  In this note, we announce a systematic analysis of continuous dependence on
the data in classical spaces for the initial-boundary-value problem of the
diffusion equation on the half-line, with data that are not necessarily
compatible at the quadrant corner. This is based on a recent approach to
rigorously analyzing integral representations derived via the unified transform
method of Fokas. No exotic phenomena were discovered in this case, yet our
findings appear to be new in the pertinent literature. These results supplement
our previous investigations on existence and (non)uniqueness within the
framework of well-posedness. The present detailed exposition elucidates the
subtleties involved while also demonstrating a generic technique. Applications
of the latter to several other IBVPs and PDEs will be reported elsewhere.
