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

Salah Djezzar,Nihed Teniou

doi:10.1155/2011/913125

Abstract

We consider in this paper an abstract parabolic backward Cauchy problem associated with an unbounded linear operator in a Hilbert space , 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 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.

Highlights

Salah Djezzar and Nihed TeniouLaboratoire Equations Differentielles, Departement de Mathematiques, Facultedes Sciences Exactes, Universite Mentouri Constantine, Constantine 25000, Algeria
Let A be a positive we suppose that A ≥ η > 0, self-adjoint unbounded linear operator which has a continuous spectrum on a Hilbert space H such that −A generates a contraction C0-semigroup on H
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 0 ≤ t < T is sought

Summary

Salah Djezzar and Nihed Teniou

Laboratoire Equations Differentielles, Departement de Mathematiques, Facultedes Sciences Exactes, Universite Mentouri Constantine, Constantine 25000, Algeria. 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 0 ≤ t < 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. Some other convergence results including some explicit convergence rates are provided

Introduction

The Convergence Results

We also have

If we put