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

A CLASS OF ABSTRACT PARABOLIC ILL-POSED PROBLEMS

Received 14 October 2004; Accepted 9 August 2005

We study a final value problem for first-order abstract differential equation with posi-tive 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 solu-tions 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.

Copyright © 2006 M. Denche and S. Djezzar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, dis-tribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

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; whereAis a positive self-adjoint operator such that 0∈ρ(A). Such problems are not well posed, that is, even if a unique so-lution 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. Such authors as Lavrentiev [8], Latt`es and Lions [7], Miller [10], Payne [11], and Showal-ter [12] have approximated (FVP) by perturbing the operatorA.

In [1,4,13] a similar problem is treated in a different way. 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)

u(T) +αu(0)=f . (1.4)

Hindawi Publishing Corporation Boundary Value Problems

Volume 2006, Article ID 37524, Pages1–8

A similar approach known as the method of auxiliary boundary conditions was given in [6,9]. Also, we have to mention that the non standard conditions of the form (1.4) for parabolic equations have been considered in some recent papers [2,3].

In this paper, we perturbe the final condition (1.2) to form an approximate nonlocal problem depending on a small parameter, with boundary condition containing a deriva-tive of the same order than the equation, as follows:

u(t) +Au(t)=0, 0< t < T, (1.5)

u(T)−αu(0)=f . (1.6)

Following [4], this method is called quasi-boundary value method, and the related approximate problem is called quasi-boundary value problem (QBVP). We show that the approximate problems are well posed and that their solutionsuαconverge inC1([0,T],H)

if and only if the original problem has a classical solution. We show that this method gives a better approximation than many other quasi reversibility type methods, for example, [1,4,7]. Finally, we obtain several other results, including some explicit convergence rates. The case where the operatorAhas discrete spectrum has been treated in [5].

2. The approximate problem

Definition 2.1. A functionu: [0,T]→His called a classical solution of the (FVP) prob-lem (resp., (QBVP) probprob-lem) ifu∈C1_{([0,T],H),u(t)∈D(A) for everyt∈[0,T] and} satisfies (1.1) and the final condition (1.2) (resp., the boundary condition (1.6)).

Now, let{Eλ}λ>0be a spectral measure associated to the operatorAin the Hilbert space

H, then for all f ∈H, we can write

f = _∞

0 dEλf . (2.1)

If the (FVP) problem (resp., (QBVP) problem) admits a solutionu(resp.,uα), then this

solution can be represented by

u(t)= _∞

0 e

λ(T−t)_dE

λf, (2.2)

respectively,

uα(t)=

_∞

0

e−λt

αλ+e−λTdEλf . (2.3)

Theorem2.2. For all f ∈H, the functionsuαgiven by (2.3) are classical solutions to the

(QBVP) problem and we have the following estimate

_uα(t)≤ T

α1 + ln(T/α)f, ∀t∈[0,T], (2.4)

Proof. If we assume that the functionsuαgiven in (2.3) are defined for allt∈[0,T], then,

it is easy to show thatuα∈C1([0,T],H) and

u_α(t)= _∞

0

−λe−λt

αλ+e−λTdEλf . (2.5)

From

_Auα(t)2 =

∞

0

λ2_e−2λt

αλ+e−λT2d

_Eλf2 ≤ 1

α2 ∞

0 d _Eλf2

= 1

α2f

2_{, (2.6)}

we getuα(t)∈D(A) and souα∈C([0,T],D(A)). This shows that the functionuα is a

classical solution to the (QBVP) problem. Now, using (2.3), we have

_uα(t)2 ≤

_∞

0

1

αλ+e−λT2d

_Eλf2

, (2.7)

if we put

h(λ)=αλ+e−λT−1_{, forλ >0, (2.8)}

then,

sup

λ>0

h(λ)=h

ln(T/α)

T

, (2.9)

and this yields _uα(t)2

≤

_T

α1 + ln(T/α) 2_∞

0 d _Eλf2

=

_T

α1 + ln(T/α) 2

f2_{. (2.10)}

This shows that the integral defininguα(t) exists for allt∈[0,T] and we have the desired

estimate.

Remark 2.3. One advantage of this method of regularization is that the order of the error, introduced by small changes in the final value f, is less than the order given in [4].

Now, we give the following convergence result.

Theorem2.4. For every f ∈H,uα(T)converges to f inH, asαtends to zero.

Proof. Letε >0, chooseη >0 for which _∞

η d

_Eλf2

< ε

2. (2.11)

From (2.3), we have

_uα(T)−f2 ≤α2

_η

0

λ2

αλ+e−λT2d

_Eλf2 +ε

so by choosingαsuch that

α2_{< ε}

2 η

0 λ 2_e2λT_E

λf2

−1

, (2.13)

we obtain the desired result.

Theorem2.5. For every f ∈H, the (FVP) problem has a classical solutionugiven by (2.2), if and only if the sequence(u_α(0))α>0converge inH. Furthermore, we then have thatuα(t)

converges tou(t)inC1_{([0,T],H)asαtends to zero.}

Proof. If we assume that the (FVP) problem has a classical solutionu, then we have _u

α(0)−u(0)

2 =

_∞

0

α2_λ4_e2λT

αλ+e−λT2

_dEλf2

≤α2 η

0 λ

4_e4λT_dE λf2+

∞

η

α2_λ4_e2λT

α2_λ2 dEλf 2

< α2

_η

0 λ

4_e4λT_dE

λf2+ε

2,

(2.14)

so by choosingαsuch thatα2_{< ε(2}η

0λ4e4λTdEλf2)−1, we obtain

_u

α(0)−u(0)

2

< ε, (2.15)

this shows thatu_α(0)−u(0)tends to zero asαtends to zero. Since _u

α(t)−u(t)

2 ≤ ∞ 0 λ 2 1

αλ+e−λT−eλT

2

dEλf2

=u_α(0)−u(0)2,

(2.16)

thenuα(t) converges tou(t) uniformly in [0,T] asαtends to zero.

Since

_{uα(0)−u(0)}2 ≤α2

_η

0 λ

2_e4λT_dE

λf2+ε

2, (2.17)

forηquite large. Then by choosingαsuch thatα2_<(2η

0λ2e4λTdEλf2)−1, we get

_{uα(0)−u(0)}2

< ε. (2.18)

Thusuα(0) converges tou(0), which in turn gives thatuα(t) converges tou(t) uniformly

in [0,T] asαtends to zero. Combining all these convergence results, we conclude that

uα(t) converges tou(t) inC1([0,T],H).

Now, assume that (u_α(0))α>0 converges in H. Since uα is a classical solution to the

(QBVP) problem, then we have _u α(0) 2 = _∞ 0 λ2

αλ+e−λT2d

_Eλf2

and it is easy to show that lim

α↓0u

α(0)

2=

∞

0 λ

2_e2λT_dE

λf2, (2.20)

and so the functionu(t) defined by

u(t)= _∞

0 e

λ(T−t)_dE

λf, (2.21)

is a classical solution to the (FVP) problem. This ends the proof of the theorem. Theorem2.6. If the functionugiven by (2.2) is a classical solution of the (FVP) problem, anduδ

αis a solution of the (QBVP) problem for f = fδ, such thatf −fδ< δ, then we have

_u(0)−uδ

α(0)≤c

1 + lnT

δ

−1

, (2.22)

wherec=T(1 +Au(0)).

Proof. Suppose that the functionugiven by (2.2) is a classical solution to the (FVP) prob-lem, and let’s denote byuδ

αa solution of the (QBVP) problem for f = fδ, such that

_{f −fδ< δ. (2.23)}

Then,uδ

α(t) is given by

uδ_α(t)= _∞

0

e−λt

αλ+e−λTdEλfδ, ∀t∈[0,T]. (2.24)

From (2.2) and (2.24), we have

_u(0)−uδ

α(0)≤Δ1+Δ2, (2.25)

whereΔ1= u(0)−uα(0), andΔ2= uα(0)−uδα(0). Using (2.9), we get

Δ1≤ T

1 + ln(T/α) ∞

0 λ

2_e2λT_dE λf2

1/2 ,

Δ2≤ T

α1 + ln(T/α)f−fδ,

(2.26)

then,

Δ1≤ TAu(0) 1 + ln(T/α),

Δ2≤ Tδ

α1 + ln(T/α).

(2.27)

From (2.27), we obtain _uα(0)−uδ

α(0)

2

≤TAu(0) 1 + ln(T/α)+

Tδ

then, for the choiceα=δ, we get _uα(0)−uδ

α(0)

2 ≤T

1 +Au(0)

1 + ln(T/α) . (2.29)

Remark 2.7. From (2.22), forT > e−1_{we get}

_u(0)−uδ

α(0)≤c

ln1

δ

−1

, (2.30)

Remark 2.8. Under the hypothesis of the above theorem, if we denote byUδ

αthe solution

of the approximate (FVP) problem for f = fδ, using the quasireversibility method [7],

we obtain the following estimate _u(0)−Uδ

α(0)≤c1

ln1

δ

−2/3

. (2.31)

Proof. A proof can be given in a similar way as in [9].

Theorem2.9. If there exists anε∈]0, 2[so that ∞

0 λ

ε_eελT_dE

λf2, (2.32)

converges, thenuα(T)converges to f with orderαεε−2asαtends to zero.

Proof. Letε∈]0, 2[ such that 0∞λεeελTdEλf2 converges, and letβ∈]0, 2[. For a fix λ >0, and if we define a functiongλ(α)=αβ/(αλ+e−λT)2. Then we can show that

gλ(α)≤gλα0

, ∀α >0, (2.33)

whereα0=βe−λT/(2−β)λ. Furthermore, from (2.3), we have _uα(T)−f2

=α2−β

_∞

0 λ 2_g

λ(α)dEλf . (2.34)

Hence from (2.33) and (2.34) we obtain _uα(T)−f2

≤α2−β

_β

2−β

β∞

0 λ

2−β_e(2−β)λT_dE λf

2

. (2.35)

If we chooseβ=(2−ε), we have _uα(T)−f2

≤αε_ε−2

4 _∞

0 λ

ε_eελT_dE λf

2

, (2.36)

hence

_uα(T)−f2

≤cεαεε−2 (2.37)

withcε=4

_∞

Now, we give the following corollary.

Corollary2.10. If there exists anε∈]0, 2[so that _∞

0 λ

(ε+2γ)_e(ε+2)λT_dE

λf2, (2.38)

whereγ=0, 1, converges, thenuαconverges touinC1([0,T],H)with order of convergence αε_ε−2_.

Proof. If we assume that (2.38) is satisfied, then _∞

0 λ

2_e2λT_dE

λf2, (2.39)

converges, and so the functionu(t) given by (2.2) is a classical solution of the (FVP) problem. Letu(αγ),u(γ)denote the derivatives of orderγ(γ=0, 1) of the functionsuαand u, respectively. Using the following inequalities

u(αγ)(0)−u(γ)(0)

2 =

_∞

0

α2_λ(2+2γ)_e2λT

αλ+e−λT2d

_Eλf2

≤α2−β β

2−β

β∞

0 λ

(2+2γ−β)_e(4−β)λT_dE λf2,

(2.40)

and settingβ=2−ε, in (2.40), we obtain

u(αγ)(0)−u(γ)(0)

2

≤cε,γαεε−2, (2.41)

wherecε,γ=4

_∞

0 λ(ε+2γ)e(ε+2)λTdEλf2.

And since

u(αγ)(t)−u(γ)(t)

2

≤u(αγ)(0)−u(γ)(0)

2

, (2.42)

thenu(αγ)(t) converges tou(γ)(t) uniformly in [0,T], with order of convergenceαεε−2, and

souαconverges touinC1([0,T],H), with orderαεε−2.

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M. Denche: Laboratoire Equations Differentielles, D´epartement de Math´ematiques, Facult´e des Sciences, Universit´e Mentouri Constantine, 25000 Constantine, Algeria E-mail address:[email protected]