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

• Published in: Taiwanese 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
• Language: English
• 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
• ISSN: 1027-5487; 2224-6851
• DOI: 10.11650/twjm/1500405803

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.