Local and Global Existence of Solutions to Initial Value Problems of Nonlinear Kaup-Kupershmidt Equations Local and Global Existence of Solutions to Initial Value Problemsof Nonlinear Kaup–Kupershmidt Equations期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

Local and Global Existence of Solutions to Initial Value Problems of Nonlinear Kaup-Kupershmidt Equations

引用本文：	Shuang Ping TAO Shang Bin CUI. Local and Global Existence of Solutions to Initial Value Problems of Nonlinear Kaup-Kupershmidt Equations[J]. 数学学报(英文版), 2005, 21(4): 881-892. DOI: 10.1007/s10114-004-0433-0

作者姓名：	Shuang Ping TAO Shang Bin CUI

作者单位：	[1]Department of Mathematics. Northwest Normal University, Lanzhou 730070, P. R. China [2]Department of Mathematics, Zhongshan University, Guangzhou 510275, P. R. China

基金项目：	Supported by NWNU-KJCXGC-212

摘要：	This paper is devoted to studying the initial value problems of the nonlinear Kaup Kupershmidt equations δu/δt ＋ α1 uδ^2u/δx^2 ＋ βδ^3u/δx^3 ＋ γδ^5u/δx^5 = 0, （x,t）∈ E R^2, and δu/δt ＋ α2 δu/δx δ^2u/δx^2 ＋ βδ^3u/δx^3 ＋ γδ^5u/δx^5 = 0, （x, t） ∈R^2. Several important Strichartz type estimates for the fundamental solution of the corresponding linear problem are established. Then we apply such estimates to prove the local and global existence of solutions for the initial value problems of the nonlinear Kaup- Kupershmidt equations. The results show that a local solution exists if the initial function u0（x） ∈ H^s（R）, and s ≥ 5/4 for the first equation and s≥301/108 for the second equation.
关键词：	Kaup-Kupershmidt方程初值问题局部存在性广义存在性
收稿时间：	2002-01-19
修稿时间：	2002-01-192002-09-20
Local and Global Existence of Solutions to Initial Value Problemsof Nonlinear Kaup–Kupershmidt Equations

Shuang Ping Tao,Shang Bin Cui. Local and Global Existence of Solutions to Initial Value Problemsof Nonlinear Kaup–Kupershmidt Equations[J]. Acta Mathematica Sinica(English Series), 2005, 21(4): 881-892. DOI: 10.1007/s10114-004-0433-0

Authors:	Shuang Ping Tao Shang Bin Cui

Affiliation:	(1) Department of Mathematics, Northwest Normal University, Lanzhou 730070, P. R. China;(2) Department of Mathematics, Zhongshan University, Guangzhou 510275, P. R. China

Abstract:	This paper is devoted to studying the initial value problems of the nonlinear Kaup–Kupershmidt equations $frac{{partial u}} {{partial t}} + a_{1} frac{{upartial ^{2} u}} {{partial x^{2} }} + beta frac{{partial ^{3} u}} {{partial x^{3} }} + gamma frac{{partial ^{5} u}} {{partial x^{5} }} = 0,$ (x, t) ∈ R ², and $frac{{partial u}} {{partial t}} + a_{2} frac{{partial u}} {{partial x}}frac{{partial ^{2} u}} {{partial x^{2} }} + beta frac{{partial ^{3} u}} {{partial x^{3} }} + gamma frac{{partial ^{5} u}} {{partial x^{5} }} = 0,$ (x, t) ∈ R ². Several important Strichartz type estimates for the fundamental solution of the corresponding linear problem are established. Then we apply such estimates to prove the local and global existence of solutions for the initial value problems of the nonlinear Kaup–Kupershmidt equations. The results show that a local solution exists if the initial function u ₀(x) ∈ H ^s (R), and s ≥ 5/4 for the first equation and s ≥ 301/108 for the second equation. Supported by NWNU-KJCXGC-212

Keywords:	Kaup-Kupershmidt equation Initial value problem Solution Local existence Global existence
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