| Abstract: |
| The classical singular Keller-Segel system with logistic growth has been shown to admit a unique traveling wave solution (up to a translation) with a minimal (critical) wave speed, yet the stability of this solution remains unproven. In contrast, in the absence of logistic growth, the system has a unique traveling wave solution with a single and well-defined wave speed whose stability has been well established in the literature. This study establishes the nonlinear stability of traveling wave solutions for the singular Keller-Segel system with logistic growth, covering both critical and supercritical wave speed scenarios. To our knowledge, this work marks the first stability result concerning traveling wave solutions of chemotaxis-growth systems. We prove our stability result by following three steps. First, we derive several novel auxiliary results regarding the traveling wave solutions that are indispensable for subsequent stability analysis. Next, by leveraging the anti-weighted function technique, we identify appropriate weight functions needed in our stability analysis. Finally, by employing the weighted energy method and conducting sophisticated coupling estimates, we show that the unique traveling wave solution is nonlinearly asymptotically stable under small perturbations. |
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