Improved Regularization Method for Backward Cauchy Problems Associated with Continuous Spectrum Operator

Abstract

We consider in this paper an abstract parabolic backward Cauchy problem associated with an unbounded linear operator in a Hilbert space $H$, where the coefficient operator in the equation is an unbounded self-adjoint positive operator which has a continuous spectrum and the data is given at the final time $t=T$ and a solution for is sought. It is well known that this problem is illposed in the sense that the solution (if it exists) does not depend continuously on the given data. The method of regularization used here consists of perturbing both the equation and the final condition to obtain an approximate nonlocal problem depending on two small parameters. We give some estimates for the solution of the regularized problem, and we also show that the modified problem is stable and its solution is an approximation of the exact solution of the original problem. Finally, some other convergence results including some explicit convergence rates are also provided.