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

Authors: Salah Djezzar; Nihed Teniou
Journal: International Journal of Differential Equations
 (2011)

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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 spaceH, 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.


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## Body

## 1. Introduction

LetA 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. Let T be a positive real number. We consider the final value problem (FVP) of finding u:[0,T]→H such that(1.1)u′(t)+Au(t)=0,0≤t<T,(1.2)u(T)=ϕ,
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 whereA 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 [6–8]. By perturbing the final value condition, they approximated the problem (1.1), (1.2) with(1.3)u′(t)+Au(t)=0,0≤t<T.αu(0)+u(T)=ϕ.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:(1.4)vσ′(t)+Aαvσ(t)=0,0≤t<T,vσ(T)+β(vσ(0)-vσ′(0))=ϕ,
where the operator A is replaced by the operator Aα=A(I+αA)-1 and u(T)=ϕ by uσ(T)+β(uσ(0)-uσ′(0))=ϕ, and σ=(α,β), where α>0, β>0.We show that the approximate problems are well posed and that their solutionsvσ 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 byH a Hilbert space, {Eλ,λ≥η>0} the resolution of the identity associated with the positive unbounded self-adjoint operator A. So the spectral representation of C0 semigroup S(t)=e-tA (resp., A) is given by S(t)=e-tA=∫η+∞e-tλdEλ (resp., A=∫η+∞λdEλ), and so for all u∈D(A),Au=∫η+∞λdEλu, and this is characterized by (1.5)u∈D(A)iff‖Au‖2=∫η+∞λ2d‖Eλu‖2<∞.
Also, throughout this work, we mean by a solution of problem (1.1), (1.2) on the interval [0,T] a function u∈C([0,T];H)∩C1(]0,T[;H) such that, for all t∈]0,T[,u(t)∈D(A) and (1.1), (1.2) hold. A useful characterization of the admissible set for which problem (1.1), (1.2) has a solution is as follows: Problem (1.1), (1.2) has a solution u if and only if ∫η+∞e2λTd∥Eλϕ∥2<∞, and this unique solution is represented by u(t)=∫η+∞e(T-t)λdEλϕ<∞ (see [19], Lemma 1]).


## 2. The Approximate Problem

We approximate the final value problem (1.1), (1.2), by the following perturbed problem:
(2.1)vσ′(t)+Aαvσ(t)=0,0⩽t<T,vσ(T)+β(vσ(0)-vσ′(0))=ϕ,
where A is as above and Aα is the Yosida approximation of operator A, and σ=(α,β) where α>0,β>0.Definition 2.1.
Define the function(2.2)vσ(t)=Sα(t)(β(I+Aα)+Sα(T))-1ϕ,
for ϕ∈H,σ=(α,β) and α>0,β>0,t∈[0,T], where Sα(t) is the semigroup generated by -Aα.Now, we give the following theorem where the proof is based on the semigroups theory [22].Theorem 2.2.
The functionvσ(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:(2.3)vσ′(t)+Aαvσ(t)=0,0<t<T,vσ(0)=(β(I+Aα)+Sα(T))-1ϕ.
It is clear that vσ(t) is the unique solution and
(2.4)vσ(T)+β(vσ(0)-vσ′(0))=Sα(T)(β(I+Aα)+Sα(T))-1ϕ+β(I+Aα)(β(I+Aα)+Sα(T))-1ϕ=(β(I+Aα)+Sα(T))(β(I+Aα)+Sα(T))-1ϕ=ϕ.
The continuous dependence of vσ on ϕ is obtained by showing that
(2.5)‖vσ(t)‖=‖Sα(t)(β(I+Aα)+Sα(T))-1ϕ‖⩽‖(β(I+Aα)+Sα(T))-1ϕ‖⩽1β+(βη/(1+αη))+e-T/α‖ϕ‖.Now, we consider the following problem:(2.6)uσ′(t)+Auσ(t)=0,uσ(0)=ϕσ=(β(I+Aα)+Sα(T))-1ϕ.Theorem 2.3.
The problem (2.6) is well posed, and its solution is given by
(2.7)uσ(t)=S(t)(β(I+Aα)+Sα(T))-1ϕ,
furthermore,
(2.8)‖uσ(t)‖⩽1(β(1+(η/(1+αη)))+e-T/α)(T-t)/T‖ϕ‖.Proof.
Since(2.9)‖uσ(t)‖2=∫η+∞e-2λT(β(1+(λ/(1+αλ)))+e-(λT)/(1+αλ))2d‖Eλϕ‖2≤∫η+∞[1(β(1+(η/(1+αη)))+e-T/α)2]T-t/Td‖Eλϕ‖2,
then,
(2.10)‖uσ(t)‖⩽[1(β(1+(η/(1+αη)))+e-T/α)]T-t/T‖ϕ‖.Now we give some convergence results.


## 3. The Convergence Results

Theorem 3.1.
For allϕ∈H,∥uσ(T)-ϕ∥→0, as |σ|→0.Proof.
We have(3.1)‖uσ(T)-ϕ‖2=∫η+∞(e-λTβ(1+(λ/(1+αλ)))+e-(λT)/(1+αλ)-1)2d‖Eλϕ‖2=∫η+∞(β(1+(λ/(1+αλ)))β(1+(λ/(1+αλ)))+e-(λT/(1+αλ))+e-(λT/(1+αλ))-e-λTβ(1+(λ/(1+αλ)))+e-(λT/(1+αλ)))2d‖Eλϕ‖2.
Putting
(3.2)Mσ(λ)=β(1+(λ/(1+αλ)))β(1+(λ/(1+αλ)))+e-(λT/(1+αλ)),Nσ(λ)=e-(λT/(1+αλ))-e-λTβ(1+(λ/(1+αλ)))+e-(λT/(1+αλ)),
and if we put
(3.3)Pσ(λ)=Mσ(λ)+Nσ(λ),
then we get
(3.4)‖uσ(T)-ϕ‖2=∫η+∞(Pσ(λ))2d‖Eλϕ‖2⩽2∫η+∞(Mσ2(λ)+Nσ2(λ))d‖Eλϕ‖2.
It is clear that, for all ɛ>0, there exists k∈ℕ*/∫k+∞d∥Eλϕ∥2<ε/8 and Mσ(λ)≤1,Nσ(λ)⩽1.
If we put(3.5)Iσ=∫ηkMσ2(λ)d‖Eλϕ‖2+∫k+∞Mσ2(λ)d‖Eλϕ‖2,Jσ=∫ηkNσ2(λ)d‖Eλϕ‖2+∫k+∞Nσ2(λ)d‖Eλϕ‖2.
then we have
(3.6)Iσ⩽ε8+β2(1+1α)2e2kT‖ϕ‖2,Jσ⩽ε8+α2T2k4‖ϕ‖2.
Finally, choosing σ such that |σ|2=α2+β2⩽(1/∥ϕ∥2)(1/T2k4+1/e2kT)(ε/8), we obtain the following estimate
(3.7)‖uσ(T)-ϕ‖2⩽(ϵ2+2(β2(1+1α)2e2kT+α2T2k4))‖ϕ‖2,
which gives the desired result.Let us denote byCθ(A), θ≥0 the following set:(3.8)Cθ(A)={h∈H:‖h‖Cθ2=∫η+∞e2Tθλd‖Eλϕ‖2<+∞}.It is clear that the following proprieties hold(3.9)Cθ1(A)⊆Cθ2(A),θ2⩾θ1,Cθ(A)⊂H,θ>0.
Now, we give some convergence results with explicit convergence rates.Theorem 3.2.
Ifϕ∈Cθ(A), then one has
(3.10)‖uσ(T)-ϕ‖2⩽2(C12(θ)β2θα2(η1+αη)2(θ-1)+C22(θ,T)α2)‖ϕ‖Cθ2,
for 0<θ<1 and
(3.11)‖uσ(T)-ϕ‖2⩽2(β2α2+C22(θ,T)α2)‖ϕ‖Cθ2,forθ⩾1,
such that
(3.12)C1(θ)=(1-θ)1-θθθ⩽1,C2(θ,T)=1Tθ2.Proof.
Using the proof of the previous theorem, we have(3.13)‖uσ(T)-ϕ‖2=∫η+∞Pσ2(λ)d‖Eλϕ‖2=∫η+∞Pσ2(λ)e-2Tθλe2Tθλd‖Eλϕ‖2⩽2(Mσ,θ,∞2+Nσ,θ,∞2)‖φ‖Cθ2,
where
(3.14)Mσ,θ,∞=sup⁡λ⩾ηMσe-Tθλ,Nσ,θ,∞=sup⁡λ⩾ηNσe-Tθλ.
If 0<θ<1, then we have
(3.15)Mσ,θ,∞≤2αβθC1(θ).
And, for θ⩾1, we have
(3.16)Mσ,θ,∞≤2βα.
We also have
(3.17)Nσ,θ,∞⩽4αTθ2e2⩽αTθ2,∀θ>0.
Then, using the above estimates, we get the desired results.Now, letF be the function defined by(3.18)F:R+×R+⟶H,σ=(α,β)⟶F(σ)={uσ(0)=ϕσ,σ≠(0,0),u(0)=ϕ0,σ=(0,0).Theorem 3.3.
For allϕ∈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, uσ(t)→u(t),as|σ|→0, uniformly in t.Proof.
We assume thatlim|σ|→0ϕσ=ϕ0 and ∥ϕ0∥<∞. Let w(t)=S(t)ϕ0. So, we have
(3.19)‖w(t)-uσ(t)‖⩽‖ϕ0-ϕσ‖.
Hence,
(3.20)sup⁡0⩽t⩽T‖w(t)-uσ(t)‖⩽‖ϕ0-ϕσ‖⟶|σ|→00.
Since lim|σ|→0uσ(T)=ϕ and lim|σ|→0uσ(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).
Now, we suppose thatu(t)=∫η+∞e(T-t)λdEλϕ is a solution to problem (1.1), (1.2). Since u(0)=S(-T)ϕ∈H (see [19], Lemma 1), then we have
(3.21)‖u(0)‖2=‖ϕ‖C12=∫η+∞e2Tλd‖Eλϕ‖2<∞.
Let k>0 and ϵ>0, such that ∫k+∞e2Tλd∥Eλϕ∥2<ϵ/8. If σ1=(α1,β1) and σ2=(α2,β2). Then,
(3.22)‖uσ1(0)-uσ2(0)‖2=∫η+∞(1β1(1+(λ/(1+α1λ)))+e-(λT/(1+α1λ))-1β2(1+(λ/(1+α2λ)))+e-(λT/1+α2λ)))2d‖Eλϕ‖2≤2∫η+∞[β1(1+(λ/(1+α2λ)))-β2(1+(λ/(1+α1λ)))(β1(1+(λ/(1+α1λ)))+e-(λT/(1+α1λ)))(β2(1+λ/1+α2λ)+e-(λT/(1+α2λ)))]2d‖Eλϕ‖2+2∫η+∞[e-λT/1+α1λ-e-λT/1+α2λ(β1(1+λ/(1+α1λ))+e-(λT/(1+α1λ)))(β2(1+λ/(1+α2λ))+e-(λT/(1+α2λ)))]2d‖Eλϕ‖2.
If we put
(3.23)Mσ1,σ2=|β1(1+(λ/(1+α2λ)))-β2(1+(λ/(1+α1λ)))|(β1(1+λ/(1+α1λ))+e-λT/1+α1λ)(β2(1+(λ/(1+α2λ)))+e-λT/1+α2λ),Nσ1,σ2=|e-(λT/(1+α1λ))-e-(λT/(1+α2λ))|(β1(1+λ/(1+α1λ))+e-(λT/(1+α1λ)))(β2(1+(λ/(1+α2λ)))+e-(λT/(1+α2λ))),
then we have
(3.24)‖uσ1(0)-uσ2(0)‖2⩽2∫η+∞Mσ1,σ22(λ)d‖Eλϕ‖2+2∫η+∞Nσ1,σ22(λ)d‖Eλϕ‖2.
Using analogous calculations as in the proof of Theorem 3.1, we obtain
(3.25)‖uσ1(0)-uσ2(0)‖2⩽ϵ2+2e2kT‖ϕ‖C12×[[β2(1+k1+α2k)-β1(1+k1+α1k)]2+T2k4(α1-α2)2].
Now, if σ0=(0,0), then by choosing σ=(α,β) such that
(3.26)|σ|2=α2+β2⩽1‖ϕ‖C121e2kT(1+1T2k4)ϵ8,
we find that ∥uσ(0)-uσ0(0)∥2=∥ϕσ-ϕ0∥2≤ϵ. This means that the function F is continuous at (0,0).We note that we can easily show that(3.27)‖uσ(0)-u(0)‖2=‖uσ(T)-ϕ‖C12,
for ϕ∈C1(A). And by using Theorem 3.1 again, we see that(3.28)‖uσ(0)-u(0)‖2⟶0as|σ|⟶0.Theorem 3.4.
Ifϕ∈Cθ+1(A), then one has
(3.29)‖uσ(0)-u(0)‖2⩽2(4α2C12(θ)β2θ+C22(θ,T)α2)‖ϕ‖Cθ+12,
for 0<θ<1. And
(3.30)‖uσ(0)-u(0)‖2⩽2(4β2α2+C22(θ,T)α2)‖ϕ‖Cθ+12,forθ⩾1
such that
(3.31)C1(θ)=(1-θ)1-θθθ≤1,C2(θ,T)=1Tθ2.Proof.
Since(3.32)‖uσ(0)-u(0)‖2=‖uσ(T)-ϕ‖C12=∫η+∞Pσ2(λ)e2(1+θ)Tλe-2Tλθd‖Eλϕ‖2,⩽2(Mσ,θ,∞2+Nσ,θ,∞2)‖ϕ‖Cθ+12,
then using Theorem 3.2, we get the required result.Corollary 3.5.
Ifϕ∈Cθ+1(A), then one has
(3.33)sup⁡0⩽t⩽T‖uσ(t)-u(t)‖2⩽‖uσ(0)-u(0)‖2⩽2(4α2C12(θ)β2θ+C22(θ,T)α2)‖ϕ‖Cθ+12,
for 0<θ<1. And
(3.34)sup⁡0⩽t⩽T‖uσ(t)-u(t)‖2⩽‖uσ(0)-u(0)‖2⩽2(4β2α2+C22(θ,T)α2)‖ϕ‖Cθ+12,forθ⩾1.Proof.
Using Theorem3.3, we have
(3.35)sup⁡0⩽t⩽T‖uσ(t)-u(t)‖2⩽‖uσ(0)-u(0)‖2.
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(ϵ). 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].Conclusion 1.
(1.1) Note that, in this work, the error factor ɛ(σ) introduced by small changes in the final value is of order 1/(β(1+(η/(1+αη)))+e-T/α) and in a recent work [21], by the same authors, the error factor given was of order 1/(βη/(1+αη)+e-T/α)≥1/(β(1+(η/(1+αη)))+e-T/α).
(1.2) We also note that the error factor e(β) given in [7] (resp. e(α) in [1]) is of order 1/β (resp., eT/α) and so 1/(βη/(1+αη)+e-T/α)≤1/β and 1/(βη/(1+αη)+e-T/α)≤eT/α.
(1.3) Also the error factor e(σ) given in Boussetila and Rebbani [19] is of order 1/(β+e-T/α) and since 1/(β(1+(η/(1+αη)))+e-T/α)≤1/(β+e-T/α), for 0<α≤1-1/η,η>1 and so for all operators A(A≥η>1), considered above, our method of approximation gives a better approximation than the methods given by Boussetila and Rebbani in [19] and other authors, for example, [1, 7].



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