Global regular solutions to three-dimensional thermo-visco-elasticity with nonlinear temperature-dependent specific heat

Irena PawŃow; Wojciech M. Zajączkowski

doi:10.3934/cpaa.2017065

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2017, Volume 16, Issue 4: 1331-1372. Doi: 10.3934/cpaa.2017065

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Global regular solutions to three-dimensional thermo-visco-elasticity with nonlinear temperature-dependent specific heat

Irena PawŃow^1,2, and
Wojciech M. Zajączkowski^3,4,

1.
Systems Research Institute, Polish Academy of Sciences, Newelska 6, 01-447 Warsaw, Poland
2.
Institute of Mathematics and Cryptology, Cybernetics Faculty, Military University of Technology, S. Kaliskiego 2, 00-908 Warsaw, Poland
3.
Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-656 Warsaw, Poland
4.
Institute of Mathematics and Cryptology, Cybernetics Faculty, Military University of Technology, S. Kaliskiego 2, 00-908 Warsaw, Poland

Received: April 2016

Revised: September 2016

Published: May 2017

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Abstract

A three-dimensional thermo-visco-elastic system for Kelvin-Voigt type material at small strains is considered. The system involves nonlinear temperature-dependent specific heat relevant in the limit of low temperature range. The existence of a unique global regular solution is proved without small data assumptions. The proof consists of two parts. First the existence of a local in time solution is proved by the Banach successive approximations method. Then a lower bound on temperature and global a priori estimates are derived with the help of the theory of anisotropic Sobolev spaces with a mixed norm. Such estimates allow to extend the local solution step by step in time. The paper generalizes the results of the previous author's publication in SIAM J. Math. Anal. 45, No. 4 (2013), pp. 1997–2045.

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Mathematics Subject Classification: Primary: 74B20, 35K50; Secondary: 35Q74, 74F05.

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