Well-posedness and long time behavior of a parabolic-hyperbolic phase-field system with singular potentials
✍ Scribed by Maurizio Grasselli; Alain Miranville; Vittorino Pata; Sergey Zelik
- Publisher
- John Wiley and Sons
- Year
- 2007
- Tongue
- English
- Weight
- 376 KB
- Volume
- 280
- Category
- Article
- ISSN
- 0025-584X
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✦ Synopsis
Abstract
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)