期刊
EUROPEAN JOURNAL OF APPLIED MATHEMATICS
卷 24, 期 -, 页码 371-413出版社
CAMBRIDGE UNIV PRESS
DOI: 10.1017/S0956792512000447
关键词
Anisotropic diffusion; Degenerate diffusion; Large time behaviour; Singularity formation; Pattern formation
We consider an anisotropic diffusion equation of the form u(t) = del del(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.
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