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

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 $\tildeσ$ and $σ^*$, on the surrounding nutrient concentration $\barσ$. If $\barσ\leq\tildeσ$, then the considered problem admits no stationary solution and all evolutionary tumors will finally vanish, while if $\barσ>\tildeσ$, then it admits a unique stationary solution and all evolutionary tumors will converge to this dormant tumor; moreover, the dormant tumor is nonnecrotic if $\tildeσ<\barσ\leqσ^*$ and necrotic if $\barσ>σ^*$. The connection and mutual transition between the nonnecrotic and necrotic phases are also given.

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