Regularity properties of solutions of linear integrodifferential equations with singular kernels
type: Item
instance of: scholarly article
title: Regularity properties of solutions of linear integrodifferential equations with singular kernels
zbMATH Open document ID: 0824.45015
author: Hans Engler
published in: Journal of Integral Equations and Applications
publication date: 1995-11-16T00:00:00Z
review text: Let \({\mathbf H}\) be a Hilbert space with norm \(|\cdot |\). Let \(A\) be a possibly unbounded linear operator in \({\mathbf H}\), and let \(a: [0,\infty)\to {\mathbf R}\) be a given locally integrable kernel function. In this note we study properties of solutions of the abstract linear integrodifferential equation \[ u'(t)+ a * Au(t)= 0,\qquad 0< t< T,\tag{1} \] in \({\mathbf H}\) . Here, the usual convolution notation is employed: \(f * g(t)= \int^ t_ 0 f(t- s) g(s) ds\), if one of the two functions is scalar-valued and the other one is vector-valued. We want to consider mild solutions of (1), i.e., continuous functions \(u(\cdot)\) for which \(1 * a * u(t)\in D(A)\) for all \(0\leq t\leq T\) and for which the integrated version \[ u(t)+ A(1 * a * u(t))= u(0) \] of (1) holds for all \(t\in [0, T]\). The goal of this note is to give conditions on the kernel function \(a\) under which such mild solutions satisfy an estimate of the form \[ | Au(t)|\leq \textstyle{{C\over t^ M}} | u(0)| \] for a suitable power \(M\). It will be proved that such an estimate is always true if the derivative \(a'\) is integrable and behaves like \(- t^{- 2\alpha}\) near zero, and that in this case \(M= 1/\alpha\) is a suitable exponent.
Mathematics Subject Classification ID: 45N05
Mathematics Subject Classification ID: 45J05
zbMATH DE Number: 784747
zbMATH Keywords: regularity
zbMATH Keywords: singular kernels
zbMATH Keywords: Hilbert space
zbMATH Keywords: abstract linear integro- differential equation
zbMATH Keywords: convolution
zbMATH Keywords: mild solutions
zbMATH Keywords: estimate
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1216/jiea/1181075571
OpenAlex ID: W1979875068
cites work: Resolvent Formulas for a Volterra Equation in Hilbert Space
cites work: An abstract Volterra equation with applications to linear viscoelasticity
cites work: Linear integro-differential equations in Banach spaces
cites work: Q3697587
cites work: Singular Relaxation Moduli and Smoothing in Three-Dimensional Viscoelasticity
cites work: Smoothing Properties of Linear Volterra Integrodifferential Equations
cites work: Analytic resolvent operators for integral equations in Banach space
cites work: On wave propagation in linear viscoelasticity
cites work: Semigroups of linear operators and applications to partial differential equations
cites work: Positivity and regularity of hyperbolic Volterra equations in Banach spaces
cites work: Q3830011
DOI: 10.1216/JIEA/1181075571
description: scientific article; zbMATH DE number 784747