The Fisher–KPP problem with doubly nonlinear “fast” diffusion

• Published in: Nonlinear analysis 2017-07, Vol.157, p.212-248
• Main Authors: Audrito, Alessandro, Vázquez, Juan Luis
• Format: Article
• Language: English
• Subjects: Diffusion; Diffusion rate; Doubly nonlinear diffusion; Fisher–KPP equation; Mathematical models; Nonlinear equations; Propagation; Propagation of level sets; Traveling waves; Wave propagation
• ISSN: 0362-546X; 1873-5215
• DOI: 10.1016/j.na.2017.03.015

The famous Fisher–KPP reaction–diffusion model combines linear diffusion with the typical Fisher–KPP reaction term, and appears in a number of relevant applications. It is remarkable as a mathematical model since, in the case of linear diffusion, it possesses a family of travelling waves that describe the asymptotic behaviour of a wide class of solutions 0≤u(x,t)≤1 of the problem posed in the real line. The existence of propagation wave with finite speed has been confirmed in the cases of “slow” and “pseudo-linear” doubly nonlinear diffusion too, see Audrito and Vázquez (2016). We investigate here the corresponding theory with “fast” doubly nonlinear diffusion and we find that general solutions show a non-TW asymptotic behaviour, and exponential propagation in space for large times. Finally, we prove precise bounds for the level sets of general solutions, even when we work in with spacial dimension N≥1. In particular, taking spatial logarithmic scale, we show that the location of the positive level sets is approximately linear for large times. This represents a strong departure from the linear case, in which the location of the level sets is not purely linear, but presents the celebrated logarithmic deviation for large times.