Abstract

We introduce rigorous mathematical results concerning the qualitative behavior of nonlinear PDE models describing the mechanistic and chemical properties of vasculogenesis which is the early stage of the formation of a vascular network on a substrate. The first PDE model consists of the compressible Euler equations and a reaction-diffusion equation through nonlocal coupling. Depending on the parametric and boundary conditions, different steady state solutions are constructed on bounded domains, which are reasonable consistent with experimental observations, and the steady states are shown to be locally exponentially stable provided a particular parameter is sufficiently large. The second model contains the incompressible and inhomogeneous Navier-Stokes equations nonlocally coupled with a reaction-diffusion equation. In this case, global stability of two-dimensional, large-data classical solutions on bounded domains subject to no-flow or dynamic Couette flow boundary condition is established. This talk is based on recent joint works with Hongyun Peng (Guangdong University of Technology) and Zhian Wang (Hong Kong Polytechnic University).