论文摘要
本文主要利用位势井方法和凹函数方法及泛函分析理论,针对具m-Laplacian的非线性抛物方程和具-Laplacian的非线性抛物方程的定解问题的适定性进行研究,以期分析上述两类问题的解关于初值的依赖性.本文在次临界初始能级、临界初始能级以及超临界初始能级下分别对上述两类方程定解的整体适定性进行研究.本文通过研究两类具变指数Laplacian项的非线性抛物方程,推广了具变指数Laplacian项在位势井理论中的应用,得到了具变指数Laplacian项的非线性发展方程定解的整体适定性,从而使得位势井理论得到进一步发展和完善.第二章针对一类具m-Laplacian的非线性抛物方程的初边值问题进行研究.该方程可用来模拟非线性扩散模型的动力行为及与线性扩散模型相关的退化特性.在次临界初始能级,借助两个不同的辅助函数证明具m-Laplacian的非线性抛物方程定解的有限时间爆破并估计爆破时间的上下界.然后通过对初值伸缩变换将次临界初始能级得到的所有结果平行地扩展到临界初始能级.在超临界初始能级,由于能量不再被位势井深所控制,不变集合将不再成立.课题通过对初始能量重新施加一个与初值有关的限制证明了不变集合,结合凹函数方法且引入新的辅助函数证明了方程定解的有限时间爆破并估计了爆破时间上下界.本章的研究意义是对具m-Laplacian项的非线性抛物方程的初边值问题的解的适定性质在全能级下进行全面的研究.第三章针对一类具m(x)-Laplacian的非线性抛物方程的初边值问题在三个不同的能级下对解的适定性进行研究.该方程被用来描述电流变流体的流动.借助经典的Faedo-Galerkin方法和有界性原理得到了解在次临界和临界能级下的整体存在性和唯一性.进一步,通过分析势能泛函和Nehari泛函之间的关系,得到了方程定解的渐近行为.借助位势井深和解的模之间的关系且利用凹函数方法,证明了该方程定解的有限时间爆破,在解有限时间爆破的基础上分别估计爆破时间.在超临界能级状态下,证明了解的有限时间爆破并且估计了爆破时间上下界.本章的研究意义是将具m(x)-Laplacian的非线性抛物方程的初边值问题解的整体适定性进行了系统化结构化的研究.
论文目录
文章来源
类型: 硕士论文
作者: 庞月
导师: 徐润章
关键词: 算子,非线性抛物方程,整体适定性,位势井
来源: 哈尔滨工程大学
年度: 2019
分类: 基础科学
专业: 数学
单位: 哈尔滨工程大学
分类号: O175
总页数: 83
文件大小: 8715K
下载量: 26
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