# American Institute of Mathematical Sciences

September  2014, 3(3): 399-410. doi: 10.3934/eect.2014.3.399

## 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, Italy 2 Istituto per le Applicazioni del Calcolo M. Picone, C.N.R. Consiglio Nazionale delle Ricerche, Rome, Italy

Received  May 2013 Revised  February 2014 Published  August 2014

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.
Citation: Sandra Carillo, Vanda Valente, Giorgio Vergara Caffarelli. Heat conduction with memory: A singular kernel problem. Evolution Equations & Control Theory, 2014, 3 (3) : 399-410. doi: 10.3934/eect.2014.3.399
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