Heat conduction with memory: A  singular  kernel problem

Sandra Carillo; Vanda Valente; Giorgio Vergara Caffarelli

doi:10.3934/eect.2014.3.399

Article Contents

2014, Volume 3, Issue 3: 399-410. Doi: 10.3934/eect.2014.3.399

This issue Previous Article On the viscoelastic coupled suspension bridge Next Article Thermal control of a rate-independent model for permanent inelastic effects in shape memory materials

Heat conduction with memory: A singular kernel problem

1.
Dipartimento di Scienze di Base e Applicate, per l'Ingegneria - Sezione Matematica, Sapienza Università di Roma, Rome
2.
Istituto per le Applicazioni del Calcolo M. Picone, C.N.R. Consiglio Nazionale delle Ricerche, Rome

Received: April 30, 2013

Revised: January 31, 2014

Published: August 2014

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Abstract

The existence and uniqueness of solution to an integro-differential problem arising in heat conduction with memory is here considered. Specifically, a singular kernel problem is analyzed in the case of a multi-dimensional rigid heat conductor. The choice to investigate a singular kernel material is suggested by applications to model a wider variety of materials and, in particular, new materials whose heat flux relaxation function may be superiorly unbounded at the initial time $t=0$. The present study represents a generalization to higher dimensions of a previous one concerning a $1$-dimensional problem in the framework of linear viscoelasticity with memory. Specifically, an existence theorem is here proved when initial homogeneous data are assumed. Indeed, the choice of homogeneous data is needed to obtain the a priori estimate in Section 2 on which the subsequent results, are based.

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Mathematics Subject Classification: Primary: 45K05, 35Q79.

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