A hyperbolic free boundary problem modeling tumor growth: Asymptotic behavior

First published in Transactions of the American Mathematical Society in volume 357, issue 12, published by the American Mathematical Society.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) = R-s. We prove that ( i) if lim(T-->infinity) integral(T+1)(T)\(r) over dot(t)\dt = 0, then lim(t-->infinity)R(t) = R-s, and ( ii) the stationary solution is linearly asymptotically stable