Asymptotic behaviour of quasi-stationary solutions of a nonlinear problem modelling the growth of tumours

Abstract

We study the global stability of quasi-steady solutions for a simple mathematical model describing the growth of a spherical vascularized tumour consisting only of living cells. By assuming the rates of proliferation and absorption to be increasing nonlinear functions of the nutrient concentration, we establish the existence of a non-trivial steady solution and conditions for the existence and uniqueness of a quasi-steady solution for each initial configuration. Also, we prove that all these quasi-steady solutions converge uniformly to a non-trivial steady solution. The quasi-steady approach is justified by the smallness of the parameter that measures the ratio between the timescales for the diffusion of nutrients and growth of the tumour.