A note on the Cauchy problem to a class of nonlinear dispersive equations with singular initial data
type: Item
instance of: scholarly article
title: A note on the Cauchy problem to a class of nonlinear dispersive equations with singular initial data
zbMATH Open document ID: 0987.35138
DOI: 10.1016/S0362-546X(98)00344-7
author: Huijiang Zhao
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2002-06-13T00:00:00Z
review text: The Cauchy problem for the following class of nonlinear equations \[ \partial_t u +\partial_x f(u)+\delta C_n \partial_x^nu - \varepsilon \partial_x^2 u=0 \] is considered. The goal of the author is to generalize results by \textit{M. E. Schonbek} [Commun. Partial Differ. Equations 7, 959-1000 (1982; Zbl 0496.35058)] on the KdV-Burgers equation and by \textit{B. Cassis} [Commun. Partial Differ. Equations 14, 619-634 (1989; Zbl 0703.35043).] on a fourth-order equation. The unique, global-in-time, smooth solution is constructed for every initial condition from \(L^2({\mathbb R})\) and for the flux function \(f(u)\) of a quadratic growth. Moreover, for convex flux functions, the limit as \(\delta,\varepsilon\to 0\) of corresponding solutions is studied.
reviewed by: Grzegorz Karch
Mathematics Subject Classification ID: 35Q53
Mathematics Subject Classification ID: 35B25
Mathematics Subject Classification ID: 35L65
zbMATH DE Number: 1491650
zbMATH Keywords: unique global-in-time smooth solution
zbMATH Keywords: admissible measure-valued solutions
zbMATH Keywords: compensated compactness
zbMATH Keywords: Cauchy problem
MaRDI profile type: Publication
full work available at URL: s0362-546x(98)00344-7
OpenAlex ID: W2065071149
description: scientific article; zbMATH DE number 1491650