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2003 Blow-up and instability of a regularized long-wave-KP equation
Yue Liu, Michael M. Tom
Differential Integral Equations 16(9): 1131-1152 (2003).

Abstract

A regularized long-wave--Kadomtsev-Petviashvili equation of the form $$(u_{t}-u_{xxt}+u_{x}+u^{p}u_{x})_{x}-u_{yy}=0, \tag*{(*)} $$ is considered. It is shown that if $p\geq4$, certain initial data can lead to a solution that blows up in finite time. More precisely, under the above condition the solution cannot remain in the Sobolev class $H^{2}(\mathbb R)$ for all time. Also demonstrated here is the solitary-wave solutions $ u(x, y, t) = \phi_c (x-ct, y) $, which exist if and only if $1\leq p <4 $ and $ c > 1 $, when considered as solutions of the initial-value problem for (*), are nonlinearly unstable to perturbations of the initial data, if ${4\over3} <p <4$ and $ 1 < c < {4p\over 4+p}$.

Citation

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Yue Liu. Michael M. Tom. "Blow-up and instability of a regularized long-wave-KP equation." Differential Integral Equations 16 (9) 1131 - 1152, 2003.

Information

Published: 2003
First available in Project Euclid: 21 December 2012

zbMATH: 1031.35125
MathSciNet: MR1989545

Subjects:
Primary: 35Q53
Secondary: 35B40 , 35Q51

Rights: Copyright © 2003 Khayyam Publishing, Inc.

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Vol.16 • No. 9 • 2003
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