tóm tắt luận án tiến sĩ tiếng anh phương trình parabolic ngược thời gian

MINISTRY OF EDUCATION AND TRAINING
VINH UNIVERSITY

—————— ——————–
—————– ——————

NGUYEN VAN DUC
PARABOLIC EQUATIONS
BACKWARD IN TIME
Subject: Mathematical Analysis
Code: 62 46 01 01
PhD THESIS SUMMARY
VINH - 2011
The thesis is completed at Vinh University
Supervisors: 1. Prof. Dr. Sc. Dinh Nho H`ao
2. Assoc. Prof. Dr. Dinh Huy Hoang
Referee 1: Prof. Dr. Nguyen Huu Du
Hanoi University of Science, VNU
Referee 2: Assoc. Prof. Dr. Ha Tien Ngoan
Institute of Mathematics - Vietnam Academy of Science and Technology
Referee 3: Assoc. Prof. Dr. Nguyen Xuan Thao
Hanoi University of Science and Technology
The thesis will be defended at the exam Committee at
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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
time. . . . . . date . . . . . . month . . . . . . 2011
The thesis is available at:
- National Library of Vietnam
- Library of Vinh University
INTRODUCTION
Parabolic equations backward in time appear frequently in the heat transfer

theory, geophysics, groundwater problems, materials science, hydrodynamics, im-
age processing This is the problem, when the initial condition is not known
and we must determine it from the final condition. These problems have been
intensively studied, but only for some special classes. Moreover, finding efficient
numerical methods for them is always desired.
Parabolic equations backward in time are ill-posed in sense Hadamard. A
problem is called well-posed if it fulfills the following properties:
a) For all admissible data, a solution exists.
b) If a solution exists, it is unique.
c) The solution depends continuously on the data.
If one of the above properties is not satisfied, then the problem is called ill-
posed. Hadamard supposed that ill-posed problems have no physical meaning.
However, many practical problems of science and technology have led to ill-posed
problems. Therefore, since 1950 many papers concerned ill-posed problems have
been published. Mathematicians such as A. N. Tikhonov, M. M. Lavrent’ev, F.
John, C. Pucci, V. K. Ivanov are pioneers in this field.
In 1955, John reported some results about a method to numerically solve the
Cauchy problem for the heat equation backward in time. Then, Krein and cowork-
ers also published some results on stability estimates and backward uniqueness for
parabolic equations backward in time. In 1963, Tikhonov proposed a regular-
ization method which is applicable for almost all inverse and ill-posed problems.
Especially, this method was applied successfully to the backward heat equation in
1974 by Franklin. Addition to these, many authors also use another methods such
as: QR method, SQR method, the backward beam equation method, the method
1
2
of non-local boundary value problems, iterative methods, finite difference meth-
ods, mollification method for parabolic equations backward in time. However,
no method is universal for all problems. For example, Tikhonov method or QR
method require to solve a equation of double of that of the original equation and

choosing regularization parameters is not easy. Further, it is very difficult to use
Tikhonov method in Banach spaces.
Until now, hundreds of papers devoted to parabolic equations backward in time
there have been published which focused mainly on
1) backward uniqueness,
2) stability estimates,
3) regularization methods, stable and efficient numerical methods.
In this thesis, we focus on obtaining stability estimates and regularization meth-
ods for parabolic equations backward in time. We regularize the problem

u
t
+ Au = 0, 0 < t < T,
u(T ) − f  ε
(0.1)
by the well-posed non-local boundary value problem

v
αt
+ Av
α
= 0, 0 < t < aT,
αv
α
(0) + v
α
(aT ) = f
with a  1 being given and α > 0, the regularization parameter. We suggest
a priori and a posteriori parameter choice rules in order to yield order-optimal
regularization metho ds. Furthermore, the method were tested on the computer

and the results are very encouraging.
To our knowledge, Vabishchevich is one of the first using this method for the
parabolic equations backward in time in 1981. He proposed an a priori method
for (0.1) but without giving the convergence rates as we do in this thesis. Further,
he suggested the following a posteriori metho d for (0.1).
Solve the well-posed problem

u
αt
+ Au
α
= 0, 0 < t < T,
αu
α
(0) + u
α
(T ) = f, α > 0,
and choose α such that
u
α
(T ) − f = ε.
3
However, we could not prove that such an approach yields an order-optimal
method. Therefore, in this thesis we propose to use the following method.
Let a > 1 be a fixed number. Consider the well-posed problem

v
αt
+ Av
α

= 0, 0 < t < aT,
αv
α
(0) + v
α
(aT ) = f, α > 0,
(0.2)
and take v
α
((a − 1)T + t) as an approximation to u(t). We suggest a priori and
a posteriori strategies for choosing the parameter α and prove that these yield
order-optimal regularization methods for (0.1). The a priori method is given in
Theorems 1.2.1, 1.2.3, 1.2.5, 1.3.1, the a posteriori method is as follows. Suppose
ε < f and let τ > 1 satisfy τε < f . Choose α > 0 such that
v
α
(aT ) − f = τε.
We note that, since f −v
α
(aT ) = αv
α
(0), the above discrepancy principle has the
very simple form: Choose α > 0 such that
αv
α
(0) = τε.
Showalter, Clark and Oppenheimer and Mel’nikova also regularized the problem
(0.1) by the non-local boundary value problem

u

t
+ Au = 0, 0 < t < T,
αu(0) + u(T) = f, α > 0.
(0.3)
Our results in case a = 1 are better than theirs. We note that Denche and Bessila
approximated the problem (0.1) by the problem

u
t
+ Au = 0, 0 < t < T,
−αu
t
(0) + u(T ) = f, α > 0.
(0.4)
They obtained an error estimate at t = 0 of logarithmic type with a strong con-
dition that Au(0) is bounded. It means u(0) has to be in the domain of A that
is not frequently met in practice. We will show that we do not need to require u
t
exist at t = 0 as these authors required but again by the problem (1.6) we can
establish stability estimates which are comparable to theirs.
The problem becomes much more difficult if the operator A depends on time
and there are very few results in this case. In this thesis, we improve the related
4
results by Krein, Agmon and Nirenberg. Furthermore, we also suggest a regu-
larization method. Our regularization method with a priori and a posteriori pa-
rameter choice yields error estimates of H¨older type. This is the only result when
a regularization metho d for backward parabolic equations with time-dependent
coefficients provides a convergence rate.
In the last part of the thesis, we use the mollification method to regularize for
the heat equation backward in time in Banach space L

p
(R) 1 < p < ∞ . Namely,
we study the following problem: Let p ∈ (1, ∞), ϕ ∈ L
p
(R) and ε, E be given
constants such that 0 < ε < E < ∞. Consider the heat equation backward in
time

u
t
= u
xx
, x ∈ R, t ∈ (0, T),
u(·, T ) − ϕ(·)
L
p
(R)
 ε,
(0.5)
subject to the constraint
u(·, 0)
L
p
(R)
 E. (0.6)
We note that the case p = 2 is much more difficult, since we do not have the
Parseval equality and in general the Fourier transform of a function in L
p
(R) with
p > 2 is a distribution.

This problem has been considered by the first author. He gave a stability
estimate of H¨older type for the case p ∈ (1, ∞]: if u
1
and u
2
are two solutions of
the problem, there is a constant c
∗
such that
u
1
(·, t)−u
2
(·, t)
L
p
(R)
≤ 4
√
3((c
∗
E)
1−t/T
ε
t/T
+(c
∗
E)
1−t/(4T )
ε

t/(4T )
), ∀t ∈ [0, T ].
One of the aims of the thesis is to improve this estimate for p ∈ (1, ∞). Namely,
for p ∈ (1, ∞), we show that there is a constant c > 0 such that
u
1
(·, t) −u
2
(·, t)
L
p
(R)
 cε
t/T
E
1−t/T
, ∀t ∈ [0, T ].
The heat equation backward in time is well-known to be ill-posed: a small
perturbation in the Cauchy data may cause a very large error in solution. To
overcome this difficulty, Dinh Nho H`ao proposed a mollification method for solving
the problem in a stable way and proved stability estimates of H¨older type for the
solutions. In this thesis we shall follow this technique to regularize the problem
5
(0.5)–(0.6). However, instead of using the de la Vall´ee Poussin kernel for mollifying
the Cauchy data ϕ, we use the Dirichlet kernel and thus work with mollified data
generated by the convolution of this kernel with ϕ. The mollified data belong to
the space of band-limited functions, in which the Cauchy problem is well-posed,
and with appropriate choices of mollification parameter we obtain error estimates
of H¨older type. Stability estimates for the solutions of the problem (0.5)–(0.6) is
the direct consequence of these error estimates and the triangle inequality.

In this thesis, supplementally to the result of Dinh Nho H`ao for p = 2, we
establish stability estimates of H¨older type for all derivatives with respect to x
and t of the solutions. It is worth to note that such estimates are very seldom in
the literature of ill-posed problems.
It is well known that with only the condition (0.6), we cannot expect any con-
tinuous dependence of the solution at t = 0. This can be recovered if an additional
condition on the smoothness of u(x, 0) is available (see Theorem 3.2.7). To this
purpose, in the literature the regularization parameters are chosen dependently
on the parameters of this ”source condition” which are in general not known. To
overcome this shortcoming, in Theorems 3.2.6 and 3.2.10 we propose a choice of
mollification parameters using only the condition (0.6) which guarantees error es-
timates of H¨older type in (0, T ] and a continuous dependence at t = 0 when a
source condition is available but without knowing its parameters. This choice of
mollification parameters seems to be quite interesting for the numerical treatment
of the problem (0.5)–(0.6).
For p = 2, since the Fourier transform of mollified data has compact support,
one has at least two equivalent forms of the mollification method: one in its original
form, another uses the frequency cut-off technique. These two forms lead to two
different numerical schemes which can be easily implemented numerically using
the fast Fourier transform technique (FFT). For p = 2, these schemes do not work
and we propose a stable marching difference scheme for (0.5). We test the methods
for different numerical examples and see that they are very stable and fast.
The thesis consists of an introduction, three chapters, conclusion and references.
Chapter 1 presents results on the regularization of parabolic equations backward
6
in time with time-independent coefficients in Hilbert Spaces. Theorems 1.2.1, 1.2.3
and 1.2.5 provide the results on an a priori parameter choice rule in case a = 1.
Theorems 1.3.1, 1.3.3 provide the results on a priori and a posteriori parameter
choice rules in case a > 1. At the end of Chapter 1, numerical results are presented
and discussed to confirm the theory.

Chapter 2 presents results on stability estimates and regularization of parabolic
equations backward in time with time-dependent coefficients in Hilbert spaces.
Chapter 3 presents stability results for the heat equation backward in time in
Hilbert and Banach spaces, namely, in L
p
(R) for p ∈ (1, +∞). Theorems 3.2.1,
3.2.3 present stability and regularization results in Banach spaces with the same
convergence rate as in case L
2
(R). A slightly modifying the choice of ν in Theorem
3.2.1 gives a stability estimate of H¨older type for t ∈ (0, T ] which guarantees a
continuous dependence of logarithmic type at t = 0 without explicitly knowing
˜
E and γ is the main result of Theorem 3.2.6. In case p = 2, Theorem 3.2.7
provides error estimates of H¨older type for all derivatives with respect to x and t
of the solutions.Theorem 3.2.10 shows that a slightly modifying the choice of ν in
Theorem 3.2.7 guarantees a continuous dependence of logarithmic type at t = 0
without explicitly knowing
˜
E and γ. At the end of Chapter 3, a stable marching
difference scheme and numerical examples are presented and discussed.
CHAPTER 1
PARABOLIC EQUATIONS BACKWARD IN TIME WITH
TIME-INDEPENDENT COEFFICIENTS
Consider the ill-posed parabolic equation backward in time

u
t
+ Au = 0, 0 < t < T,
u(T ) − f  ε

(1.1)
with the positive self-adjoint unbounded operator A that admits an orthonormal
eigenbasis {φ
i
}
i1
in Hilbert space H with norm  · , associated with the eigen-
values {λ
i
}
i1
such that 0 < λ
1
 λ
2
 . . . , and lim
i→+∞
λ
i
= +∞. In order to
regularize the problem, we suppose that there is a positive constant E > ε > 0
such that
u(0)  E. (1.2)
In this chapter, we regularize the problem (1.1), (1.2) by the well-posed non-local
boundary value problem

v
αt
+ Av
α

= 0, 0 < t < aT,
αv
α
(0) + v
α
(aT ) = f
(1.3)
with a  1 being given and α > 0, the regularization parameter. A priori and a
posteriori parameter choice rules are suggested which yield order-optimal regular-
ization methods. The results of this chapter are published in Journal of Mathe-
matical Analysis and Applications and IMA Journal of Applied Mathematics.
1.1 Some concepts and basic lemmas
Definition 1.1.1. Let H be a Hilbert space with the inner product ·, · and the
norm ·, a and T are positive number. The space C([0, aT ]; H) consist of all con-
tinuous functions u : [0, aT ] → H with the norm u
C([0,aT ];H)
= max
0taT
u(t) <
∞.
7
8
The space C
1
((0, aT ); H) consist of all continuously differentiable functions
u : (0, aT) → H. D(A) ⊂ H is domain of the operator A : D(A) ⊂ H → H.
Definition 1.1.2. A function v
α
: [0, aT ] → H is called a solution of (2.19) if
v

α
∈ C
1
((0, aT ), H) ∩C([0, aT ], H), v
α
(t) ∈ D(A), ∀t ∈ (0, aT ), and satisfies the
equation v
αt
+ Av
α
= 0 in the interval (0, aT ) and the boundary value condition
αv
α
(0) + v
α
(aT ) = f.
Definition 1.1.3. The function H(η) is defined by
H(η) =

η
η
(1 −η)
1−η
, η ∈ (0, 1),
1, η = 0 and 1.
(1.4)
It is clear that H(η) ≤ 1.
The function C(x, y) with 1 > x  0, y > 0 is defined by
C(x, y) =


y
1 −x

y
e
1−x−y
. (1.5)
1.2 Regularization of parabolic equations backward in time
by a non-local boundary value problem in case a=1
In this section, we regularize the problem (1.1), (1.2) by the non-local boundary
value problem

v
αt
+ Av
α
= 0, 0 < t < aT,
αv
α
(0) + v
α
(aT ) = f
(1.6)
We denote the solution of (1.1) by u(t), and the solution of (1.6) by v(t).
Theorem 1.2.1. The following inequality holds
v(t) −u(t)  Q(t, α)

α
t/T −1
ε + α

t/T
E

, ∀t ∈ [0, T ]. (1.7)
If we choose
α
=
ε
E
, then

v
(
t
)
−
u
(
t
)


2
Q

t,
ε
E

ε

t/T
E
1−t/T
,
∀
t
∈
[0
, T
]
.
Here, Q(t, α) = min{H(t, α), K(t)}, t ∈ [0, T ],
H(t, α) :=

(t/T )
t/T
(1 −t/T)
1−t/T
√
2α + 1
−t/T
∈ (0, 1), ∀t ∈ (0, T ), ∀α > 0,
H(0) = 1, H(T ) = 1/
√
2α + 1,
K(t) := (t/T )
t/T
(1 −t/T)
1−t/T
∈ (0, 1), ∀t ∈ (0, T ),

K(0) = K(T ) = 1.
9
Theorem 1.2.1 does not give any information about the continuous dependence
of the solution of (1.1),(1.2) at t = 0 on the data, as the condition (1.2) is too
weak. To establish this, we suppose that
∞

n=1
λ
2β
n
u(0), φ
n

2
 E
2
1
, (1.8)
or
∞

n=1
e
2γλ
n
u(0), φ
n

2

 E
2
2
, (1.9)
for some positive constants β, γ, E
1
and E
2
.
Theorem 1.2.3. Suppose that instead of (1.2), we have (1.8). Then for all t ∈
[0, T )
u(t) −v(t) 


























Q(t, α)α
t
T
−1
ε + α
t
T

T
ln

(
T λ
1
e
β(t)
)
β(t)
/α


β
C(t)

t
T
−1
E
1
if 0 < α < (
T λ
1
β(t)
)
β(t)
,
Q(t, α)α
t
T
−1
ε + α

e
λ
1
T
λ
β(t)
1
C(t)

1−
t
T

E
1
if α  (
T λ
1
β(t)
)
β(t)
,
where β(t) =
βT
T − t
, ∀t ∈ [0, T ) , C(t) = 1 if 0 < β(t) < 1 and C(t) = 2
1−β(t)
if
β(t)  1.
If we choose α = α
0
:=
ε
1−δ
E
1
with 0 < δ < 1, then for all t ∈ [0, T )
u(t)−v(t) 


























ε
t
T
E
1−
t
T
1

Q(t, α

0
)ε
δ(1−
t
T
)
+ ε
−δ
t
T

T
ln

(
T λ
1
e
β(t)
)
β(t)
E
1
/ε
1−δ


β
C(t)
t

T
−1

,
if 0 < α
0
< (
T λ
1
β(t)
)
β(t)
,
ε
t
T
E
1−
t
T
1

Q(t, α
0
)ε
δ(1−
t
T
)
+ ε

1−δ−
t
T

e
λ
1
T
λ
β(t)
1
C(t)

1−
t
T
E
t
T
−1
1

,
if α
0
 (
T λ
1
β(t)
)

β(t)
.
10
Remark 1.2.4. From final estimate of Theorem 1.2.3, at t = 0 we have an error
estimate of logarithmic type. Particularly, for β = 1, our error estimate at t = 0
is comparable to that of Denche and Bessila.
Theorem 1.2.5. Suppose that instead of (1.2), we have (1.9). Then for all t ∈
[0, T )
u(t) −v(t) 

Q(t, α)α
t/T −1
ε + α
(t+β)/T
E
2
, if 0 < β < T − t,
Q(t, α)α
t/T −1
ε + αE
2
, if β  T − t.
If we choose α = α
1
:=
ε
1−δ
E
2
for 0 < δ < 1, then for t ∈ [0, T )

u(t) −v(t) 











ε
t/T
E
1−t/T
2

Q(t, α
1
)ε
δ(1−t/T )
+ ε
(β(1−δ)−δt)/T
E
t/T −1
2

,
if 0 < β < T − t,

ε
t/T
E
1−t/T
2

Q(t, α
1
)ε
δ(1−t/T )
+ ε
1−t/T −δ
E
t/T −1
2

,
if β  T − t.
1.3 Regularization of parabolic equation backward in time
by a non-local boundary value problem in case a > 1
We denote the solution of (1.1) by u(t), and the solution of (1.3) by v
α
(t).
1.3.1 A priori parameter choice rule
Theorem 1.3.1. (i) If u(t) satisfies (1.2), then with α =

ε
E

a

we have
u(t) −v
α
((a −1)T + t) 

H

1
a

+ 1

ε
t/T
E
1−t/T
< 2ε
t/T
E
1−t/T
, ∀t ∈ [0, T].
(ii) If u(t) satisfies (1.8), then with α =

ε
E
1

1
T
ln

E
1
ε

β

a
we have, for all
t ∈ [0, T], as ε → 0
+
,
u(t) −v
α
((a −1)T + t)  C
1
(t, T, a, β)ε
t/T
E
1−t/T
1

1
T
ln
E
1
ε

−β(T −t)/T
(1 + o(1)),

11
where
C
1
(t, T, a, β) =

1 + C

1
a
, β

t/T

H

1
a

+ C (0, β)

1−t/T
.
(iii) If u(t) satisfies (1.9), then with α =

ε
E
2

aT

T +γ
we have, for all t ∈ [0, T ],
u(t) −v
α
((a −1)T + t) 













2ε
t+γ
T +γ
E
1−
t+γ
T +γ
2
, if 0 < γ  (a −1)T,

ε
aT

T +γ
E
1−
aT
T +γ
2

t/T

2ε
γ
T +γ
E
T
T +γ
2

1−t/T
(1 + o(1)), as ε → 0
+
,
if (a −1)T < γ  aT,
ε
aT
T +γ
E
1−
aT
T +γ
2

(1 + o(1)), as ε → 0
+
, if γ > aT.
1.3.2 A posteriori parameter choice rule
Theorem 1.3.3. Suppose that ε < f. Choose τ > 1 such that 0 < τε < f.
Then there exists a unique number α
ε
> 0 such that
v
α
ε
(aT ) − f = τε. (1.10)
Further,
(i) if u(t) is a solution of the problem (1.1) and (1.2), then
u(t) −v
α
ε
((a −1)T + t)  (τ + 1)
t/T

2τ − 1
τ − 1

1−t/T
ε
t/T
E
1−t/T
, ∀t ∈ [0, T ],
(1.11)

(ii) if u(t) is a solution of the problem (1.1) and (1.8), then as ε → 0
+
,
u(t)− v
α
ε
((a− 1) T +t)  C
2
(τ, t, T, a, β)ε
t/T
E
1−t/T
1

1
T
ln
E
1
ε

−β(T −t)/T
(1+o(1)),
(1.12)
where
C
2
(τ, t, T, a, β) = (τ + 1)
t/T
a

β(1−t/T )

1
τ − 1
C

1
a
, β

H

1
a

+ C (0, β)

1−
t
T
.
12
(iii) if u(t) is a solution of the problem (1.1) and (1.9), then, for t ∈ [0, T ],
u(t) −v
α
ε
((a −1)T + t) 
















(τ + 1)
t/T

(τ + 1)
γ/(T +γ)
+

1
τ − 1

T/(T +γ)

1−t/T
ε
t+γ
T +γ
E
1−

t+γ
T +γ
2
,
if 0 < γ  (a −1)T,
(τ + 1)
t/T

(τ + 1)
γ/(T +γ)
ε
(γ−(a−1)T )
a(T +γ)
E
−
(γ−(a−1)T )
a(T +γ)
2
+

1
τ − 1

1/a

1−t/T
×
×ε
1−
1

a
(1−
t
T
)
E
1
a
(1−
t
T
)
2
, if γ > (a −1)T.
Remark 1.3.5. (a) In the first and second cases, our method is order-optimal.
(b) For the third case, our method is of optimal order for γ ∈ (0, (a − 1)T].
1.4 Numerical examples
We tested on the computer for the a posteriori parameter choice rule in §1.3 with
two examples and find that that the method is stable and efficient.
1.5 Conclusion of Chapter 1
The parabolic equation backward in time

u
t
+ Au = 0, 0 < t < T,
u(T ) − f  ε
subject to the constraint u(0)  E (E > ε > 0) is regularized by the well-posed
non-local b oundary value problem

v

αt
+ Av
α
= 0, 0 < t < aT,
αv
α
(0) + v
α
(aT ) = f, a  1, α > 0.
- In case a = 1, an a priori parameter choice rule is suggested which yields
error estimates of H¨older type for all t ∈ (0; T ], error estimates of logarithm type
or H¨older type at t = 0 if an additional condition on the smoothness of u(x, 0) is
available. If only with the condition (2.2), an a posteriori parameter choice rule is
also suggested which yields order optimal regularization methods for all t ∈ (0; T ].
- In case a > 1, A priori and a posteriori parameter choice rules are suggested
which yield order optimal regularization methods.
- Numerical results based on the boundary element method are presented and
discussed to confirm the theory.
CHAPTER 2
PARABOLIC EQUATIONS BACKWARD IN TIME WITH
TIME-DEPENDENT COEFFICIENTS
In this chapter, we present results on stability estimates and regularization for
parabolic equations backward in time with time-dependent coefficients

u
t
+ A(t)u = 0, 0 < t < T,
u(T ) − f  ε
(2.1)
subject to the constraint

u(0)  E. (2.2)
Here, H is a Hilbert space with the inner product ·, · and the norm ·, f ∈ H,
A(t) (0  t  T ) is positive self-adjoint unbounded operators from D(A(t)) ⊂ H
to H, and E > ε > 0 are known. The results of this chapter are published in
Inverse Problems.
2.1 Stability estimates
2.1.1 Agmon and Nirenberg’s results
For the reader’s convenience, we summarize Agmon and Nirenberg’s results
Suppose that:
(i) A(t) : D(A(t)) ⊂ H → H is a closed densely defined operator for each
t ∈ [0, T], and u(t) belongs to the domain of A
∗
(t) as well as to that of A(t).
(ii) In addition, A(t) is smooth in its dependence on t and A is ”almost self-
adjoint”. These hypotheses are best expressed in a single condition: if u(t) is the
solution of the equation
Lu =
du
dt
+ A(t)u = 0, 0  t  T, (2.3)
13
14
then for some positive constants k, c
−
d
dt
A(t)u(t), u(t) ≥
1
2
(A(t) + A

∗
(t))u(t)
2
− c(A(t) + k)u(t), u(t).
Theorem 2.1.1. (Agmon and Nirenberg) Let conditions (i)–(ii) be satisfied. Then
the function log |e
−kt
u(t)| is a convex function of the variable s = e
ct
.
The following results are direct consequences of Theorem 2.1.1.
Proposition 2.1.2 (Stability estimates). Let conditions (i)–(ii) be satisfied. Then,
for all t ∈ [0, T ],
u(t) ≤ e
kt−kT µ(t)
u(T ) 
µ(t)
u(0)
1−µ(t)
, (2.4)
where
µ(t) =
e
ct
− 1
e
cT
− 1
. (2.5)
In case A(t) is a self-adjoint, we have the following results.

Proposition 2.1.3. Suppose that
(iii) A(t) is a self-adjoint operator for each t, and u(t) belongs to the domain
of A(t).
(iv) if u(t) is the solution of the equation (2.3), then for some positive constants
k, c,
−
d
dt
A(t)u(t), u(t) ≥ 2A(t)u
2
− c (A(t) + k)u(t), u(t). (2.6)
Then, for all t ∈ [0, T ],
u(t) ≤ e
kt−kT µ(t)
u(T ) 
µ(t)
u(0)
1−µ(t)
. (2.7)
Remark 2.1.4. We always have µ(t) <
t
T
, ∀t ∈ (0, T ). Thus, the order of
the stability estimates by Agmon and Nirenberg is not higher than that in case
time-independent coefficients.
2.1.2 An improvement of Agmon and Nirenberg’s results
Theorem 2.1.5. Suppose that
(i) A(t) is a self-adjoint operator for each t, and u(t) belongs to the domain of
A(t).
15

(ii) if u(t) is the solution of the equation
Lu =
du
dt
+ A(t)u = 0, 0 < t ≤ T,
then for some non-negative constants k, c,
−
d
dt
A(t)u(t), u(t) ≥ 2A(t)u
2
− c (A(t) + k)u(t), u(t). (2.8)
Choose a
1
(t), a Riemann integrable function on [0, T ], such that a
1
(t)  c, ∀t ∈
[0, T ] and
−
d
dt
A(t)u(t), u(t) ≥ 2A(t)u
2
− a
1
(t) (A(t) + k)u(t), u(t). (2.9)
For all t ∈ [0, T ], let
a
2
(t) = exp



t
0
a
1
(τ)dτ

, a
3
(t) =

t
0
a
2
(ξ)dξ,
ν(t) =
a
3
(t)
a
3
(T )
. (2.10)
Then, for all t ∈ [0, T ],
u(t) ≤ e
kt−kT ν(t)
u(T )
ν(t)

u(0)
1−ν(t)
. (2.11)
Remark 2.1.7. If a
1
(t) = 0, ∀t ∈ [0, T ], then ν(t) =
t
T
, ∀t ∈ [0, T]. In particular,
if A(t) is independent of t, we can choose a
1
(t) = 0. Hence ν(t) =
t
T
, ∀t ∈ [0, T].
Remark 2.1.8. If a
1
(t) < 0, ∀t ∈ (0, T ), then ν(t) >
t
T
, ∀t ∈ (0, T).
Proposition 2.1.9. The following inequality holds
ν(t)  µ(t), ∀t ∈ [0, T ].
Example 2.1.11. We consider the simple one-dimensional problem

u
t
= (a(x, t)u
x
)

x
, 0 < x < π, 0 < t < T
u(0, t) = u(π, t) = 0, 0 ≤ t ≤ T
(2.12)
where the coefficient a(x, t) is a C
1
smooth function and
a(x, t) ≥ ¯a > 0.
16
In this case, A(t)u = −(a(x, t)u
x
)
x
, A(t)u, u =

π
0
a(x, t)u
2
x
dx. Let
˙
A(t)u =
−(a
t
(x, t)u
x
)
x
. Then


˙
A(t)u, u

=

π
0
a
t
(x, t)u
2
x
dx. If we choose
a
1
(t) = sup
x∈[0,π]
a
t
(x, t)
a(x, t)
, t ∈ [0, T], then

˙
A(t)u, u

≤ a
1
(t) A(t)u, u.

Therefore, we can verify that
−
d
dt
A(t)u(t), u(t) ≥ 2A(t)u
2
− a
1
(t) Au(t), u(t).
In this case k = 0. Using Theorem 2.1.5, we obtain
u(t) ≤ u(T)
ν(t)
u(0)
1−ν(t)
, ∀t ∈ [0, T ].
We note that if a
t
(x, t) ≤ 0, ∀(x, t) ∈ (0, π) × (0, T ), then ν(t) ≥
t
T
, ∀t ∈ [0, T ].
Moreover, if c is a constant such that c ≥ a
1
(t), ∀t ∈ [0, T], then
ν(t) ≥ µ(t) =
e
ct
− 1
e
cT

− 1
, ∀t ∈ [0, T].
As a concrete example, let a(x, t) =
T π
T + t
+ x, ∀(t, x) ∈ [0, T ] × [0, π]. Then
a
1
(t) = sup
x∈[0,π]
a
t
(x, t)
a(x, t)
= −
T
(2T + t)(T + t)
and
ν(t) =
t
T
+ ln(1 +
t
T
)
1 + ln2
>
t
T
, ∀t ∈ (0, T ).

2.2 Regularization
In this section we make the following assumptions for the operators A(t).
(H
1
) For 0  t  T , the spectrum of A(t) is contained in a sectorial open
domain
σ(A(t)) ⊂ Σ
ω
= {λ ∈ C; |argλ| < ω}, 0  t  T, (2.13)
with some fixed angle 0 < ω <
π
2
, and the resolvent satisfies the estimate
(λ −A(t))
−1
 
M
|λ|
, λ ∈ Σ
ω
, 0  t  T, (2.14)
17
with some constant M  1.
(H
2
) The domain D(A(t)) is independent of t and A(t) is strongly continuously
differentiable.
(H
3
) For all t ∈ [0, T ], A(t) is a positive self-adjoint unbounded operator and if

u(t) is a solution of the equation Lu =
du
dt
+ A(t)u = 0, 0 < t ≤ T, then there are
a non-negative constant k and some Riemann integrable function on [0, T ], a
1
(t)
such that
−
d
dt
A(t)u(t), u(t) ≥ 2A(t)u
2
− a
1
(t) (A(t) + k)u(t), u(t). (2.15)
Remark 2.2.1. If assumptions (H
1
) −(H
2
) are satisfied, then
A(t)(A(t)
−1
− A(s)
−1
)  N|t − s|, 0  s, t  T, (2.16)
for some constant N > 0.
Now, let
B(t) =


A(t), if 0  t  T,
A(2T − t), if T < t  2T.
(2.17)
Then B(t) = B(2T −t), ∀t ∈ [0, 2T ]. Further B(t), (0  t  2T ) are also positive
self-adjoint unbounded operators, the domain D(B(t)) is independent of t and
B(t), (0  t  2T ) also satisfy the conditions (2.13), (2.14) and (2.16).
Denote by w the solution of the well-posed problem

w
t
+ B(t)w = 0, T < t  2T,
w(T ) = f.
(2.18)
Set
g = w(2T ).
Consider the non-local boundary value problem

v
t
+ B(t)v = 0, 0 < t  2T,
αv(0) + v(2T ) = g, α > 0.
(2.19)
In this chapter, we denote the solution of (2.1), (2.2) by u(t), and the solution
of (2.19) by v
α
(t).
Definition 2.2.2. A function v
α
: [0, 2T ] → H is called a solution of (2.19) if
v

α
∈ C
1
((0, 2T ), H) ∩ C([0, 2T ], H), v
α
(t) ∈ D(A), ∀t ∈ (0, 2T ) and satisfies
(2.19).
18
2.2.1 A priori parameter choice rule
Theorem 2.2.3. The problem (2.19) is well-posed and if u(t) is a solution of the
problem (2.1), then by choosing α =
ε
E
, we obtain, for all t ∈ [0, T ],
u(t) −v(t) ≤ 2e
kt−kT ν(t)
ε
ν(t)
2
E
1−
ν(t)
2
,
where ν(t) is defined by (2.10).
2.2.2 A posteriori parameter choice rule
Theorem 2.2.4. Suppose that ε < g. Choose τ > 1 such that 0 < τε < g.
Then there exists a unique number α
ε
> 0 such that

v
α
ε
(2T ) − g = τε. (2.20)
Further, if u(t) is a solution of the problem (2.1) satisfying (2.2), then
u(t) −v
α
ε
(t)  (1 + τ)(τ − 1)
ν(t)
2
−1
e
kt−kT ν(t)
ε
ν(t)
2
E
1−
ν(t)
2
, ∀t ∈ [0, T ]. (2.21)
2.3 Examples
Suppose that Ω is a bounded domain in R
n
with sufficiently smooth boundary,
0 < T < ∞. Set Q := Ω × (0, T ). For any multi-index p = (p
1
, p
2

, . . . , p
n
), we
define
|p| = p
1
+ p
2
+ ···+ p
n
and
D
p
=
∂
|p|
∂x
p
1
1
x
p
2
2
···x
p
n
n
.
The results and methods in §2.1 and § 2.2 are applicable to following parabolic

equations of order 2m (m ≥ 1)
u
t
= −

|p|,|q|≤m
(−1)
|p|
D
p
(a
pq
(x, t)D
q
u), (x, t) ∈ Q. (2.22)
with real functions a
pq
∈ C
1
([0, T ], L
∞
(Ω)) satisfying a
pq
= a
qp
and the uniform
ellipticity condition

|p|,|q|=m
ξ

p
a
pq
(x, t)ξ
q
≥ δ|ξ|
2m
, ∀(x, t) ∈ Q, ξ ∈ R
n
\ {0}. (2.23)
19
for a positive constant δ, independent of t.
2.4 Conclusion of Chapter 2
We prove new stability estimates for backward parabolic equations with time-
dependent coefficients (Theorem 2.1.5). Our stability estimates improve the re-
lated results by Krein, Agmon and Nirenberg. Furthermore, we propose a regu-
larization method for the problem. Our regularization method with a priori and a
posteriori parameter choice yields error estimates of H¨older type (Theorems 2.2.3
and 2.2.11).
CHAPTER 3
STABILITY RESULTS FOR THE HEAT EQUATION BACKWARD
IN TIME
In this chapter, we present stability results for the heat equation backward in
time
u
t
= u
xx
, x ∈ R, t ∈ (0, T), u(·, T ) − ϕ(·)
L

p
(R)
 ε (3.1)
subject to the constraint
u(·, 0)
L
p
(R)
 E (3.2)
with T > 0, ϕ ∈ L
p
(R), 0 < ε < E, 1 < p < ∞ being given. The results of this
chapter are published in Journal of Mathematical Analysis and Applications.
3.1 Auxiliary results
Definition 3.1.1. The function g(z), z ∈ C is called an entire function of expo-
nential type ν, if it satisfies the following properties:
(i) It is an entire function; i.e., it decomposes into a power series g(z) =

k0
a
k
z
k
with constant coefficients a
k
and the series converges absolutely for all com-
plex z ∈ C.
(ii) For every ε > 0 there exists a positive number A
ε
such that for all complex

z ∈ C the inequality
|g(z)|  A
ε
exp((ν + ε)|z|)
is satisfied.
20
21
Definition 3.1.2. M
ν,p
:= M
ν,p
(R) (1  p  ∞) is the collection of all entire
functions of exponential type ν which as functions of a real variable x ∈ R lie in
L
p
(R).
Definition 3.1.5. The function
D
ν
(x) =
sin(νx)
x
is called the Dirichlet kernel and has the following properties:
(i) It is an entire function of exponential type ν belonging to L
2
(R),
(ii)

2
π


D
ν
=

1 on [−ν, ν],
0 outside [−ν, ν],
(iii)
1
π

+∞
−∞
D
ν
(x)dx = 1 (ν > 0),
(iv) For f ∈ L
p
(R), p ∈ (1, ∞), the convolution
S
ν
(f)(x) =
1
π
D
ν
∗ f =
1
π


+∞
−∞
D
ν
(y)f(x − y)dy
belongs to M
ν,p
and D
ν
∗ f
p
 c
p
f
p
, where c
p
is a constant depending
only on p,
(v) If ω ∈ M
ν,p
, then S
ν
(ω) = ω,
(vi) F [D
v
∗ f] =

f on [−ν, ν],
(vii) f − S

ν
(f)
p
 (1 + c
p
)E
ν
(f)
p
.
Here,
E
ν
(f)
p
:= inf
g∈M
ν,p
f − g
p
.
22
3.2 Mollification method and stability results
We consider the mollified problem

u
ν
t
= u
ν

xx
, x ∈ R, t ∈ (0, T ),
u
ν
(x, T ) = S
ν
(ϕ)(x).
(3.3)
We have the following result.
Theorem 3.2.1. Let ϕ ∈ L
p
(R) with p ∈ (1, ∞). Then the problem (3.3) has a
unique solution. Furthermore, for any t ∈ [0, T ], the function u
ν
(·, t) belongs to
M
ν,p
and
u
ν
(·, t)
p

c
p
π
e
(T −t)ν
2
ϕ

p
.
If in the problem (3.3) we choose
ν =

1
T
ln
E
ε
,
then
u
ν
(·, t) −u(·, t)
p


c
p
π
+ ˜c
p

ε
t/T
E
1−t/T
, ∀t ∈ [0, T ],
where u is a solution of the problem (3.1)–(3.2) and ˜c

p
= (1 + c
p
)(1 + 2
√
3)e
3/2
.
Theorem 3.2.2. Let p ∈ (1, ∞) and u
1
and u
2
be two solutions of the problem
(3.1)–(3.2). Then
u
1
(·, t) −u
2
(·, t)
p
 2

c
p
π
+ ˜c
p

ε
t/T

E
1−t/T
, t ∈ [0, T].
Theorem 3.2.6. Let the conditions of Theorem 3.2.1 be satisfied. Let β ∈ (0, 1).
If u is a solution of the problem (3.1)–(3.2) and in the problem (3.3) we choose
ν =

β
1
T
ln
E
ε
,
then
u
ν
(·, t) −u(·, t)
p


c
p
π
E
β−1
ε
1−β
+ ˜c
p


ε
βt/T
E
1−βt/T
, ∀t ∈ [0, T ].
Additionally, if there are positive finite constants
˜
E, γ which maybe not known
such that
ω(u(·, 0), h)
p

˜
Eh
γ
, ∀h > 0, (3.4)
23
then at t = 0 we have
u
ν
(·, 0) −u(·, 0)
p

c
p
π
E
β
ε

1−β
+ ˜c
p
˜
E

β
1
T
ln
E
ε

−γ/2
which is of logarithmic type.
Theorem 3.2.7. Let p = 2 and u be a solution of the problem (3.1)–(3.2), u
ν
the
solution of the problem 3.2.1. Then with
ν =

1
T
ln
E
ε
(3.5)
we have






∂
m+n
u
ν
(·, t)
∂t
n
∂x
m
−
∂
m+n
u(·, t)
∂t
n
∂x
m





2















2ε
t/T
E
1−t/T

1
T
ln
E
ε

(m+2n)/2
if m + 2n  2t, t ∈ [0, T],

1 +

m + 2n
2t

(m+2n)/2



1
T
ln
E
ε

(m+2n)/2
ε
t/T
E
1−t/T
,
if m + 2n > 2t, t ∈ (0, T],
(3.6)
where m, n ∈ N.
If instead u(·, 0)
2
 E we have the stronger condition
u(·, 0)
H
s
:=


+∞
−∞
|u(ξ, 0)|
2

(1 + ξ
2
)
s
dξ

1/2
 E
s
(3.7)
for some s > 0 and E
s
> 0, then by choosing
ν =




ln

E
s
ε

1
T

ln
E
s

ε

−
s
2T

, (3.8)
we obtain, for ε → 0
+
,





∂
m+n
u
ν
(·, t)
∂t
n
∂x
m
−
∂
m+n
u(·, t)
∂t
n

∂x
m





2


















(
1
T
)

m/2+n
ε
t/T
E
1−t/T
(ln
E
s
ε
)
−(T −t)s/(2T )+n+m/2
(1 + T
s/2
+ o(1)),
if m + 2n −s  2t, t ∈ [0, T ],
(
1
T
)
m/2+n
ε
t/T
E
1−t/T
(ln
E
s
ε
)
−(T −t)s/(2T )+n+m/2

×
×

1 + T
s/2

m + 2n −s
2t

(m+2n−s)/2
+ o(1)

if m + 2n −s > 2t, t ∈ (0, T ].
(3.9)