Determine the fixed-chemical-diffusion strong-attraction limit

Determine the limiting behavior of solutions to the volume-filling Keller–Segel system under strong attraction χ=ε⁻¹→∞ and fixed chemical diffusion D=1, including whether convergence to a time-independent sharp-interface state occurs and how the lack of uniform control of the singular energy for u≠v affects the analysis.

Background

The paper establishes a static sharp-interface limit under the simultaneous scaling χ=ε⁻¹→∞ and D=ε²→0 for characteristic initial data of finite perimeter. It leaves unresolved what happens when the chemical diffusion coefficient is held fixed, for example at D=1. The authors point out that the singular part of the energy is then no longer uniformly controlled when the density u differs from the chemical concentration v, which obstructs a direct extension of their compactness and convergence argument.