Local bifurcation structure for a free boundary problem modeling tumor growth

Abstract

There are many papers in the literature studying a classic free boundary problem modeling 3-dimensional tumor growth, initiated by Byrne and Chaplain. One of the most important parameters is the tumor aggressiveness constant $\mu$. Friedman and Reitich (1999) showed the problem admits a unique radially symmetric solution with the free boundary $r=R_S$ when the external nutrient concentration is greater than the threshold concentration for proliferation. A sequence of papers, Fontelos and Friedman (2003), Friedman and Hu (2008) and Pan and Xing (2022) derived a sequence of symmetry-breaking branches bifurcating from the spherical state $r=R_S$ at an increasing sequence of $\mu ={\mu }_n$ ($n\ge 2$). Friedman and Hu (2008) studied the structure of the branching solution at $\mu ={\mu }_2$. These bifurcation results cover only the direction of spherical harmonic function $Y_{2,0}$. In this paper, we determine a plethora of new local bifurcation structures at $\mu ={\mu }_n$ for even $n\ge 2$ in directions involving combinations of $Y_{n,m}$ for $m\ne 0$.