Stability of the steady state for multi-dimensional thermoelastic systems ofshape memory alloys

Takashi Suzuki; Shuji Yoshikawa

doi:10.3934/dcdss.2012.5.209

Article Contents

2012, Volume 5, Issue 1: 209-217. Doi: 10.3934/dcdss.2012.5.209

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Stability of the steady state for multi-dimensional thermoelastic systems of shape memory alloys

Takashi Suzuki^1, and
Shuji Yoshikawa^2,

1.
Division of Mathematical Science, Department of System Innovation, Graduate School of Engineering Science, Osaka University, 1-3 Machikane-yama, Toyonaka, Osaka, 560-8531
2.
Department of Engineering for Production and Environment, Graduate School of Science and Engineering, Ehime University, 3 Bunkyo-cho, Matsuyama, Ehime 790-8577

Received: March 31, 2009

Revised: October 31, 2009

Published: January 2012

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Abstract

This paper studies a dynamical stability of the steady state for some thermoelastic and thermoviscoelastic systems in multi-dimensional space domain. More general nonlinear term can be taken here than the one in [6] which studied the stability for the one-dimensional system called the Falk model system. We also give applications to thermoviscoelastic systems treated in [8] and [9].

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Mathematics Subject Classification: Primary: 74N30; Secondary: 35Q72, 80A22.

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References

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[2]	F. Falk and P. Konopka, Three-dimensional Landau theory describing the martensitic phase transformation of shape memory alloys, Journal of Physics: Condensed Matter, 2 (1990), 61-77.doi: 10.1088/0953-8984/2/1/005.
[3]	I. Pawłow, Three-dimensional model of thermomechanical evolution of shape memory materials, Control Cybernet., 29 (2000), 341-365.
[4]	T. Suzuki, "Mean Field Theories and Dual Variation," Atlantis Press, Amsterdam, 2008.
[5]	T. Suzuki and S. Tasaki, Stationary solutions to the Falk system on shape memory alloys, in preparation.
[6]	T. Suzuki and S. Yoshikawa, Stability of the steady state for the Falk model system of shape memory alloys, Math. Methods Appl. Sci., 30 (2007), 2233-2245.doi: 10.1002/mma.889.
[7]	H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators," North-Holland Mathematical Library 18, North-Holland Publishing Co., Amsterdam-New York, 1978.
[8]	S. Yoshikawa, I. Pawłow and W. M. Zajączkowski, Quasilinear thermoelasticity system arising in shape memory materials, SIAM J. Math. Anal., 38 (2007), 1733-1759.doi: 10.1137/060653159.
[9]	S. Yoshikawa, I. Pawłow and W. M. Zajązkowski, A quasilinear thermoviscoelastic system for shape memory alloys with temperature dependent specific heat, Commun. Pure Appl. Anal., 8 (2009), 1093-1115.doi: 10.3934/cpaa.2009.8.1093.