Analysis of a mathematical model for the growth of tumors

In this paper we study a recently proposed model for the growth of a nonnecrotic, vascularized tumor. The model is in the form of a free-boundary problem 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 function 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 = R0 (which depends on the various parameters of the problem). Denoting by c the quotient of the diffusion time scale to the tumor doubling time scale, so that c is small, we rigorously prove that (i) lim inf s(t) > 0, i.e. once engendered, tumors persist in time. t-->infinity Indeed, we further show that (ii) If c is sufficiently small then s(t)-->R0 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 gamma determined explicitly by the parameters of the problem, then t-->infinity lim 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.