On the smoothness of a solution of the first boundary-value problem for second-order degenerate elliptic-parabolic equations
In this work, the first boundary-value problem is considered for second-order degenerate elliptic-parabolic equation with, generally speaking, discontinuous coefficients. The matrix of senior coefficients satisfies the parabolic Cordes condition with respect to space variables. We prove that the g...
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2008
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509239628267520 |
|---|---|
| author | Gadjiev, T. S. Gasimova, E. R. Гаджиїв, Т. С. Газимова, Є. Р. |
| author_facet | Gadjiev, T. S. Gasimova, E. R. Гаджиїв, Т. С. Газимова, Є. Р. |
| author_sort | Gadjiev, T. S. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:48:06Z |
| description | In this work, the first boundary-value problem is considered for second-order degenerate elliptic-parabolic equation with,
generally speaking, discontinuous coefficients. The matrix of senior coefficients satisfies the parabolic Cordes condition with respect to space variables.
We prove that the generalized solution to the problem belongs to the Holder space C 1+λ if the right-hand side f belongs to Lp, p > n. |
| first_indexed | 2026-03-24T02:37:57Z |
| format | Article |
| fulltext |
UDC 517.9
T. S. Gadjiev, E. R. Gasimova (Inst. Math. and Mech. Nat. Acad. Sci. Azerbaijan, Baku)
ON SMOOTHNESS OF SOLUTION OF THE FIRST BOUNDARY-
VALUE PROBLEM FOR SECOND-ORDER DEGENERATE
ELLIPTIC-PARABOLIC EQUATIONS
PRO HLADKIST| ROZV’QZKU PERÍO} KRAJOVO} ZADAÇI
DLQ VYRODÛENYX ELIPTYÇNO-PARABOLIÇNYX
RIVNQN| DRUHOHO PORQDKU
In this work, the first boundary-value problem is considered for second-order degenerate elliptic-parabo-
lic equation with, generally speaking, discontinuous coefficients. The matrix of senior coefficients satis-
fies the parabolic Cordes condition with respect to space variables. We prove that the generalized soluti-
on to the problem belongs to the Hölder space C1+λ if the right-hand side f belongs to Lp , p n> .
Rozhlqnuto perßu krajovu zadaçu dlq vyrodΩenoho eliptyçno-paraboliçnoho rivnqnnq druhoho
porqdku iz, vzahali kaΩuçy, rozryvnymy koefici[ntamy. Matrycq starßyx koefici[ntiv
zadovol\nq[ paraboliçnu umovu Kordesa za prostorovymy zminnymy. Dovedeno, wo uzahal\nenyj
rozv’qzok zadaçi naleΩyt\ do prostoru Hel\dera C1+λ
, qkwo prava çastyna f naleΩyt\ Lp ,
p n> .
Introduction. Investigations of boundary-value problems for second-order degenerate
elliptic-parabolic equations ascend to the work by Keldysh [1], where correct state-
ments for boundary-value problems were considered for the case of one space variable
as well as existence and uniqueness of solutions. In the work by Fichera [2], bounda-
ry-value problems were given for multidimensional case. He proved existence of ge-
neralized solutions to these boundary-value problems. In the work by Oleynik [3],
existence and uniqueness of generalized solution to these problems were proved for
smooth and piecewise smooth domains. In the case of smooth coefficients and some
weighted functions, the generalized solvability was studied in [4] and [5]. Moreover,
the smoothness of the solution was studied and the condition (15) and the example (22)
were apparently given for the first time in the paper [5].
Let Rn+1 be an ( )n + 1 -dimensional Euclidian space of points ( , )x t = ( x1, x2, …
… , xn , t ) , Ω be a bounded n -dimensional domain in Rn with the boundary ∂Ω ,
QT = Ω-× ( 0, T ) be a cylinder in Rn+1, T ∈ ( 0, ∞ ) , Q0 = { }( , ) : ,x t x t∈ =Ω 0 and
let Γ( )QT = Q T0 0∪ ( [ , ])∂ ×Ω be a parabolic boundary of the cylinder QT
.
Let us consider in Q T the first boundary-value problem for second-order degene-
rate elliptic-parabolic operator
© T. S. GADJIEV, E. R. GASIMOVA, 2008
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 6 723
724 T. S. GADJIEV, E. R. GASIMOVA
Zu =
i j
n
ij
i j i
n
i
i
a x t u
x x
x t u
t
b x t u
x
c x t u u
t,
( , ) ( , ) ( , ) ( , )
= =
∑ ∑∂
∂ ∂
+ ∂
∂
+ ∂
∂
+ − ∂
∂1
2 2
2
1
ψ = f x t( , ), (1)
u QTΓ( ) = 0. (2)
Assume that the coefficients of the operator Z satisfy the following conditions:
a x tij ( , ) is a real symmetrical matrix with elements measurable in QT and, for any
( x, t ) ∈ QT and ξ ∈ Rn , the following inequalities are true:
γ ξ 2 ≤ a x tij
i j
n
i j( , )
, =
∑
1
ξ ξ ≤ γ ξ−1 2 , (3)
where γ ∈ ( 0, 1 ] is a constant,
σ =
sup ( , )
inf ( , )
,Q
iji j
n
Q iii
n
T
T
a x t
a x t
2
1
1
2
=
=
∑
∑[ ]
< 1
1 2n − /
, (4)
c x t( , ) ≤ 0, c x t L Qn T( , ) ( )∈ +1 , (5)
b x t L Qn T( , ) ( )∈ +2 , b x t Kc x t( , ) ( , )2 + ≤ 0. (6)
Assume that the following conditions are true for the weighted function:
ψ( , )x t = λ ρ ϕ( ) ( ) ( )w t T t− ,
where
ρ = ρ( )x = dist ( , )x ∂Ω , λ ρ( ) ≥ 0, λ ρ( ) [ , ]∈C1 0 diamΩ ,
′λ ρ( ) ≤ p λ ρ( ) , w t( ) ≥ 0, w t C T( ) [ , ]∈ 1 0 ,
ϕ( )z ≥ 0, ′ϕ ( )z ≥ 0, ϕ( ) [ , ]z C T∈ 1 0 , ϕ( )0 = ′ϕ ( )0 = 0, ϕ( )z ≥ β ϕz z′( ),
(7)
where p, β are positive constants and ψ( , )x t has bounded derivatives of the second
order.
The condition (4) is called the condition of Cordes type and is taken within the ac-
curacy of a linear nonsingular transformation. This means that the Cordes condition is
taken within the accuracy of nondegenerate linear transformation, that is the domain
QT can be covered by a finite number of domains Q QM1, ,… so that in each Qi
there exists such a nondegenerate linear transformation that a matrix of sinior
coefficients of the image of the operator Z satisfies the condition (4) in the image of
Qi , i = 1, M .
Before we move to the proof of the basic result, let us consider some auxiliary pro-
blems. Let
′L u = a x t u
x x
x t u
tij
i j
n
i j
( , ) ( , )
, =
∑ ∂
∂ ∂
+ ∂
∂1
2 2
2ψ +
+ b x t u
x
b x t u
t
c x t ui
i
n
i
( , ) ( , ) ( , )
=
∑ ∂
∂
+ ∂
∂
+
1
0 = f x t( , ). (8)
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 6
ON SMOOTHNESS OF SOLUTION OF THE FIRST BOUNDARY-VALUE PROBLEM … 725
Without loss of generality, we may assume that the coefficients are smooth in QT and
their derivatives are bounded. To speak more exactly, let us say that the coefficients
and the right-hand side have the first Hölder derivatives. Let
′L uε = a x t u
x x
x t u
tij
i j
n
i j
( , ) ( , )
, =
∑ ∂
∂ ∂
+ ∂
∂1
2 2
2ψε +
+ b x t u
x
b x t u
t
c x t ui
i
n
i
( , ) ( , ) ( , )
=
∑ ∂
∂
+ ∂
∂
+
1
0 = f x t( , ), (9)
where ψε( , )x t is defined so: for any fixed ε ∈( , )0 T
ϕε( )z = ϕ ε ϕ ε ε ϕ ε
ε
( )
( ) ( )− ′ + ′
−m m
zm
m
1 for z ∈( , ]0 ε , ϕε( )z = ϕ( )z
for z T∈[ , ]ε , m = 2
β
(we denoted by z the argument of ϕ ( )T t− ). Similarly, for
any fixed ε ∈( , )0 T
w zε( ) = w
w
m
w
m
zm
m( ) ( ) ( )ε ε ε ε
ε
− ′ − ′
−1 for z ∈( , ]0 ε ,
w zε( ) = w z( ) for z T∈[ , ]ε , m = 2
β
(we denoted by z the argument of w t( ) ).
Analogously the new value of λε( )z = λ ε ε( ) + on the correspondent segment.
We multiply all the approximated functions to obtain ψε( , )x t .
Everywhere further, we consider the case where ψ( )z > 0 for z > 0. If ψ( )z ≡
≡ 0, then the equation (1) is parabolic, and the corresponding result on smoothness of
the solution ensues from [6]. But if ψ( )z = 0 for z z∈[ , ]0 0 , then the solution to the
problem (1), (2) can be obtained by composition of the solution u x t( , ) to the first bo-
undary-value problem in the cylinder Q
z0 and the solution v( , )x t to the first boun-
dary-value problem for the parabolic equation in the sylinder Ω × ( , )z T0 with the bo-
undary conditions
v( , )x z0 = u x z( , )0 ,
v ∂ ×Ω [ , ]z T0 = 0.
Note that under the conditions (3) – (6) for the coefficients, the smoothness of the solu-
tion results from [7]. Denote by Σ0 the part of QT , where ψ( , )x t = 0, i.e., where
the equation (8) degenerates: denote by Γ0 the part of intersection of Σ0 and the
boundary Γ, where a tangent plane to the surface Γ is orthogonal to the axis t, i.e.,
has a characteristical direction.
By maximum principle, the solutions u x tε( , ) of the equation (9) in the domain sa-
tisfy the following estimate:
u x tε( , ) ≤ f x t
c x t
( , )
( , )
,
that is u x tε( , ) are uniformly bounded with respect to ε.
Lemma 1. The derivatives of the solution u x tε( , ) are uniformly bounded on a
closed subset of the boundary Γ, that belongs to Γ Γ\ 0 .
Proof. Let us take a point ( , ) \′ ∈x t Γ Γ0 such that at the point a tangent plane to
Γ is not orthogonal to the axis t, i.e., the surface Γ near the point has an equation of
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 6
726 T. S. GADJIEV, E. R. GASIMOVA
the kind x1 = θ( , , , )x x tn2 … , where θ has derivatives up to second order. Let
χ( , , , )x x tn2 … be twice continuously differentiable function equal to a positive
constant β in some neighborhood of a projection ( , )′x t onto the plane ( , , , )x x tn2 …
and equal to zero in a little greater neighborhood 0 ≤ χ( , , , )x x tn2 … ≤ β. We denote
by QT
1 the part of QT being between the surfaces Γ and σ θ χ{ }x1 = + . Let Γ1
denote that part of Γ, where χ = β . Consider a function v = e xα θ χ( )− + +1 . It is ob-
vious, that on the surface σ v = 1. Then in QT
1 , for sufficiently great α, we have
′Lε( )v ≥ α γ αµ µ2
1− − >
α γ2
2
,
′ ±L uε ε( )v >
α γ2
2
− max ( , )
QT
f x t > 0, (10)
where µ , µ1 are maximums of the modules of the solution itself and its first deriva-
tives within QT . Then we choose α independent of ε so, that (10) is true and,
moreover, eαβ > 1 + max ( , )
QT
u x tε . This means that on Γ1 the values of functions
v ± uε equal to eαβ are greater than their values on σ, where v = 1 (taking into
account that u x tε( , ) Γ = 0 ). By maximum principle, it results from the following
estimate (10) that functions v ± uε within the domain QT
1 cannot take maximal
positive value. Hence, they reach maximum on the boundary Γ, i.e., on the part Γ1
too, while on the other part of Γ v ± uε = eαχ ≤ eαβ . So, at points that belong to
Γ1, we have
∂ ±
∂
( )v u
x
ε
1
≤ 0 or
∂
∂
u x t
x
ε( , )
1 1Γ
≤ –
∂
∂
v
x1 1Γ
= α αβe .
In other words on Γ1, the derivatives
∂
∂
u x t
x
ε( , )
1
are uniformly bounded. Moreover,
derivatives of u x tε( , ) with respect to directions lying in a tangent plane are equal to
zero as u x tε( , ) Γ = 0. Thus, the derivatives
∂
∂
u x t
xi
ε( , )
, i = 1, n , are uniformly bo-
unded with respect to ε on Γ1.
Let us take a point ( , ) \′ ∈x t Γ Γ0 . Let a tangent plane to Γ at this point be
orthogonal to the axis t . This case can be proved similarly.
The lemma is proved.
Remark 1. If the boundary does not contain points of Γ0, then
∂
∂
u x t
t
ε( , )
are
uniformly bounded on the entire boundary.
Lemma 2. Suppose that on Σ0 the condition
c x t
b x t
t
( , )
( , )+ ∂
∂
0 < 0 (11)
is true and Σ1 is any closed domain with a boundary σ1, which belongs to QT .
Then at ( , )x t ∈Σ1
i
n
i
u x t
x
u x t
t=
∑ ∂
∂
+ ∂
∂
1
2 2
ε ε( , ) ( , )
≤
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 6
ON SMOOTHNESS OF SOLUTION OF THE FIRST BOUNDARY-VALUE PROBLEM … 727
≤ C
u x t
x
u x t
t
C
x t i
n
i
max
( , ) ( , )
( , )∈ =
∑ ∂
∂
+ ∂
∂
+
σ
ε ε
1 1
2 2
1, (12)
where C, C1 are constants depending on a structure of the equation.
Proof. Introduce the notation Σ Σ Σ1 0 1∩ ∩( ) = Σ2. Let us prove the inequality
in some neighborhood of closed domain Σ2. The boundary of Σ2 consists of the part
σ1 of the boundary Σ1 and the surface σ2 being in the part, where ψ( , )x t > 0. At
points ( , )x t ∈σ2 the inequality
i
n
i
u x t
x
u x t
t=
∑ ∂
∂
+ ∂
∂
1
2 2
ε ε( , ) ( , )
≤
≤ C
u x t
x
u x t
t
C
x t i
n
i
max
( , ) ( , )
( , )∈ =
∑ ∂
∂
+ ∂
∂
+
σ
ε ε
1 1
2 2
1 (13)
is true.
The following estimate (13) is obtained from the fact, that derivatives of the
solution are bounded in any closed subdomain for the case of bounded derivatives up to
the boundary of a domain. Now if we show that the following estimate (13) is also true
for the domain Σ2, the from (13) and this following estimate we will get (12) for the
domain Σ1. Assume that (11) is also satisfied in Σ2. For simplicity of calculations,
we will find following estimates for one space variable and in the end show the
changes in calculations in the case of many space variables. Without loss of generality,
we take the coefficient at second derivative with respect to a space variable x equal to
unit, as it can be easily obtained by division by terms by the coefficient. Denote
z =
∂
∂
+ ∂
∂
−u x t
t
u x t
t
u x t
m m
ε ε
εα( , ) ( , )
( , )1
2
2 .
First, we show that for corresponding n, α1, we have ′L zε > 0 in Σ2, if
∂
∂
u x t
t
ε( , ) 2
> µ1. Let n be a positive even number. We get
′ ∂
∂
+ ∂
∂
−
L
u x t
t
u x t
t
u x t
m m
ε
ε ε
εα( , ) ( , )
( , )1
2
2 = ′L zε > 0,
if
∂
∂
u x t
t
ε( , ) 2
> µ1. Now if z takes its maximum within Σ2, then at this point
L zε ≤ 0. So, either
∂
∂
u x t
t
ε( , ) 2
≤ µ1 or the value of z within Σ2 is not greater
than the maximum on the boundary Σ2. Since
∂
∂
u x t
t
ε( , ) 2
≤ z m2/ ≤ C
u x t
t
C2
2
3
∂
∂
+ε( , )
and
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 6
728 T. S. GADJIEV, E. R. GASIMOVA
∂
∂
u x t
t
ε( , ) 2
2Σ
≤ z m2
2
/
Σ
≤
C z Cm
2
2
3
2 1
max /
σ σ∪
+ <
<
C
u x t
t
C4
2
5
2 1
max
( , )
σ σ
ε
∪
∂
∂
+ , (14)
∂
∂
u x t
x
ε( , )
can be estimated similarly.
The lemma is proved.
Lemma 3. Assume that on the set Σ0, the following condition is satisfied:
∂
∂
+ ∂
∂
+
2
2
02
ψ( , ) ( , )
( , )
x t
t
b x t
t
c x t < 0 (15)
and first derivatives of u x tε( , ) are uniformly bounded in a closed domain Σ1 ⊂ QT
with the boundary σ . Then
i
n
i i j
n
i j
u x t
x t
u x t
x x
u x t
t= =
∑ ∑∂
∂ ∂
+ ∂
∂ ∂
+ ∂
∂
1
2 2
1
2 2 2
2
2
ε ε ε( , ) ( , ) ( , )
,
≤
≤ C
u x t
x t
u x t
x x
u x t
t
C
x t i
n
i i j
n
i j
max
( , ) ( , ) ( , )
( , ) ,∈ = =
∑ ∑∂
∂ ∂
+ ∂
∂ ∂
+ ∂
∂
+
σ
ε ε ε
1
2 2
1
2 2 2
2
2
1, (16)
where C, C1 do not depend on ε.
Proof. As c x t( , ) < 0,
∂
∂
2
2
ψ
t
≥ 0 on Σ0, the statement of the lemma for first
derivatives results form Lemma 2. To prove the lemma, as in the proof of Lemma 2,
we have to show that in some neighborhood of Σ Σ0 1∩ : ′L zε 1 > 0 at the correspon-
ding m (an even number) and αi . Here, z1 is the same as in Lemma 2, but it con-
tains additional terms. An element
∂
∂
2
2
u x t
t
m
ε( , )
is the main in it, so we have to esti-
mate
′ ∂
∂
L
u x t
t
m
ε
ε
2
2
( , )
= m
u x t
t
L
u x t
t
m
∂
∂
′ ∂
∂
−2
2
1 2
2
ε
ε
ε( , ) ( , )
+
+ m m
u x t
t
a x t
u x t
t x
u x t
t x
m
ij
i j
n
i j
( )
( , )
( , )
( , ) ( , )
,
− ∂
∂
∂
∂ ∂
∂
∂ ∂
−
=
∑1
2
2
2
1
3
2
3
2
ε ε ε +
+ m m
u x t
t
x t
u x t
t
m c x t
u x t
t
m m
( )
( , )
( , )
( , )
( ) ( , )
( , )− ∂
∂
∂
∂
− − ∂
∂
−
1 1
2 3
3
2 2
2
ε
ε
ε εψ .
Taking into account (15) on Σ0, for sufficiently great m, we have
– m
t
b
t
c c
m
∂
∂
+ ∂
∂
+ −
2
2
02
ψε > µ1m
in some neighborhood of Σ0. Now we choose β < µ µ1 2− , where µ2 0> , and
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 6
ON SMOOTHNESS OF SOLUTION OF THE FIRST BOUNDARY-VALUE PROBLEM … 729
fix β. Then
L
u x t
t
m
ε
ε∂
∂
2
2
( , )
>
> m m
u x t
t
u x t
t x
m
u x t
t
m
i
n
i
m
( )
( , ) ( , ) ( , )− ∂
∂
∂
∂ ∂
+ ∂
∂
−
=
∑1
2
2
2
1
3
2
2
2
2
2µ µε ε ε +
+ m m x t
u x t
t
u x t
t
m
( ) ( , )
( , ) ( , )− ∂
∂
∂
∂
−
1
2
2
2 3
3
2
µψε
ε ε –
– m
u x t
t
x t
u x t
t
u x t
x x
m
i j i j
n
i j
µ ψε
ε
ε ε
3
2
2
2 3
3
2
0
2 2
∂
∂
∂
∂
+ ∂
∂ ∂
−
+ > ≠
∑( , )
( , )
( , ) ( , )
;
+
+
i
n
i i j
n
i j
u x t
x t
u x t
x x t= =
∑ ∑∂
∂ ∂
+ ∂
∂ ∂ ∂
+
1
2 2
1
3 2
1ε ε( , ) ( , )
,
.
Let us choose sufficiently great m, so that – m m mµ µ3 1+ −( ) > µ3 > 0 and fix
m. Under this condition,
L
u x t
t
m
ε
ε∂
∂
2
2
( , )
≥ µ ε ε
4
2
2
2
1
3
2
2
∂
∂
∂
∂ ∂
−
=
∑u x t
t
u x t
t x
m
i
n
i
( , ) ( , )
+
+ µ µ ψε
ε
ε ε
5
2
2 3
2
2
2 3
3
2
∂
∂
+ ∂
∂
∂
∂
−
u x t
t
x t
u x t
t
u x t
t
m m
( , )
( , )
( , ) ( , )
–
– µ ε ε ε
4
2
2
2
0
2 2
1
2 2
∂
∂
∂
∂ ∂
+ ∂
∂ ∂
−
+ > ≠ =
∑ ∑u x t
t
u x t
x x
u x t
x t
m
i j i j
n
i j i
n
i
( , ) ( , ) ( , )
;
+
+
i j
n u x t
t,
( , )
=
∑ ∂
∂
+
1
3
3
2
1ε .
Having obtained the other estimates similarly to Lemma 2, we get the statement of the
lemma.
The lemma is proved.
Lemma 4. Let the condition (15) be satisfied on the set Σ0 and the boundary
of QT have no points of Σ0. Then in the closed domain QT , derivatives of
u x tε( , ) with respect to space up to the second-order variables are uniformly boun-
ded.
Proof. Let us take a point ( , )x t∗ ∗ ∈Γ and let in its neighborhood the boundary
Γ be presented in the form x1 = ϕ ( , , , )x x tn2 … . By means of change of variables
t = t∗, ξ1 = x x x tn1 2− …ϕ ( , , , ), ξ2 = x2, … , ξn = xn in the neighborhood of
( , )x t∗ ∗ , the equation (9) is reduced to the form
L uε ε
∗ =
i j
n
ij
i j i
n
i
i
a t
u
t
u
t
b t
u
,
( , ) ( , )
( )
( , )
=
∗ ∗ ∗ ∗
∗
=
∗ ∗∑ ∑∂
∂ ∂
+ ∂
∂
+ ∂
∂1
2 2
2
1
ξ
ξ ξ
ψ ξ ξ
ξ
ε
ε
ε ε +
+ b t
u
t
c t u0
∗ ∗
∗
∗ ∗∂
∂
+( , ) ( , )ξ ξε
ε = f t∗ ∗( , )ξ , (17)
where a t11
∗ ∗( , )ξ ≥ µ > 0, c t∗ ∗( , )ξ < 0, and due to assumptions on smoothness of
the coefficients and boundary, the coefficients of (17) have uniformly bounded
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730 T. S. GADJIEV, E. R. GASIMOVA
derivatives. The boundary Γ will have the equation ξ1 = 0 in the neighborhood of
( , )x t∗ ∗ . For clarity, we take the axis ξ1 to be pointed into QT . As in Lemma 1, we
denote by χ ξ ξ( , , , )2 … ∗
n t a nonnegative twice continuously differentiable function
equal to the constant β inside some neighborhood Γ1 of the point ( , )x t∗ ∗ on the
boundary Γ and equal to zero outside a little greater neighborhood 0 ≤ ≤χ β. The
part of the domain QT lying between the boundary Γ{ }ξ1 0= and σ ξ χ ξ{ ( ,1 2= …
… , ξ γ αn t, ) / }∗ , will be denoted by QT
ε . Further, α will be chosen as depending on
ε, and γ as not depending on ε. In QT , the uniform boundedness results from
Lemma 2 for first derivatives of u x tε( , ) with respect to xi and t, and hence, with
respect to ξi , t∗ in a neighborhood of ( , )x t∗ ∗ . By Lemma 3, second derivatives of
u x tε( , ) are estimated via their values on the boundary, and as second derivatives with
respect to xi and t, as well as with respect to ξi , t∗, are mutually expressed by each
other and by first derivatives in a neighborhood of ( , )x t∗ ∗ in a uniformly bounded
way, so
∂
∂ ∂
+ ∂
∂ ∂
+ ∂
∂
∗ ∗ ∗2 2 2
2
u t u t
t
u t
ti j i
ε ε εξ
ξ ξ
ξ
ξ
ξ( , ) ( , ) ( , )
< µ ε µH( ) + 1 (18)
at ( , )ξ t QT
∗ ∈ , i, j = 1, n . Here, a maximum of second derivatives on the boundary
Γ is denoted by H( )ε .
If at the point ( , )x t∗ ∗ a tangent plane to Γ is orthogonal to the axis t, then by de-
finition of Σ0, at the point, and that’s in some its neighborhood, ψ µε( , )x t > >1 0 .
Thus, for each point ( , )x t∗ ∗ ∈Γ , a neighborhood exists on the boundary such that
∂
∂ ∂
+ ∂
∂ ∂
+ ∂
∂
2 2 2
2
u u
t
u
ti j i
ε ε ε
ξ ξ ξ
< µ ε µ6 7H( ) + , i, j = 1, n .
Taking a finite number of such neighborhoods covering Γ, and taking into account the
smoothness of change of coordinates in each of these neighborhoods, we get
∂
∂ ∂
+ ∂
∂ ∂
+ ∂
∂
2 2 2
2
u
x x
u
x t
u
ti j i
ε ε ε < µ ε µ8 9H( ) +
on entire boundary Γ or, due to definition of H( )ε , H( )ε ≤ µ ε µ10 11H( ) + .
Hence, H( )ε < µ12, i.e., we have boundedness of second derivatives on the bounda-
ry, and by Lemma 3, in the whole domain QT . Here, we used only boundedness of
first derivatives of coefficients of the equation (17).
The lemma is proved.
Now we can move to the proof of existence and the uniqueness theorem for the first
boundary-value problem for the equation (8).
Theorem 1. Let the equation (8) defined in a cylindrical domain QT with the
boundary Γ , degenerate on the set Σ0 ⊂ QT into a parabolic one, let the
condition (3) be satisfied and let all the coefficients and the right-hand side of the
equation (8) have bounded derivatives up to the first order, satisfying the Hölder
condition. Assume that, in a cylindrical domain ′ ⊃Q QT T , ψ( , )x t ≥ 0 and the
conditions (7) are satisfied. If the boundary Γ has no points of Γ0 and the
condition (15) is satisfied on Σ0, then in QT there exists a unique solution of the
equation (8) that satisfy the condition (2) and have in QT derivatives of the first
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ON SMOOTHNESS OF SOLUTION OF THE FIRST BOUNDARY-VALUE PROBLEM … 731
order satisfying the Hölder condition; and the following estimate is true:
u C QT
1+λ ( ) ≤ K f uC Q
Q
T
T
1 λ ( ) sup+
. (19)
Proof. From Lemma 4 it results that solutions of the equation
′L u x tε ε( , ) = f x t( , ), (20)
vanishing on Γ, are uniformly bounded in the closed domain QT along with their
derivatives up to the second order. In other words, it is possible to find a sequence
u x tε( , ) such that as ε → 0, it uniformly converges to some function u x t( , ) along
with its derivatives up to the first order in the closed domain QT . And it is clear that
these derivatives of u x t( , ) will be Hölder derivatives and the function u x t( , ) equals
zero on the boundary Γ. Moreover, for such solutions, the estimate (19) is true (see [6,
p. 235], Chapter 3). Passing to the limit in the equation (20) as ε → 0, we obtain that
u x t( , ) satisfies the equation (8) and the estimate (19) is true. Uniqueness of the
solution follows directly from maximum principle.
Remark 2. From the proof of Theorem 1, the convergence of the solutions of the
equation (20) to the solution of the equation (1) as ε → 0 also follows.
Remark 3. The condition (15) cannot be omitted. There exists an essential diffe-
rence from existence theorems proved for a smooth solution of the Dirichlet problem
for elliptic equation. Let us give an example.
Example 1. Let us consider the equation
t
u x t
t
u x t
x
t
u x t
t
cu2
2
2
2
2
∂
∂
+ ∂
∂
+ ∂
∂
+( , ) ( , ) ( , )β = 0 (21)
with sufficiently smooth coefficients, where β, c are constants, c ≤ 0. It is easy to
check that the equation has a solution
u x t( , ) = t pxγ sin , (22)
γ γ βγ( )− + +1 c = p2 . (23)
The equation degenerates on the axis x . The condition (15) for the equation means
that 2 2+ +β c < 0. Let the condition be not satisfied, e.g., 2 2+ +β c > 0. Then
such p, γ < 2 exist that they satisfy (23). Let us consider the domain QT containing
a segment of the axis x , whose boundary near the axis x consists of straight lines
x = 0 and x = π / p and everywhere is sufficiently smooth. Then the solution (22)
will be sufficiently smooth on the boundary (near the axis x = 0 it is zero), but,
nevertheless, its first order derivatives will not satisfy the Hölder condition for t = 0,
0 < x < π / p .
Let us give the scheme of proof of the solvability when passing from smooth coef-
ficients to coefficients satisfying (3) – (6), (8).
First, let f x t( , ) be sufficiently smooth in QT . Denote by v( , )x t a classical
solution of the first boundary-value problem
∆v v− t = f x t( , ), ( , )x t QT∈ ,
(24)
v Γ( )QT
= 0.
It is known that the solution v( , )x t to the problem exists and v( , ) , ( )x t C QT∈ 2 1 .
Now we take an operator Lε . Let u x tε( , ) be a classical solution of the Dirichlet
problem
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 6
732 T. S. GADJIEV, E. R. GASIMOVA
L u x tε ε( , ) = f x t( , ), ( , )x t QT∈ ,
u QTε Γ( ) = 0, u t Tε = = v t T= .
Such a solution u x tε( , ) exists due to smoothness of ψε( , )x t and f x t( , ). As we
have shown, { ( , )}u x tε are uniformly bounded with respect to ε in C QT0
2 1, ( ). There-
fore, it is compact in this space, i.e., there exist such a function u x t C QT( , ) ( ),∈ 0
2 1 and
a sequence εk → 0, k → ∞ , that the corresponding sequence { ( , )}u x t
kε converges
to the function u x t C QT( , ) ( ),∈ 0
2 1 as k → ∞ . Further, we can obtainthat L u0 = f in
QT . Now let f x t L Qp T( , ) ( )∈ , p > n + 2. Then a sequence { ( , )}f x tm , f x tm( , ) ∈
∈ C QT
∞( ) exists such that
lim ( )m
m L Qf f
p T→∞
− = 0.
For natural m, denote by u x tm( , ) the sequence of solutions of the first boundary-
value problem for
u x t C Qm T( , ) ( ),∈ 0
2 1 ,
L u x tm0 ( , ) = f x tm( , ), ( , )x t QT∈ .
It is proved that the limit u x t( , ) of the sequence { ( , )}u x tm in C QT0
2 1, ( ), m → ∞ ,
satisfies in QT the equation L u x t0 ( , ) = f x t( , ).
Note that as we said above, ψ ( x, t ) > 0. If ψ ( x, t ) ≡ 0, then the equation (1) is
parabolic and that is why under the conditions (3) – (6) and f x t L Qp T( , ) ( )∈ , p > n +
+ 2, for the bounded solution of the equation (1) the following estimate is true:
u C QT
1+λ ρ( ) ≤ K f uL Q
Q
p T
T
1 ( ) sup+
. (25)
If ψ ( x, t ) > 0 and the condition of Theorem 1 is satisfied for the coefficients, then for
the bounded solution of the equation (1) the estimate (25) is true. The estimate (25)
can be obtained by composition of the solution u ( x, t ) to the problem in the cylinder
Q
z0 , where ψ ( z ) = 0 for z ∈ [ 0, z
0
] , and the solution v( , )x t to the first bounda-
ry-value problem for parabolic equation in the cylinder Ω × ( z
0, T ) with boundary
conditions v ( x, z
0
) = u ( x, z
0
) , v ∂ ×Ω [ , ]z T0 = 0. It must be noted that the theorem
has been obtained for smooth coefficients, but we can pass to f x t L Qp T( , ) ( )∈ by
means of the above mentioned scheme. Further, to prove the estimate (25) under the
conditions (3) – (7), we apply the method of continuation by parameter.
Theorem 2. Suppose that the equation (1) defined in QT degenerates on the
set Σ0 ⊂ QT into a parabolic one, the conditions (3) – (7) are satisfied for the
coefficients, and the right-hand side of the equation f x t L Qp T( , ) ( )∈ , p > n + 2. If
the boundary Γ has no points of Γ
0 and on Σ
0 the condition (15) is satisfied,
then for the bounded solution u ( x, t ) of the equation (1) the following estimate is
true:
u C QT
1+λ ρ( ) ≤ K f uL Q
Q
p T
T
1 ( ) sup+
,
where λ > 0 depends only on coefficients of the operator L and n ; and K1,
moreover, depends on p, ρ, diam QT .
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ON SMOOTHNESS OF SOLUTION OF THE FIRST BOUNDARY-VALUE PROBLEM … 733
Remark 4. Theorem 2 in this formulation is also true for the equation (1), if in the
condition (15) instead of b x t0( , ) will be taken b x t1( , ) .
Proof of Theorem 2. To prove it, we consider a family of operators Z( )τ =
= ( )1 − ′ +τ τL Z for τ ∈[ , ]0 1 , where ′L is a model operator defined from the equa-
tion (8) with Laplacian main part and smooth coefficients, and the operator Z is defin-
ed from the equation (1). Let us show that the set E of points τ, at which for
solutions of the problem
Z u( )τ = f x t( , ), ( , )x t QT∈ , (26)
u QTΓ( ) = 0, (27)
the estimate (25) is true if f x t L Qp T( , ) ( )∈ , p > n + 2, is nonempty, and open and
closed simultaneously with respect to the segment [ 0, 1 ] . Hence, E = [ 0, 1 ] and, in
particular, for the solution of the problem (26), (27) the estimate (26) is true for τ = 1,
i.e., when Z Z( )1 = . The set E is nonempty by Theorem 1. Let us show that it is
open. For this purpose, we prove that for solutions of the problem (26), (27) the
estimate (25) is true for all τ ∈[ , ]0 1 such that τ τ− 0 < ε (here, τ0 ∈E and ε >
> 0 will be chosen later). Rewrite the problem (26), (27) in the equivalent form
Z u( )τ0 = f x t Z Z u( , ) ( )( ) ( )− −τ τ0 , ( , )x t QT∈ , (28)
u x t C QT( , ) ( ),∈ 0
2 1 .
We introduce an arbitrary function v( , ) ( ), ,x t C QT∈ 0
2 1 λ and consider the first bounda-
ry-value problem
Z u( )τ0 = f x t Z Z( , ) ( )( ) ( )− −τ τ0 v , ( , )x t QT∈ , (29)
u x t C QT( , ) ( ),∈ 0
2 1 .
It is clear that ( )( ) ( ) , , ( )Z Z C QT
τ τ λ− ∈0 2 1v . Indeed, note that for all operators Z( )τ
the conditions (3) and (4) are satisfied with constants γ γτ( ) min{ , }0 ≥ n and σ στ( ) ≤ ,
respectively. Let us show this. Denote by a x tij
( )( , )τ , i n= 1, , the coefficients of the
operator Z( )τ at higher derivatives with respect to space variables. Let
ι = sup
( , )
( , )
,
Q
iji j
n
T
a x t
g x t
2
1
2
=∑
, ι τ( ) = sup
( , )
( , )
( )
,
( )Q
iji j
n
iii
n
T
a x t
a x t
τ
τ
[ ]
[ ]
=
=
∑
∑
2
1
1
2 , ι τ( ) = sup ( , )( )
QT
x tι τ ,
where
g x t( , ) = a x tii
i
n
( , )
=
∑
1
.
Taking into account (4) and the fact that for any operator of Z-type the inequality
ι ≥ 1 is true, we conclude that
ι τ( )( , )x t =
n g x t a x t
n ng x t g x t
iji j
n
( ) ( ) ( , ) ( , )
( ) ( ) ( , ) ( , )
,
1 2 1
1 2 1
2 2 2
1
2 2 2 2
− + − +
− + − +
=∑τ σ τ τ
τ τ τ τ
≤
≤ 1 12 2
2 2n
n g x t
g x t
+ −τ ι
τ
( ) ( , )
( , )
/ = ι. (30)
Let now
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 6
734 T. S. GADJIEV, E. R. GASIMOVA
λ− = inf ( , )
QT
g x t , λ+ = sup ( , )
QT
g x t , λ τ( ) =
inf ( , )
sup ( , )
( )
( )
Q iii
n
Q
iii
n
T
T
a x t
a x t
τ
τ
=
=
∑
∑
1
1
.
Calculations we made before show that λ τ( ) = ( )
( )
1
1
− +
− +
−
+
τ τλ
τ τλ
n
n
. But on the other
hand, ′λ τ( ) = λ λ
τ τλ
− +
+
−
− +[ ]( )1 2n
≤ 0. That is why,
λ ι( ) ≥ λ ( )1 = λ , (31)
(30) and (31) imply that σ τ( ) = ι
λ τ
τ( )
( )
−
−
1
2n
≤ ι
λ
−
−
1
2n
= σ , and the needed
statement is obtained.
Note that all above mentioned reasonings and Lemma 4 imply that if T ≤ T
0, the
following estimate is true for any τ ∈[ , ]0 1 and any function u x t C QT( , ) ( ), ,∈ 2 1 λ :
u C QT
2 1, , ( )λ ≤ K Z u
C QT
2
0
0
( )
( ),λ( ). (32)
For the solution u x t( , ) of the boundary-value problem (29), due to the assumption
made, the estimate (25) is true for any v( , ) ( ), ,x t C QT∈ 0
2 1 λ . Thus, an operator Φ is
defined from C QT0
2 1, , ( )λ to C QT0
2 1, , ( )λ and u = Φ v . This operator is compressing
at ε chosen in an appropriate way. Indeed, let v
( ) , ,( , ) ( )i
Tx t C Q∈ 0
2 1 λ , u i( ) = Φv( )i ,
i = 1, 2. Then, takling into account that ( )( ) ( )Z Zτ τ− 0 = ( )( )τ τ− − ′0 Z L , we con-
clude that u x t u x t( ) ( )( , ) ( , )1 2− is a classical solution of the first boundary-value prob-
lem
Z u x t u x t( ) ( ) ( )( )( , ) ( , )τ0 1 2− = ( ) ( , ) ( , )( )( )( ) ( )τ τ− − ′ −0
1 2Z L x t x tv v ,
u x t u x t C QT
( ) ( ) , ,( , ) ( , ) ( )1 2
0
2 1− ∈ λ .
Using (32), we get
u x t u x t
C QT
( ) ( )
( )
( , ) ( , ) , ,
1 2
2 1− λ ≤
≤ K Z L x t x t
C QT
2 0
1 2
0τ τ λ− − ′ −( ) ( ) ( )
( )
( , ) ( , ) ,v v . (33)
On the other hand,
( ) ( ) ( )
( )
( , ) ( , ) ,Z L x t x t
C QT
− ′ −v v1 2
0 λ ≤
K Z n T x t x t
C QT
3
1 2
2 1( , , , ) ( , ) ( , )( ) ( )
( ), ,Ω v v− λ .
So,
u x t u x t
C QT
( ) ( )
( )
( , ) ( , ) , ,
1 2
2 1− λ ≤
K K x t x t
C QT
2 3
1 2
2 1ε λv v( ) ( )
( )
( , ) ( , ) , ,− .
Now taking ε = 1 2 2 3/ K K , we prove that the operator Φ is compressing. Hence, it
has a stationary point u = Φ u , that is a classical solution of the boundary-value prob-
lem (28), and of (26), (27) as well, and for the solution the estimate (25) is true. So, we
have proved that the set E is open.
Let us show that the set E is closed. Let τk E∈ , k = 1, 2, … , lim
k
k
→∞
τ = τ. For
natural k , we denote by u x tk[ ]( , ) the solution of the first boundary-value problem
Z u x tk
k
( )
[ ]( , )τ = f x t( , ), ( , )x t QT∈ , u x tk QT
[ ] ( )
( , )
Γ
= 0, for which the following
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ON SMOOTHNESS OF SOLUTION OF THE FIRST BOUNDARY-VALUE PROBLEM … 735
estimate takes place:
u x tk C Ql T
[ ] ( )
( , ) ,2 1 ≤ K f L Qp T3 ( ). (34)
So, from (34) we obtain that the family of functions { }[ ]( , )u x tk is compact in
C QT0
2 1, ( ), i.e., there exists such a subsequence of natural numbers { }kl , lim
l
lk
→∞
= ∞
and a function u x t C QT( , ) ( ),∈ 0
2 1 that, for any ϕ( , ) ( )x t C QT∈ ∞
0 ,
lim ,( )( )
[ ]
l
kZ uk
l
l
→∞
τ ϕ = ( )( ) ,Z uτ ϕ . (35)
However,
( )( )
[ ],Z u kl
τ ϕ = (( ) )( ) ( )
[ ], ( , )Z Z u fk
l
l
k
τ τ ϕ ϕ− + = J l f1( ) ( , )+ ϕ . (36)
Moreover, taking into account (33) and (34), we have
J l1( ) ≤ τ τ ϕ− − ′kl kZ L u
l
( ) [ ], ≤ τ τ ϕ λ− kl k C Q C QK u
l T T4 2 1 0[ ] ( ) ( ), , ≤
≤ K K fkl L Q C Qp T T3 4 0τ τ ϕ λ− ( ) ( ), . (37)
It follows from (37) that lim ( )
l
J l
→∞
1 = 0. From (36) and (37) we get that ( )( ) ,Z uτ ϕ =
= ( , )f ϕ , i.e., Z u( )τ = f x t( , ), everywhere in QT . Thus, we showe that τ ∈E ,
i.e., the set E is closed.
The theorem is proved.
Now we prove some estimate for the solution, which can also be taken as an inde-
pendent result.
Theorem 3. Let the conditions (3) – (7) be satisfied for the coefficients of the
operator (1). Then for any function u x t( , ) ∈
�
W QT2
2 2
,
, ( )ψ , the following estimate is
true:
u x t C QT
( , ) ( ) ≤ k f L Qn T+1( ) , (38)
where k = k n( , )γ .
Proof. Suppose that ( , )x t QT
0 0 ∈ and at this point sup
QT
u = u x t( , )0 0 = µ > 0.
Let us take an auxiliary function z = u
m, where m ≥ 2 is a natural number, which
will be chosen later. Denote by Az the set
{ ( , ) : ( , )x t x t QT∈ , u x t( , ) ≥ 0, z x tt ( , ) ≥ 0, z x ttt ( , ) ≤ 0,
z x tij ( , ) is a positively defined matrix } .
We have
µm n( )+1 ≤ K z a z dxdtt ij ij
i j
n
n
Az
1
1
1
−
=
+
∑∫
,
≤ K z a z x t z dxdtt ij ij
i j
n
tt
n
Az
− −
=
+
∑∫
,
( , )
1
1
ψ ≤
≤ K mu Zu mu u x t b x t u x t
A
m m
i
i
n
x
z
2
1 2
1
2 1 2
∫ ∑− −
=
− +
∇
( ) ( , ) ( , ) ( , )
/
+
+ c x t u m u x t dxdtx
n
( , ) ( ) ( , )2 2
1
1− − ∇
+
γ . (39)
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 6
736 T. S. GADJIEV, E. R. GASIMOVA
If ( , )x t Az∈ is such that
∇xu x t( , ) ≥ b x t
m
u x t
( , )
( )
( , )
− 1 γ
,
then
u b u x t cu m u x tx x∇ + − − ∇( , ) ( ) ( , )2 21 γ ≤ 0.
However, if
∇xu x t( , ) ≤ b x t
m
u x t
( , )
( )
( , )
− 1 γ
for ( , )x t Az∈ ,
then
u b u x t cu m u x tx x∇ + − − ∇( , ) ( ) ( , )2 21 γ ≤ u
m
b m c
2
2
1
1
( )
( )
−
+ −( )γ
γ .
Now we take max ,2 1 +
m
γ
as ψε( , )x t m . Then from (14) we get that
µm n( )+1 ≤ K m f dxdtn m n
Q
n
Tz
2
1 1 1 1+ − + +∫µ( )( ) .
Hence, the estimate (38) with K = K mn
2
1 1/( )+ is obtained in a standard way. The case
where ( , )x t0 0 = ( , )x T0 , x0 ∈ Ω is considered similarly.
Theorem 4. The conditions of Theorem 2 be satisfied and in the cylinder QT
the solution to the first boundary-value problem (1), (2) be defined, f L Qp T∈ ( ) ,
p > n + 2. Then the following estimate is true:
u x t C QT
( , ) ( )1+λ ≤ K f L Qp T4 ( ) . (40)
Proof. To prove this, we should use the estimate (25) from Theorem 2 and the
estimate (38) from Theorem 3, which implies the estimate (40).
As a consequence of the estimate (40), we get the theorem on classical solvability
of the first boundary-value problem for the operator Z, which can be proved by the
standard Lere – Schauder method [6].
Theorem 5. Let the conditions of Theorem 2 be satisfied. Then the problem (1),
(2) has a classical solution u x t C QT( , ) ( ), ,∈ 2 1 λ and λ > 0 depends only on σ, n.
Note that classical solvability can be proved analogously to Theorem 2.
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main // Dokl. AN SSSR. – 1951. – 77, #-2. – P. 181 – 183.
2. Fichera G. On a unified theory of boundary value problem for elliptic-parabolic equations of se-
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Received 11.04.06,
after revision — 04.12.06
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 6
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| id | umjimathkievua-article-3192 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:37:57Z |
| publishDate | 2008 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/2e/a07045f2da8f45a3dc15f2088c858d2e.pdf |
| spelling | umjimathkievua-article-31922020-03-18T19:48:06Z On the smoothness of a solution of the first boundary-value problem for second-order degenerate elliptic-parabolic equations Про гладкість розв'язку першої крайової задачі для вироджених еліптично-параболічних рівнянь другого порядку Gadjiev, T. S. Gasimova, E. R. Гаджиїв, Т. С. Газимова, Є. Р. In this work, the first boundary-value problem is considered for second-order degenerate elliptic-parabolic equation with, generally speaking, discontinuous coefficients. The matrix of senior coefficients satisfies the parabolic Cordes condition with respect to space variables. We prove that the generalized solution to the problem belongs to the Holder space C 1+&lambda; if the right-hand side f belongs to Lp, p > n. Розглянуто першу крайову задачу для виродженого еліптично-параболічного рівняння другого порядку із, взагалі кажучи, розривними коефіцієнтами. Матриця старших коефіцієнтів задовольняє параболічну умову Кордеса за просторовими змінними. Доведено, що узагальнений розв'язок задачі належить до простору Гельдера C 1+&lambda;, якщо права частина f належить Lp, p > n. Institute of Mathematics, NAS of Ukraine 2008-06-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3192 Ukrains’kyi Matematychnyi Zhurnal; Vol. 60 No. 6 (2008); 723–736 Український математичний журнал; Том 60 № 6 (2008); 723–736 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/3192/3130 https://umj.imath.kiev.ua/index.php/umj/article/view/3192/3131 Copyright (c) 2008 Gadjiev T. S.; Gasimova E. R. |
| spellingShingle | Gadjiev, T. S. Gasimova, E. R. Гаджиїв, Т. С. Газимова, Є. Р. On the smoothness of a solution of the first boundary-value problem for second-order degenerate elliptic-parabolic equations |
| title | On the smoothness of a solution of the first boundary-value problem for second-order degenerate elliptic-parabolic equations |
| title_alt | Про гладкість розв'язку першої крайової задачі для вироджених еліптично-параболічних рівнянь другого порядку |
| title_full | On the smoothness of a solution of the first boundary-value problem for second-order degenerate elliptic-parabolic equations |
| title_fullStr | On the smoothness of a solution of the first boundary-value problem for second-order degenerate elliptic-parabolic equations |
| title_full_unstemmed | On the smoothness of a solution of the first boundary-value problem for second-order degenerate elliptic-parabolic equations |
| title_short | On the smoothness of a solution of the first boundary-value problem for second-order degenerate elliptic-parabolic equations |
| title_sort | on the smoothness of a solution of the first boundary-value problem for second-order degenerate elliptic-parabolic equations |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3192 |
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