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

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

Song Huijuan1,Huang Qian2,Wang Zejia1,*()   

  1. 1School of Mathematics and Statistics, Jiangxi Normal University, Nanchang 330022
    2Guangfeng Middle School, Jiangxi Shangrao 334699
  • Received:2021-11-24 Revised:2022-07-05 Online:2023-02-26 Published:2023-03-07
  • Supported by:
    The NSFC(12261047);The NSFC(12161045);The NSFC(11861038);Natural Science Foundation of Jiangxi Province of China(20212BAB201016)


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.

Key words: Free boundary problem, Necrotic tumor, Angiogenesis, Periodic solution, Stability

CLC Number: 

  • O175