 One occasionally encounters a so-called "backward parabolic PDE", which takes the form formula_40 (note the absence of a minus sign).An initial-value problem for the backward heat equation,

is equivalent to a final-value problem for the ordinary heat equation,

An initial/boundary-value problem for a backward parabolic PDE is usually not well-posed
(solutions often grow unbounded in finite time, or even fail to exist).Nonetheless, these problems are important for the study of the reflection of singularities of solutions to various other PDEs.