一类分数阶扩散方程及稳态解的存在性研究

发布时间：2018-05-28 22:27

  本文选题：分数阶Laplace算子 + 延拓　； 参考：《华东师范大学》2017年硕士论文

【摘要】：分数阶Laplace算子的定义比较复杂,为了方便读者理解,文中我们详细给出了分数阶Laplace算子在Rn上的三种等价性定义以及在有界区域Ω上的两种等价性定义.目前最常用的是Luis Caffarelli-Luis Silvestre的延拓性定义,用这种定义可以将非局部问题转化为局部问题,使得我们可以用局部的方法去处理分数阶Laplace 算子.本文中我们考虑了分数阶扩散方程其中(-△)αu(x)=Cn,αP.V.∫Rn((u(x)-u(y))/(|x-y|n+2α))dy是分数阶Laplace算子,α∈(0,1).利用它的基本解和压缩映射原理,系统给出了方程解的存在性;并且我们研究了分数阶MEMS方程其中α ∈(0,1),λ＞ 0是一个常数.根据λ的取值,给出了分数阶MEMS方程解的存在性.
[Abstract]:The definition of fractional order Laplace operator is complicated. In order to facilitate readers' understanding, we give three definitions of equivalence of fractional order Laplace operator on rn and two definitions of equivalence on bounded domain 惟. At present, the extension definition of Luis Caffarelli-Luis Silvestre is the most commonly used one. With this definition, the nonlocal problem can be transformed into a local problem, so that we can deal with fractional order Laplace operators in a local way. In this paper, we consider the fractional order diffusion equation in which ~ (-) 伪 u ~ n ~ (?) C _ n, 伪 P.V. ~ (?) ~ (-) ~ (?) By using its fundamental solution and contraction mapping principle, we give the existence of the solution of the equation, and we study the fractional order MEMS equation, where 伪 鈭，

本文编号：1948421