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Attractor for a conserved phase‐field system with hyperbolic heat conduction
Author(s) -
Grasselli Maurizio,
Pata Vittorino
Publication year - 2004
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.533
Subject(s) - dissipative system , attractor , mathematics , hyperbolic partial differential equation , boundary (topology) , mathematical analysis , heat equation , thermal conduction , conserved quantity , type (biology) , field (mathematics) , boundary value problem , mathematical physics , heat flux , partial differential equation , physics , heat transfer , pure mathematics , thermodynamics , ecology , biology
We consider a conserved phase‐field system of Caginalp type, characterized by the assumption that both the internal energy and the heat flux depend on the past history of the temperature and its gradient, respectively. The latter dependence is a law of Gurtin–Pipkin type, so that the equation ruling the temperature evolution is hyperbolic. Thus, the system consists of a hyperbolic integrodifferential equation coupled with a fourth‐order evolution equation for the phase‐field. This model, endowed with suitable boundary conditions, has already been analysed within the theory of dissipative dynamical systems, and the existence of an absorbing set has been obtained. Here we prove the existence of the universal attractor. Copyright © 2004 John Wiley & Sons, Ltd.