Self-similar smoothing of a discontinuity by degenerate cross-diffusion

Abstract

We consider the dynamics of an idealised cross-diffusion model of biological invasion in which the diffusion of an invading population is inhibited by a resident population, which is in turn degraded by the former. We formulate and numerically solve the problem that describes the self-similar smoothing of an initially piecewise-constant invading population. In the limit that the height ahead of the initial discontinuity vanishes, the speed of propagation also vanishes, but only logarithmically slowly. This result is confirmed by a matched asymptotic analysis. The singular nature of this limit indicates that the system does not permit the existence of compactly supported solutions that exhibit a moving front or interface, which has important implications for the simulation of more complex models that also feature this degenerate diffusive mechanism.