Some existence results for a nonlinear hyperbolic integrodifferential equation with singular kernel
type: Item
instance of: scholarly article
title: Some existence results for a nonlinear hyperbolic integrodifferential equation with singular kernel
zbMATH Open document ID: 0765.45006
published in: Journal of Integral Equations and Applications
publication date: 1992-08-13T00:00:00Z
review text: The author studies the nonlinear Volterra integro-differential equation \[ u_ t(t,x)=\int^ t_ 0a(t-s)b(u_ x(s,x))_ xds+f(t,x),\quad t\geq 0,\;x\in\mathbb{R} \] \(u(0,x)=u_ 0(x)\) where \(a\) is of strong positive type and \(b\in C^ 3(\mathbb{R})\), \(b(0)=0\), \(b'(0)>0\). The following results are proved: if \(a'\in L^ 1(\mathbb{R}_ +)\) then there exists a global solution if the data are sufficiently small (with reference to suitable norms); morover if \(a'\in L^ 1_{loc}(\mathbb{R}_ +)\) and \(0<c\leq b'(x)\leq d\), \(x\in\mathbb{R}\) for \(c,d>0\) constant then there exists a local solution for data of arbitrary size. The method of the proof uses approximating kernels of strongly positive type and energy estimates.
reviewed by: Gabriella Di Blasio
Mathematics Subject Classification ID: 45K05
zbMATH DE Number: 40684
zbMATH Keywords: singular kernel
zbMATH Keywords: hyperbolic equations
zbMATH Keywords: nonlinear Volterra integro- differential equation
zbMATH Keywords: global solution
zbMATH Keywords: kernels of strongly positive type
zbMATH Keywords: energy estimates
author: Stig-Olof Londen
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1216/jiea/1181075599
OpenAlex ID: W2007208202
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DOI: 10.1216/JIEA/1181075599
description: scientific article; zbMATH DE number 40684