Inverse parabolic partial differential equation

An equation of the form

$$u_t+\sum _{i,j=1}^n{a}_{ij}(x,t)u_{x_i{x}_j}-\sum _{i=1}^n{a}_i(x,t)u_{x_i}-a(x,t)u=f(x,t),$$

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[edit]

The Cauchy problem for an equation $\text{(???)}$ 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)

$$u_t+\mathrm{\Delta }u=0$$

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

References[edit]