Solvability of a boundary-value problem for degenerate equations
UDC 517.9 We consider a boundary-value problem for degenerate equations with discontinuous coefficients and establish the unique strong solvability (almost everywhere) of this problem in the corresponding weighted Sobolev space.
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| Дата: | 2020 |
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Institute of Mathematics, NAS of Ukraine
2020
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860512209577181184 |
|---|---|
| author | Gadjiev, T. Kerimova, M. Gasanova, G. Gadjiev, Т. Kerimova, М. Gasanova, G. Gadjiev, Т. Kerimova, М. Gasanova, G. |
| author_facet | Gadjiev, T. Kerimova, M. Gasanova, G. Gadjiev, Т. Kerimova, М. Gasanova, G. Gadjiev, Т. Kerimova, М. Gasanova, G. |
| author_sort | Gadjiev, T. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2022-03-26T11:01:36Z |
| description | UDC 517.9
We consider a boundary-value problem for degenerate equations with discontinuous coefficients and establish the unique strong solvability (almost everywhere) of this problem in the corresponding weighted Sobolev space.
|
| doi_str_mv | 10.37863/umzh.v72i4.6000 |
| first_indexed | 2026-03-24T03:25:09Z |
| format | Article |
| fulltext |
DOI: 10.37863/umzh.v72i4.6000
UDC 517.9
T. Gadjiev, M. Kerimova, G. Gasanova (Inst. Math. and Mech. Nat. Acad. Sci. Azerbaijan, Baku)
SOLVABILITY OF A BOUNDARY-VALUE PROBLEM
FOR DEGENERATE EQUATIONS
РОЗВ’ЯЗНIСТЬ ГРАНИЧНОЇ ЗАДАЧI
ДЛЯ ВИРОДЖЕНИХ РIВНЯНЬ
We consider a boundary-value problem for degenerate equations with discontinuous coefficients and establish the unique
strong solvability (almost everywhere) of this problem in the corresponding weighted Sobolev space.
Розглянуто граничну задачу для вироджених рiвнянь з розривними коефiцiєнтами. Встановлено однозначну сильну
(майже скрiзь) розв’язнiсть цiєї задачi у вiдповiдному зваженому просторi Соболєва.
1. Introduction. The purpose of this work is to prove a unique strong (almost everywhere) sol-
vability of the first boundary-value problem for equation
Zu =
n\sum
i,j=1
aij (x, t)uij + \psi (x, t)utt - ut = f (x, t) , (1.1)
u| \Gamma (QT ) = 0, (1.2)
in cylinder QT = \Omega \times (0, T ) , T \in (0,\infty ) , where \Omega is a bounded domain in Rn with a boundary
\partial \Omega \subset C2. \Gamma (QT ) = (\partial \Omega \times [0, T ]) \cup \Omega \times \{ (x, t) : t = 0\} is a parabolic boundary of the domain
QT , \psi (x, t) and coefficients aij (x, t) tend to zero. Here
uij =
\partial 2u (x, t)
\partial xi\partial xj
, utt =
\partial 2u (x, t)
\partial t2
, ut =
\partial u
\partial t
.
Initial boundary problems for this type of degenerate equations have been studied by many
authors (see, for example, [2 – 4]). In [1], Fichera considered boundary-value problems for degenerate
equations in multidimensional case. He proved existence of generalized solutions to these boundary-
value problems. Boundary-value problems for the degenerate equations of such type were studied
in the stationary case in [5] and in the nonstationary case in [6]. In [8], coercive estimates for
this problem have been obtained. We also mention the works [2 – 4] where strong solvability of
the boundary-value problem (1.1), (1.2) was established for equations with smooth coefficients.
Similar results Cordes-type discontinuous coefficients have been established in [4]. In [9, 10], some
classes of elliptic parabolic equations are considered. In [9], well-posedness of the initial boundary-
value problem for pseudoparabolic equations is studied and estimates of the generalized solution
are obtained. In [10], solvability results have been obtained in case of Cordes-type discontinuous
coefficients. In [11], some general problem for linear and quasilinear equations of parabolic type is
considered.
c\bigcirc T. GADJIEV, M. KERIMOVA, G. GASANOVA, 2020
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 4 435
436 T. GADJIEV, M. KERIMOVA, G. GASANOVA
In our paper we consider wide classes of elliptic parabolic equations.
Assume that the coefficients satisfy the conditions: | aij (x, t)| is a symmetrical matrix with real
measurable elements in QT and, for any (x, t) \in QT , \xi \in Rn, following inequalities are true:
\gamma \omega (x) | \xi | 2 \leq
n\sum
i,j=1
aij (x, t) \xi i\xi j \leq \gamma - 1\omega (x) | \xi | 2 , (1.3)
where \gamma \in (0, 1] , \omega (x) \in Ap satisfies the Muckenhoupt condition (see [7]) and
\psi (x, t) = \omega (x)\lambda (t)\varphi (T - t) , (1.4)
where
\lambda (t) \geq 0, \lambda (t) \in C1 [0, T ] ,
\varphi (z) \geq 0, \varphi \prime (z) \geq 0, \varphi (z) \in C1 [0, T ] ,
\varphi (0) = \varphi \prime (0) = 0, \varphi (z) \geq \beta z\varphi \prime (z) ,
\beta is a positive constant.
2. Auxiliary results. Our goal is to establish a unique strong solvability of the boundary-
value problem (1.1), (1.2) by means of coercive estimate obtained in [8], using coercive continuation
method by parameter. For this purpose, let us prove the solvability for some model equation from
the class under consideration. As a model operator we considering operator
Z0 = \omega (x)\Delta + \varphi (T - t)
\partial 2
\partial t2
- \partial
\partial t
,
where \Delta =
\sum n
i=1
\partial 2
\partial x2i
is a Laplace operator and function \varphi (z) satisfies the conditions (1.4).
Throughout this paper we consider the most interesting case, where \varphi (z) > 0 for z > 0. If
\varphi (z) \equiv 0, then the equation (1.1) is parabolic and the corresponding results on solvability of the
boundary-value problem was obtained in this case in [7]. But if \varphi (z) = 0 at z \in [0, z0], then the
solution of the problem (1.1), (1.2) can be obtained by combining the solution u(x, t) of the problem
in a cylinder Qz0 with the solution v(x, t) of boundary problem for parabolic equation in a cylinder
\Omega \times (z0, T ) with boundary conditions v(x, z0) = u(x, z0), v| \partial \Omega \times [z0,T ] = 0. Let us fix an arbitrary
\varepsilon \in (0, T ) and introduce a function \varphi (z) by \varphi \varepsilon (z) = \varphi (\varepsilon ) - \varphi \prime (\varepsilon )\varepsilon
m
+
\varphi \prime (\varepsilon )
m\varepsilon m - 1
zm for z \in [0, \varepsilon ),
\varphi \varepsilon (z) = \varphi (z) for z \in [\varepsilon , T ], where m =
2
\beta
. It is easy to see that \varphi \varepsilon \in C1[0, T ]. Let us show that
for z \in [0, T ]
\varphi \varepsilon (z) \geq
1
2
\varphi (z). (2.1)
It suffices to prove (2.1) for z \in [0, \varepsilon ). It is clear that due to monotonicity of \varphi (z) the inequality
(2.1) will be fulfilled if
\varphi (\varepsilon ) - \varphi \prime (\varepsilon )\varepsilon
m
\geq 1
2
\varphi (\varepsilon )
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 4
SOLVABILITY OF A BOUNDARY-VALUE PROBLEM FOR DEGENERATE EQUATIONS 437
or
\varphi (\varepsilon ) \geq 2
m
\varphi \prime (\varepsilon )\varepsilon .
The last estimate is true by condition (1.4). Hence, the inequality (2.1) is proved. Without loss of
generality, we consider the case m > 1. Then
q\varepsilon (T ) = \mathrm{s}\mathrm{u}\mathrm{p}
[0,T ]
\varphi \prime
\varepsilon (z) \leq q(T ) = \mathrm{s}\mathrm{u}\mathrm{p}
[0,T ]
\varphi (z). (2.2)
For R > 0, x0 \in Rn, we consider a ball BR(x0) = \{ x : | x - x0| < R and a cylinder QRT =
= BR(x
0)\times (0, T ). Let BR(x
0) \subset \Omega . We say that u(x, t) \subset A
\bigl(
QRT (x
0)
\bigr)
if u(x, t) \in C\infty
\Bigl(
Q
R
T (x
0)
\Bigr)
,
u| t=0 = 0 and \mathrm{s}\mathrm{u}\mathrm{p} p u
\bigl(
Q
\rho
T (x
0)
\bigr)
for some \rho \in (0, R) . We will also use the Banach spaces
W 1,0
2 (QT ) , W
2,0
2 (QT ) , W
2,1
2 (QT ) and W 2,2
2,\psi (QT ) of functions u(x, t) given on QT with finite
norms (see also [8])
\| u\| W 1
2,\omega (QT )
=
\left( \int
QT
\omega (x)
\Biggl(
u2 +
n\sum
i=1
u2xi
\Biggr)
dxdt
\right)
1
2
,
\| u\| W 2
2,\omega (QT )
=
\left( \int
QT
\omega (x)
\left( u2 + n\sum
i=1
u2xi +
n\sum
i,j=1
u2xixj
\right) dxdt
\right)
1
2
,
\| u\|
W 2,1
2,\omega (QT )
= \| u\| W 2
2,\omega (QT )
+ \| ut\| L2(QT )
,
\| u\|
W 2,2
2,\psi (QT )
=
\left( \int
QT
\left[ \omega (x)
\left( u2 + n\sum
i=1
u2xi +
n\sum
i,j=1
u2xixj
\right) + u2t+
+\psi 2(x, t)u2tt + \psi (x, t)
n\sum
i=1
u2it
\right] dxdt
\right)
1
2
.
Let
\circ
W
2,2
2,\psi (QT ) be a subspace of W 2,2
2,\psi (QT ) consisting of all functions from C\infty \bigl( QT \bigr) which vanish
on \Gamma (QT ) and form a dense set in W 2,2
2,\psi (QT ) .
Let us consider the operator
Z\varepsilon = \omega (x)\Delta + \varphi \varepsilon (T - t)
\partial 2
\partial t2
- \partial
\partial t
.
Lemma 2.1. If \omega (x) satisfies the Muckenhoupt condition [7], then there exists T1(\varphi (z), \omega (x), n)
such that , for T \leq T1 and any function u(x, t) \in B(QRT (x
0)), the following estimate is true:\int
QRT (x
0)
\left( \omega (x) n\sum
i,j=1
u2ij + u2t + \varphi \varepsilon (T - t)
n\sum
i=1
u2it + \varphi \varepsilon (T - t)
n\sum
i=1
u2it
\right) dxdt \leq
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 4
438 T. GADJIEV, M. KERIMOVA, G. GASANOVA
\leq (1 + 2(n+ 1)q(T ))
\int
QRT (x
0)
Z\varepsilon (u)
2dxdt. (2.3)
(We say that u(x, t) \in B(QRT (x
0)) if u(x, t) \in A(QRT (x
0)), u| t=T = ut| t=T 0.)
Proof. For simplicity we shall write QT instead of QRT (x
0). We have\int
Q
(Z\varepsilon u)
2 dxdt \geq
\int
Q
\omega (x)
n\sum
i,j=1
u2ijdxdt+
\int
Q
\varphi 2
\varepsilon (T - t)u2ttdxdt+
+
\int
Q
u2tdxdt+ 2
\int
Q
\varphi \varepsilon (T - t)\Delta uuttdxdt - 2
\int
Q
\varphi \varepsilon (T - t)ututtdxdt. (2.4)
But, on the other hand,
2
\int
Q
\varphi \varepsilon (T - t)\Delta u uttdxdt - 2
\int
Q
n\sum
i,j=1
(\varphi \varepsilon (T - t)uii)t utdxdt =
= 2
\int
Q
\varphi \prime
\varepsilon (T - t)
n\sum
i=1
uiiutdxdt - 2
\int
Q
\varphi \varepsilon (T - t)
n\sum
i=1
uiitutdxdt \geq
\geq - q\varepsilon (T )
\int
Q
\omega (x)
n\sum
i,j=1
u2ijdxdt - nq\varepsilon (T )
\int
Q
u2tdxdt+
+2
\int
Q
\varphi \varepsilon (T - t)
n\sum
i=1
u2i dxdt, (2.5)
because
uii| t=0 = uii| t=T = 0.
Also we have
- 2
\int
Q
\varphi \varepsilon (T - t)ututtdxdt = -
\int
Q
\varphi \prime
\varepsilon (T - t)
n\sum
i=1
u2i dxdt+
+\varphi \varepsilon (T )
\int
B
u2t (x, 0)dxdt \geq - q\varepsilon (T )
\int
Q
u2tdxdt, (2.6)
because
uii| t=T = 0.
By virtue of conditions (1.4) for T \rightarrow 0, we have q(T ) \rightarrow 0. Choose T1 such that
(n+ 1)q(T1) \leq
1
2
.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 4
SOLVABILITY OF A BOUNDARY-VALUE PROBLEM FOR DEGENERATE EQUATIONS 439
Then, for T \leq T1, we have
1
1 - (n+ 1)q(T )
\leq 1 + 2(n+ 1)q(T ).
Now using this inequality (2.2), and Lemma 1 of [8], we get the estimate (2.3).
Lemma 2.1 is proved.
Lemma 2.2. Let the function \varphi (z) satisfies the conditions (1.4), and \omega (x) satisfies the Mucken-
houpt condition. Let the operators Z\varepsilon with \varepsilon > 0 be the same as in Lemma 2.1. Then, for
T \leq T2(\varphi , \omega , n,\Omega ) and any function u(x, t) \in
\circ
W
2,2
2,\varphi \varepsilon (QT ), the following estimate is true:
\| u\| \circ
W
2,2
2,\varphi \varepsilon
(QT )
\leq C1(\varphi , \omega , n,\Omega ) \| Z\varepsilon u - \mu u\| L2(QT )
, (2.7)
where \mu =
1
T
, W 2,2
2,\varphi \varepsilon
(QT ) is a Banach space of functions defined above with function \psi replaced
by \varphi \varepsilon .
Proof. It suffices to prove the lemma for functions u(x, t) \in C\infty \bigl( QT \bigr) , u| \partial QT = 0. Note that,
according to the above mentioned, q
\biggl(
T
T1
\biggr)
\leq 1. Then, as in the proof of coercive estimate [8], we
derive from (1.1) the existence of T3(\varphi , \omega , n,\Omega ) \leq T1 such that if T \leq T3, then for any function
v(x, t) \in C\infty \bigl( QT \bigr) , v| \Gamma (QT ) = 0, v| t=T = vt| t=T = 0 the following estimate is true:
\| v\|
W 2,2
2,\varphi \varepsilon
(QT )
\leq C2(\varphi , \omega , n,\Omega )
\Bigl(
\| Z\varepsilon v\| L2(QT )
+ \| v\| L2(QT )
\Bigr)
. (2.8)
Let T \leq T3/2. We take R = T/4 and let u(x, t) \in C\infty \bigl( QT \bigr) , u| \partial QT = 0. We consider a
function g(t) \in C\infty [0, T ] such that g(t) = 1 for t \in [0, T - R], g(t) = 0 at t \in
\biggl[
T - R
2
, T
\biggr]
,
0 \leq g(t) \leq 1 and \bigm| \bigm| g\prime (t)\bigm| \bigm| \leq C3/R,
\bigm| \bigm| g\prime (t)\bigm| \bigm| \leq C3/R
2. (2.9)
Putting in (2.8) v(x, t) = u(x, t)g(t) and taking into account (2.9), we get
\| u\|
W 2,2
2,\varphi \varepsilon
(QT - R)
\leq C4(\varphi , \omega , n,\Omega )
\Bigl(
\| Z\varepsilon (ug)\| L2(QT )
+ \| u\omega (x)\| L2(QT )
\Bigr)
\leq
\leq C5(\varphi , \omega , n,\Omega )
\biggl(
\| Z\varepsilon u\| L2(QT )
+
\biggl(
C6
R
+ 1
\biggr)
\| u\omega (x)\| L2(QT )
\biggr)
+
+
2C6
R
\| \varphi \varepsilon ut\| L2(QT )
+
2C6
R2
\| \varphi \varepsilon u\| L2(QT )
. (2.10)
From the conditions (1.4) it follows that \mathrm{s}\mathrm{u}\mathrm{p}[0,T ] \varphi (z) \leq C7(\varphi )T. So, taking into account that
\mathrm{s}\mathrm{u}\mathrm{p}[0,T ] \varphi \varepsilon (z) \leq \mathrm{s}\mathrm{u}\mathrm{p}[0,T ] \varphi (z), we conclude
\| \varphi \varepsilon u\| L2(QT )
\leq C7T \| u\omega (x)\| L2(QT )
. (2.11)
On the other hand, for any \alpha \prime > 0 the interpolation inequality
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 4
440 T. GADJIEV, M. KERIMOVA, G. GASANOVA
\| \varphi \varepsilon ut\| L2(QT )
\leq C8T\alpha
\prime \| \varphi \varepsilon utt\| L2(QT )
+
1
\alpha \prime \| u\| L2(QT )
(2.12)
holds. Indeed, let us fix an arbitrary \alpha \prime and consider for \nu > 0 the integral
k =
\int
QT
\biggl[
\nu \varphi 2
\varepsilon (T - t)utt +
1
\nu
u\omega (x)
\biggr] 2
dxdt.
It is clear that k \geq 0. At the same time
k = \nu 2
\int
QT
\varphi 2
\varepsilon (T - t)u2ttdxdt+
1
\nu 2
\int
QT
\omega (x)u2dxdt+ 2
\int
QT
\varphi 2
\varepsilon (T - t)uttudxdt \leq
\leq C2
8T
2\nu 2
\int
QT
\varphi 2
\varepsilon (T - t)u2ttdxdt+
1
\nu 2
\int
QT
\omega (x)u2dxdt - 2
\int
QT
\varphi 2
\varepsilon (T - t)u2ttdxdt+
+4
\int
QT
\varphi \varepsilon (T - t)\varphi \prime
\varepsilon (T - t)uutdxdt.
Besides, by using the fact that q(T ) \leq 1 and the inequality (2.2), we get
4
\int
QT
\varphi \varepsilon (T - t)\varphi \prime
\varepsilon (T - t)uutdxdt \leq
\int
QT
\varphi 2
\varepsilon (T - t)u2tdxdt+
+4
\int
QT
(\varphi \prime
\varepsilon (T - t))u2dxdt \leq
\int
QT
\varphi 2
\varepsilon (T - t)u2tdxdt+ 4
\int
QT
\omega (x)u2dxdt. (2.13)
From (2.12), (2.13) it follows that\int
QT
\varphi 2
\varepsilon (T - t)u2tdxdt \leq C2
8T
2\nu 2
\int
QT
\varphi 2
\varepsilon (T - t)u2ttdxdt+
+
\biggl(
1
\nu 2
+ 4
\biggr) \int
QT
\omega (x)u2dxdt.
Now putting \nu = \mathrm{m}\mathrm{i}\mathrm{n}\{ \alpha \prime , 1\} we prove the inequality (2.12).
By using (2.11) and (2.12) in (2.10), we conclude that, for any \alpha \prime > 0, the inequality
\| u\|
W 2,2
2,\varphi \varepsilon
(QT - R)
\leq C4 \| Z\varepsilon (ug)\| L2(QT )
+ 8\alpha \prime C4C5C6 \| u\| W 2,2
2,\varphi \varepsilon
(QT )
+
+
C7(\varphi , \omega , n,\Omega )
\alpha \prime R
\| u\omega (x)\| L2(QT )
(2.14)
holds.
Let us fix an arbitrary \alpha > 0 and choose \alpha \prime =
\alpha
C4C5C6
. Then from (2.14) follows that
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 4
SOLVABILITY OF A BOUNDARY-VALUE PROBLEM FOR DEGENERATE EQUATIONS 441
\| u\|
W 2,2
2,\varphi \varepsilon
(QT - R)
\leq C4 \| Z\varepsilon u)\| L2(QT )
+ \alpha \| u\|
W 2,2
2,\varphi \varepsilon
(QT )
+
+
C8(\varphi , \omega , n,\Omega )
\alpha T 2
\| u\omega (x)\| L2(QT )
. (2.15)
Similarly, we can show that if Q\prime = \Omega \times (T - 2R, T + 2R), Q\prime \prime = \Omega \times (T - R, T + R) ,
S (Q\prime ) = \partial \Omega \times [T - 2R, T + 2R] , then for any function W (x, t) \in C\infty \bigl( QT \bigr) , W\omega (x)| S(QT ) = 0
and for any \alpha > 0 the following estimate is true:
\| W\|
W 2,2
2,\varphi \varepsilon
(Q\prime \prime ) \leq C4 \| Z\varepsilon W\| L2(Q\prime ) + \alpha \| W\|
W 2,2
2,\varphi \varepsilon
(Q\prime )+
+
C9(\varphi , \omega , n,\Omega )
\alpha \prime R
\| W\omega (x)\| L2(QT )
. (2.16)
Let Q\prime
t = \Omega \times (T - 2R, T ), Q\prime
- = \Omega \times (T, T + 2R), Q\prime \prime
+ = \Omega \times (T - R, T ). Let us extend
the function \varphi \varepsilon (T - t) - ih through the hyperplane t = T from Q\prime
+ onto Q\prime
- in an even and odd
ways. Denote the extended functions also by u(x, t) and \varphi \varepsilon (T - t), respectively.
Putting w = u in (2.16) and taking into account the inequality
\| u\|
W 2,2
2,\varphi \varepsilon
(Q\prime \prime
+)
\leq
\surd
2 \| u\|
W 2,2
2,\varphi \varepsilon
(Q\prime
+)
and similar inequalities for the norms \| u\|
W 2,2
2,\varphi \varepsilon
(Q\prime ) , \| u\| L2(Q\prime ) , \| Z\varepsilon u\| L2(Q\prime ) , we get
\| u\|
W 2,2
2,\varphi \varepsilon
(Q\prime \prime
+)
\leq C10 \| Z\varepsilon u\| L2(Q\prime
+) + \alpha \| u\|
W 2,2
2,\varphi \varepsilon
(Q\prime
+)
+
+
C11(\varphi , \omega , n,\Omega )
\alpha T
\| u\| L2(Q\prime
+) . (2.17)
Combining (2.15), (2.17) and choosing the corresponding \alpha , we conclude
\| u\| 2
W 2,2
2,\varphi \varepsilon
(Q+)
\leq C12(\varphi , \omega , n,\Omega ) \| Z\varepsilon u\| 2L2(QT )
+
1
T 2
\| u\omega (x)\| 2L2(QT )
. (2.18)
On the other hand, recalling that \mu =
1
T
, we have\int
QT
(Z\varepsilon u - \mu u)2 dxdt = \| Z\varepsilon u\| 2L2(QT )
+ \mu 2 \| u\omega (x)\| 2L2(QT )
- 2\mu
\int
QT
uZ\varepsilon udxdt =
= \| Z\varepsilon u\| 2L2(QT )
+ \mu 2 \| u\omega (x)\| 2L2(QT )
+K1. (2.19)
Moreover,
K1 = - 2\mu
\int
QT
u\omega (x) (\Delta u+ \varphi \varepsilon (T - t)utt - ut)dxdt =
= 2\mu
\int
QT
n\sum
i=1
\omega (x)u2i dxdt - 2\mu
\int
QT
\varphi \varepsilon (T - t)uuttdxdt+ \mu
\int
QT
(u2)tdxdt \geq
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 4
442 T. GADJIEV, M. KERIMOVA, G. GASANOVA
\geq 2\mu
\int
QT
\varphi \varepsilon (T - t)uuttdxdt - 2\mu
\int
QT
\varphi \prime
\varepsilon (T - t)uutdxdt. (2.20)
Let us show that for z \in (0, T )
\varphi \varepsilon (z) \geq \beta z\varphi \prime
\varepsilon (z). (2.21)
Due to (1.4) it suffices to prove (2.21) only for z \in (0, \varepsilon ). But for such z the inequality (2.21)
is equivalent to the inequality
\varphi (\varepsilon ) - \varphi \prime (\varepsilon )\varepsilon
m
\geq \varphi \prime (\varepsilon )zm
m\varepsilon m - 1
, where m =
2
\beta
.
The last inequality is true if the following estimate holds:
\varphi (\varepsilon ) \geq 2
m
\varphi \prime (\varepsilon )\varepsilon . (2.22)
Note that (2.22) is follows from the conditions (1.4). Hence, from (2.20), (2.21) and (2.22), we
obtain
k1 \geq - \mu
2
\int
QT
[\varphi \prime
\varepsilon (T - t)]2
\varphi \varepsilon (T - t)
u2dxdt \geq \mu
2\beta
\int
QT
\varphi \prime
\varepsilon (T - t)
T - t
u2dxdt \geq
\geq - \mu q(T )T
2\beta
\int
QT
\omega (x)u2
(T - t)2
dxdt. (2.23)
We apply the Hardy inequality according to which\int
QT
\omega (x)u2
(T - t)2
dxdt \leq 4
\int
QT
u2tdxdt. (2.24)
Then, from (2.19), (2.23) and (2.24), we conclude
\| Z\varepsilon u\| 2L2(QT )
+ \mu 2 \| \omega (x)u\| 2L2(QT )
\leq \| Z\varepsilon u - \mu u\| 2L2(QT )
+
+
2q(T )
\beta
\| u\|
W 2,2
2,\varphi \varepsilon
(QT )
. (2.25)
Now let us choose T4(\varphi , \omega , n,\Omega ) small enough to satisfy
q(T4) \leq
\beta
4C12
and fix T2 = \mathrm{m}\mathrm{i}\mathrm{n}
\biggl\{
T3
2
, T4
\biggr\}
.
Then, from (2.18) and (2.25), we obtain the estimate (2.7).
Lemma 2.2 is proved.
Now let us establish the solvability of our problem for a model equation. Let us consider the
operator
Z \prime
0 = \omega (x)\Delta + \psi (x, t)
\partial 2
\partial t2
- \partial
\partial t
,
where \Delta =
\sum n
i=1
\partial 2
\partial x2i
is a Laplace operator.
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SOLVABILITY OF A BOUNDARY-VALUE PROBLEM FOR DEGENERATE EQUATIONS 443
Lemma 2.3. If \omega (x) satisfies the Muckenhoupt condition and \psi (x, t) satisfies the conditions
(1.4), then, for T \leq TS(\psi ), \tau \in [0, 1] and any function u(x, t) \in A
\bigl(
QRT (x
0)
\bigr)
, the following
estimate is true: \int
QRT (x
0)
\left( \omega 2(x)
n\sum
i,j=1
u2ij + u2t + \psi 2(x, t)u2tt + \psi (x, t)
n\sum
i=1
u2it
\right) dxdt \leq
\leq (1 +D(T )S2)
\int
QRT (x
0)
\Bigl(
Z \prime
0u - \tau
T
\omega (x)u
\Bigr) 2
dxdt, (2.26)
where S2 = S2 (\psi , n) is some constant, D(T ) = q(T ) + q1(T ), q1(T ) = \mathrm{s}\mathrm{u}\mathrm{p}t\in [0,T ] \varphi (t), q(T ) =
= \mathrm{s}\mathrm{u}\mathrm{p}t\in [0,T ] \varphi
\prime (t).
Proof. It suffices to consider the case \tau > 0. We denote
\tau
T
by \mu \prime . Then we have
I1 =
\int
QRT (x
0)
\bigl(
Z \prime
0u - \mu \prime \omega (x)u
\bigr) 2
dxdt =
\int
QRT (x
0)
\bigl(
Z \prime
0u
\bigr) 2
dxdt+
+
\bigl(
\mu \prime
\bigr) 2 \int
QRT (x
0)
\omega (x)u2dxdt - 2\mu \prime
\int
QRT (x
0)
\omega (x)u\Delta udxdt+
+2\mu \prime
\int
QRT (x
0)
uutdxdt - 2\mu \prime
\int
QRT (x
0)
\psi (x, t)uttudxdt. (2.27)
In Lemma 2.1 of [8], the following estimate has been obtained:\int
QRT (x
0)
\left( \omega 2(x)
n\sum
i,j=1
u2ij + u2t + \psi 2(x, t)u2tt + \psi (x, t)
n\sum
i=1
u2it
\right) dxdt \leq
\leq (1 +DS)
\int
QRT (x
0)
\bigl(
Z \prime
0u
\bigr) 2
dxdt,
where S = S (\psi , n) is some constant.
We can rewrite it as follows:\int
QRT (x
0)
\bigl(
Z \prime
0u
\bigr) 2
dxdt \geq 1
1 + SD(T )
\int
QRT (x
0)
\left( \omega 2(x)
n\sum
i,j=1
u2ij + u2t+
+ \psi 2(x, t)u2tt + \psi (x, t)
n\sum
i=1
u2it
\right) dxdt.
But
1
1 + SD(T )
= 1 - SD(T )
1 + SD(T )
\geq 1 - SD(T ) and
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444 T. GADJIEV, M. KERIMOVA, G. GASANOVA
\int
QRT (x
0)
\bigl(
Z \prime
0u
\bigr) 2
dxdt \geq (1 - SD(T ))
\int
QRT (x
0)
\left( \omega 2(x)
n\sum
i,j=1
u2ij + u2t+
+ \psi 2(x, t)u2tt + \psi (x, t)
n\sum
i=1
u2it
\right) dxdt.
We have used the last inequality to estimate the first term in (2.27). For the third term in (2.27)
we have
- 2\mu \prime
\int
QRT (x
0)
\omega 2(x)u\Delta udxdt = 2\mu \prime \mu
\int
QRT (x
0)
\omega xi(x)
n\sum
i=1
u2i dxdt \geq
\geq 2\mu \prime M
\int
QRT (x
0)
\omega (x)
n\sum
i=1
u2i dxdt \geq 0,
where M = \mathrm{s}\mathrm{u}\mathrm{p}QRT (x0)
| u(x)| .
For the fourth term we get
2\mu \prime
\int
QRT (x
0)
\omega 2(x)uutdxdt = \mu \prime
\int
QRT (x
0)
\omega 2(x)u2(x, T )dx \geq 0.
Let us consider the fifth term in (2.27) in detail:
- 2\mu \prime
\int
QRT (x
0)
\psi (x, t)uttudxdt = - 2\mu \prime
\int
QRT (x
0)
\varphi (T - t)\lambda (t)\omega (x)uttudxdt =
= - 2\mu \prime
\int
QRT (x
0)
\psi (x, t)u2tdxdt - 2\mu \prime
\int
QRT (x
0)
\varphi \prime (t - T )\lambda (t)\omega (x)utudxdt -
- 2\mu \prime
\int
QRT (x
0)
\varphi (T - t)\lambda \prime (t)\omega (x)utudxdt \geq
\geq - 2\mu \prime
\int
QRT (x
0)
\varphi \prime (T - t)\lambda (t)\omega (x) | u| | ut| dxdt -
- 2\mu \prime
\int
QRT (x
0)
\varphi (T - t) | \lambda (t)| \omega (x) | u| | ut| dxdt \geq - \mu \prime C13(\omega )C14(\lambda )2q(T )\times
\times
\int
QRT (x
0)
u2tdxdt -
\mu \prime
\alpha
C13(\omega )C14(\lambda )q(T )
\int
QRT (x
0)
\omega 2(x)u2dxdt -
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SOLVABILITY OF A BOUNDARY-VALUE PROBLEM FOR DEGENERATE EQUATIONS 445
- \mu \prime C13(\omega )C14(\lambda )\alpha q1(T )
\int
QRT (x
0)
u2tdxdt -
- \mu
\prime
\alpha
C13(\omega )C14(\lambda )q1(T )
\int
QRT (x
0)
\omega 2(x)u2dxdt. (2.28)
Let C15 = \mathrm{m}\mathrm{a}\mathrm{x}\{ C13, C14\} , C16 = C13C15. Then, from (2.28), we obtain
- 2\mu \prime
\int
QRT (x
0)
\psi (x, t)uttudxdt \geq - C15\alpha D(T )
\int
QRT (x
0)
u2tdxdt -
- \mu
\prime
\alpha
C15D(T )
\int
QRT (x
0)
\omega 2(x)u2dxdt. (2.29)
Let T \leq TS(\psi ) be so small that C15D(T ) \leq 1. Then, taking into account the above inequalities,
from (2.27) we get
I1 \geq (1 - SD(T ))
\int
QRT (x
0)
\left( \omega 2
n\sum
i,j=1
u2ij + u2t+
+ \psi 2(x, t)u2tt + \psi (x, t)
n\sum
i=1
u2it
\right) dxdt+
\bigl(
\mu \prime
\bigr) 2 \int
QRT (x
0)
\omega 2u2dxdt -
- \mu \prime C15\alpha D(T )
\int
QRT (x
0)
u2tdxdt -
\mu \prime
\alpha
\int
QRT (x
0)
\omega 2(x)u2dxdt.
If we put \alpha =
1
\mu \prime
, then we have
I1 \geq (1 - SD(T ))
\int
QRT (x
0)
\left( \omega 2(x)
n\sum
i,j=1
u2ij + u2t+
+ \psi 2(x, t)u2tt + \psi (x, t)
n\sum
i=1
u2it
\right) dxdt -
- C15D(T )
\int
QRT (x
0)
u2tdxdt = (1 - S3D(T ))
\int
QRT (x
0)
\left( \omega 2(x)
n\sum
i,j=1
u2ij+
+ u2t + \psi 2(x, t)u2tt + \psi (x, t)
n\sum
i=1
u2it
\right) dxdt,
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446 T. GADJIEV, M. KERIMOVA, G. GASANOVA
where S3 = S + C15. Hence,
\int
QRT (x
0)
\left( \omega 2(x)
n\sum
i,j=1
u2ij + u2t + \psi 2(x, t)u2tt + \psi (x, t)
n\sum
i=1
u2it
\right) dxdt \leq
\leq 1
1 - S3D(T )
I1 +
S3D(T )
1 - S3D(T )
I1.
Let T5 be so small that S3D(T ) \leq 1
2
. Then
\int
QRT (x
0)
\left( \omega 2(x)
n\sum
i,j=1
u2ij + u2t + \psi 2(x, t)u2tt + \psi (x, t)
n\sum
i=1
u2it
\right) dxdt \leq
\leq (1 + 2S3D(T )) I1 = (1 + S4D(T )) I1.
So, we get the needed estimate (2.26).
Lemma 2.3 is proved.
Lemma 2.4. Let the coefficients of the operator Z satisfy the conditions (1.3), (1.4). Then,
for any function u(x, t)C\infty \bigl( QT \bigr) , u| \Gamma (QT ) = 0, for T \leq T6 (\gamma , \psi , n,\Omega ) and any \tau \in [0, 1], the
following estimate is true:
\| u\|
W 2,2
2,\varphi \varepsilon
(Q
\prime \prime
+)
\leq C16 (\gamma , \psi , n)
\bigm\| \bigm\| \bigm\| Zu - \tau
T
\omega 2(x)u
\bigm\| \bigm\| \bigm\|
L2(QT )
.
Proof is similar to the proof of coercive estimate for the operator Z in [8].
In what follows, we will denote the operators Z0 - \mu and Z\varepsilon - \mu by M0 and M, respectively.
We will also denote T0 = \mathrm{m}\mathrm{i}\mathrm{n}\{ T9, T6\} .
3. Strong solvability of boundary-value problem. Main results.
Theorem 3.1. Let the function \varphi (z) satisfies the conditions (1.4). Then, for T \leq T 0, the
boundary-value problem
M0u = f(x, t)(x, t) \in QT , (3.1)
u| \Gamma (QT ) = 0, (3.2)
has a unique strong solution in the space W 2,2
2,\varphi (QT ) for any function f(x, t) \in L2(QT ).
Proof. First assume that f(x, t) \in C\infty (QT ) . Let v(x, t) be a classical solution of the boundary-
value problem
\omega (x)\Delta v - vt = f(x, t), (x, t) \in QT ,
v| \Gamma (QT ) = 0.
It is clear that this solution exists and due to [7, 9]
v(x, t) \in W 2,2
2,\omega (QT ),
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SOLVABILITY OF A BOUNDARY-VALUE PROBLEM FOR DEGENERATE EQUATIONS 447
and
\| v\|
W 2,2
2,\omega (QT )
\leq C17(n,\Omega , f), (3.3)
where W 2,2
2,\omega (QT ) is a Banach space of functions given on QT with finite norms of W 2,2
2,\psi (QT ) type.
For \varepsilon \in (0, T ) we have \varphi \varepsilon (z) \leq 1. Then, we conclude from (3.3) that
\| v\|
W 2,2
2,\varphi \varepsilon
(QT )
\leq C17. (3.4)
We denote by
\circ
W
2,2
2,\omega (QT ) the complement of a set of all functions from C\infty (QT ) vanishing with
respect to the norm of the space W 2,2
2,\omega (QT ), and by u\varepsilon (x, t) the strong (almost everywhere) solution
of the problem
M\varepsilon u
\varepsilon = f(x, t), (x, t) \in QT ,
(u\varepsilon (x, t) - v(x, t)) \in
\circ
W
2,2
2,\omega (QT ).
This solution exists for every \varepsilon > 0 due to [7]. It is clear that (u\varepsilon (x, t) - v(x, t)) \in W 2,2
2,\varphi \varepsilon
(QT ).
Taking into account v| \Gamma (QT ) = 0 and the inequality (2.1), we get
u\varepsilon (x, t) \in
\circ
W
2,2
2,\varphi \varepsilon (QT ).
Moreover, for F\varepsilon (x, t) =M\varepsilon v, taking into account (3.3), we have
\| F\varepsilon \| L2(QT )
\leq C18(n,\Omega , T, f). (3.5)
From Lemma 2.2 it follows that
\| u\varepsilon - v\|
W 2,2
2,\varphi \varepsilon
(QT )
\leq C1
\Bigl(
\| f\| L2(QT )
+ \| F\varepsilon \| L2(QT )
\Bigr)
.
Then, from (3.3), (3.4) and (2.1) we conclude
\| u\varepsilon \|
W 2,2
2,\varphi (QT )
\leq C15 \| u\| W 2,2
2,\varphi \varepsilon
(QT )
\leq C20(n,\Omega , T, f).
Thus, a family of functions \{ u\varepsilon (x, t)\} is bounded by the norm of the space W 2,2
2,\varphi (QT ) uniformly with
respect to \varepsilon . So, this family is weakly compact in
\circ
W
2,2
2,\varphi (QT ). This means, in particular, that there
exist the sequences of positive numbers \{ \varepsilon k\} , \mathrm{l}\mathrm{i}\mathrm{m}k\rightarrow \infty \varepsilon k = 0 and a function u0(x, t) \in W 2,2
2,\varphi (QT )
such that for any h(x, t) \in C\infty \bigl( QT \bigr)
\mathrm{l}\mathrm{i}\mathrm{m}
k\rightarrow \infty
(\mu 0u
\varepsilon k , h) = (\mu 0u0, h) , (3.6)
where (a, b) =
\int
QT
abdxdt.
But
(\mu 0u
\varepsilon k , h) = ((\mu 0 - \mu \varepsilon k)u
\varepsilon k , h) + \mu \varepsilon ku
\varepsilon k , h) = ((\mu 0 - \mu \varepsilon k)u
\varepsilon k , h) + (f, h). (3.7)
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448 T. GADJIEV, M. KERIMOVA, G. GASANOVA
Besides, taking into account (2.1) and (3.5), we have
J(k) = | (\mu 0 - \mu \varepsilon k)u
\varepsilon k , h)| \leq \| (\varphi - \varphi \varepsilon k)u
\varepsilon k
tt \| L2(Q(\varepsilon k))
\| h\| L2(Q(\varepsilon k))
\leq
\leq 3 \| u\varepsilon \|
W 2,2
2,\varphi \varepsilon k
(QT )
\| h\| L2(Q(\varepsilon k))
\leq 3C20 \| h\| L2(Q(\varepsilon k))
, (3.8)
where Q(\varepsilon ) = \Omega \times (T - \varepsilon , T ) . Thus, we have J(k) \rightarrow 0 as k \rightarrow \infty . From (3.6) – (3.8) it follows
that (\mu 0u0, h) = (f, h) and \mu 0u0 = f(x, t) almost everywhere in QT . Now let f(x, t) \in L2(QT ).
In this case there exists a sequence \{ fm(x, t)\} , m = 1, 2, . . . , such that fm(x, t) \in C\infty \bigl( QT \bigr)
and \mathrm{l}\mathrm{i}\mathrm{m}m\rightarrow \infty \| fm - f\| L2(QT )
= 0. For any positive integer m, consider a sequence \{ um(x, t)\} of
strong solutions of the boundary-value problems
M0um = fm(x, t), (x, t) \in QT ,
um| \Gamma (QT ) = 0.
Based on the above, we can say that for any m there exist the function um(x, t) such that using
the estimate obtained in the previous lemma, for the operator Z \prime
0 and \tau = 1, we get
\| um\| W 2,2
2,\varphi (QT )
\leq C21 \| fm\| L2(QT )
\leq C20 (\varphi , \omega , n,\Omega , f) . (3.9)
Thus, the sequence \{ um(x, t)\} is weakly compact in
\circ
W
2,2
2,\varphi (QT ), i.e., there exists a subsequence
\{ mk\} \in N, \mathrm{l}\mathrm{i}\mathrm{m}k\rightarrow \infty mk = \infty and a function u(x, t) \in
\circ
W
2,2
2,\varphi (QT ), such that for any h(x, t) \in
\in C\infty \bigl( QT \bigr) \mathrm{l}\mathrm{i}\mathrm{m}k\rightarrow \infty (\mu 0umk , h) = (\mu 0u, h) . But
\mathrm{l}\mathrm{i}\mathrm{m}
k\rightarrow \infty
(M0umk , h) = \mathrm{l}\mathrm{i}\mathrm{m}
k\rightarrow \infty
(fmk , h) = (f, h).
Therefore, (\mu 0umk , h) = (f, h) and \mu 0u = f(x, t) almost everywhere in QT . Thus, the existence
of strong solution of the problem (3.1), (3.2) is proved. The uniqueness of the solution follows from
Lemma 2.4.
Theorem 3.1 is proved.
Theorem 3.2. Let the coefficients of the operator Z satisfy the conditions (1.3), (1.4). Then, for
T \leq T 0, the boundary-value problem (1.1), (1.2) has a unique strong solution for f(x, t) \in L2(QT )
and the following estimate is true:
\| um\| W 2,2
2,\psi (QT )
\leq C21 \| f\| L2(QT )
. (3.10)
Proof. The estimate (3.10) and the uniqueness of the solution follow from the coercive estimate
in [8]. Therefore, we only need to prove the existence of the solution. Consider a family of operators
Z(\tau ) = (1 - \tau )\mu 0 + \tau Z for \tau \in [0, 1].
Let us show that the set E of points \tau for which the problem
Z(\tau )u = f(x, t), (x, t) \in QT , (3.11)
u| \Gamma (QT ) = 0, (3.12)
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SOLVABILITY OF A BOUNDARY-VALUE PROBLEM FOR DEGENERATE EQUATIONS 449
has a unique strong solution in
\circ
W
2,2
2,\psi (QT ) for any function f(x, t) \in L2(QT ), is nonempty and
simultaneously open and closed with respect to [0, 1] . Hence, we get E = [0, 1] and, in particular,
the problem (3.11), (3.12) is solvable at \tau = 1, i.e., when Z(1) = Z.
The nonemptiness of the set E follows directly from Theorem 3.1. Let us prove its openness.
Let \tau 0 \in E. \varepsilon > 0 will be specified later. Let us show that the problem (3.11), (3.12) is solvable.
Then, we can rewrite this problem in the following equivalent form:
Z(\tau )u = f(x, t) -
\Bigl(
Z(\tau ) - Z(\tau 0)
\Bigr)
u, (x, t) \in QT , (3.13)
where u(x, t) \in
\circ
W
2,2
2,\psi (QT ). It is clear that
\bigl(
Z(\tau ) - Z(\tau 0)
\bigr)
v(x, t) \in L2(QT ). Note that for all
operators Z(\tau ) the conditions (1.3) and (1.4) with constants \gamma \prime (\tau ) \geq \mathrm{m}\mathrm{i}\mathrm{n}\{ \gamma \prime , n\} are fulfilled. Now let
us note that from the above mentioned considerations and Lemma 2.4 it follows that for T \leq T 0,
any \tau = [0, 1] and any function u(x, t) \in W 2,2
2,\psi (QT ) the following estimate is true:
\| um\| W 2,2
2,\psi (QT )
\leq C22
\bigm\| \bigm\| \bigm\| Z(\tau )u
\bigm\| \bigm\| \bigm\|
L2(QT )
. (3.14)
By the assumption, the boundary-value problem (3.13) has a strong solution u(x, t) for any v(x, t) \in
\in W 2,2
2,\psi (QT ) . Thus, the operator F from
\circ
W
2,2
2,\psi (QT ) into
\circ
W
2,2
2,\psi (QT ) is defined and
u = Fv.
Operator F is a contration operator for properly chosen \varepsilon . Indeed, let
v(i)(x, t) \in W 2,2
2,\psi (QT ), u(i) = Fv(i), i = 1, 2.
Then, taking into account the equality\Bigl(
Z(\tau ) - Z(\tau i)
\Bigr)
= (\tau - \tau 0) (Z - \mu 0) ,
we conclude that u(1)(x, t) - u(2)(x, t) is a strong solution of the boundary-value problem
Z(\tau 0)
\Bigl(
u(1)(x, t) - u(2)(x, t)
\Bigr)
= (\tau - \tau 0) (Z - \mu 0)
\Bigl(
v(1)(x, t) - v(2)(x, t)
\Bigr)
,\Bigl(
u(1)(x, t) - u(2)(x, t)
\Bigr)
\in W 2,2
2,\psi (QT ).
By using (3.14), we get \bigm\| \bigm\| \bigm\| u(1)(x, t) - u(2)(x, t)
\bigm\| \bigm\| \bigm\|
W 2,2
2,\psi (QT )
\leq
\leq C23 | \tau - \tau 0|
\bigm\| \bigm\| \bigm\| (Z - \mu 0)
\Bigl(
v(1) - v(2)
\Bigr) \bigm\| \bigm\| \bigm\|
L2(QT )
. (3.15)
On the other hand, \bigm\| \bigm\| \bigm\| (Z - M0)
\Bigl(
v(1)(x, t) - v(2)(x, t)
\Bigr) \bigm\| \bigm\| \bigm\|
L2(QT )
\leq
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450 T. GADJIEV, M. KERIMOVA, G. GASANOVA
\leq C24 (Z, n,\Omega , T )
\bigm\| \bigm\| \bigm\| v(1)(x, t) - v(2)(x, t)
\bigm\| \bigm\| \bigm\|
W 2,2
2,\psi (QT )
. (3.16)
Thus, \bigm\| \bigm\| \bigm\| u(1)(x, t) - u(2)(x, t)
\bigm\| \bigm\| \bigm\|
W 2,2
2,\psi (QT )
\leq C23C24
\bigm\| \bigm\| \bigm\| v(1)(x, t) - v(2)(x, t)
\bigm\| \bigm\| \bigm\|
W 2,2
2,\psi (QT )
.
Now choosing \varepsilon =
1
2
C23C24 we prove that the operator F has a fixed point u = Fu, which is a
strong solution of the boundary-value problem (3.13) and, consequently, (3.11), (3.12). Therefore,
the openness of the set E is proved.
Now let the set E be closed. Let \tau k \in E, k = 1, 2, . . . , and \mathrm{l}\mathrm{i}\mathrm{m}k\rightarrow \infty \tau k = \tau . For positive
integer k, denote by u[k](x, t) a strong solution of the boundary-value problem
Z(\tau k)u[k](x, t) = f(x, t), (x, t) \in QT ,
u[k](x, t)
\Gamma (QT )
= 0.
According to (3.14), we have \bigm\| \bigm\| u[k](x, t)\bigm\| \bigm\| W 2,2
2,\psi (QT )
\leq C25 \| f\| L2(QT )
. (3.17)
So, the family of functions \{ u[k](x, t)\} is weakly compact in
\circ
W
2,2
2,\psi (QT ), i.e., there exists a subse-
quence of positive integers \{ kl\} \mathrm{l}\mathrm{i}\mathrm{m}l\rightarrow \infty ki = \infty and a function u(x, t) \in
\circ
W
2,2
2,\psi (QT ), such that for
any \psi (x, t) \in C\infty \bigl( QT \bigr)
\mathrm{l}\mathrm{i}\mathrm{m}
l\rightarrow \infty
\Bigl(
Z(\tau kl)u[k], \psi
\Bigr)
=
\Bigl(
Z(\tau )u, \psi
\Bigr)
. (3.18)
But \Bigl(
Z(\tau kl)u[kl], \psi
\Bigr)
=
\Bigl(
Z(\tau ) - Z(\tau kl)u[k], \psi
\Bigr)
+ (f, \psi ) = J1(l) + (f, \psi ). (3.19)
Moreover, taking into account (3.15) and (3.16), we have
| J1(l)| \leq | \tau - \tau kl |
\bigm| \bigm| ((Z - \mu 0)u[k], \psi )
\bigm| \bigm| \leq | \tau - \tau kl | C26
\bigm\| \bigm\| u[kl]\bigm\| \bigm\| W 2,2
2,\psi (QT )
,
\| \psi \| L2(QT )
\leq C25C26 | \tau - \tau kl | \| f\| L2(QT )
\| \psi \| L2(QT )
. (3.20)
From (3.20) it follows that \mathrm{l}\mathrm{i}\mathrm{m}l\rightarrow \infty J1(l) = 0.
Further, from (3.18) and (3.19) we conclude that\Bigl(
Z(\tau )u, \psi
\Bigr)
= (f, \psi ),
i.e.,
Z(\tau )u = f(x, t)
almost everywhere in QT . So, we have showen that \tau \in E, i.e., the set E is closed.
Theorem 3.2 is proved.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 4
SOLVABILITY OF A BOUNDARY-VALUE PROBLEM FOR DEGENERATE EQUATIONS 451
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Received 03.09.16
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 4
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| id | umjimathkievua-article-6000 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:25:09Z |
| publishDate | 2020 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/ab/7bd5ed67e1f276ece676519765b6d4ab.pdf |
| spelling | umjimathkievua-article-60002022-03-26T11:01:36Z Solvability of a boundary-value problem for degenerate equations Solvability of a boundary-value problem for degenerate equations Solvability of a boundary-value problem for degenerate equations Gadjiev, T. Kerimova, M. Gasanova, G. Gadjiev, Т. Kerimova, М. Gasanova, G. Gadjiev, Т. Kerimova, М. Gasanova, G. вісовий простір Соболєва вироджувальність еліптичні-параболічні рівняння розв'язність solvability weighted Sobolev space elliptic-parabolic equations degenerated UDC 517.9 We consider a boundary-value problem for degenerate equations with discontinuous coefficients and establish the unique strong solvability (almost everywhere) of this problem in the corresponding weighted Sobolev space. УДК 517.9 Розглянуто граничну задачу для вироджених рівнянь з розривними коефіцієнтами.Встановлено однозначну сильну (майже скрізь) розв'язність цієї задачі у відповідному зваженому просторі Соболєва. УДК 517.9 Розглянуто граничну задачу для вироджених рівнянь з розривними коефіцієнтами.&nbsp;Встановлено однозначну сильну (майже скрізь) розв'язність цієї задачі у відповідному зваженому просторі Соболєва. Institute of Mathematics, NAS of Ukraine 2020-03-28 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/6000 10.37863/umzh.v72i4.6000 Ukrains’kyi Matematychnyi Zhurnal; Vol. 72 No. 4 (2020); 435-451 Український математичний журнал; Том 72 № 4 (2020); 435-451 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/6000/8699 |
| spellingShingle | Gadjiev, T. Kerimova, M. Gasanova, G. Gadjiev, Т. Kerimova, М. Gasanova, G. Gadjiev, Т. Kerimova, М. Gasanova, G. Solvability of a boundary-value problem for degenerate equations |
| title | Solvability of a boundary-value problem for degenerate equations |
| title_alt | Solvability of a boundary-value problem for degenerate equations Solvability of a boundary-value problem for degenerate equations |
| title_full | Solvability of a boundary-value problem for degenerate equations |
| title_fullStr | Solvability of a boundary-value problem for degenerate equations |
| title_full_unstemmed | Solvability of a boundary-value problem for degenerate equations |
| title_short | Solvability of a boundary-value problem for degenerate equations |
| title_sort | solvability of a boundary-value problem for degenerate equations |
| topic_facet | вісовий простір Соболєва вироджувальність еліптичні-параболічні рівняння розв'язність solvability weighted Sobolev space elliptic-parabolic equations degenerated |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/6000 |
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