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.
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.