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On a phase-field model with a logarithmic nonlinearity. (English) Zbl 1265.35139
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.

MSC:
35K55 Nonlinear parabolic equations
35J60 Nonlinear elliptic equations
80A22 Stefan problems, phase changes, etc.
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