A MODIFIED QUASI-BOUNDARY VALUE METHOD FOR A CLASS OF ABSTRACT PARABOLIC ILL-POSED PROBLEMS M. DENCHE AND S. DJEZZAR Received 14 October 2004; Accepted 9 August 2005 We study a final value problem for first-order abstract differential equation with positive self-adjoint unbounded operator coefficient. This problem is ill-posed. Perturbing the final condition, we obtain an approximate nonlocal problem depending on a small parameter. We show that the approximate problems are well posed and that their solutions converge if and only if the original problem has a classical solution. We also obtain estimates of the solutions of the approximate problems and a convergence result. | A MODIFIED QUASI-BOUNDARY VALUE METHOD FOR A CLASS OF aBsTRACT parabolic ILL-POSED PROBLEMS M. DENCHE AND S. DJEZZAR Received 14 October 2004 Accepted 9 August 2005 We study a final value problem for first-order abstract differential equation with positive self-adjoint unbounded operator coefficient. This problem is ill-posed. Perturbing the final condition we obtain an approximate nonlocal problem depending on a small parameter. We show that the approximate problems are well posed and that their solutions converge if and only if the original problem has a classical solution. We also obtain estimates of the solutions of the approximate problems and a convergence result of these solutions. Finally we give explicit convergence rates. Copyright 2006 M. Denche and S. Djezzar. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. 1. Introduction We consider the following final value problem FVP u t Au t 0 0 t T u T f for some prescribed final value f in a Hilbert space H where A is a positive self-adjoint operator such that 0 e p A . Such problems are not well posed that is even if a unique solution exists on 0 T it need not depend continuously on the final value f. We note that this type of problems has been considered by many authors using different approaches. Such authors as Lavrentiev 8 Lattes and Lions 7 Miller 10 Payne 11 and Showalter 12 have approximated FVP by perturbing the operator A. In 1 4 13 a similar problem is treated in a different way. By perturbing the final value condition they approximated the problem with u t Au t 0 0 t T u T au 0 f. Hindawi Publishing Corporation Boundary Value Problems Volume 2006 Article ID37524 Pages 1-8 DOI BVP 2006 37524 2 Regularization of parabolic ill-posed problems A similar approach known as the method of auxiliary boundary conditions
