Stability of boundary layers for a viscous hyperbolic system arising from chemotaxis : one-dimensional case

Abstract

This paper is concerned with the stability of boundary layer solutions for a viscous hyperbolic system transformed via a Cole–Hopf transformation from a singular chemotactic system modeling the initiation of tumor angiogenesis proposed in [H. A. Levine, B. Sleeman, and M. Nilsen-Hamilton, Math. Biosci., 168 (2000), pp. 71–115]. It was previously shown in [Q. Hou, Z. Wang, and K. Zhao, J. Differential Equations, 261 (2016), pp. 5035–5070] that when prescribed with Dirichlet boundary conditions, the system possesses boundary layers at the boundaries in an bounded interval (0, 1) as the chemical diffusion rate (denoted by ε > 0) is small. This paper proceeds to prove the stability of boundary layer solutions and identify the precise structure of boundary layer solutions. Roughly speaking, we justify that the solution with ε > 0 converges to the solution with ε = 0 (outer layer solution) plus the inner layer solution with the optimal rate at order of O(ε1/2) as ε → 0, where the outer and inner layer solutions are well determined and the relation between outer and inner layer solutions can be explicitly identified. Finally we transfer the results to the original pretransformed chemotaxis system and discuss the implications of our results.