Traveling front solutions in nonlinear diffusion degenerate Fisher-KPP and Nagumo equations via the Conley index

Abstract

Existence of one dimensional traveling wave solutions $u( x,t):=\phi ( x-ct) $ at the stationary equilibria, for the nonlinear degenerate reaction-diffusion equation $u_{t}=[K( u)u_{x}]_{x}+F( u) $ is studied, where $K$ is the density coefficient and $F$ is the reactive part. We use the Conley index theory to show that there is a traveling front solutions connecting the critical points of the reaction-diffusion equations. We consider the nonlinear degenerate generalized Fisher-KPP and Nagumo equations.