On the asymptotic behavior of solutions of generalized Korteweg-de Vries equations
type: Item
instance of: scholarly article
title: On the asymptotic behavior of solutions of generalized Korteweg-de Vries equations
zbMATH Open document ID: 0686.35018
DOI: 10.1016/0022-247X(89)90104-2
published in: Journal of Mathematical Analysis and Applications
publication date: 1989-01-01T00:00:00Z
review text: The author handles the generalized Korteweg-de Vries equation \[ (GKdV)\quad u_ t+u_{xxx}+p| u|^{p-1}u_ x=0. \] In particular he is interested in the asymptotic behaviour of the solution of the corresponding Cauchy problem with initial-data \(\psi \in L^ 2({\mathbb{R}})\). These are the main results: Theorem 1. Let u(x,t) be a solution of (GKdV) with \(u(x,0)=\psi \in L^ 2({\mathbb{R}}),\) \(\| \psi \| \neq 0\). Then there does not exist any solution v(x,t) of the Airy equation \((A)\quad v_ t+v_{xxx}=0,\) with \(v(x,0)=\phi (x)\in L^ 1({\mathbb{R}}),\quad \int \phi (x)dx=0,\) such that \(\| u(t)-v(t)\|_{L^ 2({\mathbb{R}})}\to 0\) as \(t\to \infty.\) Theorem 2. Let u(x,t) be as in Theorem 1 and \(\psi\) satisfies in addition \(\int \psi (x)dx=0\). Then there does not exist any solution v(x,t) of the Airy equation with \(v(x,0)=\phi (x)\in L^ 1({\mathbb{R}})\) and \[ \| u(t)-v(t)\|_{L^ 2({\mathbb{R}})}+\| u(t)-v(t)\|_{L^ 1({\mathbb{R}})}\to 0\quad as\quad t\to +\infty. \]
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35Q99
Mathematics Subject Classification ID: 35K55
zbMATH DE Number: 4124191
zbMATH Keywords: generalized Korteweg-de Vries equation
zbMATH Keywords: Airy equation
author: Mohammad Rammaha
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4124191