Singular limit of a Fisher equation with degenerate diffusion

D. Hilhorst (Université de Paris-Sud, Orsay)
with R. Kersner, E. Logak and M. Mimura
 

2 November 1999, Department of Mathematics, Hiroshima University

The purpose of this study is to gain a better understanding of some reaction-diffusion models for bacterial colonies; such models often have the form of a noncooperative reaction-diffusion system with nonlinear diffusion and the main unknown functions are the density of a bacteria and the concentration of some nutrient.

In this talk we consider a simpler model involving a Fisher equation with degenerate diffusion, namely

$$u_t = \varepsilon \Delta u^m + {1 \over \varepsilon} u (1-u) \qquad m >1,$$

together with a homogeneous Neumann boundary condition and a suitable initial condition. We prove that, as $\varepsilon$ tends to zero, $u^\varepsilon$converges to a limiting function u which is almost everywhere equal to zero or to one and that the interface between the regions where u=0 and u=1 moves according to the law

V_n = c_*,

where c_* is the minimum velocity of the travelling waves of a related equation.