Disappearing Interfaces in Nonlinear Diffusion 1

We study the large-time behaviour and the behaviour of the interfaces of the non-linear diiusion equation (x)u t = A(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 > 0 is supposed continuous and bounded from above. If is not bounded away from zero, the large-time behaviour of solutions and their interfaces can be essentially diierent from the case when 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 is monotonic the result states that the interface (t) = supfx 2 R : u(x; t) > 0g tends to innnity in nite time if and only if R 1 0 xx(x) dx < 1.