Travelling waves and instability in a Fisher-KPP problem with a nonlinear advection and a high-order diffusion

The instability of travelling waves (TW) in high-order operators has been a source of investigation in the last years. Instability shall be understood as the existence of oscillatory exponential bundles of solutions. One of the principal topics to explore is related with the characterization of the mentioned instability phenomena in the proximity of the stationary solutions given by KPP-Fisher terms. Along this analysis, the main areas discussed cover TW instability, TW propagation speed and characterization of a local inner region with positive monotone behaviour of solutions (also called positive inner region). Furthermore, a sharp estimation of the TW propagation speed to ensure the existence of such positive inner region is obtained. Eventually, insights on regularity, existence and uniqueness of solutions are provided as supplementary information.

Automated summary

This paper examines the instability of travelling waves in high-order differential operators, focusing on the characterization of TW instability, TW propagation speed, and positive monotone behavior of solutions in a local inner region. The authors provide a sharp estimation of the TW propagation speed to ensure the existence of a positive inner region, as well as insights on regularity, existence, and uniqueness of solutions.