Analysis of a nonlinear free boundary problem modeling the radial growth of two-layer tumors

In this paper we study a nonlinear free boundary problem on the radial growth of a two-layer solid tumor with a quiescent core. The tumor surface and its inner interface separating the proliferating cells and the quiescent cells are both free boundaries. By deeply analyzing their relationship and employing the maximum principle, we show this problem is globally well-posed and prove the existence of a unique positive threshold $\sigma^*$ such that the problem admits a unique stationary solution with a quiescent core if and only if the externally supplied nutrient $\bar\sigma> \sigma^*$. The stationary solution is globally asymptotically stable. The formation of the quiescent core and its interesting connection with the necrotic core are also given.