Large time behavior of solutions of higher order nonlinear dispersive equations of KdV type with weak nonlinearity
type: Item
instance of: scholarly article
title: Large time behavior of solutions of higher order nonlinear dispersive equations of KdV type with weak nonlinearity
zbMATH Open document ID: 1022.35058
author: Nakao Hayashi
author: T. Matos
author: Pavel I. Naumkin
published in: Differential and Integral Equations
publication date: 2003-01-26T00:00:00Z
review text: The Cauchy problem for the nonlinear dispersive equation of the Korteweg-de Vries type with weak nonlinearity is studied, namely the equation \(u_t+\partial _xf(u)+Ku=0\) with the Cauchy data \(u(0,x)=u_0(x)\). Here, \(K\) is a pseudodifferential operator defined via the Fourier transformation. A particular type of nonlinearities \(f\) with polynomial growth is studied. For sufficiently small (in \(H^{1,1}\) norm) data \(u_0\) the existence of a unique global solution in \(C(\mathbb R,H^{1,1})\) is proven together with the appropriate estimates of Lebesgue norms of \(u\) and \(uu_x\). Moreover, for large time \(t\), the asymptotic formula of the behavior of \(u\) (uniformly in \(x\)) is derived. The technique of Sobolev, weighted Sobolev and Besov spaces is used throughout the proofs, which are split into several, rather technical lemmas. The result is interesting and the paper is, in spite of its technicality, clearly written.
Mathematics Subject Classification ID: 35Q53
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35S05
zbMATH DE Number: 1858734
zbMATH Keywords: Korteweg-de Vries (KdV) equation
zbMATH Keywords: weak nonlinearity
zbMATH Keywords: existence of a unique global solution
zbMATH Keywords: dispersive equation
zbMATH Keywords: Cauchy problem
zbMATH Keywords: large time solutions
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1858734