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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Ill-posedness results for the (generalized) Benjamin-Ono-Zakharov-Kuznetsov equation


Authors: Amin Esfahani and Ademir Pastor
Journal: Proc. Amer. Math. Soc. 139 (2011), 943-956
MSC (2010): Primary 35Q51, 35Q53; Secondary 35Q35
Published electronically: July 28, 2010
MathSciNet review: 2745646
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Abstract: Here we consider results concerning ill-posedness for the Cauchy problem associated with the Benjamin-Ono-Zakharov-Kuznetsov equation, namely,

\begin{displaymath} \left\{ \begin{array}{ll} u_t-\mathscr{H}u_{xx}+u_{xyy}+u^ku... ...b{R}^+, \\ u(x,y,0)=\phi(x,y). \end{array}\right.\tag*{(IVP)} \end{displaymath}

For $ k=1$, (IVP) is shown to be ill-posed in the class of anisotropic Sobolev spaces $ H^{s_1,s_2}(\mathbb{R}^2), s_1,s_2\in \mathbb{R}$, while for $ k\geq2$ ill-posedness is shown to hold in $ H^{s_1,s_2}(\mathbb{R}^2), 2s_1+s_2<3/2-2/k$. Furthermore, for $ k=2,3$, and some particular values of $ s_1,s_2$, a stronger result is also established.


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Additional Information

Amin Esfahani
Affiliation: Department of Mathematics, IME-USP, Rua do Matão 1010, Cidade Universitária, 05508-090, São Paulo, SP, Brazil
Address at time of publication: School of Mathematics and Computer Science, Damghan University of Basic Sciences, Damghan, 36716-41167, Iran
Email: amin@impa.br, esfahani@dubs.ac.ir

Ademir Pastor
Affiliation: IMECC-UNICAMP, Cidade Universitária, Rua Sérgio Buarque de Holanda, 651, 13083-859, Campinas, SP, Brazil
Email: apastor@ime.unicamp.br

DOI: http://dx.doi.org/10.1090/S0002-9939-2010-10532-4
PII: S 0002-9939(2010)10532-4
Keywords: Nonlinear PDE, Cauchy problem, ill-posedness
Received by editor(s): January 15, 2010
Received by editor(s) in revised form: March 22, 2010
Published electronically: July 28, 2010
Additional Notes: The first author was supported by FAPESP/SP-Brazil grant 2008/58892-6.
The second author was supported by CNPq-Brazil grant 152234/2007-1.
Communicated by: Hart F. Smith
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.