In other words, it is ill-posed and cannot be used to recover the initial conditions.
With precise measurement you can. Once started to be solved numerically different initial conditions (for the standard diffusion equation) are all going to yield constant function after some time due to rounding errors, so the information is lost and can’t be recovered by time-reversed process. But as a mathematical problem the diffusion equation with reversed time is well defined and has unique solution nevertheless.
From what I recall, the reverse-time diffusion u_t=-u_xx is not well posed, i.e. for a given solution u(t), if we perturb u(0) by epsilon, there is no finite t such that the deviation of the new solution from the old one is bounded by epsilon*e^t. A quick Google search confirms it: (pdf, search inside for “well posed”)
I didn’t realise that “well-posed” is a term with a technical meaning. The definition of well-posedness I have found says that the solution must exist, be unique and continuously depend on the initial data, I am not sure whether this is equivalent to your definition.
Anyway, the problem with the reverse dissipation equation is that for some initial conditions, namely discontinuous ones, the solution doesn’t exist. However, if a function u(x,t) satisfies the diffusion equation on the interval [t1,t2], we can recover it completely from knowledge of not only u(x,t1), but also from u(x,t0) with any fixed t0 lying between t1 and t2.
With precise measurement you can. Once started to be solved numerically different initial conditions (for the standard diffusion equation) are all going to yield constant function after some time due to rounding errors, so the information is lost and can’t be recovered by time-reversed process. But as a mathematical problem the diffusion equation with reversed time is well defined and has unique solution nevertheless.
From what I recall, the reverse-time diffusion u_t=-u_xx is not well posed, i.e. for a given solution u(t), if we perturb u(0) by epsilon, there is no finite t such that the deviation of the new solution from the old one is bounded by epsilon*e^t. A quick Google search confirms it: (pdf, search inside for “well posed”)
I didn’t realise that “well-posed” is a term with a technical meaning. The definition of well-posedness I have found says that the solution must exist, be unique and continuously depend on the initial data, I am not sure whether this is equivalent to your definition.
Anyway, the problem with the reverse dissipation equation is that for some initial conditions, namely discontinuous ones, the solution doesn’t exist. However, if a function u(x,t) satisfies the diffusion equation on the interval [t1,t2], we can recover it completely from knowledge of not only u(x,t1), but also from u(x,t0) with any fixed t0 lying between t1 and t2.