On a degenerate diffusion equation of the form \(c(z)_ t=\phi (z_ x)_ x\) with application to population dynamics (Q1092237)

We study the equation \(c(z)_ t=\phi (z_ x)_ x\), where \(\phi (s)=| s|^{m-1}s\) \((m>1)\) and where the positive derivative c'(z) has a discontinuity at \(z=0\). In particular we analyze the level set \({\mathcal J}=\{(x,t):\) \(z(x,t)=0\}\) and prove that after some well- defined time T, \({\mathcal J}\) becomes a smooth curve. The equation arises in a theory for two interacting populations which disperse in response to population pressure. If the populations are initially segregated, they remain segregated for all later times, and the level set \({\mathcal J}\) represents the segregation front.