Qualitative Atherosclerosis Model

In order to study many questions in the natural sciences of physics, chemistry and biology, lots of mathematical models have established among which reaction-diffusion equations are widely used. A large number of practical problems can be explained more scientifically and rigorously by studying reaction-diffusion equations. A atherosclerosis initiation model is theoretically investigated in this paper. This model has been developed in some degree, however, it is discussed in the ordinary differential systems. In this paper, we consider the effect of the diffusion and based on the theory of reaction-diffusion equations, bifurcation theory, degree theory, make it more realistically.This paper focuses on the following model whereThe main contents in this thesis are as follows:In chapter1, there are introduction and preliminaries which introduce the back-ground of the model, the maximum principle and bifurcation theorem.In chapter2, first, we discuss the non-negative constant equilibrium solution, dissipation, persistence and stability. Second, the priori estimations of positive steady states are given by the use of maximum principle solution and by using Y-oung inequality and Poincare inequality, the non-existence of non-constant positive steady-state solutions is established when, and d is large enough. Then, based on+v*/T2)>0, treating d2as the bifurcation parameter, the local bifurcation from the constant steady-state solution Ul is studied. Under appropriate conditions, it is proved that the local bifurcation can be extended to the global bifurcation. Finally, by treating d2as the bifurcation parameter, the local bifurcation of the constant steady-state solution Ul from double eigenvalues are studied and the existence of the non-constant steady-state solutions are estab-lished. The existence of non-constant positive steady-state solutions is gained by the spectrum analysis of operators and degree theory.