Paper Info
Title
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, 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.
Year
DOI
Venue
2018
10.1137/17M112748X
SIAM JOURNAL ON MATHEMATICAL ANALYSIS
Keywords
Field
DocType
boundary layers,chemotaxis,logarithmic singularity,asymptotic analysis,energy estimates
Differential equation,Mathematical analysis,Hyperbolic systems,Dirichlet boundary condition,Viscosity,Boundary layer,Asymptotic analysis,Diffusion,Mathematics,Bounded function
Journal
Volume
Issue
ISSN
50
3
0036-1410
Citations 
PageRank 
References 
0
0.34
3
Authors
4
Name
Order
Citations
PageRank
Qianqian Hou100.34
Cheng-Jie Liu211.45
Yaguang Wang3296.70
Zhi-An Wang4113.55