dimensional thermoviscoelasticity

Jerzy Gawinecki^1, and
Wojciech M. Zajączkowski²

1.
Institute of Mathematics and Cryptology, Cybernetics Faculty, Military University of Technology, S. Kaliskiego 2, 00-908 Warsaw
2.
Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-656 Warsaw

Received: February 28, 2015

Revised: October 31, 2015

Published: April 2016

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Abstract

A two-dimensional thermoviscoelastic system of Kelvin-Voigt type with strong dependence on temperature is considered. The existence and uniqueness of a global regular solution is proved without small data assumptions. The global existence is proved in two steps. First global a priori estimate is derived applying the theory of anisotropic Sobolev spaces with a mixed norm. Then local existence, proved by the method of successive approximations for a sufficiently small time interval, is extended step by step in time. By two-dimensional solution we mean that all its quantities depend on two space variables only.

Keywords:

Thermoviscoelastic system,
Kelvin-Voigt type materials,
anisotropic Sobolev spaces with a mixed norm,
a priori estimates,
global existence.

Mathematics Subject Classification: Primary: 74B20, 35K50; Secondary: 35Q72, 74F05.

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References

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