The Fisher–KPP problem with doubly nonlinear "fast" diffusion. (July 2017)

• Record Type: Journal Article
• Authors: Audrito, Alessandro; Vázquez, Juan Luis
• Is Part Of: Nonlinear analysis. Volume 157(2017)
• Page Start: 212
• Page End: 248
• Publication Date: 2017-07
• Subjects: 35K57 -- 35K65 -- 35C07 -- 35K55; Fisher–KPP equation -- Doubly nonlinear diffusion -- Propagation of level sets; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2017.03.015
• Languages: English
• ISSNs: 0362-546X

Abstract

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 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.