Stability of constant steady states of an attraction-repulsion chemotaxis system

Abstract We consider the Cauchy problem for an attraction-repulsion chemotaxis system in $$\mathbb {R}^n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:math> that admits a continuum of constant steady states. In the attractive case, positive constant steady states are stable under suitable conditions, whereas, in the repulsive case, all positive constant steady states are stable. We establish a sharp stability criterion for constant steady states in the general attraction-repulsion setting. We show that a constant steady state is asymptotically stable under sufficiently small perturbations whenever its amplitude is below an explicit threshold determined by the system parameters. This provides a refined classification of stability based on the balance between attractive and repulsive effects.