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

In this paper we study the initial-value problem associated with the dispersion generalized-Benjamin-Ono-Zakharov-Kuznetsov equation, $u_{t}+D^{a+1}_x \partial_{x}u+u_{xyy}+uu_{x}=0, \qquad a\in(0,1).$ More specifically, we study the persistence property of the solution in the weighted anisotropic Sobolev spaces $H^{(1+a)s,2s}(\R^{2})\cap L^{2}((x^{2r_1} +y^{2r_2})dxdy),$ 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.

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