On the Fisher-KPP equation with fast nonlinear diffusion

In this paper we examine the Cauchy problem for a nonlinear Fisher-Kolmogorov-Petrovski-Piskunov (Fisher-KPP) reaction-diffusion equation with fast diffusion, namely u(t) = del . (u(-n)delu) + u(1 - u) for x in R-N, 0 < n < max(1, 2/N) and t > 0. We show, in particular, that the problem does not have permanent-form travelling-wave solutions, and we derive the large-time asymptotic form of the solution.