Preprint A104/2001
Some results about a bidimensional version of the generalized BO

Aniura Milanes

**Keywords: **
nonlinear dispersive equations | Cauchy problem

For the bidimensional version of the generalized Benjamin-Ono equation:
(null)
we use the method of parabolic regularization to prove local
well-posedness in the spaces $H^s(\mathbb{R}^2),\;s>2$ and in the weighted spaces
$\mathcal{F}_r^s=H^s(\mathbb{R}^2) \cap L^2\big((1+x^2+y^2)^rdxdy\big),\;s>2$, $r\in [0,1]$
and
$\mathcal{F}_{1,k}^k=H^k(\mathbb{R}^2) \cap L^2\big((1+x^2+y^{2k})dxdy\big)$,
$k\in\mathbb{N},\;k\geq 3$.
As in the case of BO there is lack of
persistence for both the linear and nonlinear
equations (for $p$ odd) in $\mathcal{F}_2^s$. That leads to unique
continuation principles in a natural way. By standard methods based
on $L^p-L^q$ estimates of
the associated group we obtain global well-posedness for small initial
data and nonlinear scattering for $p\geq 3,\;s>3$. Nonexistence
of square integrable solitary waves of the form
$u(x,y,t)=v(x,y-ct),\;c>0,\;p\in \{1,2\}$ is obtained using the results about
existence of solitary waves of the BO and variational methods.