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Quasireversibility methods for non-well-posed problems. (English) Zbl 0811.35157

Summary: The final value problem, \[ u_ t + Au = 0, \quad 0 < t < T, \qquad u(T) = f \] 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 nonlocal 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.

MSC:

35R25 Ill-posed problems for PDEs
35A35 Theoretical approximation in context of PDEs
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