Spectral analysis for stability of bubble steady states of a Keller–Segel's minimal chemotaxis model

Abstract

A number of techniques, some of which are novel, are introduced to develop a systematic method to study a set of eigenvalue problems arising from the stability analysis of bubble steady states of a Keller-Segel's minimal chemotaxis model. Estimates of the eigenvalue with largest real part of an elliptic system without variational structure and the second eigenvalue of a corresponding subproblem possessing variational structure are obtained. These estimates provide critical information about the stability of the bubble steady state with respect to the time relaxation parameter; in particular, it is shown that the stability decreases to zero as the relaxation parameter goes to infinity.

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

“…In particular, their result implies that the stationary cellular population density of the minimal model (1.1) approaches the δ-function in this limit. The literature developed in line has also witnessed the success in Keller-Segel models with volume-filling [28,51], logarithmic sensitivity [29], etc. Moreover, though this approach is currently restricted to 1D setting, one of its advantages is the applicability in problems with cellular growth [43,44,45] and variants of reaction-advection-diffusion systems [14,41,42].…”