The Fisher-KPP problem with doubly nonlinear "fast" diffusion

TL;DR

This paper explores the Fisher-KPP reaction diffusion model with 'fast' doubly nonlinear diffusion, revealing that solutions exhibit exponential spatial propagation and non-traditional asymptotic behavior, contrasting with classical linear diffusion results.

Contribution

It provides the first analysis of the Fisher-KPP model with 'fast' doubly nonlinear diffusion, showing exponential propagation and bounds on level sets, diverging from classical wave solutions.

Findings

Solutions show exponential spatial propagation for large times.

General solutions exhibit non-traditional asymptotic behavior.

Level sets are approximately linear in logarithmic spatial scale for large times.

Abstract

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 solutions $0\leq u(x,t)\leq 1$ 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 arXiv:1601.05718. 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 $N \geq 1$. In particular, we show that location of the level sets is approximately linear for large times, when we take spatial logarithmic scale, finding a strong departure from the linear case, in which appears the famous Bramson logarithmic correction.