# Inverse parabolic partial differential equation

From Encyclopedia of Mathematics

An equation of the form

(*) |

where the form is positive definite. The variable plays the role of "inverse" time. The substitution reduces equation (*) to the usual parabolic form. Parabolic equations of "mixed" type occur, for example, is a direct parabolic equation for and an inverse parabolic equation for , with degeneracy of the order for .

#### Comments

The Cauchy problem for an equation (*) is a well-known example of an ill-posed problem (cf. Ill-posed problems). For a discussion of the backward heat equation (cf. also Thermal-conductance equation)

( being the Laplace operator) see [a1].

#### References

[a1] | L.E. Payne, "Improperly posed problems in partial differential equations" , SIAM (1975) |

**How to Cite This Entry:**

Inverse parabolic partial differential equation. A.P. Soldatov (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Inverse_parabolic_partial_differential_equation&oldid=19128

This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098