Stationary Ring and Concentric-Ring Solutions of the Keller--Segel Model with Quadratic Diffusion

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

“…This boundary spike moves to a boundary point with the maximum mean curvature as d → 0 + . Further development of stationary solutions with a single boundary or interior spike or multiple spikes can be found in [1,3,5,11,12,15,16,18,22,26,27,31,39,40,48] and the references therein.…”

“…Nonetheless, as we shall see in Section 2.1, any steady state of (1.1) must have a profile that is a replica of monotone solution in each sub-interval. Part of the work is motivated by [2,5] that consider a variant of (1.1) with quadratic diffusion. There the authors can study the global bifurcation diagram in detail thanks to the explicit formula of the steady states, however, the spike solutions are of compact support and their stabilities have not been analyzed.…”