Electron. J. Diff. Eqns., Vol. 1994(1994), No. 08, pp. 1-9.

Quasireversibility Methods for Non-Well-Posed Problems

Gordon W. Clark & Seth F. Oppenheimer

Abstract:
The final value problem,
$$\left\{ \eqalign{ u_t+Au &= 0, \quad 0 less than t less than T \cr u(T) &=f \cr}\right. $$
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.

Submitted November 14, 1994. Published November 29, 1994.
Math Subject Classification: 35A35, 35R25.
Key Words: Quasireversibility, Final Value Problems, Ill-Posed Problems.

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Gordon W. Clark
Department of Mathematics, Kennesaw State College, P O Box 444, Marietta, GA 30061, USA
e-mail: clark@math.msstate.edu

Seth F. Oppenheimer
Department of Mathematics and Statistics, Mississippi State University, Drawer MA MSU, MS 39762, USA
e-mail: seth@math.msstate.edu

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