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

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


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  • [1]

    M. Brokate and J. Sprekels, "Hysteresis and Phase Transitions," Appl. Math. Sci., 121, Springer, Berlin, 1996.


    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.


    I. Pawłow, Three-dimensional model of thermomechanical evolution of shape memory materials, Control Cybernet., 29 (2000), 341-365.


    T. Suzuki, "Mean Field Theories and Dual Variation," Atlantis Press, Amsterdam, 2008.


    T. Suzuki and S. TasakiStationary solutions to the Falk system on shape memory alloys, in preparation.


    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.


    H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators," North-Holland Mathematical Library 18, North-Holland Publishing Co., Amsterdam-New York, 1978.


    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.


    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.

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