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

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.