两个肿瘤生长偏微分方程模型的数学分析

发布时间：2018-03-03 16:00

  本文选题：肿瘤生长　切入点：偏微分方程　出处：《广东工业大学》2012年硕士论文　论文类型：学位论文

【摘要】：本文研究了两个肿瘤生长的偏微分方程模型,严格分析了其解整体解的适定性.全文共分三章. 第一章是绪论,分别介绍了肿瘤生长模型的历史背景和发展状况. 第二章研究了一个肿瘤化学治疗反应的空间结构的数学模型,这是一个动力系统模型,它是偏微分方程的自由边界问题.假设肿瘤的繁殖和死亡由局部药物浓度决定.在一些条件下,通过运用抛物方程的Lp理论、Banach不动点定理证明了这个问题局部解的存在唯一性,然后用延拓方法得到了整体解的存在唯一性.在另外一些条件下,通过运用反应扩散方程的上、下解方法,得到了：当0w≤w*时,此模型没有稳态解；当w*ww时,此模型有唯一的稳态解(ws,Rs). 第三章研究了一个大脑胶质瘤细胞生长情况的模型,这是一个交叉扩散模型.它包含基质金属蛋白酶和营养物浓度,并且考虑了由趋药性、趋触性和趋附性产生的效应.这个模型耦合了三个半线性抛物方程和一个常微分方程.通过运用Banach不动点定理、抛物方程的Schauder估计及抛物方程的Lp估计证明了这个问题局部解的存在唯一性,然后利用延拓方法得到了整体解的存在唯一性.
[Abstract]:In this paper, two partial differential equation models for tumor growth are studied, and the suitability of the global solution is strictly analyzed. There are three chapters in this paper. The first chapter introduces the historical background and development of tumor growth model. In the second chapter, we study a mathematical model of the spatial structure of tumor chemotherapeutic response, which is a dynamic system model. It is a free boundary problem for partial differential equations. It is assumed that the propagation and death of tumors are determined by local drug concentration. Under some conditions, the existence and uniqueness of the local solution of the problem are proved by using the Banach fixed point theorem of the parabolic equation. Then the existence and uniqueness of the global solution are obtained by using the continuation method. Under some other conditions, by using the method of upper and lower solutions of the reaction diffusion equation, it is obtained that the model does not have a steady solution when 0 w 鈮，

本文编号：1561706