Qualitative Analysis On Solutions Of Vascularized Tumor With Inhibitor And Necrotic Cores

In this thesis,we discuss a free boundary problem modeling the growth of vascularized necrotic tumor under the action of inhibitor.It is assumed that the tumor is spherical symmetry,the model can be described by the following free boundary problem:R(0)=R0,where σ(r,t),β(r,t),R(t)is the nutrient concentrations,the inhibitor concentrations in the tumor,tumor radius,respectively.c1,c2 is the ratio of nutrient,inhibitor absorption by tumor cells and growth time,respectively,usually c1,c2<<1;Here H(·)is Heaviside function(H(x)=1 if x>0,andH(x)=0 if x<0);The positive constant α reflects the degree of tumor vascularization,and the larger α is,the higher the degree of vascularization will be.In particular,if the tumor is completely surrounded by blood vessels,then α=∞.And the Robin boundary condition is transformed into Dirichlet boundary condition,where α,β are nutrient concentrations and inhibitor concentrations in the tissues around the tumor;σ0,β0 are the initial concentration of nutrients and inhibitors,σD is minimum nutrient concentration required for tumor cells survival,when σ≤σD,tumor cells will die;μ is the growth intensity of tumor cells,σis the threshold of nutrient concentrations required by mitosis of tumor cells,k1 is the inhibition intensity of inhibitors on tumor cells,and k2 is the decomposition rate of necrotic cells.The thesis is divided into three chapters.In chapter 1,we introduce the known research related on our problems and main results.In chapter 2,the existence and uniqueness of stationary solution under certain conditions are proved by using the power series expansion method,we calculate that explicit solutions of nutrient concentrations and inhibitor concentrations,and then substitute it into equations to study the existence and uniqueness of solutions of tumor radius and dead nucleus radius.In chapter 3,since all equations involve discontinuous terms,we prove the existence and boundedness of strong solutions in any given interval[0,T]under completely unsteady conditions by using the approximation method combined with Lp theory of parabolic partial differential equations and Schauder fixed point theorem.