The Zakharov-Kuznetsov equation in weighted Sobolev spaces

In this work we consider the initial value problem (IVP) associated to the two dimensional Zakharov-Kuznetsov equation $\left. \begin{array}{rl} u_t+\partial_x^3 u+\partial_x \partial_y^2 u +u \partial_x u &\hspace{-2mm}=0,\qquad\qquad (x,y)\in\mathbb R^2,\; t\in\mathbb R,\\ u(x,y,0)&\hspace{-2mm}=u_0(x,y). \end{array} \right\}$ We study the well-posedness of the IVP in the weighted Sobolev spaces $H^s(\mathbb R^2) \cap L^2((1+x^2+y^2)^{r} dx dy),$ with $s,r\in\mathbb R$.

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