2016/07/05 by Alessandro Audrito, Audrito, Alessandro, Vazquez, Juan Luis · 1 citation
Mathematics · Biochemistry, Genetics and Molecular Biology · Medicine · #Stochastic processes and statistical mechanics #Diffusion and Search Dynamics #Mathematical and Theoretical Epidemiology and Ecology Models
paper · pdf · doi:10.48550/arxiv.1607.01338
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≤ 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 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 ≥ 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.