Quasireversibility Methods for Non-Well-Posed Problems

The final value problem, { ut +Au = 0 , 0 < t < T u(T ) = f with positive self-adjoint unbounded A is known to be ill-posed. One approach to dealing with this has been the method of quasireversibility, where the operator is perturbed to obtain a well-posed problem which approximates the original problem. In this work, we will use a quasiboundary-value method, where we perturb the final condition to form an approximate non-local problem depending on a small parameter α. We show that the approximate problems are well posed and that their solutions uα converge on [0, T ] if and only if the original problem has a classical solution. We obtain several other results, including some explicit convergence rates.