Analysis of a radial free boundary tumor model with time-dependent absorption efficiency

Abstract

We investigate an n-dimensional free boundary problem modeling tumor growth in radial form with angiogenesis process. This process is reflected in the Robin boundary condition where time-dependent absorption rate β(t) is considered. The system consists of a reaction-diffusion equation for the nutrient u(r,t) with nonlinear consumption and an integro-differential equation for the tumor radius R(t) with nonlinear cell proliferation. It is shown that the large-time behavior of the tumor is greatly tied to the properties of β(t). We prove that for any sufficiently small c>0, the tumor radius R(t) will remain bounded or shrink to zero when β(t) is uniformly bounded or tends to zero, respectively; and if β(t)→β⁎, the solution (u,R) will converge to the unique steady state (u⁎,R⁎) associating with β⁎. We also give examples of R(t) blowing up as t→∞ even if β(t)→0 when c is not small.