A modified quasi-boundary value method for a class of abstract parabolic ill-posed problems

M Denche,S Djezzar

doi:10.1155/bvp/2006/37524

Abstract

We study a final value problem for first-order abstract differential equation with positive self-adjoint unbounded operator coefficient. This problem is ill-posed. Perturbing the final condition, we obtain an approximate nonlocal problem depending on a small parameter. We show that the approximate problems are well posed and that their solutions converge if and only if the original problem has a classical solution. We also obtain estimates of the solutions of the approximate problems and a convergence result of these solutions. Finally, we give explicit convergence rates.

Highlights

We consider the following final value problem (FVP)u (t) + Au(t) = 0, 0 ≤ t < T (1.1) u(T) = f (1.2)for some prescribed final value f in a Hilbert space H; where A is a positive self-adjoint operator such that 0 ∈ ρ(A)
We study a final value problem for first-order abstract differential equation with positive self-adjoint unbounded operator coefficient
We show that the approximate problems are well posed and that their solutions converge if and only if the original problem has a classical solution

Summary

Introduction

For some prescribed final value f in a Hilbert space H; where A is a positive self-adjoint operator such that 0 ∈ ρ(A). Such problems are not well posed, that is, even if a unique solution exists on [0, T] it need not depend continuously on the final value f. We note that this type of problems has been considered by many authors, using different approaches.

Regularization of parabolic ill-posed problems

The approximate problem