Symmetry‐breaking bifurcations of a free boundary problem modeling tumor growth with angiogenesis by Stokes equation

Abstract

In this paper, we consider bifurcation solutions of a free boundary problem modeling tumor growth with angiogenesis by Stokes equation. In which, the vasculature supplies nutrients to the tumor at a rate α, so that ∂σ∂truen→+αfalse(σ−trueσ‾false)=0 holds on the boundary. For each α, we first establish the existence and uniqueness of radially symmetric stationary solutions, then prove that there exist a positive integer n∗∗ and a sequence (μ/γ )n such that symmetry‐breaking stationary solutions bifurcate from the radially symmetric one for every (μ/γ )n (even n ≥ n∗∗), where μ and γ denote the proliferation rate and the cell‐to‐cell adhesiveness, respectively. Particularly, for small α, we show that (μ/γ )2, (μ/γ )4, (μ/γ )6, (μ/γ )8, … are all bifurcation points, which includes the smallest bifurcation point that is the most significant one biologically; moreover, (μ/γ )n is monotone decreasing with respect to α for every n ≥ 2, which implies that inhibiting angiogenesis has a positive impact in limiting the ability of the invasion of tumors, at least during very early stages of angiogenesis.