Qualitative Analysis Of Free Boundary Problems Modeling The Growth Of Spherically Symmetric Necrotic Tumors With Angiogenesis

In this thesis,we investigate the free boundary problems modeling the growth of spherically symmetric necrotic tumors with angiogenesis.Based on rigorous mathematical analysis,we discuss the existence and uniqueness of stationary so-lution and the global existence of transient solutions and their asymptotic behavior,in both quasi-stationary and completed nonstationary cases.The thesis is organized as follows.In Chapter 1,the introduction part,we recall some relevant existing results on free boundary problems modeling the growth of tumors,and elaborate the main results of this thesis and list some preliminary knowledge that will be used.In Chapter 2,we discuss a quasi-stationary Robin problem modeling the growth of vascularized tumors in the general case where the nutrient consumption function and the proliferation function are nonlinear and monotonically increasing.It is shown that there exist two thresholds,denoted by (?) and ?*,on the surrounding nutrient concentration ?,such that when ???,the considered problem admits no nontrivial stationary solution and all evolutionary tumors will finally vanish;whereas when (?)>(?),the considered problem admits a unique nontrivial stationary solution,and for any initial tumor radius R₀>0,it admits a unique global transient solution which converges to the stationary solution.Furthermore,the dormant tu-mor is nonnecrotic if (?)<(?)??*and necrotic if (?)>?*.Compared with the linear case[42],it is usually impossible to write down the explicit solution,thus we need to establish the results on the monotonicity of the solution with respect to some parameters.While on the other hand,the comparison principle may not work some-times,by contrast with the Dirichlet problem[37]in nonlinear case.Inspired by[31],we first transform the original problem into an equivalent one with fixed bound-ary,and get the results mentioned above by employing the relationship between the transformed function and its original function,and applying the comparison principle to problems involved with original functions.In Chapter 3,we discuss a completely nonstationary Robin problem modeling the growth of vascularized tumors in the special case where the nutrient consumption function and the proliferation function are linear.We prove the global existence of transient solutions by applying the approximation method,the Schauder fixed point theorem together with the L^p theory.