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A hyperbolic free boundary problem modeling tumor growth: Asymptotic behavior

Authors:	Xinfu Chen Shangbin Cui Avner Friedman

Institution:	Department of Mathematics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260 ; Department of Mathematics, Zhongshan University, Guangzhou, Guangdong 510275, People's Republic of China ; Department of Mathematics, The Ohio State University, 231 West 18th Avenue, Columbus, Ohio 43210-1174

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 . The nutrient concentration satisfies a diffusion equation, and 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.

Keywords:	Tumor growth free boundary problem stationary solution asymptotic behavior

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