Electronic Journal of Differential Equations,
Vol. 1994(1994), No. 08, pp. 1-9.

Title: Quasireversibility methods for non-well-posed problems

Authors: Gordon W. Clark (Kennesaw State College, Marietta, GA, USA)
         Seth F. Oppenheimer (Mississippi State Univ., MS, USA)

Abstract: The final value problem,
$$\displaylines{
  u_t+Au = 0, \quad 0<t<T \cr
  u(T)   =f}
$$
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.

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