Stationary Solutions and Stability of Constant Equilibrium Solutions for a Chemotaxis System

We consider a one-dimensional Chemotaxis model describing the dynamics of the cell density n, cell velocity u, chemoattractant c1, and chemorepellent c2, respectively. The model is related to angiogenesis in cancer cells. n represents the cell density of the vessel, the chemoattractant c1 is vascular endothelial growth factor (VEGF) and the chemorepellent c2 is the antiangiogenic drug. We study the existence of a nonconstant steady-state solution using the singular perturbation method. We also provide numerical examples of steady-state solutions for the fast system to illustrate the idea. For the dynamical problem in part II, we study the stability of constant stationary solutions for the initial boundary value problem. We discuss the existence of global solutions based on the existence of local solution and the a-priori estimates