### 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.

$u( x,t)$ $:=\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.

#### Keywords

Traveling front; degenerate nonlinear diffusion; Conley index; connected simple systems

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