Persistence properties for the dispersion generalized BO-ZK equation in weighted anisotropic Sobolev spaces

Abstract

In this paper we study the initial-value problem associated with the dispersion generalized-Benjamin-Ono-Zakharov-Kuznetsov equation, $$u_t+D_x^{a+1}{\partial }_{x}u+u_{xyy}+u{u}_x=0,a\in (0,1).$$ More specifically, we study the persistence property of the solution in the weighted anisotropic Sobolev spaces $$H^{(1+a)s,2s}({\mathbb{R}}^2)\cap L^2((x^{2r_1}+y^{2r_2})dxdy),$$ for appropriate $s$, $r_1$ and $r_2$. By establishing unique continuation properties we also show that our results are sharp with respect to the decay in the $x$-direction.

35 pages