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

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.