Analyze the vanishing-chemical-diffusion regime beyond the parabolicity threshold

Investigate the asymptotic behavior of the Keller–Segel system with volume-filling sensitivity when the parabolicity condition for the scalar nonlinear diffusion limit fails, including whether aggregation-dominated dynamics produce oscillatory behavior and how density compactness can be established in that regime.

Background

The paper proves convergence as the chemical diffusion coefficient scales as D=ε²→0 only for 1≤m<2 and chemotactic sensitivity χ below the explicit threshold ensuring parabolicity of the limiting scalar diffusion equation. The authors identify the complementary parameter regime, in which this condition fails, as unresolved. They note that aggregation may dominate there and that oscillations could occur, creating additional difficulty in obtaining compactness of the density needed to pass to the nonlinear chemotactic flux.