Analysis of a free boundary problem modeling the growth of nonnecrotic tumors with time delays in proliferation

In this paper we study a free boundary problem modeling the growth of nonnecrotic tumors with time delays in proliferation. The establishment of the model is based on the diffusion of nutrient and mass conservation for the two processes: proliferation and apoptosis (cell death due to aging). It is assumed that the process of proliferation is delayed compared to apoptosis. By LpLp theory of parabolic equations and the Banach fixed point theorem, under some conditions we prove the existence and uniqueness of a local solution and apply the continuation method to get the existence and uniqueness of a global solution. We also prove that in case cc (the ratio of the nutrient diffusion time scale to the tumor growth (e.g. tumor doubling) time scale) is sufficiently small, the volume of the tumor cannot expand unlimitedly.