Differential and Integral Equations

Blow-up and instability of a regularized long-wave-KP equation

Yue Liu and Michael M. Tom

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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}$.

Article information

Source
Differential Integral Equations, Volume 16, Number 9 (2003), 1131-1152.

Dates
First available in Project Euclid: 21 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356060561

Mathematical Reviews number (MathSciNet)
MR1989545

Zentralblatt MATH identifier
1031.35125

Subjects
Primary: 35Q53: KdV-like equations (Korteweg-de Vries) [See also 37K10]
Secondary: 35B40: Asymptotic behavior of solutions 35Q51: Soliton-like equations [See also 37K40]

Citation

Liu, Yue; Tom, Michael M. Blow-up and instability of a regularized long-wave-KP equation. Differential Integral Equations 16 (2003), no. 9, 1131--1152. https://projecteuclid.org/euclid.die/1356060561


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