Contents lists available atScienceDirectNonlinear Analysis journal homepage:www.elsevier.com/locate/na On a backward Cauchy problem associated with continuous spectrum operator aDepartme
Trang 1Contents lists available atScienceDirect
Nonlinear Analysis
journal homepage:www.elsevier.com/locate/na
On a backward Cauchy problem associated with continuous
spectrum operator
aDepartment of Mathematics, Saigon University, 273 An Duong Vuong, Ho Chi Minh City, Viet Nam
bDepartment of Mathematics, University of Natural Sciences, Vietnam National University, 227 Nguyen Van Cu, Q.5, Ho Chi Minh City, Viet Nam
a r t i c l e i n f o
Article history:
Received 12 March 2009
Accepted 12 May 2010
MSC:
35K05
35K99
47J06
47H10
Keywords:
Nonlinear parabolic problem
Backward problem
Semigroup of operator
Ill-posed problem
Contraction principle
a b s t r a c t
The nonlinear backward Cauchy problem
u t+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
1 Introduction
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
u
in [2] This error is given a form in Holder type
ku(t) −u(t)k ≤Mβ()t/T.
∗Corresponding author.
E-mail address:tuanhuy_bs@yahoo.com (N.H Tuan).
0362-546X/$ – see front matter © 2010 Elsevier Ltd All rights reserved.
doi:10.1016/j.na.2010.05.025
Trang 2We note that stability estimates of the Holder type for the nonlinear heat-parabolic equation backwards in time have been
of zero To our knowledge, the case where the operator A has a discrete spectrum has been treated in many recent papers,
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
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
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
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}the spectral resolution of the identity associated to A.
We denote by S(t) =e−tA= R∞
0 e−tλdEλ ∈L(H),t ≥0, the C0-semigroup generated by−A Some basic definitions are
listed in the following theorem
1 kS(t)k ≤1, for all t≥0;
2 the function t7−→S(t),t>0, is analytic;
3 for every real r≥0 and t>0, the operator S(t) ∈L(H,D(A r));
4 for every integer k≥0 and t>0,kS(k)(t)k = kA k S(t)k ≤c(k)t−k;
5 for every x∈D(A r),r≥0 we have S(t)A r x=A r 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
Au=
Z +∞
0
for all u∈D(A).
In this connection, u∈D(A)iff the integral(5)exists, i.e.,
Z +∞
0
λ2dkEλuk2< ∞.
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)) =
u∈H:
Z +∞
0
|f(λ)|2dkEλuk2< ∞
f(A)u=
Z +∞
0
f(λ)dEλu
for all u∈D(f(A))
Trang 33 The main results
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.
u(t) =
Z ∞
0
eλ(T−t)dE
λϕ −
Z T t
Z ∞
0
eλ(s−t)dE
λf(u(s))ds.
Since t < T , we know from(6)that, the terms e− (t−T)λand e− (t−s)λare 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
α→ 0Rα(λ,t) =e−(t−T)λ
and
lim
α→ 0Rα(λ,T +t−s) =e−(t−s)λ.
uα(t) = Z ∞
0
Rα(λ,t)dEλϕ − Z T
t
Z ∞
0
Rα(λ,T+t−s)dEλf(uα(s))ds. (6)
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 ≤kk w − 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 forλ >0
Rα(λ,0) = (αλ +e−λT)− 1.
It is easy to prove that for 0< α <eT then
Rα(λ,0) ≤Rα
ln Tα
T ,0
!
=Tα− 1 ln(Teα− 1)− 1
Rα(λ,T+t−s) =exp((s−t−T)λ)(αλ +e−λT)t−s
T (αλ +e−λT)s−t−T
T
≤exp((s−t−T)λ)(αλ +e−λT)t−s
T (e−λT)s−t−T
T
≤ (Rα(λ,0))s−t
T
≤ αt−s
T T−1ln(Teα− 1)t−T s
where
M(α,t) = αt
T T−1ln(Teα− 1)T t− 1
, t ∈ [0,T]
Forw ∈C([0,T];H), we define the operator F by
F(w)(t) = Z
∞
0
Rα(λ,t)dEλϕ − Z
∞
0
Z T t
Now we prove that for allw, v ∈C([0,T];H)the following inequality holds
kF m(w)(t) −F m(v)(t)k ≤ (kTα−1C)m
Trang 4where 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) ≤ αt−T s T−1ln(Teα− 1)t−T s < α1,
we have
kF(w)(t) −F(v)(t)k2 =
Z T t
Z ∞
0
Rα(λ,T+t−s)dEλ(f(w(s)) −f(v(s)))ds
2
≤
Z ∞
0
Z T t
(Rα(λ,T+t−s))2ds
Z T t
dkEλ(f(w(s)) −f(v(s)))k2ds
≤
T−t
α
2 Z T t
Z ∞
0
dkEλ(f(w(s)) −f(v(s)))k2ds
=
T−t
α
2 Z T t
kf(w(s)) −f(v(s))kds
≤
(T−t)k
α
2
k w(s) − v(s)k2ds
kF j+1(w)(t) −F j+1(v)(t)k2 =
Z T t
Z ∞
0
Rα(λ,T+t−s)dEλ(f(F jw)(s) −f(F jv)(s))ds
2
≤
(T −t)k
α
2 Z T t
kF j(w)(s) −F j(v)(s)k2ds
≤ (kTα− 1C)2j+ 2 (j+1)! ||| w − v|||
2.
Therefore, by the induction principle, we have(10)for allw, v ∈C([0,T];H) We consider F :C([0,T];H) →C([0,T];H) Since limm→∞ (kTα − 1C)m
equation F m0(w) = whas 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) = whas 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 Letαbe 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 andvare two solutions of problem(6)corresponding to the final valueϕandωrespectively then
ku(t) − v(t)k ≤ √2ek2(T−t) 2
αt−T
T−1ln(Teα− 1)T t− 1
k ϕ − ωk.
Proof of Theorem 2 It is well known that, for all t∈ [0,T],
u(t) − v(t) =
Z ∞
0
Rα(λ,t)dEλ(ϕ − ω) −
Z ∞
0
Z T t
Rα(λ,T+t−s)dEλ(f(u(s)) −f(v(s)))ds.
Using Lemmas 2 and 3 and the Lipchitz property of f we get
ku(t) − v(t)k2 ≤2α− 2M2(α,t)
Z ∞
0
dkEλ(ϕ − ω)k2
+2
Z ∞
0
Z T t
(Rα(λ,T+t−s))2ds
Z T t
dkEλ(f(w(s)) −f(v(s)))k2ds
≤2α− 2M2(α,t)kϕ − ωk +2k2(T−t) Z T M2(α,t)M−2(α,s)ku(s) − v(s)k2ds.
Trang 5M−2(α,t)ku(t) − v(t)k2 ≤2α− 2k ϕ − ωk +2k2(T−t)
Z T t
M−2(α,s)ku(s) − v(s)k2ds.
Applying Gronwall’s inequality, we have
M−2(α,t)ku(t) − v(t)k2≤2e2k2(T−t) 2
α− 2k ϕ − ωk2.
Hence, we get
ku(t) − v(t)k ≤ √2ek2(T−t) 2
αt−T
T−1ln(Teα− 1)t T− 1
k ϕ − ωk.
Remark 1 In [7] (see p 238), and in [1], the authors give better stability estimates than the latter discussed methods They show that the stability estimate is of ordert
T− 1 In [13] (see Theorem 2.2 page 2), the authors give another stability bound
α( 1 + lnTα ).
In our paper, we give a better estimation of the stability order, which is
Cαt
T− 1 T−1ln(Teα− 1)T t− 1
.
Comparing our results with the above related results, we see that the order of the error, introduced by small changes in the final valueϕ, 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) = R∞
0 dEλu(t)
satisfying
Z ∞
0
λ2e2tλdkE
for every t∈ (0,T].
Then, for anyα ∈ (0,Te),
ku(t) −uα(t)k ≤N exp
k2T2
2
αt
T T−1ln(Teα− 1)T t− 1
where
N= sup
t∈[ 0 ,T]
s
2
Z ∞
0
λ2e2tλdkEλu(t)k2,
and uαis the unique solution of Problem(6).
Proof of Theorem 3 The function u(t),uα(t)has the expansion
u(t) = Z
∞
0
eλ(T−t)dE
λϕ − Z T
t
Z ∞
0
eλ(s−t)dE
λf(u(s))ds,
uα(t) = Z ∞
0
Rα(λ,t)dEλϕ − Z T
t
Z ∞
0
Rα(λ,T+t−s)dEλf(uα(s))ds.
Hence, we get
u(t) −uα(t) =
Z ∞
0 (eTλ− 1
αλ +e− λT)
e−λt dEλϕ −
Z T t
Z ∞
0
eλ(s−t−T)dE
λf(u(s))ds
+
Z T t
Z ∞
0
Rα(λ,T+t−s)dEλ(f(uα(s)) −f(u(s)))ds
=
Z ∞
0
αλRα(λ,t)
eTλdE
λϕ − Z T
t
Z ∞
0
esλdE
λf(u(s))ds
+
Z T t
Z ∞
0
Rα(λ,T+t−s)dEλ(f(uα(s)) −f(u(s)))ds
=
Z ∞
αRα(λ,t)λetλdE
λu(t) + Z TZ ∞αλRα(λ,T+t−s)dEλ(f(uα(s)) −f(u(s)))ds
Trang 6ku(t) −uα(t)k2 ≤2M2(α,t)
Z ∞
0 λ2e2tλdkE
λu(t)k2 + (T−t)M2(α,t)
Z T t
M−2(α,s)
Z ∞
0
dkEλ(f(w(s)) −f(v(s)))k2ds
≤2M2(α,t) Z
∞
0
λ2e2tλdkE
λu(t)k2+ (T−t)M2(α,t) Z T
t
M−2(α,s)k(f(w(s)) −f(v(s)))k2ds.
So, we obtain
M−2(α,t)ku(t) −uα(t)k2≤N+k2(T −t)
Z T t
M−2(α,s)ku(s) −uα(s)kds.
Using Gronwall’s inequality, we get
ku(t) −uα(t)k2≤N2ek2T2α2t
T T−1ln(Teα− 1)2t T− 2
.
Remark 2 1 One superficial advantage of this method is that there is an error estimation in the original time t=0, which
ku(0) −uα(0)k ≤NT e kT
T
α
− 1 .
This error is similar to Theorem 2.6, page 5 in [13]
2 If f(u) =0, we have
u(0) =
Z ∞
0
etλdE
λu(t).
Taking the derivative of u, we get
u0(0) = −
Z ∞
0 λe−tλdE
λu(t).
Then, we get
ku0(0)k2= kAu(0)k2=
Z ∞
0
λ2e2λtdkEλu(t)k2.
T And in this article, the convergence rate is in a slightly different form than given in [4,
8,13,1], defined by V(α,t) = Dαt
T T− 1ln(Teα− 1)T t− 1
We note that limα→ 0V(α,t)
U(α,t) = 0 Hence, this error is the optimal
limα→ 0
D T− 1ln(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 quasi-reversibility studied in [7,1], are not useful to derive the error estimations in the case where t is in the neighborhood of zero.
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
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