Disappearing Interfaces In Nonlinear Diffusion

We study the large-time behaviour and the behaviour of the interfaces of the nonlinear diffusion equation ae(x)u t = \DeltaA(u) in one and two space dimensions. The function A is of porous media type, smooth but with a vanishing derivative at some values of u, and ae ? 0 is supposed continuous and bounded from above. If ae is not bounded away from zero, the large-time behaviour of solutions and their interfaces can be essentially different from the case when ae is constant. We extend results by Rosenau and Kamin [13] and derive the large-time asymptotic behaviour of solutions, as well as a precise characterisation of the behaviour of the interfaces of solutions in one space dimension and in some cases in two space dimensions. In one space dimension and when ae is monotonic the result states that the interface i(t) = supfx 2 R : u(x; t) ? 0g tends to infinity in finite time if and only if R 1 0 xae(x) dx ! 1. 1 Introduction In this article we study some properties of solutions of ..