  In this paper, we announce a rigorous approach to establishing uniqueness
results, under certain conditions, for initial-boundary-value problems for a
class of linear evolution partial differential equations (PDEs) formulated in a
quarter-plane. We also effectively propose an algorithm for constructing
non-uniqueness counter-examples which do not satisfy the said conditions. Our
approach relies crucially on the rigorous analysis of regularity and asymptotic
properties of integral representations derived formally via the celebrated
Unified Transform Method for each such PDE. For uniqueness, this boundary
behavior analysis allows for a careful implementation of an energy-estimate
argument on the semi-unbounded domain. Our ideas are elucidated via application
of the present technique to two concrete examples, namely the heat equation and
the linear KdV equation with Dirichlet data on the positive quadrant, under a
particular set of conditions at the domain boundary and at infinity.
Importantly, this is facilitated by delicate refinement of previous results
concerning the boundary behavior analysis of these two celebrated models. In
addition, 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 recast
as a Dirichlet problem. To the best of our knowledge, such well-posedness
results appear for the first time in either the classical or the weak sense.
Extensions to other classes of equations are underway and will appear
elsewhere.
