Phragmén-lindelöf alternative for an exact heat conduction equation with delay

Abstract

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 Phragmèn-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 Phragmèn-Lindelöf type alternative is obtained for the forward and backward in time equations.