Symmetry-breaking bifurcation analysis of a free boundary problem modeling 3-dimensional tumor cord growth

In this paper, we study a free boundary problem modeling the growth of 3-dimensional tumor cords. Since tumor cells grow freely in both the longitudinal and cross-sectional directions of blood vessels, the investigation of symmetry-breaking phenomena in both directions is biologically very reasonable. This forces the possible bifurcation value ${\gamma }_{m,n}$ to be dependent on two variables m and n. Some monotonicity properties of the possible bifurcation value ${\mu }_n$ or ${\mu }_j$ obtained in Friedman and Hu (2008) [1] and He and Xing (2023) [2] no longer hold here, which brings a great challenge to the bifurcation analysis. The novelty of this paper lies in determining the order of ${\gamma }_{m,n}$ for $\sqrt{m^2+n^2}$. Together with periodicity and symmetry, we propose an effective method to avoid the need for the monotonicity of ${\gamma }_{m,n}$. We give symmetry-breaking bifurcation results for every ${\gamma }_{m,n}>0$.