Sharp profiles in degenerate and doubly degenerate Fisher-KPP equations

This paper investigates the effects of a degenerate diffusion term in reaction-diffusion models u(t) - [D(u)u(x)](x) + g(u) with Fisher-KPP type g. Both in the case when D(0) = 0 and when D(0) = D(l) = 0, with D(u) > 0 elsewhere, we obtain a continuum of travelling wave solutions having wave speed c greater than a threshold value c* and we show the appearance of a sharp-type profile when c = c*. These results solve recent conjectures formulated by Sanchez-Garduno and Maim (J. Differential Equations 117 (1995) 281) and Satnoianu et al. (Discrete Continuous Dyn. Systems (Series B) 1 (2000) 339). (C) 2003 Elsevier Inc. All rights reserved.