Research On Stationary Solutions For A Free Boundary Problem Modeling Growth Of Tumor With Inhibitors

In this thesis, we investigate the existence of stationary solutions and bifurca-tion phenomenon for a free boundary problems modeling the growth of tumor with inhibitors. This thesis is divided into three chapters.In chapter 1, we introduce the known research related on our problems and main results of the thesis.In chapter 2, we discuss the free boundary problem modeling growth of tumor cord with inhibitors, in which, the inhibitors inhibit the tumor cells indirectly and directly, respectively. The forms are as follows: and the above problems are used to describe the stationary or dormant state of a solid tumor cord with inhibitor which is a cylindrical arrangement of tumor cells growing around a central blood vessel. These models assume that the tumor cord is homo-geneous in the length direction of the central blood vessel, so that it only concerns the cross-section ? of the tumor cord vertical to the length direction of the central blood vessel, therefore ? is an annular-like bounded domain in R2, whose boundary consists of two disjoint parts. J is the section of the central blood vessel wall, ? is the section of the outer surface of the tumor cord,?=?(x),?=?(x),p= p(x) rep-resenting the concentration of nutrient, the concentration of inhibitor and the tumor tissue pressure. n and v respectively denote the unit outward normal field of the interior boundary J and the exterior boundary ?. ?, ? and ? are positive contants, ? is the surface tension coefficient.f1,f2 are the nutrient and inhibitor consump-tion rate function, and g is the tumor cell proliferation rate function, respectively. f1, f2,g are smooth functions.In chapter 3, we discuss the free boundary problem modeling growth of spherical tumor with inhibitors. i.e., where ?1,?2,?,? and ? are positive constants, among which ?1 and ?2 represent the consumption rate coefficient of the nutrient and the inhibitor, respectively, by tumour cells, ? and ? reflect amounts of the nutrient and the inhibitor, respectively, that the tumor receives from its surface, k and n denote the mean curvature and the unit outward normal,? is the surface tension coefficient.The above problems are studied by the analysis of the corresponding radialized problem, by using the theory on ordinary differential equation, we prove the exis-tence of radially symmetric stationary solutions. Moreover, using the linearization at the radially symmetric stationary solutions, by reducing this problem into an abstract operator equation in a certain Banach space and applying the well-known bifurcation theorem of Crandall and Rabinowitz, we obtain the bifurcation phe-nomenon. The results show that the existence form of the stationary solution of the tumor model in addition to the radially symmetric shape, the shape of the steady state can also be provided with a certain number of bulge.