Juxtacrine signalling is the process of cell communication in which ligand and receptors are both anchored in the cell membrane. We develop three mathematical models for this process, involving different mathematical representations of the dynamics of membrane-bound ligand and free and bound receptors, within an epithelial sheet. We consider the dynamics of this system following a localized disturbance, such as would be provided by a source of ligand or by the generation of a free edge via wounding. We study the ability of the juxtacrine mechanism to transmit a signal away from this disturbance, and show analytically that the spatial half-life of the signal can in fact be arbitrarily large. This result is quite general, since we use a generic reaction kinetic scheme; the key assumption is that ligand and receptor production are both upregulated by binding. Moreover, the result applies to all three of our model formulations. We conclude by discussing applications of the result to the particular case of the transforming growth factor alpha binding to epidermal growth factor receptor in epidermal wound healing.