  We study Dirichlet-type problems for the simplest third-order linear
dispersive PDE, often referred to as the Airy equation. Such problems have not
been extensively studied, perhaps due to the complexity of the spectral
structure of the spatial operator. Our specific interest is to determine
whether the peculiar phenomenon of revivals, also known as Talbot effect, is
supported by these boundary conditions, which for third order problems are not
reducible to periodic ones. We prove that this is the case only for a very
special choice of the boundary conditions, for which a new type of weak cusp
revival phenomenon has been recently discovered. We also give some new results
on the functional class of the solution for other cases.
