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Title: Stability of boundary layers for a viscous hyperbolic system arising from chemotaxis : one-dimensional case
Authors: Hou, QQ 
Liu, CJ
Wang, YG
Wang, ZA 
Keywords: Boundary layers
Logarithmic singularity
Asymptotic analysis
Energy estimates
Issue Date: 2018
Publisher: Society for Industrial and Applied Mathematics
Source: SIAM journal on mathematical analysis, 2018, v. 50, no. 3, p. 3058-3091 How to cite?
Journal: SIAM journal on mathematical analysis 
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, T. 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 epsilon > 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 epsilon > 0 converges to the solution with epsilon = 0 (outer layer solution) plus the inner layer solution with the optimal rate at order of O(epsilon(1/2)) as epsilon -> 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.
ISSN: 0036-1410
EISSN: 1095-7154
DOI: 10.1137/17M112748X
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