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In this paper, we announce a rigorous approach to establishing uniquenessresults, under certain conditions, for initial-boundary-value problems for aclass of linear evolution partial differential equations (PDEs) formulated in aquarter-plane. We also effectively propose an algorithm for constructingnon-uniqueness counter-examples which do not satisfy the said conditions. Ourapproach relies crucially on the rigorous analysis of regularity and asymptoticproperties of integral representations derived formally via the celebratedUnified Transform Method for each such PDE. For uniqueness, this boundarybehavior analysis allows for a careful implementation of an energy-estimateargument on the semi-unbounded domain. Our ideas are elucidated via applicationof the present technique to two concrete examples, namely the heat equation andthe linear KdV equation with Dirichlet data on the positive quadrant, under aparticular set of conditions at the domain boundary and at infinity.Importantly, this is facilitated by delicate refinement of previous resultsconcerning the boundary behavior analysis of these two celebrated models. Inaddition, we announce a uniqueness theorem for the linearized BBM equation,whose proof, in a similar spirit, will appear in a forthcoming paper. Finally,we briefly demonstrate how the general case of oblique Robin data can be recastas a Dirichlet problem. To the best of our knowledge, such well-posednessresults appear for the first time in either the classical or the weak sense.Extensions to other classes of equations are underway and will appearelsewhere.

A note on uniqueness for linear evolution PDEs posed on the quarter-plane