On the Spatial Behaviour for Solutions to a Phase Transition Model Involving Two Temperatures

The aim, in this paper, is to study the spatial behaviour, in an unbounded domain, of solutions for a phase transition system involving two temperatures. When we want to determine the magnitude of the solution of an elliptic or parabolic partial differential align on a bounded domain, we generally use the maximum principle. This principle states that a solution function of such aligns has its maximum value on the boundary of the domain. Unfortunately, this property is no longer true when the domain of study of the function is not bounded. This leads us to apply a generalisation of the maximum principle known as the Phragmén-Lindelöf alternative. To apply it, we place ourselves in a domain comprising a bounded region and an unbounded region. If we can show that the solution does not explode in the bounded region, we can conclude that the solution does not explode in the whole domain.