Global Well-Posedness and Boundary Layer Effects of Radially Symmetric Solutions for the Singular Keller–Segel Model

This paper is concerned with the global well-posedness and diffusion limit (as ε→ 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 the global well-posedness of the transformed system with ε equaling to 0 or being suitably small. Moreover, the transformed system is justified to possess boundary layer effects as ε→ 0 , where the boundary layer thickness is of O(ε^α) with 0<α<12. 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 ε> 0.