Convergence to Traveling Waves to a Parabolic–Parabolic Chemotaxis System on the Half Space

Abstract

We are interested in the nonlinear stability of traveling waves for a singular chemotaxis model on the half space with nonzero chemical diffusion. This model describes the dynamics of vascular endothelial cell density attracted by the concentration of vascular endothelial growth factor during the initiation of tumor angiogenesis. We show that under the nonzero flux boundary condition for cells and the Robin boundary condition for the chemical signal, solutions to the system with suitable initial data converge to a shifted traveling wave, whose profile and wave speed are uniquely determined by the parameters involved in the boundary conditions.