Travelling-wave behaviour in problems with degenerate diffusion

Abstract

We review some recent results on the large-time behaviour of solutions to certain reaction-diffusion equations involving a diffusion operator that degenerates at the level 0. Nonnegative solutions with compactly supported initial data have a compact support for any later time, so that the problem has a free boundary whose asymptotic location one would like to determine. Problems in this family have a unique (up to translations) travelling wave solution with a finite front. When the initial datum is bounded, radially symmetric and compactly supported, we prove that solutions converging to 1 (which exist for all the reaction terms under consideration) do so by approaching a translation of this unique traveling wave in the radial direction, but with a logarithmic correction in the position of the front when the dimension is bigger than one. As a corollary we obtain the asymptotic location of the free boundary and level sets in the non-radial case up to an error term of size $O(1)$. A main technical tool of independent interest is an estimate for the flux. This is a collaboration with Y. Du, A. Gárriz and M. Zhou.