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Backward parabolic equation

One occasionally encounters a so-called backward parabolic PDE, which takes the form u t = L u {\displaystyle u_{t}=Lu} (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,

Similarly to a final-value problem for a parabolic PDE, an initial-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.