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.

本文涉及一个非线性自由边界问题，该问题以Robin边界条件为模型，模拟了具有血管生成的球形对称肿瘤的生长，其中考虑了非坏死性肿瘤和坏死性肿瘤。研究了解的适定性和渐近行为。结果表明存在两个阈值，用$\tilde{\sigma }$ 和 ${\sigma }^{\ast }$，对周围养分浓度 $\bar{\sigma }$。如果$\bar{\sigma }\le \tilde{\sigma }$，那么考虑到的问题就不会有固定的解决方案，所有进化的肿瘤最终都会消失，而如果 $\bar{\sigma }>\tilde{\sigma }$，然后它接受了唯一的固定解，所有进化性肿瘤都将收敛到该休眠的肿瘤；此外，如果$\tilde{\sigma }<\bar{\sigma }\le {\sigma }^{\ast }$ 如果坏死 $\bar{\sigma }>{\sigma }^{\ast }$。还给出了非坏死相和坏死相之间的连接和相互过渡。