Global Existence and Boundedness for the Attraction-repulsion Keller-Segel Model with Volume Filling Effect

Abstract

This paper is concerned with the attraction-repulsion Keller-Segel model with volume filling effect. We consider this problem in a bounded domain Ω ⊂ ℝ³ under zero-flux boundary condition, and it is shown that the volume filling effect will prevent overcrowding behavior, and no blow up phenomenon happen. In fact, we show that for any initial datum, the problem admits a unique global-in-time classical solution, which is bounded uniformly. Previous findings for the chemotaxis model with volume filling effect were derived under the assumption 0 ≤ u₀(x) ≤ 1 with ρ(x,t) ≡ 1. However, when the maximum size of the aggregate is not a constant but rather a function ρ(x,t), ensuring the boundedness of the solutions becomes significantly challenging. This introduces a fundamental difficulty into the analysis.