ON THE ILL-POSEDNESS OF THE REGULARIZED
rBO-ZK EQUATION IN SOBOLEV SPACES
Fabian Sanchez Salazar
Department of Mathematics
National University
Kra. 30 No. 45-03
Bogotá-110110, COLOMBIA

Abstract

In this paper, we study the ill-posedness of the regularized rBO-ZK equation, defined by

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

where $\textit{H}$ is the Hilbert transform with respect to $x$. The values $a$, $b$, $\mu$ and $\alpha$ are real numbers, with $b>0$, $\mu>0$ and $\alpha > \frac{1}{2}$ and $(-\Delta)^{\alpha}=(-\partial_x^2 - \partial_y^2)^{\alpha}$.

We show that the associated Cauchy problem is ill posed in Sobolev space $H^s(\mathbf{R}^2)$ for $s<0$ and $n=2$. The lack of local well-posedness is in the sense that the dependence of solutions upon initial data fails to be continuous.

Citation details of the article



Journal: International Journal of Applied Mathematics
Journal ISSN (Print): ISSN 1311-1728
Journal ISSN (Electronic): ISSN 1314-8060
Volume: 36
Issue: 1
Year: 2023

DOI: 10.12732/ijam.v36i1.10

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