七阶非线性色散方程初值问题解的局部和整体存在性

数学物理学报 ›› 2005, Vol. 25 ›› Issue (4): 451-460.

七阶非线性色散方程初值问题解的局部和整体存在性

陶双平, 崔尚斌

西北师范大学数学与信息科学学院兰州 730070 北京师范大学数学系北京 100875 中山大学数学系广州 510275

出版日期:2005-08-25 发布日期:2005-08-25
基金资助:
国家自然科学基金（10271095）和NWNU-KJCXGC-212基金资助

The Initial Value Problem of the Seventh Order Weakly Dispersi ve Equations

DAO Shuang-Beng, CUI Chang-Bin

Online:2005-08-25 Published:2005-08-25
Supported by:
国家自然科学基金（10271095）和NWNU-KJCXGC-212基金资助

摘要/Abstract

摘要：

该文研究七阶非线性弱色散方程:∂u/∂t + au(∂u/∂x) +β(∂^3 u /∂x^3) +γ(∂^5 u/∂x^5) + μ(∂^7 u/∂x^7)=0, (x,t)∈R^2的初值问题,通过运用震荡积分衰减估计的最近结果, 首先对相应线性方程的基本解建立了几类Strichartz型估计. 其次, 应用这些估计证明了七阶非线性弱色散方程初值问题解的局部与整体存在性和唯一性. 结果表明, 当初值u_0(x)∈H^s(R), s≥2/13 时, 存在局部解; 当s≥1时, 存在整体解.

关键词: 色散方程；初值问题；解；局部存在性；整体存在性.

Abstract:

This paper is devoted to studying the initial value problem of a class of seventh order weakly nonlinear dispersive equations:∂u/∂t + au(∂u/∂x) +β(∂^3 u /∂x^3) +γ(∂^5 u/∂x^5) + μ(∂^7 u/∂x^7)=0, (x,t)∈R^2.By using recently established decay estimates for oscillatory integrals, the authors firs t establish several Strichartz type estimates for the fundamental solution of the corresponding linear problem. Then the authors prove that a local solution exists if t he initial function u_0(x)∈H^s(R), s≥2/13, and a global solution exists if s≥1.

Key words: Dispersive equation, Initial value problem; , Solution; , Local existence, Global existence

中图分类号:

35Q30

陶双平, 崔尚斌. 七阶非线性色散方程初值问题解的局部和整体存在性[J]. 数学物理学报, 2005, 25(4): 451-460.

DAO Shuang-Beng, CUI Chang-Bin. The Initial Value Problem of the Seventh Order Weakly Dispersi ve Equations[J]. Acta mathematica scientia,Series A, 2005, 25(4): 451-460.

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