Stability of travelling waves for degenerate reaction-diffusion equations of KPP-type

The aim of this paper is to investigate the asymptotic behaviour as t → ∞ of the solutions to the Cauchy problem for the nonlinear degenerate KPP-type diffusion-reaction equation u t = (u m ) xx + u p - u q , where m,p and q are positive parameters. Our result is similar to the corresponding one of Kolmogorov, Petrowsky and Piscunov; namely, we prove that for a wide class of initial functions, the solution approaches V(x - c*t) as t → ∞ uniformly in x, where V is a travelling-wave solution with a minimal speed c*.