A Free Boundary Problem Modeling Tumor Growth With A Necrotic Core

In this thesis,we focus on a free boundary problem describing the formation mech-anism of necrotic cores in the growth of solid tumors.The tumor model contains a nonlinear elliptic equation reflecting the changes of nutrient concentration in the tumor and a first-order differential equation describing the motion of the tumor's outer surface The nutrient consumption function and tumor growth function are discontinuous at the necrosis boundary of the tumor.This thesis firstly considers the case that the exter-nal nutrient concentration is a constant ?.By upper and lower solution method and the comparison principle,we obtain the existence and uniqueness of the spherically sym-metric stationary solutions of the non-necrotic tumor model.On this basis,we further prove that there is a threshold value of external nutrient concentration ?*.If and only if ?>?*,the tumor model has a unique spherically symmetric stationary solution with necrosis.Then we use the classical theory of differential equations prove the existence and uniqueness of the global solution.After that we analyze the asymptotic behavior of the global solution and then we establish the global asymptotic stability of the spherically symmetric stationary solution.Finally,this thesis considers the case that the external nutrient supply is a periodic function.Using the fixed point method and we obtain the existence and uniqueness of the spherically symmetric periodic solution.And we prove that for arbitrary non-zero initial values,the global spherically symmetric solution of this problem will eventually converge to the periodic solution.