Infinity of solutions to initial-boundary value problems for linear constant-coefficient evolution PDEs on semi-infinite intervals

TL;DR

This paper introduces an algorithmic method to construct non-uniqueness examples for classical solutions of linear evolution PDEs on semi-infinite domains, using complex analysis and the Fokas transform method.

Contribution

It presents a novel approach for generating non-uniqueness counterexamples and establishes new uniqueness theorems for specific linear PDE models.

Findings

Constructed explicit non-uniqueness examples for heat and KdV equations.

Developed a systematic procedure using complex-analytic techniques.

Proved new uniqueness theorems for classical solutions.

Abstract

In this short communication, we announce an algorithmic procedure for constructing non-uniqueness counter-examples of classical solutions to initial-boundary-value problems for a wide class of linear evolution partial differential equations, of any order and with constant coefficients, formulated in a quarter-plane. Our approach relies on analysis of regularity and asymptotic properties, near the boundary of the spatio-temporal domain, of closed-form integral-representation formulae derived via complex-analytic techniques and rigorous implementation of the modern PDE technique known as Fokas unified transform method. In order to elucidate the novel idea and demonstrate the proposed technique in a self-contained fashion, we explicitly present its application to two concrete examples, namely the heat equation and the linear KdV equation with Dirichlet data. New uniqueness theorems for these two models are also presented herein.