Analysis of the growth of a radial tumor with triple-layered structure

. We study a free boundary problem on the evolution of a radial tumor with triple-layered structure. Based on the nutrient thresholds sigma(Q) and sigma(D), cells in the tumor spheroid are divided into three types, i.e. proliferating cells, quiescent cells and necrotic cells, hence it forms a layered structure from tumor's surface to its center. The novelty of this paper is that the growth of the triple-layered tumor is first taken into rigorous analysis, and the difficulty arises from the two unknown moving interfaces n and p between different cell layers inside the tumor. We show existence and uniqueness of the stationary solution to the system for different external nutrient supply a over bar . It is proved that there exist two constants a* and a* with the relation a* > a* > sigma, such that if a over bar > a*, the dormant tumor forms a triple-layered structure; if a* < a over bar < a*, the dormant tumor has double layers with proliferating cells and quiescent cells; if sigma < sigma <= a*, the dormant tumor is filled with only proliferating cells and appears as a single layer; and if sigma <= sigma, the dormant tumor vanishes. The asymptotic stability of the stationary solution above is also studied. We can show the tendency of the evolutionary tumor towards its dormant state by using only a over bar and the initial radius R0 as well.