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Global Existence of Solutions for the Cauchy Problem of the Kawahara Equation with L^2 Initial Data
作者姓名:Shang  Bin  CUI  Dong  Gao  DENG  Shuang  Ping  TAO
作者单位:[1]Department of Mathematics, Sun Yat-Sen University, Guangzhou 510275, P. R. China [2]Department of Mathematics, Northwest Normal University, Lanzhou 730070, P. R. China
基金项目:Project supported by the China National Natural Science Foundation (Grants 10171111, 10171112).
摘    要:In this paper we study solvability of the Cauchy problem of the Kawahara equation 偏导dtu + au偏导dzu + β偏导d^3xu +γ偏导d^5xu = 0 with L^2 initial data. By working on the Bourgain space X^r,s(R^2) associated with this equation, we prove that the Cauchy problem of the Kawahara equation is locally solvable if initial data belong to H^r(R) and -1 〈 r ≤ 0. This result combined with the energy conservation law of the Kawahara equation yields that global solutions exist if initial data belong to L^2(R).

关 键 词:Kawahara方程  Cauchy问题  整体解  数学分析
收稿时间:2004-04-19
修稿时间:2004-04-192004-07-21

Global Existence of Solutions for the Cauchy Problem of the Kawahara Equation with L 2 Initial Data
Shang Bin CUI Dong Gao DENG Shuang Ping TAO.Global Existence of Solutions for the Cauchy Problem of the Kawahara Equation with L^2 Initial Data[J].Acta Mathematica Sinica,2006,22(5):1457-1466.
Authors:Shang Bin Cui  Dong Gao Deng  Shuang Ping Tao
Institution:(1) Department of Mathematics, Sun Yat-Sen University, Guangzhou 510275, P. R. China;(2) Department of Mathematics, Northwest Normal University, Lanzhou 730070, P. R. China
Abstract:In this paper we study solvability of the Cauchy problem of the Kawahara equation $$
\partial _{t} u + au\partial _{x} u + \beta \partial ^{3}_{x} u + \gamma \partial ^{5}_{x} u = 0
$$ with L 2 initial data. By working on the Bourgain space X r,s (R 2) associated with this equation, we prove that the Cauchy problem of the Kawahara equation is locally solvable if initial data belong to H r (R) and −1 < r ≤ 0. This result combined with the energy conservation law of the Kawahara equation yields that global solutions exist if initial data belong to L 2(R). Project supported by the China National Natural Science Foundation (Grants 10171111, 10171112)
Keywords:Kawahara equation  Cauchy problem  global solution
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