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Anisotropic diffusion in oriented environments can lead to singularity formation

Published online by Cambridge University Press:  20 December 2012

THOMAS HILLEN ,
KEVIN J. PAINTER  and
MICHAEL WINKLER
THOMAS HILLEN
Affiliation:
Centre for Mathematical Biology, Department of Mathematical and Statistical Sciences, University of Alberta, Canada email: thillen@ualberta.ca
KEVIN J. PAINTER
Affiliation:
Department of Mathematics and Maxwell Institute for Mathematical Sciences, Heriot-Watt University, Edinburgh, UK email: K.Painter@hw.ac.uk
MICHAEL WINKLER
Affiliation:
Institut für Mathematik, Universität Paderborn, Germany email: michael.winkler@math.uni-paderborn.de
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Abstract

We consider an anisotropic diffusion equation of the form ut = ∇∇(D(x)u) in two dimensions, which arises in various applications, including the modelling of wolf movement along seismic lines and the invasive spread of certain brain tumours along white matter neural fibre tracts. We consider a degenerate case, where the diffusion tensor D(x) has a zero-eigenvalue for certain values of x. Based on a regularisation procedure and various pointwise and integral a priori estimates, we establish the global existence of very weak solutions to the degenerate limit problem. Moreover, we show that in the large time limit these solutions approach profiles that exhibit a Dirac-type mass concentration phenomenon on the boundary of the region in which diffusion is degenerate, which is quite surprising for a linear diffusion equation. The results are illustrated by numerical examples.

Keywords

Anisotropic diffusionDegenerate diffusionLarge time behaviourSingularity formationPattern formation
Type
Papers
Information
European Journal of Applied Mathematics , Volume 24 , Issue 3 , June 2013 , pp. 371 - 413
Copyright
Copyright © Cambridge University Press 2012

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