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非线性 Kirchhoff 型波动方程解的真空隔离
Vacuum Isolating of Solutions for the Nonlinear Wave Equations of Kirchhoff Type

宋志华
SONG Zhihua

华北水利水电学院数学与信息科学学院, 郑州, 河南, 450011
College of Mathematics and Information Science, North China University of Water Resources and Electric Power, Zhengzhou, Henan, 450011, P. R. China

收稿日期: 2011-04-11
出版日期: 2013-08-25
DOI: 10.11845/sxjz.20130410b

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摘要 本文研究初边值问题
$$\left\{\!\!\!\begin{array}{lll} u_{tt}+\|A^{\frac{1}{2}}u\|^{2\gamma}Au+u_t=|u|^{p-1}u,x\in\Omega,\ t>0,\nonumber u(x,0)=u_0,\ u_t(x,0)=u_1,\nonumber\\ u=0,x\in\partial\Omega,\ t\geq 0\nonumber \end{array}\right.
其中$\Omega$是 $\mathbb{R}^N$中的有界区域, $A=-\Delta$是定义在$A=-\Delta$上的Laplace算子.利用位势井方法得到了解的存在性定理, 并且证明了当$e\in(0,d)$时,以$E(0)\in(0,e]$为初始能量的所有解只能位于空间$D(A^{\frac{1}{2}})$中小球的外部和大球的内部, 其中$d=\frac{p-2\gamma-1}{(2\gamma+2)(p+1)}\big(\frac{1}{C_\ast^{p+1}}\big)^{\frac{2\gamma+2}{p-1-2\gamma}}$, $C_\ast$是空间$D(A^{\frac{1}{2}})$到$L^{p+1}(\Omega)$的嵌入常数.
关键词 Kirchhoff 型波动方程真空隔离位势井    
Abstract:In this paper, the initial boundary value problem $$\left\{\!\!\!\begin{array}{lll} u_{tt}+\|A^{\frac{1}{2}}u\|^{2\gamma}Au+u_t=|u|^{p-1}u,x\in\Omega,\ t>0,\nonumber\\ u(x,0)=u_0,\ u_t(x,0)=u_1,\nonumber u=0,x\in\partial\Omega,\ t\geq 0\nonumber \end{array}\right.is studied, where $\Omega\subset \mathbb{R}^N$ is a bounded domain, $A=-\Delta$ is the Laplace operator with the domain $D(A)=H^{2}(\Omega)\cap H_{0}^{1}(\Omega)$. By using the potential well method, one obtains some existence theorems of solutions, and proves that for any given $e\in(0,d)$ all solutions with initial energy $E(0)\in(0,e]$ can only lie either inside of some smaller ball or outside of some bigger ball of space $D(A^{\frac{1}{2}}),$ where $d=\frac{p-2\gamma-1}{(2\gamma+2)(p+1)}\big(\frac{1}{C_\ast^{p+1}}\big)^{\frac{2\gamma+2}{p-1-2\gamma}}$ and $ C_\ast$ is the imbedding constant from $D(A^{\frac{1}{2}})$ into $L^{p+1}(\Omega)$.
Key wordsvacuum isolating    potential well
基金资助:This work is supported by NSFC (No. 11271336).
通讯作者: songzhihua@ncwu.edu.cn   
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