## International Journal of Differential Equations

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

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

Salah Djezzar and Nihed Teniou

#### 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.

#### 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