Necessary and sufficient condition for uniqueness of solution to the first boundary value problem for the diffusion equation in unbounded domains

This paper introduces a notion of regularity of t=-∞t=-∞ for the diffusion (or heat) equation and establishes a necessary and sufficient condition for the existence of a unique bounded solution to the first boundary value problem for the diffusion equation in a general domain Ω⊂RN+1Ω⊂RN+1 which extends up to t=-∞t=-∞.