Identifying a BV-kernel in a hyperbolic integrodifferential equation

Alfredo Lorenzi; Eugenio Sinestrari

doi:10.3934/dcds.2008.21.1199

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2008, Volume 21, Issue 4: 1199-1219. Doi: 10.3934/dcds.2008.21.1199

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Identifying a BV-kernel in a hyperbolic integrodifferential equation

Alfredo Lorenzi^1, and
Eugenio Sinestrari^2,

1.
Dipartimento di Matematica “F. Enriques”, Università degli Studi di Milano, Via Saldini 50, 20133, Milano
2.
Dipartimento di Matematica, Università di Roma “La Sapienza”, P.le A. Moro 5, 00185 Roma

Received: July 31, 2007

Revised: January 31, 2008

Published: September 2008

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Abstract

This paper is devoted to determining the scalar relaxation kernel $a$ in a second-order (in time) integrodifferential equation related to a Banach space when an additional measurement involving the state function is available. A result concerning global existence and uniqueness is proved.
The novelty of this paper consists in looking for the kernel $a$ in the Banach space $BV(0,T)$, consisting of functions of bounded variations, instead of the space $W^{1,1}(0,T)$ used up to now to identify $a$.
An application is given, in the framework of $L^2$-spaces, to the case of hyperbolic second-order integrodifferential equations endowed with initial and Dirichlet boundary conditions.

Keywords:

Second-order linear integro-differential equations Recovering a scalar unknown convolution kernel. An existence and uniqueness result. Application to hyperbolic linear integro-differential equations..

Mathematics Subject Classification: Primary: 45Q05; Secondary: 45N05, 45D05, 45K05, 35L20.

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