LOCAL WELL-POSEDNESS OF THE REGULARIZED
rBO-ZK EQUATION IN SOBOLEV SPACES
OF NEGATIVE INDICES
Fabián Sánchez Salazar
Department of Mathematics
Universidad Nacional de Colombia
Kra. 30 No. 45-03 Bogotá, D.C., COLOMBIA
Abstract. In this paper, we study the local well-posedness of the regularized rBO-ZK equation, defined by

\begin{displaymath}u_{t}+a(u^{2})_{x}+(b\mathscr{H}u_{t}+u_{yy})_{x}-\mu (-\Delta)^{\alpha}u=0,\end{displaymath}

where $\mathscr{H}$ is the Hilbert transform with respect to $x$ and $a$, $b$, $\mu$ and $\alpha$ are real numbers, with $b>0$ , $\mu>0$ and $\alpha > \frac{1}{2}$, We show that the associated Cauchy problem is locally well posed in Sobolev space $H^s(\re^2)$ for $s>-2\alpha+1$.
AMS Subject classification: 35A01, 35G25, 42B35
Keywords and phrases: regularized Benjamin-Ono-Zakharov-Kuznetsov equation, Cauchy problem, negative indices, global and local well-posedness


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DOI: 10.12732/ijam.v28i4.7

Volume: 28
Issue: 4
Year: 2015