Title: Boundary Layer Problems in Chemotaxis Models
Qianqian Hou The Hong Kong Polytechnic University
Abstract: This talk is concerned with the zero-diffusion limit of a viscous hyperbolic system transformed via a Cole-Hopf transformation from a singular chemotactic system modeling the initiation of tumor angiogenesis. It was previously found by Li and Zhao (2015) 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 $\va>0$) is small, however the rigorously mathematical justification is left open. In this talk, we fist rigorously justify the existence of boundary layers (BLs), where outside the BLs the solution with $\va>0$ converges to the one with $\va=0$, but inside the BLs the convergence no longer holds. We then proceed 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 $\va>0$ converges to the solution with $\va=0$ (outer layer) plus the (inner) boundary layer solutions with the optimal rate at order of $O(\va^{1/2})$, where the outer and inner layer solutions are well determined by explicit equations. Finally, we covert the result for the transformed system to the original pre-transformed chemotaxis system and discuss the biological implications of our results.
Time and Place: 16:00-17:00, April 19, 2018,Gewubuilding, Room 522
