广东工业大学学报 ›› 2021, Vol. 38 ›› Issue (02): 60-65.doi: 10.12052/gdutxb.200109

• 综合研究 • 上一篇    下一篇

一个具有Robin自由边界的双曲肿瘤生长模型解的定性分析

周云, 卫雪梅   

  1. 广东工业大学 应用数学学院, 广东 广州 510520
  • 收稿日期:2020-08-28 出版日期:2021-03-10 发布日期:2021-01-13
  • 通信作者: 卫雪梅(1972-),女,教授,硕士生导师,主要研究方向为偏微分方程,E-mail:wxm_gdut@163.com E-mail:wxm_gdut@163.com
  • 作者简介:周云(1995-)，女，硕士研究生，主要研究方向为偏微分方程
  • 基金资助:
    国家自然科学基金资助项目(11101095)；广东省高校特色创新类项目(2016KTSCX028)

A Qualitative Analysis of a Hyperbolic Tumor Growth Model with Robin Free Boundary

Zhou Yun, Wei Xue-mei   

  1. School of Applied Mathematics, Guangdong University of Technology, Guangzhou 510520, China
  • Received:2020-08-28 Online:2021-03-10 Published:2021-01-13

HTML

PDF (PC)

2576

摘要： 研究了一个具有Robin自由边界的双曲肿瘤生长数学模型, 该模型包含了一个描述营养物浓度变化的椭圆型方程, 一个描述肿瘤半径的常微分方程和描述肿瘤细胞生长的两个双曲型偏微分方程。本文通过特征线方法结合${\rm{Banach}}$不动点定理证明了该模型整体解的存在性和唯一性。最后证明当${K_R} = 0$时, 有$\mathop {\lim }\limits_{t \to \infty } R\left( t \right) = \infty $。

关键词: 肿瘤生长, 自由边界问题, 整体解

Abstract: A hyperbolic tumor growth model with Robin free boundary is studied, which contains an elliptic partial differential equation describing the concentration of nutrients, an ordinary differential equation describing the radius of tumor and two hyperbolic equations describing the growth of tumor cells. In this paper, by applying the method of characteristic curves and the Banach fixed point theorem, the existence and uniqueness of the global solution of the model are proven. Finally, it is proven that while ${K_R} = 0$, $\mathop {\lim }\limits_{t \to \infty } R\left( t \right) = \infty$.

Key words: tumor growth, free boundary problem, global solution

中图分类号: 

  • O175
[1] 崔尚斌. 肿瘤生长的自由边界问题[J]. 数学进展, 2009, 38(1): 1-18.
CUI S B. Free boundary problems modeling tumor growth [J]. Advances in Mathematics, 2009, 38(1): 1-18.
[2] DRASDO D, HÖHME S. A single cell based model of tumor growth in vitro: monolayers [J]. Physical Biology, 2005, 2(3): 133.
[3] KIM J B, STEIN R, O'HARE M J. Three-dimensional in vitro tissue culture models for breast cancer-a review [J]. Breast Cancer Research & Treatment, 2004, 85(3): 281-291.
[4] KYLE A H, CHAN C T O, MINCHINTON A I. Characterization of three-dimensional tissue cultures using impedance spectroscopy [J]. Biophysical Journal, 1999, 76(5): 2640-2648.
[5] MUELLER-KLIESER W. Three-dimensional cell cultures: from molecular mechanisms to clinical applications [J]. American Journal of Physiology Cell Physiology, 1997, 273(4): 1109-1123.
[6] ADAM J A, BELLOMO N. A survey of models for tumor-immune system dynamics[M]. Boston: Birkhäuser, 1997: 89-133.
[7] SUTHERLAND R M. Cell and enviroment interactions in tumor microregions: the multicell spheroid model [J]. Science, 1998, 240(4849): 177-184.
[8] CUI S B, FRIEDMAN A. A hyperbolic free boundary problem modeling tumor growth [J]. Interfaces and Free Boundaries, 2003, 5(2): 159-182.
[9] CUI S B, WEI X M. Global existence for a parabolic-hyperbolic free boundary problem modelling tumor growth [J]. Acta Mathematicae Applicate Sinica, English Series, 2005, 21(4): 597-614.
[10] WEI X M, CUI S B. Global well-posedness for a drug transport model in tumor multicell spheroids [J]. Mathematical & Computer Modelling, 2007, 45(5-6): 553-563.
[11] FRIEDMAN A, LAM K. Analysis of a free boundary tumor model with angiogenssis [J]. Journal of Differential Equations, 2015, 259(12): 7636-7661.
[12] 沈海双. 带有Robin 自由边界条件的肿瘤生长模型的定性分析[D]. 广州: 广东工业大学, 2019.
[13] PETTET G J, PLEASE C P, TINDALL M J, et al. The migration of cells in multicell tumor spheroids [J]. Bulletin of Mathematical Biology, 2001, 63(2): 231-257.
[14] PROTTER M H, WEINBERGER H F. Maximum Principles in Differential Equations[M]. New York: Springer, 1967: 1-3.
[1] 梁小珍, 卫雪梅. 结肠癌细胞代谢模型解的存在性[J]. 广东工业大学学报, 2019, 36(05): 38-42.
[2] 陈美癸, 卫雪梅. 视网膜氧分布与脑红蛋白作用模型解的存在唯一性[J]. 广东工业大学学报, 2018, 35(05): 45-50.
[3] 卢创业， 卫雪梅. 一个视网膜血管肿瘤数学模型整体解的存在唯一性[J]. 广东工业大学学报, 2016, 33(03): 70-75.
[4] 王琳. 随机延迟微分方程的正整体解及矩有界[J]. 广东工业大学学报, 2013, 30(1): 73-75.
[5] 杨淑伶； . 一类反应扩散方程的整体解[J]. 广东工业大学学报, 2003, 20(2): 93-96.
Viewed
Full text


Abstract

Cited
  Shared   
  Discussed   
No Suggested Reading articles found!
版权所有 © 2017 《广东工业大学学报》编辑部
地址：广州市东风东路729号广东工业大学 邮编：510090
电话：020-37626653,020-37626139,020-37626396
邮箱：xbzrb@gdut.edu.cn
本系统由北京玛格泰克科技发展有限公司设计开发