Abstract: The stability of non-constant steady states in chemotaxis systems remains a challenging problem. In this paper, we employ local bifurcation theory to establish the existence of near-constant steady states in bounded planar domains. Building on the local bifurcation structure, together with asymptotic expansions of solutions and inequalities for Neumann eigenvalues, we investigate the asymptotic stability of these bifurcating steady states for the minimal model. In particular, we provide a detailed stability analysis for near-rectangular and near-disk planar domains with two axes of symmetry.

Key words: Pattern formation, Bifurcation theory, Asymptotic stability, Chemotaxis model, Eigenvalue problem