Improved Local Wellposedness of Cauchy Problem for Generalized KdV-BO Equation
Received:December 04, 2006  Revised:October 28, 2007
Key Word: KdV-BO equation   Cauchy problem   local wellposedness.
Fund ProjectL:the Natural Science Foundation of Zhejiang Province (No.Y6080388); the Science and Technology Research Foundation of Zhejiang Ocean University (Nos.X08M014; X08Z04).
 Author Name Affiliation ZHAO Xiang Qing Department of Mathematics, Zhejiang Ocean University, Zhejiang 316000, China GUO Ai School of Mathematical Sciences, South China University of Technology, Guangdong 510640, China
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In this paper we prove that the Cauchy problem associated with the generalized KdV--BO equation $u_t u_{xxx} \lambda{\cal H}(u_{xx}) u^2u_x=0$, $x\in R$, $t\ge 0$ is locally wellposed in $\widehat{H^s_r} (R)$ for $\frac43 \frac 1 r$ and $s \ge s(r)=\frac 12-\frac 1{2r}$. In particular, for $r=2$, we reobtain the result in [3].