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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Blow up and instability of solitary-wave solutions to a generalized Kadomtsev-Petviashvili equation


Author: Yue Liu
Journal: Trans. Amer. Math. Soc. 353 (2001), 191-208
MSC (2000): Primary 35Q53, 35B60, 76B25
Published electronically: June 8, 2000
MathSciNet review: 1653363
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Abstract: In this paper we consider a generalized Kadomtsev-Petviashvili equation in the form \begin{equation*}( u_{t} + u_{xxx} + u^{p} u_{x} )_{x} = u_{yy} \quad (x, y) \in R^{2}, t \ge 0. \end{equation*} It is shown that the solutions blow up in finite time for the supercritical power of nonlinearity $ p \ge 4/3 $ with $ p $ the ratio of an even to an odd integer. Moreover, it is shown that the solitary waves are strongly unstable if $ 2 < p < 4$; that is, the solutions blow up in finite time provided they start near an unstable solitary wave.


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Additional Information

Yue Liu
Affiliation: Department of Mathematics, The University of Texas at Arlington, Arlington, Texas 76019
Email: liu@math.uta.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-00-02465-X
PII: S 0002-9947(00)02465-X
Received by editor(s): April 6, 1998
Received by editor(s) in revised form: September 2, 1998
Published electronically: June 8, 2000
Article copyright: © Copyright 2000 American Mathematical Society