Multiple Stable Traveling Wave Profiles of a System of Conservation Laws Arising from Chemotaxis
DOI:
https://doi.org/10.4208/csiam-ls.SO-2024-0005aKeywords:
Chemotaxis, conservation laws, traveling waves, nonlinear stability, weighted energy estimates.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 Poincaré-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.