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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

A hyperbolic free boundary problem modeling tumor growth: Asymptotic behavior


Authors: Xinfu Chen, Shangbin Cui and Avner Friedman
Journal: Trans. Amer. Math. Soc. 357 (2005), 4771-4804
MSC (2000): Primary 34B15; Secondary 35C10, 35Q80, 92C15
Published electronically: July 20, 2005
MathSciNet review: 2165387
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Abstract: In this paper we study a free boundary problem modeling the growth of radially symmetric tumors with two populations of cells: proliferating cells and quiescent cells. The densities of these cells satisfy a system of nonlinear first order hyperbolic equations in the tumor, and the tumor's surface is a free boundary $r=R(t)$. The nutrient concentration satisfies a diffusion equation, and $R(t)$ satisfies an integro-differential equation. It is known that this problem has a unique stationary solution with $R(t)\equiv R_s$. We prove that (i) if $\lim _{T\to \infty} \int^{T+1}_T \vert\dot R(t)\vert\,dt=0$, then $\lim_{t\to \infty}R(t)=R_s$, and (ii) the stationary solution is linearly asymptotically stable.


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Additional Information

Xinfu Chen
Affiliation: Department of Mathematics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260
Email: xinfu@pitt.edu

Shangbin Cui
Affiliation: Department 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: http://dx.doi.org/10.1090/S0002-9947-05-03784-0
PII: S 0002-9947(05)03784-0
Keywords: Tumor growth, free boundary problem, stationary solution, asymptotic behavior
Received by editor(s): September 24, 2002
Published electronically: July 20, 2005
Article copyright: © Copyright 2005 American Mathematical Society