Boundary layer problem on the chemotaxis model with Robin boundary conditions

This paper is concerned with the boundary layer problem on a chemotaxis system modelling boundary layer formation of aerobic bacteria in fluid. Completing the system with physical Robin-type boundary conditions for oxygen and no-flux boundary conditions for bacteria, we show that the gradients of its radial solutions in a region between two concentric spheres possessing boundary layer effects as the oxygen diffusion rate $ \varepsilon $ goes to zero and the boundary-layer thickness is of order $ \mathcal{O}(\varepsilon^\alpha) $ with $ 0<\alpha<\frac{1}{2} $.