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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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A hyperbolic free boundary problem modeling tumor growth: Asymptotic behavior
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by Xinfu Chen, Shangbin Cui and Avner Friedman PDF
Trans. Amer. Math. Soc. 357 (2005), 4771-4804 Request permission

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.
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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