A hyperbolic free boundary problem modeling tumor growth: Asymptotic behavior
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- by Xinfu Chen, Shangbin Cui and Avner Friedman PDF
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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 $\lim _{T\to \infty } \int ^{T+1}_T |\dot R(t)| dt=0$, then $\lim _{t\to \infty }R(t)=R_s$, and (ii) the stationary solution is linearly asymptotically stable.References
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Additional Information
- Xinfu Chen
- Affiliation: Department of Mathematics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260
- MR Author ID: 261335
- 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
- Received by editor(s): September 24, 2002
- Published electronically: July 20, 2005
- © Copyright 2005 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 357 (2005), 4771-4804
- MSC (2000): Primary 34B15; Secondary 35C10, 35Q80, 92C15
- DOI: https://doi.org/10.1090/S0002-9947-05-03784-0
- MathSciNet review: 2165387