Homogenization and solvability in a chemotaxis-convection angiogenesis model with leakage boundary conditions

Abstract

In a finite one-dimensional interval, this work considers an inhomogeneous Neumann boundary value problem for a chemotaxis-convection system modeling the early phase of tumor-related angiogenesis. In this phase, the endothelial cells produce matrix and adhesive chemicals, both of which could decay or be degraded. In addition to primary random motion of all the above-mentioned three components, the endothelial cells move up toward the concentration gradients of adhesive chemicals, and moreover the former and the latter undergo convection with the spreading of matrix. Since the endothelial cells remain completely within the domain, zero-flux boundary conditions are imposed for them. However, to some extent, there is leaking at the boundaries as matrix and adhesive chemicals spread out beyond the domain. For any given suitably regular initial data, it is shown that the corresponding inhomogeneous initial-boundary value problem possesses a unique classical solution that is global-in-time and uniformly bounded via homogenization and a priori estimates.