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.