A CLASS OF ABSTRACT PARABOLIC ILL-POSED PROBLEMS

A CLASS OF ABSTRACT PARABOLIC ILL-POSED PROBLEMS

M. DENCHE AND S. DJEZZAR

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 DOI10.1155/BVP/2006/37524

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 solutionsu_αconverge 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) problem) 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 dE_λf . (2.1)

If the (FVP) problem (resp., (QBVP) problem) admits a solutionu(resp.,u_α), then this solution can be represented by

u(t)= _∞

0 e^λ(T−t)dEλf, (2.2)

respectively,

u_α(t)= _∞

0

e^−λt

αλ+e^−λTdE_λf . (2.3)

Theorem 2.2. For all f ∈H, the functionsu_αgiven by (2.3) are classical solutions to the (QBVP) problem and we have the following estimate

u_α(t)≤ T

α1 + ln(T/α)f, ∀t∈[0,T], (2.4) whereα < eT.

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^−λTdE_λf . (2.5)

From

Auα(t)²= _∞

0

λ²e^−2λt

αλ+e^−λT2dEλf²≤ 1 α²

_∞

0 dEλf²= 1

α²f², (2.6) we getu_α(t)∈D(A) and sou_α∈C([0,T],D(A)). This shows that the functionu_α is a classical solution to the (QBVP) problem.

Now, using (2.3), we have u_α(t)²≤

_∞

0

1

αλ+e^−λT2dE_λf², (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 dEλf²=

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.

Theorem 2.4. For every f _∈H,u_α(T) converges tof inH, asαtends to zero.

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

η dEλf²< ε

2. (2.11)

From (2.3), we have

uα(T)−f²≤α² _η

0

λ²

αλ+e^−λT2dEλf²+ε

2, (2.12)

so by choosingαsuch that

α²< ε

2 _η

0 λ²e^2λTEλf² −1

, (2.13)

we obtain the desired result.

Theorem 2.5. For everyf ∈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 thatu_α(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

α²λ⁴e^2λT αλ+e^−λT2

dEλf²

≤α² _η

0 λ⁴e^4λTdEλf²+ _∞

η

α²λ⁴e^2λT

α²λ² dEλf²

< α² _η

0 λ⁴e^4λTdEλf²+ε 2,

(2.14)

so by choosingαsuch thatα²< ε(2₀^ηλ⁴e^4λTdEλf²)⁻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 λ² 1

αλ+e^−λT−e^λT 2

dEλf²

=u_α(0)−u(0)²,

(2.16)

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

Since

uα(0)−u(0)²≤α² _η

0 λ²e^4λTdEλf²+ε

2, (2.17)

forηquite large. Then by choosingαsuch thatα²<(2₀^ηλ²e^4λTdEλf²)⁻¹, we get u_α(0)−u(0)²< ε. (2.18) Thusu_α(0) converges tou(0), which in turn gives thatu_α(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 u_α is a classical solution to the (QBVP) problem, then we have

u_α(0)²= _∞

0

λ²

αλ+e^−λT2dEλf², (2.19)

and it is easy to show that lim

α↓0u_α(0)

2

= _∞

0 λ²e^2λTdEλf², (2.20) and so the functionu(t) defined by

u(t)= _∞

0 e^λ(T−t)dEλf, (2.21)

is a classical solution to the (FVP) problem. This ends the proof of the theorem.

Theorem 2.6. If the functionugiven by (2.2) is a classical solution of the (FVP) problem, andu^δ_αis a solution of the (QBVP) problem for f = fδ, such thatf −fδ< δ, then we have

u(0)−u^δ_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 byu^δ_αa solution of the (QBVP) problem for f = f_δ, such that

f −fδ< δ. (2.23)

Then,u^δ_α(t) is given by u^δ_α(t)=

_∞

0

e^−λt

αλ+e^−λTdEλfδ, ∀t∈[0,T]. (2.24) From (2.2) and (2.24), we have

u(0)−u^δ_α(0)≤Δ1+Δ2, (2.25) whereΔ1= u(0)−u_α(0), andΔ2= u_α(0)−u^δ_α(0). Using (2.9), we get

Δ1≤ T 1 + ln(T/α)

_∞

0 λ²e^2λTdEλf² 1/2

, Δ2≤ T

α1 + ln(T/α)f−f_δ,

(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)−u^δ_α(0)²≤ TAu(0)

1 + ln(T/α)+ Tδ

α1 + ln(T/α), (2.28)

then, for the choiceα=δ, we get

u_α(0)−u^δ_α(0)²≤T1 +Au(0)

1 + ln(T/α) . (2.29)

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

u(0)−u^δ_α(0)≤c

ln1 δ

₋1

, (2.30)

Remark 2.8. Under the hypothesis of the above theorem, if we denote byU_α^δthe solution of the approximate (FVP) problem for f = fδ, using the quasireversibility method [7], we obtain the following estimate

u(0)−U_α^δ(0)≤c1

ln1

δ −2/3

. (2.31)

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

Theorem 2.9. If there exists anε_∈]0, 2[ so that _∞

0 λ^εe^ελTdEλf², (2.32)

converges, thenuα(T) converges to f with orderα^εε⁻²asαtends to zero.

Proof. Letε∈]0, 2[ such that ₀^∞λ^εe^ελTdEλf² converges, and letβ∈]0, 2[. For a fix λ >0, and if we define a functiong_λ(α)=α^β/(αλ+e^−λT)². Then we can show that

gλ(α)≤gλ α0

, ∀α >0, (2.33)

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

_∞

0 λ²g_λ(α)dE_λf . (2.34) Hence from (2.33) and (2.34) we obtain

u_α(T)−f²≤α^2−β β

2−β β_∞

0 λ^2−βe^(2−β)λTdE_λf². (2.35) If we chooseβ=(2−ε), we have

u_α(T)−f²≤α^εε⁻²

4 _∞

0 λ^εe^ελTdE_λf²

, (2.36)

hence

uα(T)−f²≤cεα^εε⁻² (2.37)

withcε=4₀^∞λ^εe^ελTdEλf².

Now, we give the following corollary.

Corollary 2.10. If there exists anε∈]0, 2[ so that _∞

0 λ^(ε+2γ)e^(ε+2)λTdEλf², (2.38)

whereγ=0, 1, converges, thenu_αconverges touinC¹([0,T],H) with order of convergence α^εε⁻².

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

0 λ²e^2λTdEλf², (2.39)

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

u^(γ)α (0)−u^(γ)(0)²= _∞

0

α²λ^(2+2γ)e^2λT

αλ+e^−λT2dEλf²

≤α^2−β β

2−β β_∞

0 λ^(2+2γ−β)e^(4−β)λTdEλf²,

(2.40)

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

u^(γ)α (0)−u^(γ)(0)²≤c_ε,γα^εε⁻², (2.41)

wherecε,γ=4₀^∞λ^(ε+2γ)e^(ε+2)λTdEλf². And since

u^(γ)α (t)−u^(γ)(t)²≤u^(γ)α (0)−u^(γ)(0)², (2.42)

thenu^(γ)α (t) converges tou^(γ)(t) uniformly in [0,T], with order of convergenceα^εε⁻², and souαconverges touinC¹([0,T],H), with orderα^εε⁻². References

[1] M. Ababna, Regularization by nonlocal conditions of the problem of the control of the initial condi- tion for evolution operator-differential equations, Vestnik Belorusskogo Gosudarstvennogo Uni- versiteta. Seriya 1. Fizika, Matematika, Informatika (1998), no. 2, 60–63, 81 (Russian).

[2] K. A. Ames and L. E. Payne, Asymptotic behavior for two regularizations of the Cauchy problem for the backward heat equation, Mathematical Models & Methods in Applied Sciences 8 (1998), no. 1, 187–202.

[3] K. A. Ames, L. E. Payne, and P. W. Schaefer, Energy and pointwise bounds in some non-standard parabolic problems, Proceedings of the Royal Society of Edinburgh. Section A. Mathematics 134 (2004), no. 1, 1–9.

[4] G. W. Clark and S. F. Oppenheimer, Quasireversibility methods for non-well-posed problems, Elec- tronic Journal of Differential Equations 1994 (1994), no. 8, 1–9.

[5] M. Denche and K. Bessila, A modified quasi-boundary value method for ill-posed problems, Jour- nal of Mathematical Analysis and Applications 301 (2005), no. 2, 419–426.

[6] V. K. Ivanov, I. V. Mel’nikova, and A. I. Filinkov, Operator-Differential Equations and Ill-Posed Problems, Fizmatlit “Nauka”, Moscow, 1995.

[7] R. Latt`es and J.-L. Lions, M´ethode de Quasi-R´eversibilit´e et Applications, Travaux et Recherches Math´ematiques, no. 15, Dunod, Paris, 1967.

[8] M. M. Lavrentiev, Some Improperly Posed Problems of Mathematical Physics, Springer Tracts in Natural Philosophy, vol. 11, Springer, Berlin, 1967.

[9] I. V. Mel’nikova, Regularization of ill-posed differential problems, Sibirski˘ı Matematicheski˘ı Zhur- nal 33 (1992), no. 2, 125–134, 221 (Russian), translated in Siberian Math. J. 33 (1992), no. 2, 289–298.

[10] K. Miller, Stabilized quasi-reversibility and other nearly-best-possible methods for non-well-posed problems, Symposium on Non-Well-Posed Problems and Logarithmic Convexity (Heriot-Watt Univ., Edinburgh, 1972), Lecture Notes in Mathematics, vol. 316, Springer, Berlin, 1973, pp.

161–176.

[11] L. E. Payne, Some general remarks on improperly posed problems for partial differential equations, Symposium on Non-Well-Posed Problems and Logarithmic Convexity (Heriot-Watt Univ., Ed- inburgh, 1972), Lecture Notes in Mathematics, vol. 316, Springer, Berlin, 1973, pp. 1–30.

[12] R. E. Showalter, The final value problem for evolution equations, Journal of Mathematical Analysis and Applications 47 (1974), no. 3, 563–572.

[13] , Cauchy problem for hyperparabolic partial differential equations, Trends in the Theory and Practice of Nonlinear Analysis (Arlington, Tex, 1984), North-Holland Math. Stud., vol. 110, North-Holland, Amsterdam, 1985, pp. 421–425.

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]

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