On local existence of solutions of initial boundary value problems for the ``bad'' Boussinesq-type equation
type: Item
instance of: scholarly article
title: On local existence of solutions of initial boundary value problems for the ``bad'' Boussinesq-type equation
zbMATH Open document ID: 1022.35052
DOI: 10.1016/S0362-546X(01)00894-X
author: Zhijian Yang
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2002-11-28T00:00:00Z
review text: The paper deals with the differential equation \[ u_{tt}-u_{xx}-bu_{xxxx}= \sigma(u)_{xx} \] together with certain appropriate initial and boundary conditions and where \(\sigma(s)\) is a given nonlinear function. In the case \(b < 0\) this equation is called the ``bad'' Boussinesq-equation since in this case even the linear part behaves bad. There are examples of not well-posed problems having this linear part. \textit{H. A. Levine} and \textit{B. D. Sleeman} [J. Math. Anal. Appl. 107, 206-210 (1985; Zbl 0591.35010)] showed the nonexistence of global positive solutions. In this interesting paper the existence of local generalized solutions is shown under rather weak conditions on \(\sigma(u)\) and the initial data, like \(\sigma\) twice continuously differentiable and \( \sigma'' (s) \) is locally Lipschitz continuous. If \(\sigma(s)\) is a concave function then, under certain conditions on the initial data, the solutions have a finite escape time (blow up).
Mathematics Subject Classification ID: 35Q53
Mathematics Subject Classification ID: 35D05
Mathematics Subject Classification ID: 35B40
zbMATH DE Number: 1836644
zbMATH Keywords: bad Boussinesq equation
zbMATH Keywords: initial boundary value problem
zbMATH Keywords: local solutions
zbMATH Keywords: blow-up of solutions
reviewed by: G. Jank
MaRDI profile type: Publication
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full work available at URL: s0362-546x(01)00894-x
OpenAlex ID: W1991899226
description: scientific article; zbMATH DE number 1836644