Bifuration Analysis For A Free Boundary Problem Modeling Growth Of Tumor

In this thesis, we investigate the bifurcation phenomenon for a free boundary problem modeling growth of tumor, namely, the existence of the radially symmetric stationary solutions and non-radially symmetric stationary solutions. This thesis is divided into four chapters.In chapter 1, we introduce the known research related on our problems, and our main results.In chapter 2, we study a free boundary problem modeling tumor growth with constant nutrient consumption rate, namely: where Ω is the domain occupied by the tumor, a denotes the concentration of nu-trient within the tumor, p is the tumor’s tissue pressure function, a and μ(σ-σ) are the nutrient consumption rate and the tumor-cell proliferation rate function, re-spectively, a is a threshold concentration, and μ is a positive parameter measuring the aggressiveness of the tumor. The condition σ=σ on аΩ means that the tumor receives constant nutrient supply from the exterior surface. In addition, κ is the mean curvature of the free boundary аQ, аn is the outward normal. For any given μ> 0, we show that there exists R> 0, which makes this problem have a unique radially symmetric stationary solution, and for m> 2, there also exists a sequence of μm, such that for each μm(m> 2 even), symmetric-breaking stationary solutions bifurcate from the radially symmetric stationary solutions.In chapter 3, we study a free boundary problem modeling tumor growth with necrotic core, namely: where σ=σ(x), p= p(x) denote the concentration of nutrient and pressure with-in the tumor, respectively. Ω is the domain occupied by the tumor, D C Ω is the necrotic core within the tumor domain, x(x) is the indicator function of Ω\D, σ= σ in D means the density of cells in the necrotic core remains constant. The aggressiveness of tumor is modeled by a positive parameter μ, we show that for any given μ> 0, when necrotic core radius is r=|x|= p, there exist R > p, which makes this problem have a unique radially symmetric stationary solution, and there exists a positive even number m** R and a sequence of μm, such that for each μm(m≥m**), symmetric-breaking stationary solutions bifurcate from the radially symmetric stationary solution.In chapter 4, we study a free boundary problem modeling tumor growth with inhibitors, namely: where Ω(?)R3 is the domain occupied by the tumor,σ, β denote the concentration of nutrient and inhibitor within the tumor, respectively. The pressure p within the tumor comes from the proliferation of the tumor cells,∫(σ), g(β) and h(σ, β) are the nutrient consumption rate, inhibitor consumption rate and tumor-cell proliferation rate function, respectively, σ and β are positive constants, σ=σ and β=β mean that the tumor receives constant nutrient and inhibitor supply from the exterior surface, respectively, v is the outward normal of the free boundary аΩ,γ is the surface tension coefficient, and κ is the mean curvature of the free boundary аΩ.According to the medicine and biology, as well the need of the mathematics, we assume that f, g, h are functions satisfying the following conditions: We prove that there exists a sequence of γk, such that for large k, nonradially stationary solutions bifurcate from the radially symmetric stationary solutions.