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 $\tilde{\sigma }$ and ${\sigma }^{\ast }$, on the surrounding nutrient concentration $\bar{\sigma }$. If $\bar{\sigma }\le \tilde{\sigma }$, then the considered problem admits no stationary solution and all evolutionary tumors will finally vanish, while if $\bar{\sigma }>\tilde{\sigma }$, then it admits a unique stationary solution and all evolutionary tumors will converge to this dormant tumor; moreover, the dormant tumor is nonnecrotic if $\tilde{\sigma }<\bar{\sigma }\le {\sigma }^{\ast }$ and necrotic if $\bar{\sigma }>{\sigma }^{\ast }$. The connection and mutual transition between the nonnecrotic and necrotic phases are also given.

References in corpus (1)

• Asymptotic stability for a free boundary tumor model with angiogenesis