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

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. 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.


Introduction
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 C 0 -semigroup on H. Let T be a positive real number.We consider the final value problem FVP of finding u : 0, T → H such that u t Au t 0, 0 ≤ t < T, 1.1 International Journal of Differential Equations for some prescribed final value φ in H.Such problems are not well posed; that is, even if a unique solution exists on 0, T , it need not depend continuously on the finial value φ.This type of problems, in the case where A has a discrete spectrum, has been considered by many authors using different approaches.Such authors as Lattès and Lions 1 , Lavrentiev 2 , Miller 3 , Payne 4 , and Showalter 5 have approximated the final value problem FVP 1.1 , 1.2 by perturbing the operator A.
A similar problem is treated in a different way; see [6][7][8] .By perturbing the final value condition, they approximated the problem 1.1 , 1.2 with u t Au t 0, 0 ≤ t < T.

1.3
A similar approach known as the method of auxiliary boundary conditions was given in 9-11 .Also, the nonstandard conditions of the form 1.3 for parabolic equations have been considered in some recent papers 12, 13 .For further results related to these type of problems, we can also see 14, 15 .It is also worth reading the recent paper by Campbell Hetrick and Hunhes 16 dealing with inhomogeneous ill-posed problems in Banach space.We also mention the very recent papers by Tuan 17 and Tuan et al. 18 which deal with similar ill-posed problems using different approaches.For some comments on the results presented in paper 17 using a different regularization approach the truncation regularization method , see Remark 3.6 at the end of this paper.
In this paper, we perturb both 1.1 and the final condition 1.2 to form an approximate nonlocal problem depending on two small parameters α and β, with boundary condition containing a derivative of the same order than the equation as follows: where the operator A is replaced by the operator A α A I αA −1 and u T φ by u σ T β u σ 0 − u σ 0 φ, and σ α, β , where α > 0, β > 0. We show that the approximate problems are well posed and that their solutions v σ converge if and only if the original problem has a classical solution.We also show that this method gives a better approximation than many other quasireversibility and quasiboundary type methods, for example, 1, 6, 7, 19-21 .Finally, we obtain several other results, including some explicit convergence rates.
Throughout this paper, we will denote by H a Hilbert space, {E λ , λ ≥ η > 0} the resolution of the identity associated with the positive unbounded self-adjoint operator A. So the spectral representation of C 0 semigroup S t e −tA resp., A is given by S t e −tA ∞ η e −tλ dE λ resp., A ∞ η λdE λ , and so for all u ∈ D A , Au ∞ η λ dE λ u, and this is characterized by

The Approximate Problem
We approximate the final value problem 1.1 , 1.2 , by the following perturbed problem: where A is as above and A α is the Yosida approximation of operator A, and σ α, β where α > 0, β > 0.
Definition 2.1.Define the function for φ ∈ H, σ α, β and α > 0, β > 0, t ∈ 0, T , where S α t is the semigroup generated by −A α .Now, we give the following theorem where the proof is based on the semigroups theory 22 .

Theorem 2.2. The function v σ t is the unique solution of the perturbed problem 2.1 , and it depends continuously on φ.
Proof.We consider the following classical Cauchy problem: It is clear that v σ t is the unique solution and

International Journal of Differential Equations
The continuous dependence of v σ on φ is obtained by showing that βη/ 1 αη e −T/α φ .

2.5
Now, we consider the following problem: Theorem 2.3.The problem 2.6 is well posed, and its solution is given by furthermore, Proof.Since 2.9 then,

2.10
Now we give some convergence results.
Proof.We have

3.1
Putting and if we put

3.4
It is clear that, for all ε > 0, there exists 3.5 then we have
Let us denote by C θ A , θ ≥ 0 the following set: It is clear that the following proprieties hold 3.9 Now, we give some convergence results with explicit convergence rates.

3.12
Proof.Using the proof of the previous theorem, we have

3.13
International Journal of Differential Equations 7 where N σ e −Tθλ .

3.16
We also have 3.17 Then, using the above estimates, we get the desired results.Now, let F be the function defined by

3.18
Theorem 3.3.For all φ ∈ H, the problem 1.1 , 1.2 has a solution u t if and only if the function F is continious at 0, 0 .Furthermore, u σ t → u t , as |σ| → 0, uniformly in t.
Proof.We assume that lim |σ| → 0 φ σ φ 0 and φ 0 < ∞.Let w t S t φ 0 .So, we have Since lim |σ| → 0 u σ T φ and lim |σ| → 0 u σ T w T and so by the unicity of the limit, we obtain that w t is a solution to problem 1.1 , 1.2 .

International Journal of Differential Equations
Now, we suppose that u t ∞ η e T −t λ dE λ φ is a solution to problem 1.1 , 1.2 .Since u 0 S −T φ ∈ H see 19 , Lemma 1 , then we have

3.22
If we put

3.24
Using analogous calculations as in the proof of Theorem 3.1, we obtain
We note that we can easily show that for φ ∈ C 1 A .And by using Theorem 3.1 again, we see that And

3.31
Proof.Since 3.32 then using Theorem 3.2, we get the required result.

3.34
Proof.Using Theorem 3.3, we have

3.35
And, by Theorem 3.4, we obtain the desired result.
Remark 3.6.We note that in a very recent paper by Tuan 17 a new use of a different regularization method the truncation method is introduced for dealing with a similar class of problems.This truncation method consists in eliminating all high frequencies from the solution of the considered ill-posed problem to get an approximate regularized solution together with some stability and error estimates that he indicates to be of Holder type.In particular, the author gives some estimates which hold at t 0 and so he gets the convergence of the approximate solution at t 0. For a significant comparison with these results obtained by this truncation regularization method, one needs the determination and selection, for each case, of a possible appropriate regularization parameter β .However, the method of regularization presented in our work still gives a better approximation than many other quasireversibility and quasi-boundary type methods, for example, 1, 6, 7, 19-21 .