Stability, free energy and dynamics of multi-spikes in the minimal Keller-Segel model

Abstract

<p style='text-indent:20px;'>One of the most impressive findings in chemotaxis is the aggregation that randomly distributed bacteria, when starved, release a diffusive chemical to attract and group with others to form one or several stable aggregates in a long time. This paper considers pattern formation within the minimal Keller–Segel chemotaxis model with a focus on the stability and dynamics of its multi-spike steady states. We first show that any steady-state must be a periodic replication of the spatially monotone one and they present multi-spikes when the chemotaxis rate is large; moreover, we prove that all the multi-spikes are unstable through their refined asymptotic profiles, and then find a fully-fledged hierarchy of free entropy energy of these aggregates. Our results also complement the literature by finding that when the chemotaxis is strong, the single boundary spike has the least energy hence is the most stable, the steady-state with more spikes has larger free energy, while the constant has the largest free energy and is always unstable. These results provide new insights into the model's intricate global dynamics, and they are illustrated and complemented by numerical studies which also demonstrate the metastability and phase transition behavior in chemotactic movement.</p>