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Local and Global Existence of Solutions to Initial Value Problems of Nonlinear Kaup-Kupershmidt Equations
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 Problems of Nonlinear Kaup–Kupershmidt Equations
Shuang Ping TAO Shang Bin CUI.Local and Global Existence of Solutions to Initial Value Problems of Nonlinear Kaup-Kupershmidt Equations[J].Acta Mathematica Sinica,2005,21(4):881-892.
Authors:Shuang Ping Tao  Shang Bin Cui
Institution:(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 $$
\frac{{\partial u}}
{{\partial t}} + a_{1} \frac{{u\partial ^{2} u}}
{{\partial x^{2} }} + \beta \frac{{\partial ^{3} u}}
{{\partial x^{3} }} + \gamma \frac{{\partial ^{5} u}}
{{\partial x^{5} }} = 0,$$ (x, t) ∈ R 2, 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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