Traveling front solutions in nonlinear diffusion degenerate Fisher-KPP and Nagumo equations via the Conley index
In the reaction diffusion equation \[ {\partial u\over \partial t}={\partial \over \partial x}\left(K(u){\partial u\over \partial x}\right)+F(u) \quad\text{for } x\in \mathbb{R},\;t\geq 0, \] let \(u(x,0)=u_0(x)\) with \(0\leq u_0(x)\leq 1\) for all \(x\in \mathbb{R}\). The density coefficient \(K\) and the reaction function \(F\) are assumed to satisfy certain smoothness and positivity conditions on the interval \([0,1]\). Two cases are considered: the degenerate generalized Fisher-KPP equation in which both \(F\)and \(K\) vanish at 0 and 1, and the degenerate generalized Nagumo equation in which both \(F\) and \(K\) also vanish at some \(\alpha \in (0,1)\). The authors use the Conley index theory to show that in each case, there exist traveling front solutions that connect the zeros of \(F\) (and \(K\)).
scientific article; zbMATH DE number 5794097