On a phase-field model with a logarithmic nonlinearity. (Q1928168)

The paper deals with a phase-field system based on the Maxwell-Cattaneo heat conduction law which couples a semilinear parabolic equation for the order parameter \(u\) and a hyperbolic equation for the relative temperature. In the introductory chapter the author gives a detailed overview on the physical background and existing literature for this system. The aim of the paper is to prove existence of a solution to the system where the often used regular double-well potential is replaced by a singular logarithmic nonlinearity for the order parameter. To this end the author derives a priori estimates which separate the solution from the singularity. This works in one and two space dimensions. Owing to the separation property existence of a solution then follows from results on regular problems. Some remarks on more general nonlinearities, three space dimensions, uniqueness, and a quasistatic model conclude the paper.