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Convergence to equilibrium for a fully hyperbolic phase‐field model with Cattaneo heat flux law
Author(s) -
Jiang Jie
Publication year - 2009
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.1092
Subject(s) - mathematics , dissipative system , heat flux , mathematical analysis , infinity , neumann boundary condition , hyperbolic partial differential equation , boundary value problem , boundary (topology) , convergence (economics) , heat transfer , physics , partial differential equation , thermodynamics , economics , economic growth
We consider a fully hyperbolic phase‐field model in this paper. Our model consists of a damped hyperbolic equation of second order with respect to the phase function χ( t ) , which is coupled with a hyperbolic system of first order with respect to the relative temperature θ( t ) and the heat flux vector q ( t ). We prove the well‐posedness of this system subject to homogeneous Neumann boundary condition and no‐heat flux boundary condition. Then, we show that this dynamical system is a dissipative one. Finally, using the celebrated Łojasiewicz–Simon inequality and by constructing an auxiliary functional, we prove that the solution of this problem converges to an equilibrium as time goes to infinity. We also obtain an estimate of the decay rate to equilibrium. Copyright © 2008 John Wiley & Sons, Ltd.