Boundary layer problem on a hyperbolic system arising from chemotaxis

This paper is concerned with the boundary layer problem for a hyperbolic system transformed via a Cole–Hopf type transformation from a repulsive chemotaxis model with logarithmic sensitivity proposed in [23] and [34] modeling the biological movement of reinforced random walkers which deposit a non-diffusible (or slowly moving) signal that modifies the local environment for succeeding passages. By prescribing the Dirichlet boundary conditions to the transformed hyperbolic system in an interval (0,1)(0,1), we show that the system has the boundary layer solutions as the chemical diffusion coefficient ε→0ε→0, and further use the formal asymptotic analysis to show that the boundary layer thickness is ε1/2ε1/2. Our work justifies the boundary layer phenomenon that was numerically found in the recent work [25]. However we find that the original chemotaxis system does not possess boundary layer solutions when the results are reverted to the pre-transformed system.