THE CAUCHY PROBLEM FOR A GENERALIZED KORTEWEG-DE VRIES EQUATION IN HOMOGENEOUS SOBOLEV SPACES

Considered in this article is the Cauchy problem of a generalized Korteweg-de Vries equation $\left\{ \matrix {u_t} + {u_{xxx}} + u{u_x} + {\left| {{D_x}} \right|^{2\alpha }}u = 0,t \in {{\Cal R}^ + },x \in {\Cal R}, \hfill \cr u\left( {x,0} \right) = \varphi \left( x \right) \hfill \cr \endmatrix \...

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Published inTaiwanese journal of mathematics Vol. 14; no. 2; pp. 479 - 499
Main Authors Xue, Ruying, 薛儒英, Hu, Sufen, 胡素芬
Format Journal Article
LanguageEnglish
Published Mathematical Society of the Republic of China (Taiwan) 01.04.2010
Subjects
Burger equation
Cauchy problem
Differential equations
Dyadics
Estimation methods
Mathematical functions
Mathematical induction
Sobolev spaces
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ISSN1027-5487
2224-6851
DOI10.11650/twjm/1500405803

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Summary:Considered in this article is the Cauchy problem of a generalized Korteweg-de Vries equation $\left\{ \matrix {u_t} + {u_{xxx}} + u{u_x} + {\left| {{D_x}} \right|^{2\alpha }}u = 0,t \in {{\Cal R}^ + },x \in {\Cal R}, \hfill \cr u\left( {x,0} \right) = \varphi \left( x \right) \hfill \cr \endmatrix \right.$ with 0 ≤ α ≤ 1. The local well-posedness of the Cauchy problem in the homogeneous Sobolev space Hs (ℝ) for $s \in \left( {\frac{{\alpha - 3}}{{2\left( {2 - \alpha } \right)}},0} \right)$ is proved. In addition, the mapping that associated to appropriate initial-data the corresponding solution is analytic as a function between appropriate Banach spaces.
ISSN:1027-5487
2224-6851
DOI:10.11650/twjm/1500405803