Transition between extinction and blow-up in a generalized Fisher-KPP model

TL;DR

This paper characterizes stationary solutions of a generalized Fisher-KPP equation with nonlinear diffusion, providing criteria to distinguish between blow-up and extinction behaviors, thus advancing understanding of solution dynamics in reaction-diffusion models.

Contribution

It introduces a comprehensive analysis of stationary solutions and a criterion for predicting blow-up or extinction in generalized Fisher-KPP models.

Findings

Stationary solutions separate blow-up from extinction.

A criterion for solution behavior based on parameters.

Characterization of solution types in generalized models.

Abstract

Stationary solutions of the Fisher-KPP equation with general nonlinear diffusion and arbitrary reactional kinetic orders terms are characterized. Such stationary (separatrix-like) solutions disjoint the blow-up solutions from those showing extinction. In addition a criterion for general parameter values is presented, which allows determining the blow-up or vanishing character of the solutions.