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This paper is concerned with a nonlinear free boundary problem modeling thegrowth of spherically symmetric tumors with angiogenesis, set with a Robinboundary condition. In which, both nonnecrotic tumors and necrotic tumors aretaken into consideration. The well-posedness and asymptotic behavior ofsolutions are studied. It is shown that there exist two thresholds, denoted by$\tilde\sigma$ and $\sigma^*$, on the surrounding nutrient concentration$\bar\sigma$. If $\bar\sigma\leq\tilde\sigma$, then the considered problemadmits no stationary solution and all evolutionary tumors will finally vanish,while if $\bar\sigma>\tilde\sigma$, then it admits a unique stationary solutionand all evolutionary tumors will converge to this dormant tumor; moreover, thedormant tumor is nonnecrotic if $\tilde\sigma<\bar\sigma\leq\sigma^*$ andnecrotic if $\bar\sigma>\sigma^*$. The connection and mutual transition betweenthe nonnecrotic and necrotic phases are also given.

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