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)

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

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

References[edit]