On a phase-field model with a logarithmic nonlinearity.
type: Item
instance of: scholarly article
title: On a phase-field model with a logarithmic nonlinearity.
zbMATH Open document ID: 1265.35139
published in: Applications of Mathematics
publication date: 2013-01-02T00:00:00Z
full work available at URL: http://hdl.handle.net/10338.dmlcz/142851
review text: 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.
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 80A22
zbMATH DE Number: 6121190
zbMATH Keywords: phase field system
zbMATH Keywords: Maxwell-Cattaneo law
zbMATH Keywords: well-posedness
zbMATH Keywords: logarithmic potential
author: Alain Miranville
MaRDI profile type: Publication
OpenAlex ID: W1965338230
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DOI: 10.1007/S10492-012-0014-Y
description: scientific article; zbMATH DE number 6121190