Analysis of a Mathematical Model of the Growth of Necrotic Tumors

In this paper we study a model of necrotic tumor growth. The tumor comprises necrotic cells which occupy a radially symmetric core and life proliferating cells which occupy a radially symmetric shell adjacent to the core. The proliferating cells receive nutrients through diffusion from the outer boundary as well as by means of blood flow through a network of capillary vessels. The mathematical model describes the evolution of the nutrient concentration between the boundary of the necrotic core r Žt. and the outer boundary of the tumor r RŽt.; within the core itself the concentration is a constant nec , a level under which life cells cannot be sustained. Both of the surfaces r Žt. and r RŽt. are free boundaries, which are unknown in advance. Under some assumptions on the parameters, we prove that Ži. there exists a stationary solution with radii r s , r Rs; Žii. for any initial data near the stationary solution, the time dependent model has a unique solution Žr, t. with free boundaries r Žt., r RŽt.; and Žiii. Žt. sand RŽt. Rsas t .