Analysis of a nonlinear free-boundary tumor model with angiogenesis and a connection between the nonnecrotic and necrotic phases

Abstract

This paper is concerned with a nonlinear free boundary problem modeling the growth of spherically symmetric tumors with angiogenesis, set with a Robin boundary condition, in which, both nonnecrotic tumors and necrotic tumors are taken into consideration. The well-posedness and asymptotic behavior of solutions are studied. It is shown that there exist two thresholds, denoted by σ̃ and σ∗, on the surrounding nutrient concentration σ̄. If σ̄≤σ̃, then the considered problem admits no stationary solution and all evolutionary tumors will finally vanish, while if σ̄>σ̃, then it admits a unique stationary solution and all evolutionary tumors will converge to this dormant tumor; moreover, the dormant tumor is nonnecrotic if σ̃<σ̄≤σ∗ and necrotic if σ̄>σ∗. The connection and mutual transition between the nonnecrotic and necrotic phases are also given.