Analysis of a mathematical model for the growth of tumors

In this paper we study a recently proposed model for the growth of a nonnec rotic, vascularized tumor. The model is in the form of a free-boundary prob lem whereby the tumor grows (or shrinks) due to cell proliferation or death according to the level of a diffusing nutrient concentration. The tumor is assumed to be spherically symmetric, and its boundary is an unknown functi on r = s(t). We concentrate on the case where at the boundary of the tumor the birth rate of cells exceeds their death rate, a necessary condition for the existence of a unique stationary solution with radius r - R-0 (which d epends on the various parameters of the problem). Denoting by c the quotien t of the diffusion time scale to the tumor doubling time scale, so that c i s small, we rigorously prove that (i) lim(t-->infinity) inf s(t) > 0, i.e. once engendered, tumors persist in time. Indeed, we further show that (ii) If c is sufficiently small then s(t)--> R-0 exponentially fast as t -- >infinity, i.e. the steady state solution is globally asymptotically stable . Further, (iii) If c is not "sufficiently small" but is smaller than some constant ga mma determined explicitly by the parameters of the problem, then lim(t-->in finity) sup s(t) < infinity; if however c is ""somewhat" larger than gamma then generally s(t) does not remain bounded and, in fact, s(t) -->infinity exponentially fast as t-->infinity.