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Local and global well-posedness for the 2D generalized Zakharov-Kuznetsov equation
This paper addresses well-posedness issues for the initial value problem
(IVP) associated with the generalized Zakharov-Kuznetsov equation, namely,
\{equation*} \quad \left\{\{array}{lll} {\displaystyle u_t+\partial_x \Delta
u+u^ku_x = 0,}\qquad (x,y) \in \mathbb{R}^2, \,\,\,\, t>0, {\displaystyle
u(x,y,0)=u_0(x,y)}. \{array} \right. \{equation*} For , the IVP
above is shown to be locally well-posed for data in ,
. For , local well-posedness is shown to hold for data in
, , where . Furthermore, for
, if and satisfies , then
the solution is shown to be global in . For , if
, , and satisfies , where is the corresponding ground state solution, then
the solution is shown to be global in
On the support of solutions to the Zakharov-Kuznetsov equation
In this article we prove that sufficiently smooth solutions of the
Zakharov-Kuznetsov equation that have compact support for two different times
are identically zero.Comment: Version of Dec 17/2010 contains simpler proof of Theorem 1.3 and new
references are adde
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