On the problem of determining the parameter of a parabolic equation
We study the boundary-value problem of determining the parameter p of a parabolic equation v′(t) + Av(t) = f(t)+p(0 ≤ t ≤ 1), v(0) = φ, v(1) = ψ, with strongly positive operator A in an arbitrary Banach space E. The exact estimates are established for the solution of this problem in Hölder norms...
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irk-123456789-1662852020-02-24T17:25:34Z On the problem of determining the parameter of a parabolic equation Ashyralyev, A. Статті We study the boundary-value problem of determining the parameter p of a parabolic equation v′(t) + Av(t) = f(t)+p(0 ≤ t ≤ 1), v(0) = φ, v(1) = ψ, with strongly positive operator A in an arbitrary Banach space E. The exact estimates are established for the solution of this problem in Hölder norms. In applications, the exact estimates are obtained for the solutions of the boundary-value problems for parabolic equations. Розглянуто крайову задачу визначення параметра р параболічного рівняння v′(t) + Av(t) = f(t)+p(0 ≤ t ≤ 1), v(0) = φ, v(1) = ψ, у довільному банаховому просторі Е із сильно додатним оператором а. Встановлено точні за нормами Гельдера оцінки для розв'язку цієї задачі. У застосуваннях одержано точні оцінки для розв'язків крайових задач для параболічних рівнянь. 2010 Article On the problem of determining the parameter of a parabolic equation / A. Ashyralyev // Український математичний журнал. — 2010. — Т. 62, № 9. — С. 1200–1210. — Бібліогр.: 26 назв. — англ. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/166285 517.5 en Український математичний журнал Інститут математики НАН України |
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Статті Статті Ashyralyev, A. On the problem of determining the parameter of a parabolic equation Український математичний журнал |
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We study the boundary-value problem of determining the parameter p of a parabolic equation
v′(t) + Av(t) = f(t)+p(0 ≤ t ≤ 1), v(0) = φ, v(1) = ψ,
with strongly positive operator A in an arbitrary Banach space E. The exact estimates are established for the solution of this problem in Hölder norms. In applications, the exact estimates are obtained for the solutions of the boundary-value problems for parabolic equations. |
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Ashyralyev, A. |
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Ashyralyev, A. |
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Ashyralyev, A. |
| title |
On the problem of determining the parameter of a parabolic equation |
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On the problem of determining the parameter of a parabolic equation |
| title_full |
On the problem of determining the parameter of a parabolic equation |
| title_fullStr |
On the problem of determining the parameter of a parabolic equation |
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On the problem of determining the parameter of a parabolic equation |
| title_sort |
on the problem of determining the parameter of a parabolic equation |
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Інститут математики НАН України |
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2010 |
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| citation_txt |
On the problem of determining the parameter of a parabolic equation / A. Ashyralyev // Український математичний журнал. — 2010. — Т. 62, № 9. — С. 1200–1210. — Бібліогр.: 26 назв. — англ. |
| series |
Український математичний журнал |
| work_keys_str_mv |
AT ashyralyeva ontheproblemofdeterminingtheparameterofaparabolicequation |
| first_indexed |
2025-07-14T21:06:54Z |
| last_indexed |
2025-07-14T21:06:54Z |
| _version_ |
1837657977741377536 |
| fulltext |
UDC 517.5
A. Ashyralyev (Fatih Univ., Istanbul, Turkey)
ON A PROBLEM OF DETERMINING THE PARAMETЕR
OF A PARABOLIC EQUATION
ПРО ЗАДАЧУ ВИЗНАЧЕННЯ ПАРАМЕТРА
ПАРАБОЛIЧНОГО РIВНЯННЯ
The boundary-value problem of determining the parameter p of a parabolic equation
v′(t) +Av(t) = f(t) + p(0 ≤ t ≤ 1), v(0) = ϕ, v(1) = ψ
in arbitrary Banach space E with the strongly positive operator A is considered. The exact estimates in Hölder
norms for the solution of this problem are established. In applications, exact estimates for the solution of the
boundary-value problems for parabolic equations are obtained.
Розглянуто крайову задачу визначення параметра p параболiчного рiвняння
v′(t) +Av(t) = f(t) + p(0 ≤ t ≤ 1), v(0) = ϕ, v(1) = ψ
у довiльному банаховому просторi E iз сильно додатним оператором A. Встановлено точнi за норма-
ми Гельдера оцiнки для розв’язку цiєї задачi. У застосуваннях одержано точнi оцiнки для розв’язкiв
крайових задач для параболiчних рiвнянь.
1. Introduction. Methods of solutions of the nonlocal boundary-value problems for
evolution equations with a parameter have been studied extensively by many researchers
(see, e.g., [1 – 21] and the references given therein).
We consider the following local boundary-value problem for the differential equation
v′(t) +Av(t) = f(t) + p(0 ≤ t ≤ 1), v(0) = ϕ, v(1) = ψ (1.1)
in an arbitrary Banach space with linear (unbounded) operator A and an unknown pa-
rameter p.
In the paper [1] the solvability of the problem (1.1) in the space C(E) of the con-
tinuous E-valued functions ϕ(t) defined on [0, 1], equipped with the norm
‖ϕ‖C(E) = max
0≤t≤1
‖ϕ(t)‖E
was studied under the necessary and sufficient conditions for the operator A. The solu-
tion depends continuously on the initial and boundary data. Namely:
Theorem 1.1. Assume that −A is the generator of the analytic semigroup
exp{−tA}(t ≥ 0) and all points 2πik, k ∈ Z, k 6= 0 are not belongs to the spec-
trum σ(A). Let v(0) ∈ E, v(1) ∈ D(A) and f(t) ∈ Cβ(E), 0 < β ≤ 1. Then for the
solution (v(t), p) of problem (1.1) in C(E)× E the estimates
‖p‖E ≤M
[
‖v(0)‖E + ‖v(1)‖E + ‖Av(1)‖E +
1
β
‖f‖Cβ(E)
]
,
‖v‖C(E) ≤M [‖v(0)‖E + ‖v(1)‖E + ‖f‖C(E)]
hold, where M does not depend on β, v(0), v(1) and f(t). Here Cβ(E) is the space
obtained by completion of the space of all smooth E-valued functions ϕ(t) on [0, 1] in
c© A. ASHYRALYEV, 2010
1200 ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 9
ON A PROBLEM OF DETERMINING THE PARAMETR OF A PARABOLIC EQUATION 1201
the norm
‖ϕ‖Cβ(E) = max
0≤t≤1
‖ϕ(t)‖E + sup
0≤t<t+τ≤1
‖ϕ(t+ τ)− ϕ(t)‖E
τβ
.
We say (v(t), p) is the solution of the problem (1.1) in Cβ,γ0 (E)×E1 if the following
conditions are satisfied:
i) v′(t), Av(t) ∈ Cβ,γ0 (E), p ∈ E1 ⊂ E,
ii) (v(t), p) satisfies the equation and boundary conditions (1.1).
Here Cβ,γ0 (E), (0 ≤ γ ≤ β, 0 < β < 1) is the Hölder space with weight obtained
by completion of the space of all smooth E-valued functions ϕ(t) on [0, 1] in the norm
‖ϕ‖Cβ,γ0 (E) = max
0≤t≤1
‖ϕ(t)‖E + sup
0≤t<t+τ≤1
(t+ τ)γ‖ϕ(t+ τ)− ϕ(t)‖E
τβ
.
In the present paper the exact estimates in Hölder norms for the solution of problem
(1.1) are proved. In applications, exact estimates for the solution of the boundary-value
problems for parabolic equations are obtained.
2. Cβ,γ
0 (E)-estimates for the solution of problem (1.1). We study the problem
(1.1) in the spaces Cβ,γ0 (E). To these spaces there correspond the spaces of traces Eβ,γ1 ,
which consist of the elements w ∈ E for which the following norm is finite:
|w|β,γ1 = max
0≤z≤1
‖A exp{−zA}w‖E+
+ sup
0≤z<z+τ≤1
(z + τ)γ‖A (exp{− (z + τ)A} − exp{−zA})w‖E
τβ
.
Assume that −A is the generator of the analytic semigroup exp{−tA}(t ≥ 0) with
exponentially decreasing norm, when t→ +∞, i.e., the following estimates hold:
‖ exp{−tA}‖E→E ≤Me−δt, t‖A exp{−tA}‖E→E ≤M, t > 0, M > 0, δ > 0.
(2.1)
From (2.1) it follows that
‖T‖E→E ≤M(δ). (2.2)
Here T = (I − exp{−A})−1. We have that
v(t) = exp{−tA}v(0) +
t∫
0
exp{−(t− s)A}f(s)ds+ (I − exp{−tA})A−1p,
p = T{Av(1)−A exp{−A}v(0)−
1∫
0
A exp{−(1− s)A}f(s)ds}
(2.3)
for the solution of problem (1.1) in the space Cβ,γ0 (E) (see, for example, [1, 22]).
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 9
1202 A. ASHYRALYEV
Theorem 2.1. Let v(0)−A−1f(0), v(1)−A−1f(1) ∈ Eβ,γ1 and f(t) ∈ Cβ,γ0 (E),
0 ≤ γ ≤ β, 0 < β < 1. Then for the solution (v(t), p) of problem (1.1) in Cβ,γ0 (E) ×
× Eβ,γ1 the estimates
‖v′‖Cβ,γ0 (E) + ‖Av − p‖Cβ,γ0 (E) + max
0≤t≤1
∣∣v(t)−A−1f(t)
∣∣β,γ
1
≤
≤M
[∣∣v(0)−A−1f(0)
∣∣β,γ
1
+
∣∣v(1)−A−1f(1)
∣∣β,γ
1
+ β−1(1− β)−1‖f‖Cβ,γ0 (E)
]
,
(2.4)
∣∣A−1p
∣∣β,γ
1
≤M
[∣∣v(0)−A−1f(0)
∣∣β,γ
1
+
+
∣∣v(1)−A−1f(1)
∣∣β,γ
1
+ β−1(1− β)−1‖f‖Cβ,γ0 (E)
]
(2.5)
hold, where M does not depend on γ, β, v(0), v(1) and f(t).
Proof. Using formula (2.3), we can write
Av(t)− f(t) = exp{−tA} (Av(0)− f(0)) + exp{−tA} (f(0)− f(t)) +
+
t∫
0
A exp{−(t− s)A} (f(s)− f(t)) ds+ (I − exp{−tA}) p =
= exp{−tA} (Av(0)− f(0)) + (I − exp{−tA}) p+ J(t), (2.6)
p = T{Av(1)− f(1)− exp{−A} (Av(0)− f(0))−
−
1∫
0
A exp{−(1− s)A} (f(s)− f(1)) ds+ exp{−A} (f(1)− f(0))} =
= T{Av(1)− f(1)− exp{−A} (Av(0)− f(0))− J(1)}, (2.7)
where
J(t) = exp{−tA} (f(0)− f(t)) +
t∫
0
A exp{−(t− s)A} (f(s)− f(t)) ds.
By [22], Theorems 5.1 and 5.2 in Chapter 1,
‖J‖Cβ,γ0 (E) ≤Mβ−1(1− β)−1‖f‖Cβ,γ0 (E), (2.8)
∣∣A−1J(t)
∣∣β,γ
1
≤Mβ−1(1− β)−1‖f‖Cβ,γ0 (E). (2.9)
From the definition of the space Eβ,γ1 and the estimate (2.1) it follows that∣∣exp{−tA}
(
v(0)−A−1f(0)
)
+ (I − exp{−tA})A−1p
∣∣β,γ
1
≤
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 9
ON A PROBLEM OF DETERMINING THE PARAMETR OF A PARABOLIC EQUATION 1203
≤ ‖exp{−tA}‖E→E
∣∣v(0)−A−1f(0)
∣∣β,γ
1
+ ‖I − exp{−tA}‖E→E
∣∣A−1p
∣∣β,γ
1
≤
≤M
[∣∣v(0)−A−1f(0)
∣∣β,γ
1
+
∣∣A−1p
∣∣β,γ
1
]
(2.10)
for all t, t ∈ [0, 1]. By (2.2), (2.9), (2.10) and the triangle inequality
max
0≤t≤1
∣∣v(t)−A−1f(t)
∣∣β,γ
1
≤
≤M
[∣∣v(0)−A−1f(0)
∣∣β,γ
1
+
∣∣A−1p
∣∣β,γ
1
+ β−1(1− β)−1‖f‖Cβ,γ0 (E)
]
, (2.11)
∣∣A−1p
∣∣β,γ
1
≤ ‖T‖E→E
[∣∣v(1)−A−1f(1)
∣∣β,γ
1
+
+ ‖exp{−A}‖E→E
∣∣v(0)−A−1f(0)
∣∣β,γ
1
+
∣∣A−1J(1)
∣∣β,γ
1
]
≤
≤M
[∣∣v(1)−A−1f(1)
∣∣β,γ
1
+
∣∣v(0)−A−1f(0)
∣∣β,γ
1
+
∣∣A−1J(1)
∣∣β,γ
1
]
. (2.12)
Estimate for max
0≤t≤1
∣∣v(t)−A−1f(t)
∣∣β,γ
1
and estimate (2.5) are proved. Using formula
(2.6) and equation (1.1), we can write
v′(t) = exp{−tA} (Av(0)− f(0)− p) + J(t). (2.13)
Applying the triangle inequality and the definition of the space Cβ,γ0 (E), we obtain
‖v′‖Cβ,γ0 (E) ≤
∣∣v(0)−A−1f(0)
∣∣β,γ
1
+
∣∣A−1p
∣∣β,γ
1
+ ‖J‖Cβ,γ0 (E).
From this estimate and (2.12), (2.12) it follows that
‖v′‖Cβ,γ0 (E) ≤M
[∣∣v(0)−A−1f(0)
∣∣β,γ
1
+
+
∣∣v(1)−A−1f(1)
∣∣β,γ
1
+ β−1(1− β)−1‖f‖Cβ,γ0 (E)
]
.
The estimate for Av(t) − p in the norm Cβ,γ0 (E) follows from this estimate and the
triangle inequality. Theorem 2.1 is proved.
Remark 2.1. The spaces Cβ,γ0 (E) in which exact estimates has been established,
depend on parameters β and γ. However, the constants in these inequalities depend only
on β. Hence, we can be choose the parameter γ freely, which increases the number of
spaces.
With the help of A we introduce the fractional space Eα(E,A), 0 < α < 1, con-
sisting of all v ∈ E for which the following norms are finite:
‖v‖α = sup
λ>0
∥∥λ1−αA exp{−λA}v
∥∥
E
+ ‖v‖E .
3. Cβ,γ
0 (Eα−β)-estimates for the solution of problem (1.1). We study the problem
(1.1) in the spaces Cβ,γ0 (Eα−β). To these spaces there correspond the spaces of traces
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 9
1204 A. ASHYRALYEV
Eβ,γ1+α−β , which consist of the elements w ∈ E for which the following norm is finite:
|w|β,γ1+α−β = max
0≤z≤1
‖A exp{−zA}w‖
α−β+
+ sup
0≤z<z+τ≤1
(z + τ)γ‖A (exp{− (z + τ)A} − exp{−zA})w‖
α−β
τβ
.
Theorem 3.1. Let v(0) − A−1f(0), v(1) − A−1f(1) ∈ Eβ,γ1+α−β and f(t) ∈
∈ Cβ,γ0 (Eα−β), 0 ≤ γ ≤ β ≤ α, 0 < α < 1. Then for the solution (v(t), p) of problem
(1.1) in Cβ,γ0 (Eα−β)× Eβ,γ1+α−β the estimates
‖v′‖Cβ,γ0 (Eα−β) + ‖Av − p‖Cβ,γ0 (Eα−β) + max
0≤t≤1
∣∣v(t)−A−1f(t)
∣∣β,γ
1
≤
≤M
[∣∣v(0)−A−1f(0)
∣∣β,γ
1+α−β +
+
∣∣v(1)−A−1f(1)
∣∣β,γ
1+α−β + α−1(1− α)−1‖f‖Cβ,γ0 (Eα−β)
]
, (3.1)
∣∣A−1p
∣∣β,γ
1+α−β ≤M
[∣∣v(0)−A−1f(0)
∣∣β,γ
1+α−β +
+
∣∣v(1)−A−1f(1)
∣∣β,γ
1+α−β + α−1(1− α)−1‖f‖Cβ,γ0 (Eα−β)
]
(3.2)
hold, where M does not depend on γ, β, α, v(0), v(1) and f(t).
Proof. By [22], Theorem 5.3 in Chapter 1,
‖J‖Cβ,γ0 (Eα−β) ≤Mα−1(1− α)−1‖f‖Cβ,γ0 (Eα−β), (3.3)
∣∣A−1J(t)
∣∣β,γ
1+α−β ≤Mα−1(1− α)−1‖f‖Cβ,γ0 (Eα−β). (3.4)
From the definition of the space Eβ,γ1+α−β and the estimate (2.1) it follows that
∣∣exp{−tA}
(
v(0)−A−1f(0)
)
+ (I − exp{−tA})A−1p
∣∣β,γ
1+α−β ≤
≤ ‖ exp{−tA}‖E→E
∣∣v(0)−A−1f(0)
∣∣β,γ
1+α−β +
+‖ − exp{−tA}‖E→E
∣∣A−1p
∣∣β,γ
1+α−β ≤
≤M
[∣∣v(0)−A−1f(0)
∣∣β,γ
1+α−β +
∣∣A−1p
∣∣β,γ
1+α−β
]
(3.5)
for all t, t ∈ [0, 1]. By (2.2), (3.4), (3.5) and the triangle inequality
max
0≤t≤1
∣∣v(t)−A−1f(t)
∣∣β,γ
1+α−β ≤
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 9
ON A PROBLEM OF DETERMINING THE PARAMETR OF A PARABOLIC EQUATION 1205
≤M
[∣∣v(0)−A−1f(0)
∣∣β,γ
1+α−β +
∣∣A−1p
∣∣β,γ
1+α−β + α−1(1− α)−1‖f‖Cβ,γ0 (Eα−β)
]
,
(3.6)
∣∣A−1p
∣∣β,γ
1+α−β ≤ ‖T‖E→E
[∣∣v(1)−A−1f(1)
∣∣β,γ
1+α−β +
+ ‖exp{−A}‖E→E
∣∣v(0)−A−1f(0)
∣∣β,γ
1+α−β +
∣∣A−1J(1)
∣∣β,γ
1+α−β
]
≤
≤M
[∣∣v(1)−A−1f(1)
∣∣β,γ
1+α−β +
∣∣v(0)−A−1f(0)
∣∣β,γ
1+α−β +
∣∣A−1J(1)
∣∣β,γ
1+α−β
]
.
(3.7)
Estimate for max
0≤t≤1
∣∣v(t)−A−1f(t)
∣∣β,γ
1+α−β and estimate (3.2) are proved. Applying
(2.13), the triangle inequality and the definition of the space Cβ,γ0 (Eα−β), we obtain
‖v′‖Cβ,γ0 (Eα−β) ≤
∣∣v(0)−A−1f(0)
∣∣β,γ
1
+
∣∣A−1p
∣∣β,γ
1
+ ‖J‖Cβ,γ0 (Eα−β).
From this estimate and (3.7), (3.7) it follows that
‖v′‖Cβ,γ0 (Eα−β) ≤M
[∣∣v(0)−A−1f(0)
∣∣β,γ
1+α−β +
+
∣∣v(1)−A−1f(1)
∣∣β,γ
1+α−β + α−1(1− α)−1‖f‖Cβ,γ0 (Eα−β)
]
.
The estimate for Av(t)− p in the norm Cβ,γ0 (Eα−β) follows from this estimate and the
triangle inequality.
Theorem 3.1 is proved.
Note that applying the definition of the space Eβ,γ1+α−β , we can obtain
|w|β,γ1+α−β ≤M‖Aw‖Eα−γ , Aw ∈ Eα−γ .
We have not been able to establish the opposite inequality necessary for the equivalence
of norms. Nevertheless, we have the following result.
Theorem 3.2. Let v(0) − A−1f(0), v(1) − A−1f(1) ∈ Eα−γ and f(t) ∈
∈ Cβ,γ0 (Eα−β), 0 ≤ γ ≤ β ≤ α, 0 < α < 1. Then for the solution (v(t), p) of
problem (1.1) in Cβ,γ0 (Eα−β)× Eα−γ the estimates
‖v′‖Cβ,γ0 (Eα−β) + ‖Av − p‖Cβ,γ0 (Eα−β) + max
0≤t≤1
‖Av(t)− f(t)‖α−γ ≤
≤M
[
‖Av(0)− f(0)‖α−γ + ‖Av(1)− f(1)‖α−γ + α−1(1− α)−1‖f‖Cβ,γ0 (Eα−β)
]
,
(3.8)
‖p‖α−γ ≤M
[
‖Av(0)− f(0)‖α−γ +
+ ‖Av(1)− f(1)‖α−γ + α−1(1− α)−1‖f‖Cβ,γ0 (Eα−β)
]
(3.9)
hold, where M does not depend on γ, β, α, v(0), v(1) and f(t).
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1206 A. ASHYRALYEV
Remark 3.1. The spaces Cβ,γ0 (Eα−β) in which exact estimates has been estab-
lished, depend on parameters α, β and γ. However, the constants in these inequalities
depend only on α. Hence, we can be choose parameters β, γ freely, which increases the
number of spaces. In particular, Theorems 3.1 and 3.2 imply the well-posedness theorem
in C(Eα).
Remark 3.2. Theorems 2.1 and 3.1, 3.2 hold for the following boundary-value
problems:
v′(t) +Av(t) = f(t) + p(0 ≤ t ≤ 1), v(0) = ϕ, v(λ) = ψ, 0 < λ ≤ 1,
v′(t)−Av(t) = f(t) + p(0 ≤ t ≤ 1), v(1) = ϕ, v(λ) = ψ, 0 ≤ λ < 1
in an arbitrary Banach space with positive operator A and an unknown parameter p.
Applications. First, the boundary-value problem on the range {0 ≤ t ≤ 1, x ∈ Rn}
for the 2m-order multidimensional parabolic equation is considered:
∂v(t, x)
∂t
+
∑
|r|=2m
ar(x)
∂|r|v(t, x)
∂xr11 . . . ∂xrnn
+ σv(t, x) = f(t, x) + p(x), 0 < t < 1,
∑
|r|=2m
ar(x)
∂|r|v(0, x)
∂xr11 . . . ∂xrnn
+ σv(0, x) = f(0, x), x ∈ Rn,
∑
|r|=2m
ar(x)
∂|r|v(1, x)
∂xr11 . . . ∂xrnn
+ σv(1, x) = f(1, x), x ∈ Rn,
|r| = r1 + . . .+ rn,
(4.1)
where ar(x) and f(t, x) are given as sufficiently smooth functions. Here, σ is a suffi-
ciently large positive constant.
It is assumed that the symbol
Bx(ξ) =
∑
|r|=2m
ar(x) (iξ1)
r1 . . . (iξn)
rn , ξ = (ξ1, . . . , ξn) ∈ Rn
of the differential operator of the form
Bx =
∑
|r|=2m
ar(x)
∂|r|
∂xr11 . . . ∂xrnn
(4.2)
acting on functions defined on the space Rn, satisfies the inequalities
0 < M1|ξ|2m ≤ (−1)mBx(ξ) ≤M2|ξ|2m <∞
for ξ 6= 0.
The problem (4.1) has a unique smooth solution. This allows us to reduce the prob-
lem (4.1) to the problem (1.1) in a Banach space E = Cµ(Rn) of all continuous bounded
functions defined on Rn satisfying a Hölder condition with the indicator µ ∈ (0, 1) with
a strongly positive operator A = Bx + σI defined by (4.2) (see [25] and [26]).
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ON A PROBLEM OF DETERMINING THE PARAMETR OF A PARABOLIC EQUATION 1207
Theorem 4.1. For the solution of the boundary problem (4.1) the following esti-
mates are satisfied:
‖v‖C1+β,γ
0 (Cµ(Rn)) ≤
M(µ)
β(1− β)
‖f‖Cβ,γ0 (Cµ(Rn)), 0 ≤ γ ≤ β < 1, 0 < µ < 1,
‖v‖C1+β,γ
0 (C2m(α−β)(Rn)) ≤M(α, β)‖f‖Cβ,γ0 (C2m(α−β)(Rn)),
‖p‖C2m(α−γ)(Rn) ≤M(α, β, γ)‖f‖Cβ,γ0 (C2m(α−β)(Rn)),
0 ≤ γ ≤ β, 0 < 2m(α− β) < 1,
where M(µ), M(α, β) and M(α, β, γ) does not depend on f(t, x).
The proof of Theorem 4.1 is based on the abstract Theorems 2.1, 3.2 and on the
following theorem on the structure of the fractional spaces Eα(A,Cµ(Rn)).
Theorem 4.2. Eα(A,Cµ(Rn)) = C2mα+µ(Rn) for all 0 < α <
1
2m
, 0 < µ <
< 1 [22].
Second, let Ω be the unit open cube in the n-dimensional Euclidean space Rn,
0 < xk < 1, 1 ≤ k ≤ n, with boundary S,Ω = Ω ∪ S. In [0, 1] × Ω we consider the
mixed boundary-value problem for the multidimensional parabolic equation
∂v(t, x)
∂t
−
n∑
r=1
αr(x)
∂2v(t, x)
∂x2
r
+ σv(t, x) = f(t, x) + p(x),
x = (x1, . . . , xn) ∈ Ω, 0 < t < 1,
−
n∑
r=1
αr(x)
∂2v(0, x)
∂x2
r
+ σv(0, x) = f(0, x), x ∈ Ω,
−
n∑
r=1
αr(x)
∂2v(1, x)
∂x2
r
+ σv(1, x) = f(1, x), x ∈ Ω,
v(t, x) = 0, x ∈ S,
(4.3)
where αr(x) (x ∈ Ω) and f(t, x) (t ∈ (0, 1), x ∈ Ω) are given smooth functions and
αr(x) ≥ a > 0. Here, σ is a sufficiently large positive constant.
We introduce the Banach spaces Cβ01(Ω), β = (β1, . . . , βn), 0 < xk < 1, k =
= 1, . . . , n, of all continuous functions satisfying a Hölder condition with the indicator
β = (β1, . . . , βn), βk ∈ (0, 1), 1 ≤ k ≤ n and with weight xβkk (1 − xk − hk)βk ,
0 ≤ xk < xk + hk ≤ 1, 1 ≤ k ≤ n, which equipped with the norm
‖f‖Cβ01(Ω) = ‖f‖C(Ω) + sup
0≤xk<xk+hk≤1,
1≤k≤n
∣∣f(x1, . . . , xn)−
−f(x1 + h1, . . . , xn + hn)
∣∣ n∏
k=1
h−βkk xβkk (1− xk − hk)βk ,
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 9
1208 A. ASHYRALYEV
where C(Ω)-is the space of the all continuous functions defined on Ω, equipped with
the norm
‖f‖C(Ω) = max
x∈Ω
|f(x)| .
It is known that the differential expression [24]
Av = −
n∑
r=1
αr(t, x)
∂2v(t, x)
∂x2
+ δv(t, x)
defines a positive operator A acting on Cβ01(Ω) with domain D(A) ⊂ C2+β
01 (Ω) and
satisfying the condition v = 0 on S.
Therefore, we can replace the mixed problem (4.3) by the abstract boundary problem
(1.1). Using the results of Theorem 2.1, we can obtain that the following theorem.
Theorem 4.3. For the solution of the mixed boundary-value problem (4.3) the
following estimate is valid:
‖v‖C1+β,γ
0 (Cµ01(Ω))) ≤
M(µ)
β(1− β)
‖f‖Cβ,γ0 (Cµ01(Ω))),
0 ≤ γ ≤ β < 1, µ = {µ1, . . . , µn}, 0 < µk < 1, 1 ≤ k ≤ n,
where M(µ) is independent of β, γ and f(t, x).
Third, we consider the mixed boundary-value problem for parabolic equation
∂v(t, x)
∂t
− a(x)
∂2v(t, x)
∂x2
+ σv(t, x) = f(t, x) + p(x), 0 < t < 1, 0 < x < 1,
−a(x)
∂2v(0, x)
∂x2
+ σv(0, x) = f(0, x), 0 ≤ x ≤ 1,
−a(x)
∂2v(1, x)
∂x2
+ σv(1, x) = f(1, x), 0 ≤ x ≤ 1,
u(t, 0) = u(t, 1), ux(t, 0) = ux(t, 1), 0 ≤ t ≤ 1,
(4.4)
where a(t, x) and f(t, x) are given sufficiently smooth functions and a(t, x) ≥ a > 0.
Here, σ is a sufficiently large positive constant.
We introduce the Banach spaces Cβ [0, 1], 0 < β < 1 of all continuous functions
ϕ(x) satisfying a Hölder condition for which the following norms are finite
‖ϕ‖Cβ [0,1] = ‖ϕ‖C[0,1] + sup
0≤x<x+τ≤1
|ϕ(x+ τ)− ϕ(x)|
τβ
,
where C[0, 1] is the space of the all continuous functions ϕ(x) defined on [0,1] with the
usual norm
‖ϕ‖C[0,1] = max
0≤x≤1
|ϕ(x)|.
It is known that the differential expression [23]
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ON A PROBLEM OF DETERMINING THE PARAMETR OF A PARABOLIC EQUATION 1209
Av = −a(x)v′′(x) + δv(x)
define a positive operator A acting in Cβ [0, 1] with domain Cβ+2[0, 1] and satisfying
the conditions v(0) = v(1), vx(0) = vx(1).
Therefore, we can replace the mixed problem (4.4) by the abstract boundary value
problem (4.4). Using the results of Theorems 2.1, 3.2, we can obtain that
Theorem 4.4. For the solution of the mixed problem (4.4) the following estimates
are valid:
‖v‖C1+β,γ
0 (Cµ[0,1]) ≤
M(µ)
β(1− β)
‖f‖Cβ,γ0 (Cµ[0,1]), 0 ≤ γ ≤ β < 1, 0 < µ < 1,
‖v‖C1+β,γ
0 (C2(α−β)[0,1]) ≤M(α, β)‖f‖Cβ,γ0 (C2(α−β)[0,1]),
‖p‖C2(α−γ)[0,1] ≤M(α, β, γ)‖f‖Cβ,γ0 (C2(α−β)[0,1]),
0 ≤ γ ≤ β, 0 < 2m(α− β) < 1,
where M(µ), M(α, β) and M(α, β, γ) does not depend on f(t, x).
The proof of Theorem 4.4 is based on the abstract Theorems 2.1, 3.2 and on the
following theorem on the structure of the fractional spaces Eα(A,C[0, 1]).
Theorem 4.5 [23]. Eα(A,C[0, 1]) = C2α[0, 1] for all 0 < α <
1
2
.
Acknowledgement. The author would like to thank Prof. Pavel Sobolevskii (Jerusa-
lem, Israel), for his helpful suggestions to the improvement of this paper.
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Received 16.04.10
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