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

Authors:
Shangbin Cui and Avner Friedman

Journal:
Trans. Amer. Math. Soc. **355** (2003), 3537-3590

MSC (2000):
Primary 34B15; Secondary 35C10, 35Q80, 92C15

DOI:
https://doi.org/10.1090/S0002-9947-03-03137-4

Published electronically:
May 29, 2003

MathSciNet review:
1990162

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Abstract | References | Similar Articles | Additional Information

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 . Because of the singularity at , 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.

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Additional Information

**Shangbin Cui**

Affiliation:
Institute of Mathematics, Zhongshan University, Guangzhou, Guangdong 510275, People’s Republic of China

Email:
mcinst@zsu.edu.cn

**Avner Friedman**

Affiliation:
Department of Mathematics, The Ohio State University, 231 West 18th Avenue, Columbus, Ohio 43210-1174

Email:
afriedman@mbi.osu.edu

DOI:
https://doi.org/10.1090/S0002-9947-03-03137-4

Keywords:
Free boundary problem,
stationary solutions,
singular differential equations,
tumor growth.

Received by editor(s):
March 15, 2002

Published electronically:
May 29, 2003

Article copyright:
© Copyright 2003
American Mathematical Society