Well-Posedness of the Right-Hand Side Identification Problem for a Parabolic Equation
We study the inverse problem of reconstruction of the right-hand side of a parabolic equation with nonlocal conditions. The well-posedness of this problem in Hölder spaces is established.
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| author | Ashyralyev, A. Erdogan, A. S. Аширалієв, A. Ердоган, А. С. |
| author_facet | Ashyralyev, A. Erdogan, A. S. Аширалієв, A. Ердоган, А. С. |
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| description | We study the inverse problem of reconstruction of the right-hand side of a parabolic equation with nonlocal conditions. The well-posedness of this problem in Hölder spaces is established. |
| first_indexed | 2026-03-24T02:19:04Z |
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UDC 517.9
A. Ashyralyev (Fatih Univ., Istanbul, Turkey; ITTU, Ashgabat, Turkmenistan),
A. S. Erdogan (Fatih Univ., Istanbul, Turkey)
WELL-POSEDNESS OF THE RIGHT-HAND SIDE
IDENTIFICATION PROBLEM FOR A PARABOLIC EQUATION
КОРЕКТНIСТЬ ПРОБЛЕМИ ПРАВОСТОРОННЬОЇ IДЕНТИФIКАЦIЇ
ПАРАБОЛIЧНОГО РIВНЯННЯ
We study the inverse problem of reconstruction of the right-hand side of a parabolic equation with nonlocal conditions.
The well-posedness of this problem in Hölder spaces is established.
Дослiджено обернену задачу вiдновлення правої частини параболiчного рiвняння з нелокальними умовами. Вста-
новлено коректнiсть цiєї задачi у просторах Гьольдера.
1. Introduction. The inverse problems take an important place in various fields of science and engi-
neering and have been studied by different authors [1 – 7]. The optimal overdetermination conditions
are analyzed in some classical boundary conditions or and similar conditions given at a point. The
literature review and various approaches for the approximate solution are given in [8 – 11]. Moreover,
the generalized overdetermination conditions such as nonlocal, integral, and final overdetermination
conditions are used [12 – 15].
The importance of well-posedness has been widely recognized by the researchers in the field of
partial differential equations [16 – 22]. Moreover, the well-posedness of the right-hand side identifi-
cation problem for a parabolic equation where the unknown function p is in space variable is also
well investigated [23 – 28].
Let us give a brief summary of papers with investigations on the right-hand side identification
problem for a parabolic equation where the unknown function p is in time variable and the motivation
of our present paper.
In article [29], a conditional stability of Hölder type (estimate in Lp-norm) of the inverse problem
of determining p(t), 0 < t < T, in the heat source of the heat equation
∂tu(x, t) = 4u(x, t) + p(t)q(x), x ∈ Rn, t > 0, (1)
from the observation u(x0, t), 0 < t < T, at a remote point x0 away from the support of q is
established by using Hölder type inequalities.
The numerical algorithm for solving inverse problem of reconstructing a distributed right-hand
side of a parabolic equation with local boundary conditions is studied in [30] and [31]. In these
articles, the numerical solution of the identification problem and well-posedness of the algorithm
is presented. For reconstructing the right-hand side function f(t, x) = p(t)q(x), where p(t) is the
unknown function, the solution is observed in the form of u(t, x) = η(t)q(x) + w(t, x), where
η(t) =
∫ t
0
p (s) ds. Then, an approximation is given for w(t, x) via fully implicit difference scheme.
c© A. ASHYRALYEV, A. S. ERDOGAN, 2014
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 2 147
148 A. ASHYRALYEV, A. S. ERDOGAN
The solution of system constructed by the difference scheme is searched in the form
wn+1
i = yi + wn+1
k zi, i = 0, 1, . . . ,M,
where k is an interior grid point and the well-posedness of the algorithm is given by a priori estimate
max
0≤i≤M
|zi| ≤ τ max
0<i<M
∣∣∣∣ 1
ψk
(aψx)x,i
∣∣∣∣
which is based on maximum principle. Thus, in [30] |zi| < 1 at small enough τ = O(1), i.e., it is
necessary to use a sufficiently small time step.
In the papers [32 – 35], inverse problems of reconstructing a distributed right-hand side of a
parabolic equation were studied by using the methods of differential equations in Banach and Hilbert
spaces. The solvability of these inverse problems is dealt with by reducing it to an abstract Volterra
operator of the second kind. In the paper [34], the inverse problem of the form
u′(t)−Au(t) = Φ(t)p(t) + f(t), 0 ≤ t ≤ T,
u(0) = u0, ϕ
(
u(t)
)
= ψ(t), 0 ≤ t ≤ T,
(2)
was considered. Here u0 is an element, A and ϕ are linear operators and Φ(∆), f(∆), ψ(∆) are
given functions and u(∆) and p(∆) are the unknowns. It was assumed that A generates a strongly
continuous semigroup in a Banach space X, u0 ∈ E, ϕ ∈ L(E,F ), Φ(∆) ∈ C([0, T ], L(F,E)),
F (∆) ∈ C([0, T ], E) and ψ(·) ∈ C1([0, T ], E). The author gives conditions on the coefficients that
insure existence, uniqueness and continuous dependence of solutions (u(∆), p(∆)) on the data.
In the paper [35], a problem of determining the right-hand side of a uniformly multidimensional
parabolic equation in a bounded domain with the Dirichlet boundary condition was studied. Addi-
tional information was given in integral form. An existence and uniqueness theorem for the solution
of the inverse problem in the Hölder class was proven and a sufficient condition for the differentia-
bility of the solution was given.
In the present paper, we investigate the well-posedness of the inverse problem of reconstructing
the right-hand side of a one dimensional parabolic equation with nonlocal conditions
∂u(t, x)
∂t
= a(x)
∂2u(t, x)
∂x2
− σu(t, x) + p(t)q(x) + f(t, x),
0 < x < l, 0 < t ≤ T,
u(t, 0) = u(t, l), ux(t, 0) = ux(t, l), 0 ≤ t ≤ T,
u(0, x) = ϕ(x), 0 ≤ x ≤ l,
u (t, x∗) = ρ(t), 0 ≤ x∗ ≤ l, 0 ≤ t ≤ T,
(3)
where u(t, x) and p(t) are unknown functions, a(x) ≥ δ > 0 and σ > 0 is a sufficiently large number
with assuming that
(a) q(x) is a sufficiently smooth function,
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 2
WELL-POSEDNESS OF THE RIGHT-HAND SIDE IDENTIFICATION PROBLEM . . . 149
(b) q(x) and q′(x) are periodic with length l,
(c) q (x∗) 6= 0.
At the end of the section, we give a brief comparison of them with our work.
The aim of the paper is to give well-posedness theorems and their proof for the inverse problem of
reconstructing the right-hand side of a parabolic equation with nonlocal conditions. In contrast to [29],
we establish the coercive inequality for the solution of problem (3) in
(
C
(
[0, T ],
◦
C2α[0, l]
)
, C[0, T ]
)
.
Comparing to stability results of [30] for the solution of difference schemes, we give the well-
posedness in differential case and for the solution of problem (3) with nonlocal conditions. The
application of operator tools permits to investigate inverse problems of reconstructing a distributed
right-hand side of a multidimensional parabolic equation more general case than problem (3) with
classical boundary conditions (see, for example, [32 – 35]).
In general, it would be interesting to establish the coercive inequality for the solution of problems
of reconstructing a distributed right-hand side of a multidimensional parabolic equation more general
case than problem (3). Of course, it will be possible after establishing theorem on well-posedness
of abstract problem (2) in
(
C
(
[0, T ], Ea
)
, C[0, T ]
)
and theorem on structure of interpolation spaces
Ea = Ea(E,A) generated by multidimensional space operator A. Unfortunately (see [36]), struc-
ture of interpolation spaces Ea = Ea(E,A) generated by multidimensional space operator A with
boundary local and nonlocal conditions is not well investigated.
2. Main results. To formulate our results, we introduce the Banach space
◦
Cα[0, l], α ∈ (0, 1) ,
of all continuous functions φ(x) defined on [0, l] with φ(0) = φ(l) satisfying a Hölder condition for
which the following norm is finite:
‖φ‖ ◦
Cα[0,l]
= max
0<x<l
|φ(x)|+ sup
0<x<x+h<l
|φ (x+ h)− φ(x)|
hα
.
With the help of A we introduce the fractional spaces Eα, 0 < α < 1, consisting of all v ∈ E
for which the following norm is finite:
‖v‖Eα = ‖v‖E + sup
λ>0
λ1−α ‖A exp {−λA} v‖E . (4)
It is known that under the assumption that the operator −A generates an analytic semigroup
exp {−tA}, t > 0, with exponentially decreasing norm, when t → +∞, i.e., the following esti-
mates hold:
‖exp {−tA}‖E→E ≤Me−δt, (5)
‖Aα exp {−tA}‖E→E ≤Me−δtt−α, (6)
where t, δ,M > 0 [36].
Positive constants, which can be differ in time will be indicated with an M. On the other hand
M (α, β, . . .) is used to focus on the fact that the constant depends only on α, β, . . . .
Theorem 1. Let ϕ(x) ∈
◦
C2α+2[0, l], f(t, x) ∈ C
(
[0, T ],
◦
C2α[0, l]
)
and ρ′(t) ∈ C[0, T ]. Then
for the solution of problem (3), the following coercive stability estimates:
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 2
150 A. ASHYRALYEV, A. S. ERDOGAN
‖ut‖
C
(
[0,T ],
◦
C
2α
[0,l]
) + ‖u‖
C
(
[0,T ],
◦
C2α+2[0,l]
) ≤M (x∗, q)
∥∥ρ′∥∥
C[0,T ]
+
+M (a, δ, σ, α, x∗, q, T )
(
‖ϕ‖ ◦
C2α+2[0,l]
+ ‖f‖
C
(
[0,T ],
◦
C2α[0,l]
) + ‖ρ‖C[0,T ]
)
,
‖p‖C[0,T ] ≤M (x∗, q)
∥∥ρ′∥∥
C[0,T ]
+
+M (a, δ, σ, α, x∗, q, T )
[
‖ϕ‖ ◦
C2α+2[0,l]
+ ‖f‖
C
(
[0,T ],
◦
C2α[0,l]
) + ‖ρ‖C[0,T ]
]
hold.
Proof. Let us search for the solution of the inverse problem in the following form:
u(t, x) = η(t)q(x) + w (t, x) , (7)
where
η(t) =
t∫
0
p (s) ds.
Taking derivatives from (7), we have
∂u(t, x)
∂t
= p(t)q (x) +
∂w(t, x)
∂t
and
∂2u(t, x)
∂x2
= η(t)
d2q(x)
dx2
+
∂2w(t, x)
∂x2
.
Moreover if we substitute x = x∗ in equation (7), we get
u (t, x∗) = η(t)q (x∗) + w (t, x∗) = ρ(t)
and
η(t) =
ρ(t)− w (t, x∗)
q (x∗)
. (8)
Taking derivative of both sides, we obtain
p(t) =
ρ′(t)− wt (t, x∗)
q (x∗)
. (9)
Using the triangle inequality and the identity (9), we have
|p(t)| =
∣∣∣∣ρ′(t)− wt (t, x∗)
q (x∗)
∣∣∣∣ ≤M(x∗, q)
(∣∣ρ′(t)∣∣+ |wt (t, x∗)|
)
≤
≤M(x∗, q)
(
max
0≤t≤T
∣∣ρ′(t)∣∣+ max
0≤t≤T
max
0≤x≤l
|wt(t, x)|
)
≤
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 2
WELL-POSEDNESS OF THE RIGHT-HAND SIDE IDENTIFICATION PROBLEM . . . 151
≤M(x∗, q)
(
max
0≤t≤T
∣∣ρ′(t)∣∣+ max
0≤t≤T
‖wt(t)‖ ◦
C2α[0,l]
)
(10)
for any t, t ∈ [0, T ]. Using equations (7), (8) and under the same assumptions on q(x), one can show
that w(t, x) is the solution of the following problem:
∂w(t, x)
∂t
= a(x)
∂2w(t, x)
∂x2
+ a(x)
ρ(t)− w (t, x∗)
q (x∗)
d2q(x)
dx2
−
− σρ(t)− w (t, x∗)
q (x∗)
q(x)− σw(t, x) + f(t, x), 0 < x < l, 0 < t ≤ T,
w(t, 0) = w(t, l), wx(t, 0) = wx(t, l), 0 ≤ t ≤ T,
w(0, x) = ϕ(x), 0 ≤ x ≤ l.
(11)
So, the end of proof of Theorem 1 is based on estimate (10) and the following theorem.
Theorem 2. For the solution of problem (11), the following coercive stability estimate:
‖wt‖ ◦
C2α[0,l]
≤M (a, δ, σ, α, x∗, q, T )
(
‖ϕ‖ ◦
C2α+2[0,l]
+ ‖f‖
C
(
[0,T ],
◦
C2α[0,l]
) + ‖ρ‖C[0,T ]
)
holds.
Proof. We can rewrite problem (11) in the abstract form
wt +Aw =
(
aq′′ − σq
) ρ(t)− w (t, x∗)
q (x∗)
+ f(t), 0 < t ≤ T,
w(0) = ϕ
in a Banach space E =
◦
C[0, l] with the positive operator A defined by
Au = −a(x)
∂2u(t, x)
∂x2
+ σu
with
D (A) =
{
u(x) : u, u′, u′′ ∈ C[0, l], u(0) = u(l), ux(0) = ux(l)
}
.
Here, f(t) = f(t, x) and w(t) = w (t, x) are known and unknown abstract functions defined on
[0, T ] with values in E =
◦
C[0, l], w (t, x∗) is unknown scalar function defined on [0, T ], q = q(x),
q′′ = q′′(x), ϕ = ϕ(x) and a = a(x) are elements of E =
◦
C[0, l] and q (x∗) is a number.
By the Cauchy formula, the solution can be written as
w(t) = e−tAϕ−
t∫
0
e−(t−s)A
aq′′ − σq
q∗
w (s, x∗) ds+
+
t∫
0
e−(t−s)A
ρ (s) (aq′′ − σq)
q∗
ds+
t∫
0
e−(t−s)Af (s) ds.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 2
152 A. ASHYRALYEV, A. S. ERDOGAN
Taking the derivative of both sides, we obtain that
wt(t) = −Ae−tAϕ+
t∫
0
Ae−(t−s)A
aq′′ − σq
q∗
w (s, x∗) ds−
−
t∫
0
Ae−(t−s)A
ρ (s) (aq′′ − σq)
q∗
ds−
t∫
0
Ae−(t−s)Af (s) ds.
Applying the formula
t∫
0
Ae−(t−s)A
aq′′ − σq
q∗
w (s, x∗) ds =
=
t∫
0
Ae−(t−s)A
aq′′ − σq
q∗
s∫
0
wz (z, x∗) dzds+
t∫
0
Ae−(t−s)A
aq′′ − σq
q∗
ϕ (x∗) ds
and changing the order of integration, we obtain that
t∫
0
Ae−(t−s)A
aq′′ − σq
q∗
w (s, x∗) ds =
=
t∫
0
t∫
z
Ae−(t−s)A
aq′′ − σq
q∗
wz (z, x∗) dsdz +
t∫
0
Ae−(t−s)A
aq′′ − σq
q∗
ϕ (x∗) ds.
Then, the following presentation of the solution of (11):
wt(t) = Ae−tAϕ+
t∫
0
t∫
z
Ae−(t−s)A
aq′′ − σq
q∗
wz (z, x∗) dsdz+
+
t∫
0
Ae−(t−s)A
aq′′ − σq
q∗
ϕ (x∗) ds−
t∫
0
Ae−(t−s)A
ρ (s) (aq′′ − σq)
q∗
ds−
−
t∫
0
Ae−(t−s)Af (s) ds =
5∑
k=1
Gk(t)
is obtained. Here,
G1(t) = Ae−tAϕ,
G2(t) =
t∫
0
t∫
z
Ae−(t−s)A
aq′′ − σq
q∗
wz (z, x∗) dsdz,
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 2
WELL-POSEDNESS OF THE RIGHT-HAND SIDE IDENTIFICATION PROBLEM . . . 153
G3(t) =
t∫
0
Ae−(t−s)A
aq′′ − σq
q∗
ϕ (x∗) ds,
G4(t) = −
t∫
0
Ae−(t−s)A
ρ (s) (aq′′ − σq)
q∗
ds,
G5(t) = −
t∫
0
Ae−(t−s)Af (s) ds.
It is very well known that, from the fact that the operators R, exp {−λA} and A commute, it follows
that [36]
‖R‖Eα→Eα ≤ ‖R‖E→E . (12)
Now, let us estimate Gk(t) for any k = 1, 2, 3, 4, 5 separately. Applying the definition of norm
of the spaces Eα and (12), we get
‖G1(t)‖Eα =
∥∥Ae−tAϕ∥∥
Eα
≤
∥∥e−tA∥∥
Eα→Eα ‖Aϕ‖Eα ≤
∥∥e−tA∥∥
E→E ‖Aϕ‖Eα .
Using estimate (5), we get
‖G1(t)‖Eα ≤M1 ‖Aϕ‖Eα (13)
for any t, t ∈ [0, T ]. Let us estimate G2(t)
‖G2(t)‖Eα =
∥∥∥∥∥∥
t∫
0
t∫
z
Ae−(t−s)A
aq′′ − σq
q∗
wz (z, x∗) dsdz
∥∥∥∥∥∥
Eα
≤
≤
t∫
0
t∫
z
∥∥∥∥Ae−(t−s)Aaq′′ − σqq∗
∥∥∥∥
Eα
ds |wz (z, x∗)| dz.
By equation (4), we have that
t∫
z
∥∥∥∥Ae−(t−s)Aaq′′ − σqq∗
∥∥∥∥
Eα
ds =
t∫
z
∥∥∥∥Ae−(t−s)Aaq′′ − σqq∗
∥∥∥∥
E
ds+
+ sup
λ>0
t∫
z
∥∥∥∥λ1−αAe−λAAe−(t−s)Aaq′′ − σqq∗
∥∥∥∥
E
ds.
By the definition of norm of the spaces Eα, we get
t∫
z
∥∥∥∥Ae−(t−s)Aaq′′ − σqq∗
∥∥∥∥
E
ds =
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 2
154 A. ASHYRALYEV, A. S. ERDOGAN
=
t∫
z
(t− s)α−1
∥∥∥∥(t− s)1−αAe−(t−s)Aaq
′′ − σq
q∗
∥∥∥∥
E
ds ≤
≤
t∫
z
(t− s)α−1 ds
∥∥∥∥aq′′ − σqq∗
∥∥∥∥
Eα
≤ Tα
α
∥∥∥∥aq′′ − σqq∗
∥∥∥∥
Eα
=
= M2 (a, σ, α, x∗, q, T ) .
Using estimate (6), we can obtain that
t∫
z
∥∥∥∥λ1−αAe−λAAe−(t−s)Aaq′′ − σqq∗
∥∥∥∥
E
ds ≤
≤
t∫
z
22−αλ1−α
(λ+ t− s)2−α
ds
∥∥∥∥λ+ t− s
2
Ae−
λ+t−s
2 A
∥∥∥∥
E→E
×
×
∥∥∥∥∥
(
λ+ t− s
2
)1−α
Ae−
λ+t−s
2 Aaq
′′ − σq
q∗
∥∥∥∥∥
E
ds ≤
≤M3 (α)
∥∥∥∥aq′′ − σqq∗
∥∥∥∥
Eα
t∫
z
λ1−α
(λ+ t− s)2−α
ds ≤
≤M3 (α)
∥∥∥∥aq′′ − σqq∗
∥∥∥∥
Eα
(
λ1−α
(1− α) (λ+ t− z)1−α
)
for any λ > 0. Then,
sup
λ>0
t∫
z
∥∥∥∥λ1−αAe−λAAe−(t−s)Aaq′′ − σqq∗
∥∥∥∥
E
ds ≤
≤M3 (α)
∥∥∥∥aq′′ − σqq∗
∥∥∥∥
Eα
1
(1− α)
= M4 (a, σ, α, x∗, q) .
Then, we get
t∫
z
∥∥∥∥Ae−(t−s)Aaq′′ − σqq∗
∥∥∥∥
Eα
ds ≤M5 (a, σ, α, x∗, q, T ) (14)
for any s, 0 ≤ z ≤ s ≤ t and
‖G2(t)‖Eα ≤M6 (a, σ, α, x∗, q, T )
t∫
0
|wz (z, x∗)| dz. (15)
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WELL-POSEDNESS OF THE RIGHT-HAND SIDE IDENTIFICATION PROBLEM . . . 155
G3(t) is estimated as follows:
‖G3(t)‖Eα =
∥∥∥∥∥∥
t∫
0
Ae−(t−s)A
aq′′ − σq
q∗
ϕ (x∗) ds
∥∥∥∥∥∥
Eα
≤
≤
∥∥∥∥∥∥
t∫
0
Ae−(t−s)A
aq′′ − σq
q∗
ds
∥∥∥∥∥∥
Eα
|ϕ (x∗)| .
Since
|ϕ (x∗)| ≤ ‖ϕ‖E ≤ ‖ϕ‖Eα ≤M ‖Aϕ‖Eα
and using estimate (14) and choosing z = 0, we obtain
‖G3(t)‖Eα ≤M7 (a, σ, α, x∗, q, T ) ‖Aϕ‖Eα (16)
for any t ∈ [0, T ].
By estimate (14), the estimation of G4(t) is as follows:
‖G4(t)‖Eα =
∥∥∥∥∥∥
t∫
0
Ae−(t−s)Aρ (s)
aq′′ − σq
q∗
ds
∥∥∥∥∥∥
Eα
≤
≤
t∫
0
∥∥∥∥Ae−(t−s)Aaq′′ − σqq∗
∥∥∥∥
Eα
ds ‖ρ‖C[0,T ] ≤
≤M8 (a, σ, α, x∗, q, T ) ‖ρ‖C[0,T ] . (17)
Now, let us estimate G5(t). By the definition of the norm of the spaces Eα, we get
‖G5(t)‖Eα =
∥∥∥∥∥∥
t∫
0
Ae−(t−s)Af (s) ds
∥∥∥∥∥∥
Eα
=
=
∥∥∥∥∥∥
t∫
0
Ae−(t−s)Af (s) ds
∥∥∥∥∥∥
E
+ sup
λ>0
λ1−α
∥∥∥∥∥∥Ae−λA
t∫
0
Ae−(t−s)Af (s) ds
∥∥∥∥∥∥
E
.
Using equation (4), we have that∥∥∥∥∥∥
t∫
0
Ae−(t−s)Af (s) ds
∥∥∥∥∥∥
E
=
t∫
0
(t− s)α−1
∥∥∥(t− s)1−αAe−(t−s)Af (s)
∥∥∥
E
ds ≤
≤
t∫
0
(t− s)α−1 ds ‖f‖C(Eα)
=
tα
α
‖f‖C(Eα)
≤M9 (α, T ) ‖f‖C(Eα)
. (18)
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 2
156 A. ASHYRALYEV, A. S. ERDOGAN
Now, we consider the second term. Using equation (4), we obtain
λ1−α
∥∥∥∥∥∥Ae−λA
t∫
0
Ae−(t−s)Af (s) ds
∥∥∥∥∥∥
E
≤
≤ λ1−α
t∫
0
(
t− s+ λ
2
)α−1( t− s+ λ
2
)−1 ∥∥∥∥ t− s+ λ
2
Ae−
t−s+λ
2 A
∥∥∥∥
E→E
×
×
∥∥∥∥∥
(
t− s+ λ
2
)1−α
Ae−
t−s+λ
2 Af (s)
∥∥∥∥∥
E
ds ≤
≤M10λ
1−α
t∫
0
(
t− s+ λ
2
)α−2
‖f‖Eα ds ≤M10λ
1−α
t∫
0
(
t− s+ λ
2
)α−2
ds ‖f‖C(Eα)
for any λ > 0. Then,
sup
λ>0
λ1−α
∥∥∥∥∥∥Ae−λA
t∫
0
Ae−(t−s)Af (s) ds
∥∥∥∥∥∥
E
≤ M102
1−α
1− α
‖f‖C(Eα)
= M11 (α) ‖f‖C(Eα)
. (19)
By estimates (18) and (19), we get
‖G5(t)‖Eα ≤M12 (α, T ) ‖f‖C(Eα)
. (20)
Combining estimates (13), (15), (16), (17) and (20) we have
‖wt‖Eα ≤M1 ‖Aϕ‖Eα +M6 (a, σ, α, x∗, q, T )
t∫
0
|wz (z, x∗)| dz+
+M7 (a, σ, α, x∗, q, T ) ‖Aϕ‖Eα +M8 (a, σ, α, x∗, q, T ) ‖ρ‖C[0,T ] +
+M12 (α, T ) ‖f‖C(Eα)
.
Using integral inequality, we can write,
‖wt‖Eα ≤ e
M6(a,σ,α,x∗,q,T )
[
M1 ‖Aϕ‖Eα +M7 (a, σ, α, x∗, q, T ) ‖Aϕ‖Eα +
+M8 (a, σ, α, x∗, q, T ) ‖ρ‖C[0,T ] +M12 (α, T ) ‖f‖C(Eα)
]
.
Then, the following theorem finishes the proof of Theorem 2.
Theorem 3 [37]. For 0 < α <
1
2
the norms of the spaces Eα (C[0, l], A) and C2α[0, l] are
equivalent.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 2
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Received 13.11.10,
after revision — 09.11.13
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| id | umjimathkievua-article-2119 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:19:04Z |
| publishDate | 2014 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/f4/353e0efba1e7664dd328fe4c31132af4.pdf |
| spelling | umjimathkievua-article-21192019-12-05T10:24:43Z Well-Posedness of the Right-Hand Side Identification Problem for a Parabolic Equation Коректність проблеми правосторонньої iдентифiкацiї параболічного рівняння Ashyralyev, A. Erdogan, A. S. Аширалієв, A. Ердоган, А. С. We study the inverse problem of reconstruction of the right-hand side of a parabolic equation with nonlocal conditions. The well-posedness of this problem in Hölder spaces is established. Досліджєно обернену задачу відновлення правої частини параболiчного рівняння з нелокальними умовами. Встановлено коректність цієї задачі у просторах Гьольдера. Institute of Mathematics, NAS of Ukraine 2014-02-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2119 Ukrains’kyi Matematychnyi Zhurnal; Vol. 66 No. 2 (2014); 147–158 Український математичний журнал; Том 66 № 2 (2014); 147–158 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/2119/1240 https://umj.imath.kiev.ua/index.php/umj/article/view/2119/1241 Copyright (c) 2014 Ashyralyev A.; Erdogan A. S. |
| spellingShingle | Ashyralyev, A. Erdogan, A. S. Аширалієв, A. Ердоган, А. С. Well-Posedness of the Right-Hand Side Identification Problem for a Parabolic Equation |
| title | Well-Posedness of the Right-Hand Side Identification Problem for a Parabolic Equation |
| title_alt | Коректність проблеми правосторонньої iдентифiкацiї параболічного рівняння |
| title_full | Well-Posedness of the Right-Hand Side Identification Problem for a Parabolic Equation |
| title_fullStr | Well-Posedness of the Right-Hand Side Identification Problem for a Parabolic Equation |
| title_full_unstemmed | Well-Posedness of the Right-Hand Side Identification Problem for a Parabolic Equation |
| title_short | Well-Posedness of the Right-Hand Side Identification Problem for a Parabolic Equation |
| title_sort | well-posedness of the right-hand side identification problem for a parabolic equation |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2119 |
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