An eigenvalue problem arising from spiky steady states of a minimal chemotaxis model

Asymptotic expansion of singular integrals is presented for the study of the existence and uniqueness of spiky steady states of Keller-Segel's minimal chemotaxis model. A tool is developed to find asymptotic expansion of eigenpairs of a nonlocal singular eigenvalue problem arising from the study of stability of the spiky steady state. New insights are given to simplify significantly the proofs of the main results in [1].