The problem is called the backward heat problem BHP, the backward Cauchy problem or the final value problem.. In the present paper, a classical Fourier method is used for solving the non
Trang 1Contents lists available atScienceDirect Nonlinear Analysis
journal homepage:www.elsevier.com/locate/na
Sharp estimates for approximations to a nonlinear backward
heat equation
aDepartment of Mathematics, SaiGon University, 273 AnDuongVuong, HoChiMinh City, Viet Nam
bDepartment of Mathematics, University of Natural Science, Vietnam National University, 227 Nguyen Van Cu, Q.5, HoChiMinh City, Viet Nam
a r t i c l e i n f o
Article history:
Received 28 May 2009
Accepted 1 June 2010
MSC:
35K05
35K99
47J06
47H10
Keywords:
Backward heat problem
Ill-posed problem
Nonlinear heat
Truncation method
Error estimate
a b s t r a c t
A nonlinear backward heat problem for an infinite strip is considered The problem is ill-posed in the sense that the solution (if it exists) does not depend continuously on the data
In this paper, we use the Fourier regularization method to solve the problem Some sharp estimates of the error between the exact solution and its regularization approximation are given
© 2010 Elsevier Ltd All rights reserved
1 Introduction
Let T be a positive number We consider the problem of finding the temperature u(x,t), (x,t) ∈R× [0,T], such that
u t−u xx=f(x,t,u(x,t)), (x,t) ∈R× (0,T),
whereϕ(x),f(x,t,z)are given The problem is called the backward heat problem (BHP), the backward Cauchy problem or the final value problem
It is known in general that the backward problem is ill-posed, i.e., a solution does not always exist, and in the case of exis-tence, it does not depend continuously on the given datum In fact, from a small noise contaminated physical measurement, the corresponding solutions may have a large error This makes the numerical computation difficult Hence, a regularization
is in order
The special case where the function f is independent of u, namely f(x,t,u) =0 or f(x,t,u) =f(x,t), has been studied
by many authors in recent years As a few examples, we mention Lattes and Lions [1], Showalter [2], Ames and Payne [3] who approximated the BHP by a quasi-reversibility method; Tautenhahn and Schroter [4] who established an optimal error estimate for a BHP; Seidman [5] who established an optimal filtering method; and Hao [6] who studied a modification method We also refer the reader to various other works of Fu et al [7–10], Campbell et al [11], Lien et al [12], Murniz
et al [13], Dokuchaev et al [14], Gilliam et al [15] and Engl et al [16]
∗Corresponding author.
E-mail address:tuanhuy_bs@yahoo.com (N.H Tuan).
0362-546X/$ – see front matter © 2010 Elsevier Ltd All rights reserved.
Trang 2Although there have been many works on the homogeneous case and the linear inhomogeneous cases, and the backward heat problem(2), literature on the nonlinear case of the backward heat problem is quite scarce In 2005, Quan and Dung [17] offered a regularized solution by a semi-group method However, they were able to give error estimates only for the very
special case where the exact solution has a finite Fourier series expansion and the Lipschitz constant k>0 is small enough
In 2006, Trong and Quan [18] used the integral transform method to treat the nonlinear case and attained an error estimate
of ordert
T for each t >0 This estimate is good at any fixed t>0 but useless at t=0 Very recently, Trong and Tuan [19] improved this method to give an error estimate of ordert
T(ln(1/))t
T− 1for all t∈ [0,T]
In the present paper, a classical Fourier method is used for solving the nonlinear backward heat problem, which will improve the results of [18,19] in two ways (although our approach is different from that of [18,19]):
(1) We shall give a criterion for choosing the regularization parameter with a rigorous mathematical proof, which itself possesses an independent significance
(2) We shall provide some sharp error estimates Under some suitable conditions on the exact solution u, we shall introduce
the error estimate of order t
2T+1 This is a significant improvement in comparison with the results of [7,6,18,20,4,12,10] Some comments on the usefulness of this method are given in some remarks
This paper is organized as follows: In Section2, we give some auxiliary results In Section3, we give the regularization solution by using the truncated method and give stable estimates of the error between the regularization solution and the exact solution
2 Some auxiliary results
Letgˆ (ξ)denote the Fourier transform of function g∈L2(R)defined formally as
ˆ
g(ξ) = √1
2π
Z +∞
−∞
Let H1=W1 , 2(R), H2=W2 , 2(R)be the Sobolev spaces which are defined by
H1(R) = {g ∈L2(R), ξ ˆg(ξ) ∈L2(R)},
H2(R) = {g ∈L2(R), ξ2gˆ (ξ) ∈L2(R)}.
We denote byk k, k.kH1, k.kH2the norms in L2(R),H1(R),H2(R)respectively, namely
kgk2
H1 = kgk2+ kg xk2= k (1+ ξ2)1
ˆ
g(ξ)k2,
kgk2
H2 = kgk2+ kg xk2+ kg xxk2= k (1+ ξ2+ ξ4)1
ˆ
g(ξ)k2.
Let us first make clear what a weak solution to the problem(1)is
Lemma 1 Let f :R× [0,T] ×R→R be a function such that
for all(x,t) ∈ R× [0,T]and for some constant K > 0 independent of x,t,u, v Let us haveϕ ∈ L2(R) Assume that
u∈C([0,T] ,H2(R)) ∩C1([0,T] ,L2(R))is a solution of the equation
ˆ
u(ξ,t) =e(T−t)ξ 2
ˆ ϕ(ξ) − Z T
t
e−(t−s)ξ 2
ˆ
Then u t,u xx,f(x,t,u) ∈ C([0,T] ,L2(R))and u is a solution to the heat equation(1) where the main equation holds in
C([0,T] ,L2(R)).
Proof By letting t=T in the equation
ˆ
u(ξ,t) =e(T−t)ξ 2
ˆ ϕ(ξ) −
Z T
t
e−(t−s)ξ 2
ˆ
f(ξ,s,u)ds, 0≤t ≤T (5)
we have immediatelyuˆ (ξ,T) = ˆϕ(ξ) Therefore, we get u(x,T) = ϕ(x)in L2(R)
Multiplying the above equation with etξ 2
we obtain
etξ 2
ˆ
u(ξ,t) =eTξ 2
ˆ ϕ(ξ) − Z T
t
esξ 2
ˆ
f(ξ,s,u)ds, t∈ [0,T]
Differentiating the latter equation w.r.t the time variable t we get
etξ 2
ξ2uˆ (ξ,t) + d
dt uˆ (ξ,t)
=etξ 2
ˆ
f(ξ,t,u),
Trang 3
ξ2uˆ (ξ,t) + d
dt uˆ (ξ,t)
= ˆf(ξ,t,u), t ∈ [0,T]
Since u∈C([0,T] ,H2(R))∩C1([0,T] ,L2(R))we haveξ2uˆ (ξ,t) = ˆu xx(ξ)anddtduˆ (ξ,t)belongs to C([0,T] ,L2(R)) Therefore,
ˆ
f(ξ,t,u)also belongs to C([0,T] ,L2(R))which is equivalent to f(x,t,u) belonging to C([0,T] ,L2(R)) Thus the latter equation means
u t−u xx=f(x,t,u)
in the sense of C([0,T] ,L2(R))
3 Regularization and error estimates
3.1 The approximation problem and the main results
Since t < T , we know from(4)that, whenξ becomes large, the term exp{ (T −t)ξ2}increases rather quickly Thus foruˆ (ξ,t) ∈L2(R)with respect toξ, the exact dataϕ(ξ) ˆ must decay rapidly as| ξ| → ∞ Small errors in high-frequency components can blow up and completely destroy the solution for 0≤t <T Define the Fourier regularization solution of
problem(1)as follows:
u(x,t) = √1
2π
Z + A
− A
e(T−t)ξ 2
ˆ ϕ(ξ)eiξxdξ − √1
2π
Z + A
− A
Z T
t
e−(t−s)ξ 2ˆf(ξ,s,u)eiξxdξds, (6)
where Ais a positive constant which will be selected appropriately as a regularization parameter such that lim→ 0A= ∞
We now study the properties of(6)considered as an approximation to(1), i.e., we will give some stability estimates
Theorem 1 Let f be the function defined by(3) Letϕ ∈L2(R) Then the problem(6)has a unique solution u ∈C([0,T];L2(R))∩
C1((0,T);L2(R)).Furthermore, this approximate solution depends continuously on the final valueϕ, i.e., letw, vbe the solutions
of the problem(6)corresponding to the final valuesϕandφ; then
k w(.,t) − v(.,t)k ≤e(T−t)AeK2(T−t) 2
k ϕ − φk.
Remark 1 (1) If A = 1, then the stability of the regularized solution is of order e
C1
It is of the same order as the results
in [21]
(2) If A = 1
Tln 1
, then the stability of the regularized solution is of ordert
T− 1 It is of the same order as the results
in [22,23]
(3) If A = 1
Tln
T
1 + ln
T
,then the stability of the regularized solution is of order T
1 + ln
T
This is better than the results in [22,23]
Theorem 2 Let f, ϕ,u be as in Theorem 1 Suppose thatϕ ∈L2(R)and letϕ∈L2(R)be measured data such thatk ϕ− ϕk ≤ .
Suppose problem(4)has a unique solution u ∈ C([0,T];L2(R))and let w ∈ C([0,T];L2(R))be the unique solution of problem(6)corresponding toϕ.
(a)If u is such that
Z +∞
−∞
e2tξ 2
| ˆu(ξ,t)|2dξ < ∞,
for all t∈ [0,T]then
k w(.,t) −u(.,t)k ≤e−tA
eK2(T−t) 2
where
N(,t) =
s
3
K e 3K2T(T−t)
Z T
0
M(,t)dt+M(,t)
(8)
and
M(,t) =3
Z − A
−∞
e2tξ 2
| ˆu(ξ,t)|2dξ + Z√∞
A
e2tξ 2
| ˆu(ξ,t)|2dξ
!
Trang 4(b)If there exists a positive number s>0 such that
Z +∞
−∞
(ξ2se2tξ 2
| ˆu(ξ,t)) |2dξ < ∞
then
k w(.,t) −u(.,t)k ≤e−tA eK2(T−t) 2
eTA +e3KT(T−t)/ 2
s
6Q
(A)s
!
(10)
where
Q = sup
0 ≤t≤T
Z +∞
−∞
(ξ2se2tξ 2
| ˆu(ξ,t)) |2dξ
(c)If there exists a positive number p>0 such that
R p= sup
0 ≤t≤T
Z +∞
−∞
e2(t+p)ξ 2
| ˆu(ξ,t)|2dξ
< ∞
then
k w(.,t) −u(.,t)k ≤e−tA
eK2(T−t) 2
Corollary 1 Let us select A = r Tln 1
(0<r<1) Then the following estimates hold true:
k w(.,t) −u(.,t)k ≤ rt
T
eK2(T−t) 2
and
k w(.,t) −u(.,t)k ≤ rt
T eK2(T−t) 2
1 −r+e3KT(T−t)/ 2
T r
s/ 2s
6Q
ln 1
s
!
Corollary 2 Let us select A = T+1pln 1
Then we get
where
H p=eK2(T−t) 2
+e3KT(T−t)/ 2p
6R p.
In particular, if we choose p=T then the error is of order t
2T+1
.
Remark 2 1 In [20], Tautenhahn gave the error estimate
ku(.,t) −u(.,t)k ≤2E1−T tt
T
and he also proved that this is the order optimal stability estimate in L2(R) Similarly, the convergence estimates in [18,23]
were also of the form Ct
T , which does not give any useful information on the continuous dependence of the solution at t=0
Actually, when t →0+, the accuracy of the regularized solution becomes progressively lower At t =0, it merely implies
that the error is bounded by C , i.e., the convergence of the regularization solution at t=0 is not obtained theoretically This
is common in the theory of ill-posed problems if we do not have additional conditions on the smoothness of the solution
To retain the continuous dependence of the solution at t=0, one has to introduce a stronger a priori assumption on exact solution
2 InTheorem 2, we give a new error estimation in the original time t = 0, which does not appear in [18,23] Notice that
the term N(,t)converges to 0 whentends to zero Thus, the right hand side of(13)converges to 0 However, the term
N(,0)is not often computed in practice This is a weak point of the error(13)
3 Fu et al [7] (see Remark 3.6, p 570) gave the error estimation of order 1
( ln ( 1 /))8
+max{1,T} β1
E whereβ = T
ln 1
ln1
− 8 ! Hao et al [6] gave the error of ln E
−2s
Trong and Tuan [19] obtained an error estimate of order T
1 + ln
T
Comparing the results inCorollary 2with the results in [7,6,19] we know that t
2T+1
is the optimal-order error estimate
Trang 53.2 Proof of the main result
Proof of Theorem 1 We divide the proof ofTheorem 1into two steps
Step 1 The existence and uniqueness of a solution of problem(6)
Forw ∈C([0,T];L2(R)), define
G(w)(x,t) = √1
2π ψ(x,t) −
1
√
2π
Z + A
− A
Z T
t
e−(t−s)ξ 2fˆ (ξ,s, w)eiξxdξds
and
ψ(x,t) =
Z + A
− A
e(T−t)ξ 2
ˆ ϕ(ξ)eiξxdξ.
Since we have the Lipschitzian property of f(x,t, w)with respect tow, we get G(w) ∈ C([0,T];L2(R))for everyw ∈
C([0,T];L2(R)) We claim that, for everyw, v ∈C([0,T];L2(R)),m≥1, we have
|||G m(w) −G m(v)||| ≤ K e TAm T
m/ 2
√
where C=max{T,1}and||| |||is the sup norm in C([0,T];L2(R)) We shall prove the latter inequality by induction
If m=1, we have
kG(w)(.,t) −G(v)(.,t)k2 = k ˆG(w)(.,t) − ˆG(v)(.,t)k2
=
Z + A
− A
Z T
t
e(s−t)ξ 2
ˆ
f(ξ,s, w) − ˆf(ξ,s, v) ds
2
dξ
≤
Z + A
− A
Z T
t
e2(s−t)ξ 2
ds
Z T
t
ˆ
f(ξ,s, w) − ˆf(ξ,s, v)
2
dsdξ
≤
Z + A
− A
Z T
t
e2TA ds
Z T
t
ˆ
f(ξ,s, w) − ˆf(ξ,s, v)
2
dsdξ
≤ e2TA(T−t)
Z + A
− A
Z T
t
ˆ
f(ξ,s, w) − ˆf(ξ,s, v)
2
dsdξ
= e2TA(T−t)
Z T
t
kf(.,s, w(.,s)) −f(.,s, v(.,s))k2ds
= K2e2TA(T−t)
Z T
t
k w(.,s) − v(.,s)k2ds
≤ CK2e2TA(T −t)|||w − v|||2.
Therefore(16)holds Suppose that(16)holds for m=p We prove that(16)holds for m=p+1 We have
kG p+1(w)(.,t) −G p+1(v)(.,t)k2 = k ˆG(G p(w))(.,t) − ˆG(G p(v))(.,t)k2
=
Z + A
− A
Z T
t
e(s−t)ξ 2
ˆ
f(ξ,s,G p(w)) − ˆf(ξ,s,G p(v)) ds
2
dξ
≤
Z + A
− A
Z T
t
e2(s−t)ξ 2
ds
Z T
t
ˆ
f(ξ,s,G p(w)) − ˆf(ξ,s,G p(v))
2
dsdξ.
≤ e2TA(T−t)
Z T
t
kf(.,s,G p(w)(.,s)) −f(.,s,G p(v)(.,s))k2ds
≤ K2e2TA(T−t) Z T
t
kG p(w)(.,s) −G p(v)(.,s)k2ds
≤ K2e2TA(T−t) K e TA2p
Z T
t
(T−s)p
p! dsC p||| w − v|||2
≤ K e TA2(p+ 1 )(T−t)(p+ 1 )C(p+ 1 )
(p+1)! ||| w − v|||
2.
Trang 6Therefore, by the induction principle, it holds for every m that
|||G m(w) −G m(v)||| ≤ K e TAm T m/ 2
√
m!
C m||| w − v|||
for everyw, v ∈C([0,T];L2(R)) Consider G:C([0,T];L2(R)) →C([0,T];L2(R))
Since
lim
m→∞ K e TAm T
m/ 2C m
√
m!
=0,
there exists a positive integer number m0such that G m0is a contraction It follows that G m0(w) = whas a unique solution
u ∈C([0,T];L2(R))
We claim that G(u) =u In fact, one has G(G m0(u)) = G(u) Hence G m0(G(u)) = G(u) By the uniqueness of the
fixed point of G m0, one has G(u) =u, i.e., the equation G(w) = whas a unique solution u ∈C([0,T];L2(R))
Step 2 Letwandvbe two solutions of the problem(6)corresponding to the final valuesϕandφrespectively We have
k w(.,t) − v(.,t)k2 ≤2
Z + A
− A
e(T−t)ξ 2
ˆ ϕ(ξ) − ˆφ(ξ)
2
dξ
+2
Z + A
− A
Z T
t
e(s−t)ξ 2
ˆ
f(ξ,s, w) − ˆf(ξ,s, v)
ds
2
dξ
=A1+A2.
The term A1can be estimated as follows:
A1 =2
Z + A
− A
e(T−t)ξ 2
ˆ ϕ(ξ) − ˆφ(ξ)
2
dξ
≤2e(T−t)A
Z +∞
−∞
ϕ(ξ) − ˆφ(ξ) ˆ
2
dξ
≤2e(T−t)Ak ˆ ϕ − ˆφk2
=2e(T−t)Ak ϕ − φk2.
We get
A2 =2
Z + A
− A
Z T
t
e(s−t)ξ 2
ˆ
f(ξ,s, w) − ˆf(ξ,s, v) ds
2
dξ
≤2(T−t)e−2tA
Z +∞
−∞
Z T
t
e2sA
ˆ
f(ξ,s, w) − ˆf(ξ,s, v)
2
dsdξ
≤2(T−t)e−2tA K2
Z T
t
e2sAk w(.,s) − v(.,s)k2ds.
It follows that
k w(.,t) − v(.,t)k2≤2e(T−t)Ak ϕ − φk2+2(T−t)e−2tA K2
Z T
t
e2sAk w(.,s) − v(.,s)k2ds.
Hence
e2tAk w(.,t) − v(.,t)k2≤e2TAk ϕ − φk2+2K2(T−t)
Z T
t
e2sAk w(.,s) − v(.,s)k2ds.
Using Gronwall’s inequality, we obtain
k w(.,t) − v(.,t)k ≤e(T−t)AeK2(T−t) 2
k ϕ − φk.
Proof of Theorem 2 Proof 2a We recall the formula of the solutions:
u(x,t) = √1
2π
Z +∞
−∞
e(T−t)ξ 2
ˆ ϕ(ξ)eiξxdξ − √1
2π
Z +∞
−∞
Z T
t
e−(t−s)ξ 2ˆf(ξ,s,u)eiξxdξds, (17) and
Trang 7u(x,t) = √1
2π
Z + A
− A
e(T−t)ξ 2
ˆ ϕ(ξ)eiξxdξ − √1
2π
Z + A
− A
Z T
t
e−(t−s)ξ 2
ˆ
f(ξ,s,u)eiξxdξds.
Hence,
u(x,t) −u(x,t) = √1
2π
Z − A
−∞
e(T−t)ξ 2
ˆ ϕ(ξ)eiξxdξ + √1
2π
Z +∞
+ A
e(T−t)ξ 2
ˆ ϕ(ξ)eiξxdξ
√
2π
Z − A
−∞
Z T
t
e−(t−s)ξ 2
ˆ
f(ξ,s,u)eiξxdξds
√
2π
Z +∞
√
A
Z T
t
e−(t−s)ξ 2
ˆ
f(ξ,s,u)eiξxdξds
√
2π
Z + A
− A
Z T
t
e−(t−s)ξ 2
ˆ
f(ξ,s,u) − ˆf(ξ,s,u) eiξxdξds
where
I1= 1
√
2π
Z − A
−∞
e(T−t)ξ 2
ˆ ϕ(ξ)eiξxdξ − √1
2π
Z − A
−∞
Z T
t
e−(t−s)ξ 2
ˆ
f(ξ,s,u)eiξxdξds,
I2= 1
√
2π
Z +∞
+ A
e(T−t)ξ 2
ˆ ϕ(ξ)eiξxdξ − √1
2π
Z +∞
√
A
Z T
t
e−(t−s)ξ 2
ˆ
f(ξ,s,u)eiξxdξds,
I3= − 1
√
2π
Z + A
− A
Z T
t
e−(t−s)ξ 2
ˆ
f(ξ,s,u) − ˆf(ξ,s,u) eiξxdξds.
Since(19), we get
etξ 2
ˆ
u(ξ,t) =eTξ 2
ˆ ϕ(ξ) −
Z T
t
esξ 2
ˆ
f(ξ,s,u)ds.
Using the Parseval equality, we estimate the term I1as follows:
kI1k2 =
Z − A
−∞
e(T−t)ξ 2
ˆ ϕ(ξ) − Z T
t
e−(t−s)ξ 2
ˆ
f(ξ,s,u)ds
2
dξ
=
Z − A
−∞
e−2tξ 2
eTξ 2
ˆ ϕ(ξ) − Z T
t
esξ 2ˆf(ξ,s,u)ds
2
dξ
≤ e−2tA
Z − A
−∞
eTξ 2
ˆ ϕ(ξ) −
Z T
t
esξ 2
ˆ
f(ξ,s,u)ds
2
dξ
≤ e−2tA
Z − A
−∞
e2tξ 2
By using an argument similar to those given before, the term I2can be estimated as follows:
kI2k2≤e−2tA
Z ∞
√
A
e2tξ 2
Using the Lipschitzian property of f , we have, after some rearrangements,
kI3k2 =
Z + A
− A
Z T
t
e−(t−s)ξ 2
ˆ
f(ξ,s,u) − ˆf(ξ,s,u) ds
2
dξ
≤
Z + A
− A
(T−t) Z T
t
e−2(t−s)A|ˆf(ξ,s,u) − ˆf(ξ,s,u)|2dsdξ
≤ T
Z T
t
e−2(t−s)Akf(ξ,s,u) −f(ξ,s,u)k2
ds
≤ KT
Z T
Trang 8From(18)–(21), we get
ku(.,t) −u(.,t)k2 ≤3(kI1k2+ kI2k2+ kI3k2)
≤3e−2tA
Z − A
−∞
e2tξ 2
| ˆu(ξ,t)|2dξ + Z
∞
√
A
e2tξ 2
| ˆu(ξ,t)|2dξ
!
+3KT
Z T
t
e−2(t−s)Aku(.,s) −u(.,s)k2ds.
It follows that
e2tAku(.,t) −u(.,t)k2 ≤3
Z − A
−∞
e2tξ 2
| ˆu(ξ,t)|2dξ + Z√∞
A
e2tξ 2
| ˆu(ξ,t)|2dξ
!
+3KT
Z T
t
e2sAku(.,s) −u(.,s)k2ds
≤M(,t) +3KT
Z T
t
where we recall that
M(,t) =3
Z − A
−∞
e2tξ 2
| ˆu(ξ,t)|2dξ +
Z ∞
√
A
e2tξ 2
| ˆu(ξ,t)|2dξ
!
Using the Gronwall’s inequality, we get
ku(.,t) −u(.,t)k2 ≤3
K e 3k2T(T−t)Z T
0
M(,t)dt+M(,t)
e−2tA
where
N(,t) =
s
3
K e 3k2T(T−t)Z T
0
M(,t)dt+M(,t)
Letw be the approximated solution of problem(6)corresponding to the final valueϕ By step 2 ofTheorem 1and(24),
we obtain
k w(.,t) −u(.,t)k ≤ kw(.,t) −u(.,t)k + ku(.,t) −u(.,t)k (25)
≤e(T−t)AeK2(T−t) 2
k ϕ − ϕk +N(,t)e−tA
≤e−tA
eK2(T−t) 2
Proof 2b Since(23), we get
M(,t) =3
Z − A
−∞
ξ−2sξ2se2tξ 2
| ˆu(ξ,t)|2dξ + Z
∞
√
A
ξ−2sξ2se2tξ 2
| ˆu(ξ,t)|2dξ
!
(A)s
Z − A
−∞
ξ2se2tξ 2
| ˆu(ξ,t)|2dξ + Z
∞
√
A
ξ2se2tξ 2
| ˆu(ξ,t)|2dξ
!
(A)s
Z +∞
−∞ (ξ2se2tξ 2
| ˆu(ξ,t)) |2dξ
≤ 6Q
(A)s.
Using again(22), we have
e2tAku(.,t) −u(.,t)k2 ≤ 6Q
(A)s+3KT
Z T
Trang 9Applying Gronwall’s inequality, we obtain
ku(.,t) −u(.,t)k2≤e−2tAe3KT(T−t) 6Q
(A)s.
Using(25), we prove(10)
Proof 2c Since(23), we estimate M(,t)as follows:
M(,t) =3
Z − A
−∞
e−2pξ 2
e2(t+p)ξ 2
| ˆu(ξ,t)|2dξ +
Z ∞
√
A
e−2pξ 2
e2(t+p)ξ 2
| ˆu(ξ,t)|2dξ
!
≤ 3e−2pA
Z − A
−∞
e2(t+p)ξ 2
| ˆu(ξ,t)|2dξ +
Z ∞
√
A
e2(t+p)ξ 2
| ˆu(ξ,t)|2dξ
!
≤ 6e−2pA
Z +∞
−∞
e2(t+p)ξ 2
| ˆu(ξ,t)|2dξ
≤ 6e−2pA R p.
Using(22), we have
e2tAku(.,t) −u(.,t)k2 ≤6e−2pA R p+3KT
Z T
t
Applying Gronwall’s inequality, we get
ku(.,t) −u(.,t)k ≤e−tAe−pAe3KT(T−t/ 2 )p
6R p.
By using an argument similar to that for(25), we obtain
k w(.,t) −u(.,t)k ≤ kw(.,t) −u(.,t)k + ku(.,t) −u(.,t)k
≤e(T−t)AeK2(T−t) 2
k ϕ − ϕk +e−tAe−pAe3KT(T−t/ 2 )p
6R p
≤e−tA
eK2(T−t) 2
eTA +e−pAe3KT(T−t/ 2 )p
Acknowledgements
The authors would like to thank the anonymous referees for their valuable comments leading to the improvement of our manuscript The first author would like to thank Phan Thanh Nam for fruitful discussions on backward heat problems
References
[1] R Lattés, J.-L Lions, Méthode de Quasi-réversibilité et Applications, Dunod, Paris, 1967.
[2] R.E Showalter, The final value problem for evolution equations, J Math Anal Appl 47 (1974) 563–572.
[3] K.A Ames, L.E Payne, Asymptotic for two regularizations of the Cauchy problem for the backward heat equation, Math Models Methods Appl Sci (1998) 187–202.
[4] U Tautenhahn, T Schröter, On optimal regularization methods for the backward heat equation, Z Anal Anwend 15 (1996) 475–493.
[5] T.I Seidman, Optimal filtering for the backward heat equation, SIAM J Numer Anal 33 (1996) 162–170.
[6] D.N Hao, N.V Duc, Stability results for the heat equation backward in time, J Math Anal Appl 353 (2) (2009) 627–641.
[7] C.-L Fu, Z Qian, R Shi, A modified method for a backward heat conduction problem, Appl Math Comput 185 (2007) 564–573.
[8] C.-L Fu, X.-T Xiong, Z Qian, Fourier regularization for a backward heat equation, J Math Anal Appl 331 (1) (2007) 472–480.
[9] W Cheng, C.-L Fu, A spectral method for an axisymmetric backward heat equation, Inverse Probl Sci Eng 17 (2009) 1085–1093.
[10] X.-T Xiong, C.L Fu, Z Qian, X Gao, Error estimates of a difference approximation method for a backward heat conduction problem, Internat J Math Math Sci (2006) 1–9.
[11] B.M Campbell H, R Hughes, E McNabb, Regularization of the backward heat equation via heatlets, Electron J Differential Equations (130) (2008) 1–8.
[12] D.D Trong, T.N Lien, Regularization of a discrete backward problem using coefficients of truncated Lagrange polynomials, Electron J Differential Equations 51 (2007) 1–14.
[13] B.W Muniz, A comparison of some inverse methods for estimating the initial condition of the heat equation, J Comput Appl Math 103 (1999) 145–163.
[14] N Dokuchaev, Regularity for some backward heat equations, J Phys A: Math Theor 43 (2010) 085201.
[15] D Gilliam, J Lund, C.R Vogel, Quantifying information content for ill-posed problems, Inverse Problems 6 (1990) 205–217.
[16] H.W Engl, M Hanke, A Neubauer, Regularization of Inverse Problems, in: Mathematics and its Applications, 2000.
[17] P.H Quan, N Dung, A backward nonlinear heat equation: regularization with error estimates, Appl Anal 84 (4) (2005) 343–355.
[18] P.H Quan, D.D Trong, A nonlinearly backward heat problem: uniqueness, regularization and error estimate, Appl Anal 85 (6–7) (2006) 641–657 [19] D.D Trong, N.H Tuan, Regularization and error estimate for the nonlinear backward heat problem using a method of integral equation, Nonlinear Anal 71 (2009) 4167–4176.
[20] U Tautenhahn, Optimality for ill-posed problems under general source conditions, Numer Funct Anal Optim 19 (1998) 377–398.
Trang 10[21] D.D Trong, N.H Tuan, Regularization and error estimates for nonhomogeneous backward heat problems, Electron J Differential Equations 4 (2006) 1–10.
[22] G.W Clark, S.F Oppenheimer, Quasireversibility methods for non-well posed problems, Electron J Differential Equations 1994 (8) (1994) 1–9 [23] D.D Trong, P.H Quan, T.V Khanh, N.H Tuan, A nonlinear case of the 1-D backward heat problem: regularization and error estimate, Z Anal Anwend.
26 (2) (2007) 231–245.
... method for a backward heat conduction problem, Internat J Math Math Sci (2006) 1–9.[11] B.M Campbell H, R Hughes, E McNabb, Regularization of the backward heat equation via heatlets,... Mathematics and its Applications, 2000.
[17] P.H Quan, N Dung, A backward nonlinear heat equation: regularization with error estimates, Appl Anal 84 (4) (2005) 343–355.... Math Anal Appl 47 (1974) 563–572.
[3] K .A Ames, L.E Payne, Asymptotic for two regularizations of the Cauchy problem for the backward heat equation, Math Models Methods Appl