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June  2003, 2(2): 233-249. doi: 10.3934/cpaa.2003.2.233

Some results about a bidimensional version of the generalized BO

1. 

Departamento de Matematica, IMECC-UNICAMP, 13081-970, Campinas, SP, Brazil

Received  February 2002 Revised  January 2003 Published  March 2003

For the bidimensional version of the generalized Benjamin-Ono equation:

$u_t-H^{(x)}u_{x y}+u^p u_y=0, \quad t\in \mathbb R,\quad (x,y)\in \mathbb R^2,$

we use the method of parabolic regularization to prove local well-posedness in the spaces $H^s(\mathbb R^2), \quad s>2$ and in the weighted spaces $\mathcal F_r^s=H^s(\mathbb R^2) \cap L^2((1+x^2+y^2)^rdxdy), \quad s>2,\quad r\in [0,1]$ and $\mathcal F_{1,k}^k=H^k(\mathbb R^2) \cap L^2((1+x^2+y^{2k})dxdy), \quad k\in\mathbb N, \quad k\geq 3. \quad $ 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 $p\geq 3,\quad s>3$. Nonexistence of square integrable solitary waves of the form $u(x,y,t)=v(x,y-ct),\quad c>0, \quad p\in \{1,2\}$ is obtained using the results about existence of solitary waves of the BO and variational methods.

Citation: Aniura Milanés. Some results about a bidimensional version of the generalized BO. Communications on Pure & Applied Analysis, 2003, 2 (2) : 233-249. doi: 10.3934/cpaa.2003.2.233
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