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May  2013, 12(3): 1221-1235. doi: 10.3934/cpaa.2013.12.1221

## Phragmén-Lindelöf alternative for an exact heat conduction equation with delay

 1 Departament de Matemàtica Aplicada 2, ETSEIAT–UPC, C. Colom 11, 08222 Terrassa, Barcelona, Spain 2 Matemática Aplicada 2, E.T.S.E.I.T.-U.P.C., Colom 11, 08222 Terrassa, Barcelona, Spain

Received  November 2011 Revised  May 2012 Published  September 2012

In this paper we investigate the spatial behavior of the solutions for a theory for the heat conduction with one delay term. We obtain a Phragmén-Lindelöf type alternative. That is, the solutions either decay in an exponential way or blow-up at infinity in an exponential way. We also show how to obtain an upper bound for the amplitude term. Later we point out how to extend the results to a thermoelastic problem. We finish the paper by considering the equation obtained by the Taylor approximation to the delay term. A Phragmén-Lindelöf type alternative is obtained for the forward and backward in time equations.
Citation: M. Carme Leseduarte, Ramon Quintanilla. Phragmén-Lindelöf alternative for an exact heat conduction equation with delay. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1221-1235. doi: 10.3934/cpaa.2013.12.1221
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