Symmetry-breaking bifurcations for free boundary problems modeling tumor growth - Pan, Hongjing; Xing, Ruixiang (Topological Methods in Nonlinear Analysis, 2022)
We study a classic free boundary problem modeling solid tumor growth. The problem contains a parameter $μ$.It is well known that the problem admit a unique radially symmetric solution with free boundary $r=R_S$ and a sequenceof symmetry-breaking branches of axisymmetric solutions bifurcating from the spherical state $r=R_S$ at anincreasing sequence of $μ= μ_\ell(R_S)$ ($\ell≥ 2$ even) with free boundary $r= R_S + ε Y_{\ell,0}(θ) + O(ε^2)$,where $Y_{\ell,0}$ is the spherical harmonic of mode $(\ell,0)$. In this paper, we use group-theoretic ideas to obtain a plethora of new branchesof non-axisymmetric solutions bifurcating at $μ= μ_\ell(R_S)$ $(\ell≥ 2)$. New solutions can model more complex shapesof tumor tissues than the known axisymmetric solutions. The approach is also applicable to many other free boundary problems arising in tumor growth,including a model involving fluid-like tissue.