Acta mathematica scientia,Series A ›› 2023, Vol. 43 ›› Issue (1): 261-273.

### Asymptotic Analysis of a Tumor Model with Angiogenesis and a Periodic Supply of External Nutrients

Huijuan Song1,Qian Huang2,Zejia Wang1,*()

1. 1School of Mathematics and Statistics, Jiangxi Normal University, Nanchang 330022
2Guangfeng Middle School, Jiangxi Shangrao 334699
In this paper, we consider a free boundary problem modeling the growth of tumors with angiogenesis and a $\omega$-periodic supply of external nutrients $\phi(t)$. Denote by $S(\sigma)$ the proliferation rate of tumor cells. We first establish the well-posedness and then give a complete classification of asymptotic behavior of solutions according to the sign of $\frac1{\omega}\int_0^\omega S(\phi(t)){\rm d}t$. It is shown that if $\frac1{\omega}\int_0^\omega S(\phi(t)){\rm d}t\le0$, then all evolutionary tumors will finally vanish; the converse is also true. If instead $\frac1{\omega}\int_0^\omega S(\phi(t)){\rm d}t>0$, then there exists a unique and stable positive periodic solution.