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Miranville, Alain

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

Appl. Math., Praha 57, No. 3, 215-229 (2012).
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.
Reviewer: Volker Pluschke (Halle)

Cited in 7 Documents

MSC:

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

Keywords:

phase field system; Maxwell-Cattaneo law; well-posedness; logarithmic potential
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\textit{A. Miranville}, Appl. Math., Praha 57, No. 3, 215--229 (2012; Zbl 1265.35139)
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