International Journal of Differential Equations

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

Salah Djezzar and Nihed Teniou

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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 0t<T 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.

Article information

Source
Int. J. Differ. Equ., Volume 2011 (2011), Article ID 913125, 11 pages.

Dates
Received: 29 May 2011
Revised: 17 August 2011
Accepted: 26 September 2011
First available in Project Euclid: 25 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.ijde/1485313261

Digital Object Identifier
doi:10.1155/2011/913125

Mathematical Reviews number (MathSciNet)
MR2854951

Zentralblatt MATH identifier
1242.34107

Citation

Djezzar, Salah; Teniou, Nihed. Improved Regularization Method for Backward Cauchy Problems Associated with Continuous Spectrum Operator. Int. J. Differ. Equ. 2011 (2011), Article ID 913125, 11 pages. doi:10.1155/2011/913125. https://projecteuclid.org/euclid.ijde/1485313261


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