Rearch On Free Boundary Problems Modeling The Growth Of Tumor Cord

Thesis is concerned with the free boundary problems of the partial differential equations modeling the growth of vascularized tumor cord,we mainly study the asymptotic behavior of the free boundary and the bifurcation of steady-state solution.The full text is divided into three chapters.In chapter 1,we introduce related research on the problems considered and our main results.In chapter 2,we discuss the existence of the steady-state solution and the asymptotic behavior of the free boundary of the free-boundary problem of a radial symmetrical ring-cylindrical vascularized tumor growth model,namely:dR(t)/dt=μ/R(t)∫aR(t)r(u(r,t))-udr,u∈(0,u),u(r,0)=u0(r),where ,the cross-section of circular cylindrical tumor area is composed of two circles,a represents the radius of the inner boundary of the tumor,R(t)represents the outer boundary radius of the tumor,u represents the concentration of nutrients in the tumor,α(t)is a positive function,which is related to the degree of vascularization,the deeper the degree of vascularization,the larger α(t),u represents the threshold of the concentration of nutrients which ensures the tumor cell to mitosis,and u0(r)∈C2(a,R(0)),0≤u0(r)≤u.For this problem,we prove that(i)there is a radially stationary solution to the problem;(ii)If the angiogenesis function a is uniformly bounded,then the free boundary R(t)is uniformly bounded;(iii)If ,the free boundary will shrink to the inner boundary,that is,the tumor disappears.In chapter 3,we discuss the bifurcation phenomenon of the steady-state solution of the free boundary problem of an approximately circular columnar vascularized tumor growth model,namely:Δu=u,x ∈Ω,Δp=-μ(u-u),x∈Ω,where p represents the internal pressure of the tumor cord,Ω,represents the space area occupied by the steady-state situation of vascularized tumors,which is a priori unknown,n represents the unit outer normal vector of the tumor region boundary,K represents the curvature of the outer boundary Γ,γ represents sur-face tension between cells at the boundary,u,u and u are constants.Utilizing the modified Bessel function,theoretical analysis of abstract operator equations in Banach space and Crandall-Rabinowitz bifurcation theorem,We prove there exists a radial symmetric steady-state solution of the free boundary problem of the partial differential equation of the tumor growth model,and a positive the integer k*and the sequence {γk},when k>k*,γ=γk,the above problem has symmetry-breaking bifurcation branches at the radial symmetric steady-state solution.