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

TL;DR

This paper develops a rigorous method to establish uniqueness for linear evolution PDEs in a quarter-plane, analyzing boundary behavior and providing algorithms for non-uniqueness counterexamples, with applications to heat, KdV, and BBM equations.

Contribution

It introduces a novel approach combining integral representation analysis and energy estimates to prove uniqueness and construct counterexamples for PDEs on a quarter-plane.

Findings

Established uniqueness results for heat and KdV equations with specific boundary conditions.

Provided an algorithm to construct non-uniqueness counterexamples.

Extended analysis to the linearized BBM equation and recast oblique Robin data as Dirichlet problems.

Abstract

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.