A free boundary problem for a singular system of differential equations: An application to a model of tumor growth

Abstract

In this paper we consider a free boundary problem for a nonlinear system of two ordinary differential equations, one of which is singular at some points, including the initial point r = 0 r=0 . Because of the singularity at r = 0 r=0 , the initial value problem has a one-parameter family of solutions. We prove that there exists a unique solution to the free boundary problem. The proof of existence employs two “shooting” parameters. Analysis of the profiles of solutions of the initial value problem and tools such as comparison theorems and weak limits of solutions play an important role in the proof. The system considered here is motivated by a model in tumor growth, but the methods developed should be applicable to more general systems.