Radial boundary layers for the singular Keller-Segel model

TL;DR

This study analyzes the boundary layer behavior of radial solutions to a chemotaxis model with singular sensitivity, demonstrating uniform convergence and boundary layer profiles as the diffusion parameter approaches zero.

Contribution

It introduces a Cole-Hopf transformation approach to handle the logarithmic singularity and characterizes the boundary layer profile in the singular Keller-Segel model.

Findings

Boundary layer thickness is of order (^lpha) with 0<lpha<1/2.

Solutions are uniformly convergent as   0.

Boundary layer appears at the gradient of solutions.

Abstract

This paper is concerned with the diffusion limit (as $\va\rightarrow 0$) of radial solutions to a chemotaxis system with logarithmic singular sensitivity in a bounded interval with mixed Dirichlet and Robin boundary conditions. We use a Cole-Hopf type transformation to resolve the logarithmic singularity and prove that the solution of the transformed system has a boundary-layer profile as $\va \to 0$, where the boundary layer thickness is of $\mathcal{O}(\va^{\alpha})$ with $0<\alpha<\frac{1}{2}$. By transferring the results back to the original chemotaxis model via Cole-Hopf transformation, we find that boundary layer profile is present at the gradient of solutions and the solution itself is uniformly convergent with respect to $\va>0$.