Quasireversibility methods for non-well-posed problems

Abstract

$$displaylines{ u_t+Au = 0, quad 0<t<T cr u(T) =f} $$ 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 quasi-boundary-value method, where we perturb the final condition to form an approximate non-local problem depending on a small parameter $alpha$. We show that the approximate problems are well posed and that their solutions $u_alpha$ 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.

Keywords

• Quasireversibility
• Final Value Problems
• Ill-Posed Problems.