In this note, we investigate the spatial behaviour of the solutions for a theory for the heat conduction with a delay term. We obtain an alternative of the Phragmen-Lindelof type. That is de solutions either decay in a exponential way or blow-up at infinity way. We also describe how to obtain an upper bound for the amplitude term. It is worth nothing yhas this is the first contribution on spatial behaviour for partial differential equations involving a delay term. We use the energy arguments to obtain our main results. The main point of the contribution is the use of a suitable weighted energy function.