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.
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.