Sharp profiles in degenerate and doubly degenerate Fisher-KPP equations

This paper investigates the effects of a degenerate diffusion term in reaction–diffusion models <f>ut=[D(u)ux]x+g(u)</f> with Fisher-KPP type <f>g</f>. Both in the case when <f>D(0)=0</f> and when <f>D(0)=D(1)=0</f>, with <f>D(u)>0</f> elsewhere, we obtain a continuum of travelling wave solutions having wave speed <f>c</f> greater than a threshold value <f>c&ast;</f> and we show the appearance of a sharp-type profile when <f>c=c&ast;</f>. These results solve recent conjectures formulated by Sa´nchez-Gardun˜o and Maini (J. Differential Equations 117 (1995) 281) and Satnoianu et al. (Discrete Continuous Dyn. Systems (Series B) 1 (2000) 339). [Copyright &y& Elsevier]