Minimal speed of unbounded traveling wave solutions for a 1D reaction-diffusion equation and their relationship with the dynamics at infinity

TL;DR

This paper investigates the minimal speed and unboundedness of traveling wave solutions in a 1D reaction-diffusion equation with an asymptotically linear reaction term, using a Poincaré compactification to analyze dynamics at infinity.

Contribution

It introduces a novel approach to determine the minimal wave speed by analyzing the system's behavior at infinity, differing from traditional linear determinacy methods.

Findings

Classified trajectories in the phase plane for traveling waves.

Proved positivity and unboundedness of certain wave profiles.

Derived explicit minimal wave speed from dynamics at infinity.

Abstract

This paper presents results on the unboundedness and minimal speed of traveling wave solutions for a one-dimensional spatial reaction-diffusion equation with an asymptotically linear reaction term and a saturation parameter. By applying a Poincar\'e-type compactification, we reveal the full dynamics (including infinity) of the two-dimensional system of ordinary differential equations satisfied by traveling wave solutions. This yields essential information characterizing traveling wave solutions: the classification of trajectories in the phase plane, the positivity and unboundedness of front-type and sign-changing profiles, and the explicit form of the minimal speed. This paper examines a special equation with an asymptotically linear reaction term. While, our results differ from those of conventional linear determinacy. We claim that the minimal speed is derived from information at infinity within the traveling wave system.