Bifurcation for a Free Boundary Problem Modeling Tumor Growth by Stokes Equation

Abstract

We consider a free boundary problem modeling tumor growth in fluid‐like tissue. The model equations include a diffusion equation for the nutrient concentration, and the Stokes equation with a source which represents the proliferation density of the tumor cells. The proliferation rate μ and the cell‐to‐cell adhesiveness γ which keeps the tumor intact are two parameters which characterize the “aggressiveness” of the tumor. For any positive radius R there exists a unique radially symmetric stationary solution with radius $r=R$. We prove that for a sequence $\mu/\gamma = M_n(R)$ there exist symmetry‐breaking bifurcation branches of solutions with free boundary $r=R+\varepsilon Y_{n,0}(\theta)+O(\varepsilon^2)$ (n $\text{even} \ge 2$) for small |ε|, where $Y_{n,0}$ is the spherical harmonic of mode $(n,0)$. Furthermore, the smallest $M_n(R)$, say, $M_{n_*}(R)$, is such that $n_*=n_*(R)\to\infty$ as $R\to\infty$. The biological implications of this result are discussed at the end of the paper.