On a backward Cauchy problem associated with continuous spectrum operator

Nonlinear Analysis 73 (2010) 1966–1972

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On a backward Cauchy problem associated with continuous spectrum operator Nguyen Huy Tuan a,∗ , Dang Duc Trong b , Pham Hoang Quan a a

Department of Mathematics, Saigon University, 273 An Duong Vuong, Ho Chi Minh City, Viet Nam

b

Department of Mathematics, University of Natural Sciences, Vietnam National University, 227 Nguyen Van Cu, Q.5, Ho Chi Minh City, Viet Nam

article

abstract

info

Article history: Received 12 March 2009 Accepted 12 May 2010

The nonlinear backward Cauchy problem ut + Au(t ) = f (u(t )), u(T ) = ϕ, where A is a positive self-adjoint unbounded operator, which has a continuous spectrum and f is a Lipschitz function being given is regularized by the well-posed problem. The new error estimates of the regularized solution are obtained. This work extends to the nonlinear case earlier results by the authors [7,1] and by Denche and Bessila [8,13]. © 2010 Elsevier Ltd. All rights reserved.

MSC: 35K05 35K99 47J06 47H10 Keywords: Nonlinear parabolic problem Backward problem Semigroup of operator Ill-posed problem Contraction principle

1. Introduction Recently, in 2008, in [1] we have considered the nonlinear parabolic backward in time problem ut + Au = f (u(t )),

0 < t < T,

u(T ) = ϕ,

(1) (2)

where A is a self-adjoint operator, having a discrete spectrum on a Hilbert space H, with the inner product h, i and the norm k, k such that −A generates a compact contraction semi-group on H. This problem is well known to be severely ill-posed and regularization methods for it are required. We have established, under the hypothesis that f is a global Lipschitzian function from H to H, the existence of a unique solution for the approximated problem ut + f (A)u = f (u (t )),

0 < t < T,



u (T ) = ϕ,

(3) (4)

with 0 <  < 1 and f (A) being chosen later under some better conditions. We assume that (1)–(2) has a unique smooth solution u(t ). Under this assumption, we obtain an error estimate on the smoothing of u, generalizing a previous work of in [2]. This error is given a form in Holder type

ku(t ) − u (t )k ≤ M β()t /T . ∗

Corresponding author. E-mail address: [email protected] (N.H. Tuan).

0362-546X/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2010.05.025

N.H. Tuan et al. / Nonlinear Analysis 73 (2010) 1966–1972

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We note that stability estimates of the Holder type for the nonlinear heat-parabolic equation backwards in time have been established in some recent papers, such as [3–7,1]. These results give no information on the continuous dependence of the solution on the data at t = 0. And so, the convergence of the approximate solution is very slow when t is in the neighborhood of zero. To our knowledge, the case where the operator A has a discrete spectrum has been treated in many recent papers, such as [8,9]. Once more, we also note that there are many works on the linear homogeneous case of the backward Cauchy problem, but the literature on the nonlinear case of the problem is quite scarce. So, it is not easy to regularize the nonlinear problems. Recently, the nonhomogeneous and nonlinear backward problem in Banach space has been considered by Hetrict and Hughes [10,11]. As we know, the position operator usually has a continuous spectrum, much like the momentum operator in an infinite space. But the momentum in a compact space, the angular momentum, and the Hamiltonian of various physical systems, especially bound states, tend to have a discrete (quantized) spectrum—that is where the name quantum mechanics comes from. The formal scattering theory has a strong overlap with the theory of continuous spectra. In this paper, we shall use new methods to extend the continuous dependence results of [12,13] to more general nonlinear problems. We also improve some related results given in [8,13,7,1] with two objectives. First, the present work is a first step in the nonlinear backward Cauchy problem, in which the operator A has a continuous spectrum. Thus, for some related questions on homogeneous parabolic equations backwards in time, as in the case A where has a continuous spectrum, we refer the reader to [12,13]. As far as we know, there are few nonlinear backward Cauchy problems until now. Second, we give some new error estimates, which are not of the Holder type. The major object of this paper is to provide a quite simple and convenient new regularization method. Meanwhile, some more faster convergence error estimates are given. Especially, the convergence of the approximate solution at t = 0 is also proved. This paper is organized as follows. In the next section, for ease of the reading, we summarize some well-known facts in semigroup of operator. The stability estimates of the regularized solution will be presented in Section 2. 2. The basic results In this section we present the notation and the functional setting which will be used in this paper and prepare some material which will be used in our analysis. We denote by {Eλ , λ ≥ 0} theRspectral resolution of the identity associated to A. ∞ We denote by S (t ) = e−tA = 0 e−t λ dEλ ∈ L(H ), t ≥ 0, the C0 -semigroup generated by −A. Some basic definitions are listed in the following theorem. See Chap. 2, Theorem 6.13, in [14]. For this family of operators we have: 1. 2. 3. 4. 5.

kS (t )k ≤ 1, for all t ≥ 0; the function t 7−→ S (t ), t > 0, is analytic; for every real r ≥ 0 and t > 0, the operator S (t ) ∈ L(H , D (Ar )); for every integer k ≥ 0 and t > 0, kS (k) (t )k = kAk S (t )k ≤ c (k)t −k ; for every x ∈ D (Ar ), r ≥ 0 we have S (t )Ar x = Ar S (t )x.

Theorem 1. Let A : D(A) ⊂ H → H be a self-adjoint operator on the Hilbert space X over K . Then there exists exactly one spectral family {Eλ } such that +∞

Z

λdEλ u

Au =

(5)

0

for all u ∈ D(A). In this connection, u ∈ D(A) iff the integral (5) exists, i.e., +∞

Z

λ2 dkEλ uk2 < ∞. 0

Definition. Let A : D(A) ⊂ H → H be a self-adjoint operator on the Hilbert space H over K and let f , g : R → K be a piecewise continuous function. We set D(f (A)) =



+∞

Z

|f (λ)|2 dkEλ uk2 < ∞

u∈H :



0

and define the linear operator f (A) : D(A) ⊂ H → H by the formula f (A)u =

+∞

Z

f (λ)dEλ u 0

for all u ∈ D(f (A)).

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N.H. Tuan et al. / Nonlinear Analysis 73 (2010) 1966–1972

3. The main results Now we are ready to state and prove the main results of this paper. For 0 ≤ t ≤ s ≤ T , we denote Rα (λ, t ) = e−λt (αλ + e−λT )−1 . This also means that Rα (λ, T + t − s) = e(s−t −T )λ (αλ + e−λT )−1 . If the problem (1)–(2) admits a solution u then this solution can be represented by u(t ) =

∞

Z

eλ(T −t ) dEλ ϕ −

T

Z

eλ(s−t ) dEλ f (u(s))ds.

0

t

0

∞

Z

Since t < T , we know from (6) that, the terms e−(t −T )λ and e−(t −s)λ are the sources of instability. So, we replace them by approximation terms such as Rα (λ, t ), Rα (λ, T + t − s). Thus, it is easy to see that lim Rα (λ, t ) = e−(t −T )λ

α→0

and lim Rα (λ, T + t − s) = e−(t −s)λ .

α→0

Hence, the ill-posed problem (1)–(2)can be approximated by uα (t ) =

Z

∞

Rα (λ, t )dEλ ϕ −

T

Z

0

∞

Z

t

Rα (λ, T + t − s)dEλ f (uα (s))ds.

(6)

0

Noting that if f = 0, (6) is also the problem (2.2) given in page 2, [13]. Our first main theorem is the following, Theorem 1. Let 0 < α < Te, ϕ ∈ H and let f : H → H be a continuous operator satisfying

kf (w) − f (v)k ≤ kkw − vk, for a k > 0 independent of w, v ∈ H , t ∈ R. Then problem (6) has a unique solution uα ∈ C ([0, T ]; H ). Proof of Theorem 1. First, we consider the following function for λ > 0 Rα (λ, 0) = (αλ + e−λT )−1 . It is easy to prove that for 0 < α < eT then




Rα (λ, 0) ≤ Rα

!

ln αT ,0 T


−1 = T α −1 ln(Teα −1 ) .

(7)

For 0 ≤ t ≤ s ≤ T , then Rα (λ, T + t − s) = exp((s − t − T )λ)(αλ + e−λT )

t −s T

(αλ + e−λT )

≤ exp((s − t − T )λ)(αλ + e−λT )

t −s T

(e−λT )

≤ (Rα (λ, 0)) ≤α

t −s T

s−t −T T

s−t −T T

s−t T


t −T s

T −1 ln(Teα −1 )

= M (α, t )M −1 (α, s)

(8)

where


Tt −1

t

M (α, t ) = α T T −1 ln(Teα −1 )

,

t ∈ [0, T ].

For w ∈ C ([0, T ]; H ), we define the operator F by F (w)(t ) =

∞

Z

Rα (λ, t )dEλ ϕ − 0

∞

Z 0

T

Z

Rα (λ, T + t − s)dEλ f (w(s))ds.

(9)

t

Now we prove that for all w, v ∈ C ([0, T ]; H ) the following inequality holds

kF m (w)(t ) − F m (v)(t )k ≤

(kT α −1 C )m |||w − v|||, m!

(10)

N.H. Tuan et al. / Nonlinear Analysis 73 (2010) 1966–1972

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where C = max{T , 1} and |||.||| is sup norm in C ([0, T ]; H ). In fact, for m = 1, using (8), the Lipschitz property of f and noting that Rα (λ, T + t − s) ≤ α


t −T s

t −s T

T −1 ln(Teα −1 )

<

1

α

,

we have

Z T Z ∞

2

kF (w)(t ) − F (v)(t )k2 = R (λ, T + t − s ) dE ( f (w( s )) − f (v( s ))) ds α λ

t 0 Z T Z ∞Z T dkEλ (f (w(s)) − f (v(s)))k2 ds ≤ (Rα (λ, T + t − s))2 ds ≤

t

t

0



T −t

2 Z

α 

=

T −t

2 Z

≤

∞

Z

dkEλ (f (w(s)) − f (v(s)))k2 ds 0

t

α 

T

T

kf (w(s)) − f (v(s))kds t

(T − t )k α

2

kw(s) − v(s)k2 ds

≤ (kTC )2 α −2 |||w − v||| |2 .

(11)

Suppose that (11) holds for m = j. We prove that (11) holds for m = j + 1. In fact, we have

kF

j +1

(w)(t ) − F

2

Z T Z ∞

j j

Rα (λ, T + t − s)dEλ (f (F w)(s) − f (F v)(s))ds (v)(t )k =

t 0  2 Z T (T − t )k ≤ kF j (w)(s) − F j (v)(s)k2 ds α t (kT α −1 C )2j+2 |||w − v|||2 . ≤ (j + 1)!

j +1

2

Therefore, by the induction principle, we have (10) for all w, v ∈ C ([0, T ]; H ). We consider F : C ([0, T ]; H ) → C ([0, T ]; H ). (kT α −1 C )m

Since limm→∞ = 0, there exists a positive integer number m0 such that F m0 is a contraction. It follows that the m! equation F m0 (w) = w has a unique solution uα ∈ C ([0, T ]; H ). We claim that F (uα ) = uα . In fact, one has F (F m0 (uα )) = F (uα ). Hence F m0 (F (uα )) = F (uα ). By the uniqueness of the fixed point of F m0 , one has F (uα ) = uα , i.e., the equation F (w) = w has a unique solution uα ∈ C ([0, T ]; H ).  Now we have the following theorem in which, we show that the stability magnitude of our method is less than order one in the previous methods. Theorem 2. Let α be as in Theorem 1. Then the (unique) solution of problem (1)–(2) depends continuously (in C ([0, T ]; H )) on ϕ , this means that, if u and v are two solutions of problem (6) corresponding to the final value ϕ and ω respectively then

ku(t ) − v(t )k ≤

√

t


−1 t −T 2 2 2ek (T −t ) α T T −1 ln(Teα −1 ) T kϕ − ωk.

Proof of Theorem 2. It is well known that, for all t ∈ [0, T ], u(t ) − v(t ) =

∞

Z

Rα (λ, t )dEλ (ϕ − ω) − 0

∞

Z 0

Z

T

Rα (λ, T + t − s)dEλ (f (u(s)) − f (v(s)))ds.

t

Using Lemmas 2 and 3 and the Lipchitz property of f we get

ku(t ) − v(t )k2 ≤ 2α −2 M 2 (α, t )

∞

Z

dkEλ (ϕ − ω)k2 0

∞

Z

T

Z

(Rα (λ, T + t − s))2 ds

+2 0

t

T

Z

dkEλ (f (w(s)) − f (v(s)))k2 ds t

≤ 2α −2 M 2 (α, t )kϕ − ωk + 2k2 (T − t )

T

Z

M 2 (α, t )M −2 (α, s)ku(s) − v(s)k2 ds. t

1970

N.H. Tuan et al. / Nonlinear Analysis 73 (2010) 1966–1972

Hence M −2 (α, t )ku(t ) − v(t )k2 ≤ 2α −2 kϕ − ωk + 2k2 (T − t )

T

Z

M −2 (α, s)ku(s) − v(s)k2 ds. t

Applying Gronwall’s inequality, we have 2 2 M −2 (α, t )ku(t ) − v(t )k2 ≤ 2e2k (T −t ) α −2 kϕ − ωk2 .

Hence, we get

ku(t ) − v(t )k ≤

√

t


−1 t −T 2 2 kϕ − ωk. 2ek (T −t ) α T T −1 ln(Teα −1 ) T

This inequality follows the solution of the problem (6) depending continuously on ϕ and Theorem 2 is proved.



Remark 1. In [7] (see p. 238), and in [1], the authors give better stability estimates than the latter discussed methods. They t

show that the stability estimate is of order  T −1 . In [13] (see Theorem 2.2 page 2), the authors give another stability bound T having the form of T . α(1+ln α )

In our paper, we give a better estimation of the stability order, which is


Tt −1

t

C α T −1 T −1 ln(Teα −1 )

.

Comparing our results with the above related results, we see that the order of the error, introduced by small changes in the final value ϕ , is less than the order given in [4,8,13,7,1]. This is among of the best advantages of our method. Theorem 3. Let u ∈ C ([0, T ]; H ) be a solution of (1)–(2). Assume that u has the eigenfunction expansion u(t ) = satisfying ∞

Z

λ2 e2t λ dkEλ u(t )k2 < ∞,

R∞ 0

dEλ u(t )

(12)

0

for every t ∈ (0, T ]. Then, for any α ∈ (0, Te),

ku(t ) − uα (t )k ≤ N exp



k2 T 2



2

t

α T T −1 ln(Teα −1 )


Tt −1

,

∀t ∈ (0, T ],

(13)

where

s Z N = sup

∞

λ2 e2t λ dkEλ u(t )k2 ,

2

t ∈[0,T ]

0

and uα is the unique solution of Problem (6). Proof of Theorem 3. The function u(t ), uα (t ) has the expansion u(t ) =

∞

Z

eλ(T −t ) dEλ ϕ −

0

uα (t ) =

T

Z t

Z

∞

Rα (λ, t )dEλ ϕ −

0

∞

Z 0 T

Z

eλ(s−t ) dEλ f (u(s))ds, ∞

Z

t

Rα (λ, T + t − s)dEλ f (uα (s))ds. 0

Hence, we get u(t ) − uα (t ) =

∞

Z

(eT λ −

0 T

Z

 Z αλRα (λ, t ) eT λ dEλ ϕ −

Z

T

0

∞

Z 0

∞

eλ(s−t −T ) dEλ f (u(s))ds



0

∞

Z

esλ dEλ f (u(s))ds



0

∞

Z

Rα (λ, T + t − s)dEλ (f (uα (s)) − f (u(s))) ds

+ =

T t

t

t

Z

0

∞

=

T

Rα (λ, T + t − s)dEλ (f (uα (s)) − f (u(s))) ds

+ Z

αλ + e

 Z −λt ) e dE ϕ − λ −λT

∞

Z

t

1

0

α Rα (λ, t )λet λ dEλ u(t ) +

T

Z t

Z 0

∞

αλRα (λ, T + t − s)dEλ (f (uα (s)) − f (u(s))) ds

N.H. Tuan et al. / Nonlinear Analysis 73 (2010) 1966–1972

1971

then

ku(t ) − uα (t )k2 ≤ 2M 2 (α, t )

∞

Z

λ2 e2t λ dkEλ u(t )k2

0

+ (T − t )M 2 (α, t )

T

Z

M −2 (α, s)

∞

Z

dkEλ (f (w(s)) − f (v(s)))k2 ds 0

t

≤ 2M 2 (α, t )

∞

Z

λ2 e2t λ dkEλ u(t )k2 + (T − t )M 2 (α, t )

Z

T

M −2 (α, s)k(f (w(s)) − f (v(s)))k2 ds.

t

0

So, we obtain M

−2

(α, t )ku(t ) − uα (t )k ≤ N + k (T − t ) 2

2

T

Z

M −2 (α, s)ku(s) − uα (s)kds. t

Using Gronwall’s inequality, we get 2T 2

ku(t ) − uα (t )k2 ≤ N 2 ek


2t −2 2t α T T −1 ln(Teα −1 ) T .

This completes the proof of Theorem 3.



Remark 2. 1. One superficial advantage of this method is that there is an error estimation in the original time t = 0, which is not given in many recently known results in [7,1]. We have the following estimate

ku(0) − uα (0)k ≤ NT e

kT

 −1

 1 + ln

T

α

.

This error is similar to Theorem 2.6, page 5 in [13]. 2. If f (u) = 0, we have u(0) =

Z

∞

et λ dEλ u(t ).

0

Taking the derivative of u, we get u (0) = − 0

∞

Z

λe−t λ dEλ u(t ).

0

Then, we get

ku0 (0)k2 = kAu(0)k2 =

∞

Z

λ2 e2λt dkEλ u(t )k2 .

0

Hence, the condition (12) is natural and acceptable. 3. In [7] (see p. 241), and in [1] the error estimates between the exact solution and the approximate solution are of t

the same order U (α, t ) = C α T . And in this article, the convergence rate is in a slightly different form than given in [4, t


t −1

V (α,t )

8,13,1], defined by V (α, t ) = Dα T T −1 ln(Teα −1 ) T . We note that limα→0 U (α,t ) = 0. Hence, this error is the optimal error estimate to our knowledge.   Moreover, we also have limα→0 (limt →0 U (α, t )) = C and limα→0 (limt →0 V (α, t )) = limα→0 D T −1 ln(Teα −1 )


−1

= 0. So, it is easy to see that if the time t is near to the original time t = 0, the convergence

of the approximation solution is very slow. This implies that the methods such as Quasi-boundary value and stabilized quasireversibility studied in [7,1], are not useful to derive the error estimations in the case where t is in the neighborhood of zero. This also proves that our method give a better approximation than the previous case which we know. Comparing (13) with the results obtained in [8,13,7,1], we realize that (13) is sharp and the best known estimate. This is a generalization of many previous results. Acknowledgements The authors would like to thank Professor Ravi P. Agarwal for his valuable help in the presentation of this paper. The authors are also grateful to the anonymous referees for their valuable comments leading to the improvement of our paper. References [1] D.D. Trong, N.H. Tuan, Stabilized quasi-reversibility method for a class of nonlinear ill-posed problems, Electron. J. Differential Equations (84) (2008) 1–12. [2] K. Miller, Stabilized quasi-reversibility and other nearly-best-possible methods for non-well posed problems, in: Symposium on Non-Well Posed Problems and Logarithmic Convexity, in: Lecture Notes in Mathematics, vol. 316, Springer-Verlag, Berlin, 1973, pp. 161–176.

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