Well‐posedness and long time behavior of a parabolic‐hyperbolic phase‐field system with singular potentials

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)