Anisotropic diffusion in oriented environments can lead to singularity formation

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. © Cambridge University Press 2012.