Improved Local Wellposedness of Cauchy Problem for Generalized KdV-BO Equation

Abstract:

      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].