Blow-up conditions for nonlinear Volterra integral equations with power nonlinearity
type: Item
instance of: scholarly article
title: Blow-up conditions for nonlinear Volterra integral equations with power nonlinearity
zbMATH Open document ID: 1177.45002
author: Tomasz Małolepszy
author: Wojciech Okrasiński
published in: Applied Mathematics Letters
publication date: 2008-12-10T00:00:00Z
review text: The authors study the existence of blow-up (i.e., nontrivial) solutions of nonlinear Volterra equations of the form \[ u(t)=\int_0^t k(t-s) g(u(s))\,ds, \quad t\geq 0,\tag{1} \] where \(K:(0,\infty)\to[0,\infty)\) is a locally integrable positive function and \(g:[0,\infty)\to[0,\infty)\) is a strictly increasing absolutely continuous function of the form \[ g(t)=\begin{cases} g_1(t)&\text{if}\;\;t\in[0,d),\\ bt^p&\text{if}\;\;t\geq d,\end{cases} \] where \(d>0,\;b>0,\;p>1\), \(g_1(0)=0\) and \(\lim_{u\to 0^+}(u/g_1(u))=0\). The following criterion is established: equation (1) has a blow-up solution if and only if the integral \[ \int_0^t K^{-1}(s)\frac{ds}{s(-\ln s)},\quad\text{where}\;\;K(t)=\int_0^t k(s)\,ds, \] is convergent for any \(t\in(0,1)\).
reviewed by: Yuri I. Karlovich
Mathematics Subject Classification ID: 45G10
zbMATH DE Number: 5380073
zbMATH Keywords: nonlinear Volterra integral equation
zbMATH Keywords: power nonlinearity
zbMATH Keywords: blow-up solutions
MaRDI profile type: Publication
full work available at URL: j.aml.2007.03.020
OpenAlex ID: W2071291203
cites work: Coupled Volterra equations with blow-up solutions
cites work: Analysis of explosion for nonlinear Volterra equations
cites work: Volterra equations which model explosion in a diffusive medium
cites work: Nonlinear Volterra Integral Equations and the Apéry Identities
cites work: Unique solutions of some Voltera integral equations.
cites work: Remarks on a nonlinear Volterra equation
cites work: The existance of nontrival solutions of Volterra equations.
cites work: A Volterra inequality with the power type nonlinear kernel
cites work: Nontrivial solutions for a class of nonlinear Volterra equations with convolution kernel
cites work: Nontrivial Solutions to Nonlinear Volterra Integral Equations
cites work: Conditions for blow-up of solutions of some nonlinear Volterra integral equations
DOI: 10.1016/J.AML.2007.03.020
description: scientific article; zbMATH DE number 5380073