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Well‐posedness and long time behavior of a parabolic‐hyperbolic phase‐field system with singular potentials
Author(s) -
Grasselli Maurizio,
Miranville Alain,
Pata Vittorino,
Zelik Sergey
Publication year - 2007
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.200510560
Subject(s) - mathematics , uniqueness , logarithm , mathematical analysis , attractor , inertial frame of reference , singular perturbation , hyperbolic partial differential equation , parabolic partial differential equation , heat equation , exponential function , singular solution , partial differential equation , classical mechanics , physics
In this article, we study the long time behavior of a parabolic‐hyperbolic system arising from the theory of phase transitions. This system consists of a parabolic equation governing the (relative) temperature which is nonlinearly coupled with a weakly damped semilinear hyperbolic equation ruling the evolution of the order parameter. The latter is a singular perturbation through an inertial term of the parabolic Allen–Cahn equation and it is characterized by the presence of a singular potential, e.g., of logarithmic type, instead of the classical double‐well potential. We first prove the existence and uniqueness of strong solutions when the inertial coefficient ε is small enough. Then, we construct a robust family of exponential attractors (as ε goes to 0). (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)