Wave Front Propagation for KPP-Type Equations

Abstract

The following equation was considered in [15]:

$$ \frac{{\partial u\left( {t,x} \right)}}{{\partial t}} = \frac{D}{2}\frac{{{\partial ^2}u}}{{\partial {x^2}}} + f\left( u \right),t > 0,x \in {R^1},u\left( {0,x} \right) = \chi - \left( x \right) = \left\{ {\begin{array}{*{20}{c}} {1,x \leqslant 0} \\ {0,x > 0.} \end{array}} \right. $$

(1.1.1)

Here D > 0 and f (u) = c(u)u,where the function c(u) is supposed to be Lipschitz continuous, positive for u < 1 and negative for u > 1, and such that c = c(0) = max_0≤u≤1 c(u). Let us denote the class of such functions f (u) by F ₁.