Inverse parabolic partial differential equation

An equation of the form

$$ \tag{* } u _ {t} + \sum _ {i , j = 1 } ^ { n } a _ {ij} ( x , t ) u _ {x _ {i} x _ {j} } - \sum _ { i= } 1 ^ { n } a _ {i} ( x , t ) u _ {x _ {i} } - a ( x , t ) u = $$

$$ = \ f ( x , t ) , $$

where the form $ \sum a _ {ij} \xi _ {i} \xi _ {j} $ is positive definite. The variable $ t $ plays the role of "inverse" time. The substitution $ t = - t ^ \prime $ reduces equation (*) to the usual parabolic form. Parabolic equations of "mixed" type occur, for example, $ u _ {t} = x u _ {xx} $ is a direct parabolic equation for $ x > 0 $ and an inverse parabolic equation for $ x < 0 $, with degeneracy of the order for $ x = 0 $.

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)

$$ u _ {t} + \Delta u = 0 $$

( $ \Delta $ being the Laplace operator) see [a1].

References

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