30 Novembre, 2016 13:15 in punto

Sezione di Analisi

The Fisher-KPP problem with doubly nonlinear diffusion

Alessandro Audrito, Università degli Studi di Torino, Universidad Autonoma de Madrid

Aula seminari 3° piano

The famous Fisher-KPP reaction-diffusion model combines linear diffusion with the typical KPP reaction term, and appears in a number of relevant applications in biology and chemistry. It is remarkable as a mathematical model since it possesses a family of travelling waves that describe the asymptotic behaviour of a large class solutions 0 <= u(x, t) <= 1 of the problem posed in the real line. The existence of propagation waves with finite speed has been confirmed in some related models and disproved in others. We investigate here the corresponding theory when the linear diffusion is replaced by the “slow” and “fast” doubly nonlinear diffusion. In the first case, we find travelling waves that represent the wave propagation of more general solutions even when we extend the study to several space dimensions (we show a TW asymptotic behaviour also in the critical case that we call “pseudo-linear”, i.e., when the operator is still nonlinear but has homogeneity one. In the second one, we prove the non-existence of travelling waves while the propagation is exponentially fast in space for large times. Moreover, in the “fast” diffusion case, we show precise bounds for the level sets of general solutions.