Multiple Stable Traveling Wave Profiles of a System of Conservation Laws Arising from Chemotaxis

Abstract

In this paper, we establish the existence and nonlinear stability of a hyperbolic system of conservation laws derived from a repulsive singular chemotaxis model. By the phase plane analysis alongside Poincar´e-Bendixson theorem, we first prove that this hyperbolic system admits three different types of traveling wave profiles, which are explicitly illustrated with numerical simulations. Then using a unified weighted energy estimates and technique of taking anti-derivatives, we prove that all types of traveling wave profiles, including non-monotone pulsating wave profiles, are nonlinearly and asymptotically stable if the initial data are small perturbations with zero mass from the spatially shifted traveling wave profiles.

Keywords

Chemotaxis / conservation laws / traveling waves / nonlinear stability / weighted energy estimates

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