On solvability of one class of third order differential equations
UDC 517.9 One-dimensional mixed problem for one class of third order partial differential equation with nonlinear right-hand side is considered. The concept of generalized solution for this problem is introduced. By the Fourier method, the problem of existence and uniqueness of generalized solution...
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| author | Bilalov, B. T. Ismailov, M. I. Kasumov, Z. A. Bilalov, B. T. Ismailov, M. I. Kasumov, Z. A. |
| author_facet | Bilalov, B. T. Ismailov, M. I. Kasumov, Z. A. Bilalov, B. T. Ismailov, M. I. Kasumov, Z. A. |
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| description | UDC 517.9
One-dimensional mixed problem for one class of third order partial differential equation with nonlinear right-hand side is considered. The concept of generalized solution for this problem is introduced. By the Fourier method, the problem of existence and uniqueness of generalized solution for this problem is reduced to the problem of solvability of the countable system of nonlinear integro-differential equations. Using Bellman's inequality, the uniqueness of generalized solution is proved. Under some conditions on initial functions and the right-hand side of the equation, the existence theorem for the generalized solution is proved using the method of successive approximations. |
| doi_str_mv | 10.37863/umzh.v73i3.195 |
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DOI: 10.37863/umzh.v73i3.195
UDC 517.9
B. T. Bilalov (Inst. Math. and Mech. NAS Azerbaijan, Baku),
M. I. Ismailov (Baku State Univ., Azerbaijan),
Z. A. Kasumov (Inst. Math. and Mech. NAS Azerbaijan, Baku)
ON SOLVABILITY OF ONE CLASS OF THIRD ORDER
DIFFERENTIAL EQUATIONS*
ПРО РОЗВ’ЯЗНIСТЬ ОДНОГО КЛАСУ ДИФЕРЕНЦIАЛЬНИХ РIВНЯНЬ
ТРЕТЬОГО ПОРЯДКУ
One-dimensional mixed problem for one class of third order partial differential equation with nonlinear right-hand side
is considered. The concept of generalized solution for this problem is introduced. By the Fourier method, the problem of
existence and uniqueness of generalized solution for this problem is reduced to the problem of solvability of the countable
system of nonlinear integro-differential equations. Using Bellman’s inequality, the uniqueness of generalized solution is
proved. Under some conditions on initial functions and the right-hand side of the equation, the existence theorem for the
generalized solution is proved using the method of successive approximations.
Розглянуто одновимiрну мiшану задачу для одного класу диференцiальних рiвнянь третього порядку з частинними
похiдними з нелiнiйною правою частиною. Введено поняття узагальненого розв’язку для цiєї задачi. За допомогою
методу Фур’є задачу iснування та єдиностi узагальненого розв’язку зведено до задачi розв’язностi злiченної системи
нелiнiйних iнтегро-диференцiальних рiвнянь. З використанням нерiвностi Беллмана доведено єдинiсть узагальне-
ного розв’язку. При деяких умовах на початковi функцiї та праву частину рiвняння на основi методу послiдовних
iтерацiй доведено теорему про iснування узагальненого розв’язку.
1. Introduction. In the last century, there has been considerable interest in local and nonlocal
boundary-value problems for partial differential equations with time and spatial variables. The theory
and applications of local and nonlocal boundary-value problems for third order PDEs have been
studied by many mathematicians (see [1 – 12]).
Many problems of elasticity theory such as the problem of longitudinal oscillations of non uni-
form viscoelastic rod, the problem of longitudinal impact of perfectly rigid body on non uniform
finite-length viscoelastic rod with variable cross section, the problem of wave propagation in a visco-
elastic body, etc. are reduced to the solution in the domain D = (0, T ) \times (0, \pi ) (T is any positive
number) to the mixed problem for the equation
utt(t, x) - \alpha utxx(t, x) = F (u)(t, x), (1.1)
with initial and boundary conditions
u(0, x) = \varphi (x), ut(0, x) = \psi (x), 0 \leq x \leq \pi , (1.2)
u(t, 0) = 0, u(t, \pi ) = 0, 0 \leq t \leq T, (1.3)
where 0 < \alpha is a fixed number, F is in general a given nonlinear operator, \varphi , \psi are the given
functions from certain space of functions. Definition of the solution of problem (1.1) – (1.3) is given
* This work was supported by the Science Development Foundation under the President of the Republic of Azerbai-
jan (grant No. EIF-ETL-2020-2(36)-16/04/1-M04) and by the Scientific and Technological Research Council of Turkey
(TUBITAK) with Azerbaijan National Academy of Sciences (project No. P19042020).
c\bigcirc B. T. BILALOV, M. I. ISMAILOV, Z. A. KASUMOV, 2021
314 ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 3
ON SOLVABILITY OF ONE CLASS OF THIRD ORDER DIFFERENTIAL EQUATIONS 315
on the next section. Earlier in [13 – 15], the existence and uniqueness of the classical solution of
the problem (1.1) – (1.3) for t > 0 and 0 < x < \pi as well as its behavior for t \rightarrow +\infty have been
considered. In [16 – 18], existence and uniqueness theorems for classical solution of this problem
have been proved under some conditions on problem data and the properties of the solution have been
explored. A priori estimates which allow obtaining the conditions for the existence, uniqueness and
asymptotic stability of the solution have been treated in [19, 20]. Similar matters have been studied
in [21 – 24]. The existence and uniqueness of classical, generalized and almost everywhere solutions
of (1.1) – (1.3) have been considered in [25 – 27]. Using the method of successive approximations
and the principles of Krasnoselski, Schauder and Leray – Schauder, in [27] proved local and global
existence and uniqueness theorems for classical, generalized and almost everywhere solutions of the
problem (1.1) – (1.3).
Fourier’s well-known method of separation of variables is applicable also for solving the prob-
lem (1.1) – (1.3). Among the works devoted to justification of the Fourier method for solving such
problems we can mention the works [27 – 31]. Note that in [31] introduced the Banach spaces B2,1
2,2,T
of the functions u(t, x) of the form
u(t, x) =
\infty \sum
n=1
un(t) \mathrm{s}\mathrm{i}\mathrm{n}nx, (1.4)
considered in the set D, with un(t) \in C(2)
\bigl(
[0, T ]
\bigr)
, equipped with the finite norm
\| u\|
B2,1
2,2,T
=
\Biggl( \infty \sum
n=1
\biggl(
n2 \mathrm{m}\mathrm{a}\mathrm{x}
0\leq t\leq T
\bigm| \bigm| un(t)\bigm| \bigm| \biggr) 2\Biggr)
1
2
+
\Biggl( \infty \sum
n=1
\biggl(
n \mathrm{m}\mathrm{a}\mathrm{x}
0\leq t\leq T
\bigm| \bigm| u\prime n(t)\bigm| \bigm| \biggr) 2
\Biggr) 1
2
.
The generalized solutions of (1.1) – (1.3) are considered in the space B2,1
2,2,T . Using Fourier method,
they reduced the problem (1.1) – (1.3) to the countable system of nonlinear integro-differential equa-
tions, which, in turn, was reduced to finding a fixed point of some nonlinear operator in corresponding
space. In [32, 33], the continuous dependence (in some sense) of solution of the problem (1.1) – (1.3)
on F, \varphi , \psi has been considered. In [34, 35], the estimates for the classical, generalized and almost
everywhere solutions of (1.1) – (1.3) have been treated. In works [1, 12, 14, 16 – 19] using the ope-
rator approach, it is established stability estimates for the solution of the boundary-value problem for
third order partial differential equations.
In this paper, we consider the problem (1.1) – (1.3) in the Banach space B
1+ 2
q
, 2
q
p,p,T of functions of
the form (1.4) with the coefficients un(t) \in C(2)
\bigl(
[0, T ]
\bigr)
, equipped with the norm
\| u\|
B
1+2
q , 2q
p,p,T
=
\left( \Biggl( \infty \sum
n=1
\biggl(
n
1+ 2
q \mathrm{m}\mathrm{a}\mathrm{x}
0\leq t\leq T
\bigm| \bigm| un(t)\bigm| \bigm| \biggr) p
\Biggr) 1
p
+
\Biggl( \infty \sum
n=1
\biggl(
n
2
q \mathrm{m}\mathrm{a}\mathrm{x}
0\leq t\leq T
\bigm| \bigm| u\prime n(t)\bigm| \bigm| \biggr) p
\Biggr) 1
p
\right) ,
where p > 2 and p, q are the numbers conjugate of each other. We prove the existence and
uniqueness theorems for the generalized solution of (1.1) – (1.3). To justify our method, we use the
analog of Littlewood – Paley theorem for vector-valued coefficients of decomposition of a function
with respect to the system \{ \mathrm{s}\mathrm{i}\mathrm{n}nx\} in the sense of b-basis [36 – 38]. The matter of existence of
the solution is reduced to finding the coefficients of the sought solution in concrete Banach space of
sequences of functions.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 3
316 B. T. BILALOV, M. I. ISMAILOV, Z. A. KASUMOV
2. Some basic notations and auxiliary facts. In this section, we introduce some that we need
notations and the definition of the generalized solution of the problem (1.1) – (1.3). By Lp,p - 2(0, \pi ),
p \geq 2, we denote the Banach space of functions f(x) \in Lp(0, \pi ), with
\{ fn\} n\in N \in lp,p - 2, fn =
2
\pi
\pi \int
0
f(x) \mathrm{s}\mathrm{i}\mathrm{n}nxdx,
equipped with the norm
\| f\| Lp,p - 2(0,\pi )
=
\Biggl( \infty \sum
n=1
np - 2 | fn| p
\Biggr) 1
p
,
where lp,p - 2 the Banach space of sequences of scalars \lambda = \{ \lambda n\} n\in N , with the norm \| \lambda \| lp,p - 2
=
=
\Bigl( \sum \infty
n=1
np - 2| \lambda n| p
\Bigr) 1
p
, Lp,p - 2
\bigl(
[0, T ], Lp(0, \pi )
\bigr)
, p \geq 2, will denote the set of vector-functions
u(t) : [0, T ] \rightarrow Lp(0, \pi ), such that
\| u\| p
Lp,p - 2
\bigl(
[0,T ],Lp(0,\pi )
\bigr) = \infty \sum
n=1
np - 2
T\int
0
\bigm| \bigm| un(t)\bigm| \bigm| pdt < +\infty ,
where un(t) =
2
\pi
\int \pi
0
u(t, x) \mathrm{s}\mathrm{i}\mathrm{n}nxdx. The space Lp,p - 2
\bigl(
[0, T ], Lp(0, \pi )
\bigr)
is a Banach space with
respect to the norm \| u\|
Lp,p - 2
\bigl(
[0,T ],Lp(0,\pi )
\bigr) .
Let X be some Banach space. Denote by the W (1)
p ([a, b], X) set of vector-functions u : [a, b] \rightarrow
\rightarrow X such that for all t \in [0, T ] there exist in X a strong limit \mathrm{l}\mathrm{i}\mathrm{m}\Delta t\rightarrow 0
u(t+\Delta t) - u(t)
\Delta t
= u\prime (t)
and it holds
\| u\| p
W
(1)
p ([a,b],X)
=
b\int
a
\| u(t)\| pXdt+
b\int
a
\bigm\| \bigm\| u\prime (t)\bigm\| \bigm\| p
X
dt < +\infty .
The space W (1)
p ([a, b], X) is a Banach space with respect to the norm \| u\|
W
(1)
p ([a,b],X)
.
In the sequel, the elements of the space W
(1)
1
\bigl(
[0, T ], Lq(0, \pi )
\bigr)
will be denoted in the form
u = u(t, x), where t \in (0, T ), x \in (0, \pi ).
Before giving the definition for the generalized solution of the problem (1.1) – (1.3), we make
some remarks.
Remark 2.1. If p > 2,
1
p
+
1
q
= 1, then the space B
1+ 2
q
, 2
q
p,p,T is continuously embedded in the space
B2,1
2,2,T and the following relation is true:
\| u\|
B2,1
2,2,T
\leq
\biggl(
\pi 2
6
\biggr) 2 - q
2q
\| u\|
B
1+2
q , 2q
p,p,T
.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 3
ON SOLVABILITY OF ONE CLASS OF THIRD ORDER DIFFERENTIAL EQUATIONS 317
In fact, for u(t, x) \in B
1+ 2
q
, 2
q
p,p,T , we have
\| u\|
B2,1
2,2,T
=
\Biggl( \infty \sum
n=1
\biggl(
n2 \mathrm{m}\mathrm{a}\mathrm{x}
0\leq t\leq T
\bigm| \bigm| un(t)\bigm| \bigm| \biggr) 2\Biggr)
1
2
+
\Biggl( \infty \sum
n=1
\biggl(
n \mathrm{m}\mathrm{a}\mathrm{x}
0\leq t\leq T
\bigm| \bigm| u\prime n(t)\bigm| \bigm| \biggr) 2
\Biggr) 1
2
=
=
\Biggl( \infty \sum
n=1
\biggl(
n
1+ 2
q \mathrm{m}\mathrm{a}\mathrm{x}
0\leq t\leq T
\bigm| \bigm| un(t)\bigm| \bigm| \biggr) 2 n 2(q - 2)
q
\Biggr) 1
2
+
\Biggl( \infty \sum
n=1
\biggl(
n
2
q \mathrm{m}\mathrm{a}\mathrm{x}
0\leq t\leq T
\bigm| \bigm| u\prime n(t)\bigm| \bigm| \biggr) 2 n 2(q - 2)
q
\Biggr) 1
2
.
Applying Hölder’s inequality with index
p
2
\Bigl(
whose conjugate is
q
2 - q
\Bigr)
to every sum in the last
equality, we obtain
\| u\|
B2,1
2,2,T
\leq
\Biggl( \infty \sum
n=1
\biggl(
n
1+ 2
q \mathrm{m}\mathrm{a}\mathrm{x}
0\leq t\leq T
\bigm| \bigm| un(t)\bigm| \bigm| \biggr) p
\Biggr) 1
p
\Biggl( \infty \sum
n=1
n - 2
\Biggr) 2 - q
2q
+
+
\Biggl( \infty \sum
n=1
\biggl(
n
2
q \mathrm{m}\mathrm{a}\mathrm{x}
0\leq t\leq T
\bigm| \bigm| u\prime n(t)\bigm| \bigm| \biggr) p
\Biggr) 1
p
\Biggl( \infty \sum
n=1
n - 2
\Biggr) 2 - q
2q
=
=
\Biggl( \infty \sum
n=1
n - 2
\Biggr) 2 - q
2q
\left( \Biggl( \infty \sum
n=1
\biggl(
n
1+ 2
q \mathrm{m}\mathrm{a}\mathrm{x}
0\leq t\leq T
\bigm| \bigm| un(t)\bigm| \bigm| \biggr) p
\Biggr) 1
p
+
\Biggl( \infty \sum
n=1
\biggl(
n
2
q \mathrm{m}\mathrm{a}\mathrm{x}
0\leq t\leq T
\bigm| \bigm| u\prime n(t)\bigm| \bigm| \biggr) p
\Biggr) 1
p
\right) =
=
\biggl(
\pi 2
6
\biggr) 2 - q
2q
\| u\|
B
1+2
q , 2q
p,p,T
.
Obviously, if u(t, x) \in B
1+ 2
q
, 2
q
p,p,T , then the convergence of the series
\sum \infty
n=1
n\| un\| C[0,T ] and\sum \infty
n=1
\| u\prime n\| C[0,T ] implies u(t, x), ut(t, x), ux(t, x) \in C(D).
Remark 2.2. For u(t, x) \in B
1+ 2
q
, 2
q
p,p,T there exist partial derivatives uxx, utx and
uxx(t, x) = -
\infty \sum
n=1
n2un(t) \mathrm{s}\mathrm{i}\mathrm{n}nx, utx(t, x) =
\infty \sum
n=1
nu\prime n(t) \mathrm{c}\mathrm{o}\mathrm{s}nx.
Besides uxx, utx \in C
\bigl(
[0, T ];Lp,p - 2(0, \pi )
\bigr)
. In fact, we have
\infty \sum
n=1
\bigl(
n2| un(t)|
\bigr) p
np - 2 \leq
\infty \sum
n=1
\Bigl(
n
1+ 2
q \| un\| C[0,T ]
\Bigr) p
< +\infty \forall t \in [0, T ].
Then, by Paley theorem (see [39, p. 182]), there exists f(t, x) \in Lp(0, \pi ) such that f(t, x) =
= -
\sum \infty
n=1
n2un(t) \mathrm{s}\mathrm{i}\mathrm{n}nx and
\pi \int
0
\bigm| \bigm| f(t, x)\bigm| \bigm| pdx \leq Ap
\infty \sum
n=1
np - 2
\bigl(
n2| un(t)|
\bigr) p
.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 3
318 B. T. BILALOV, M. I. ISMAILOV, Z. A. KASUMOV
Hence we obtain f(t, x) \in C
\bigl(
[0, T ];Lp,p - 2(0, \pi )
\bigr)
. Moreover, f(t, x) = uxx(t, x). In fact,
f(t, x) =
\partial
\partial x
x\int
0
f(t, y)dy =
\partial
\partial x
\left( -
\infty \sum
n=1
n2un(t)
x\int
0
\mathrm{s}\mathrm{i}\mathrm{n}nydy
\right) =
=
\partial 2
\partial x2
x\int
0
\Biggl( \infty \sum
n=1
nun(t) \mathrm{c}\mathrm{o}\mathrm{s}ny
\Biggr)
dy =
\partial 2
\partial x2
\left( \infty \sum
n=1
nun(t)
x\int
0
\mathrm{c}\mathrm{o}\mathrm{s}nydy
\right) =
=
\partial 2
\partial x2
\Biggl( \infty \sum
n=1
un(t) \mathrm{s}\mathrm{i}\mathrm{n}nx
\Biggr)
= uxx(t, x).
The derivative utx is treated similarly.
We introduce the following definition the generalized solution of (1.1) – (1.3).
Definition 2.1. The function u(t, x) \in B
1+ 2
q
, 2
q
p,p,T satisfying condition (1.2), is called a generalized
solution of (1.1) – (1.3) if for any function v(t, x) \in W
(1)
1
\bigl(
[0, T ], Lq(0, \pi )
\bigr)
such that v(T, x) = 0 on
a.e. [0, \pi ] the integral identity
T\int
0
\pi \int
0
\{ ut(t, x)vt(t, x) - \alpha uxx(t, x)vt(t, x) + F (u)(t, x)v(t, x)\} dxdt -
- \alpha
\pi \int
0
\varphi \prime \prime (x)v(0, x)dx+
\pi \int
0
\psi (x)v(0, x)dx = 0 (2.1)
is fulfilled.
When obtaining the main results we will need the following lemma.
Lemma 2.1 (Gronwall – Bellman). Let the functions u(t), f(t) be continuous and nonnegative
for t \geq t0 and it hold
u(t) \leq a+
t\int
t0
f(\tau )u(\tau )d\tau , t \geq t0,
where a is some positive constant. Then
u(t) \leq ae
\int t
t0
f(\tau )d\tau
, t \geq t0.
3. Existence and uniqueness of generalized solution. In this section, we prove the existence
and uniqueness theorem for the generalized solution of the problem (1.1) – (1.3).
Lemma below can be proved similarly to the one proved in [31] for p = 2.
Lemma 3.1. Let u(t, x) be a generalized solution of the problem (1.1) – (1.3) and F (u) \in
\in Lp,p - 2
\bigl(
[0, T ], Lp(0, \pi )
\bigr)
. Then the coefficients un(t) are the solutions of the following countable
of nonlinear integro-differential equations:
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 3
ON SOLVABILITY OF ONE CLASS OF THIRD ORDER DIFFERENTIAL EQUATIONS 319
un(t) = \varphi n +
1
\alpha n2
\bigl(
1 - e - \alpha n2t
\bigr)
\psi n +
1
\alpha n2
t\int
0
Fn(u, \tau )
\bigl(
1 - e - \alpha n2(t - \tau )
\bigr)
d\tau , (3.1)
where \varphi n, \psi n and Fn(u, t) are the Fourier coefficients by system \{ \mathrm{s}\mathrm{i}\mathrm{n}nx\} of the functions \varphi (x), \psi (x)
and F (u)(t, x), respectively.
Proof. Substituting the function of the form
v\tau ,n(t, x) =
\left\{
2
\pi
(t - \tau ) \mathrm{s}\mathrm{i}\mathrm{n}nx, 0 \leq t \leq \tau , 0 \leq x \leq \pi ,
0, \tau < t \leq T, 0 \leq x \leq \pi ,
into (2.1) with fixed n \in N and \tau \in [0, T ], we obtain
\tau \int
0
\bigl\{
u\prime n(t) + \alpha n2un(t) + (t - \tau )Fn(u, t)
\bigr\}
dt - \alpha n2\varphi n\tau - \psi n\tau = 0.
On differentiating the last equality twice in \tau , we have
u\prime \prime n(\tau ) + \alpha n2u\prime n(\tau ) - Fn(u, \tau ) = 0, \tau \in [0, T ].
Taking into account the conditions un(0) = \varphi n and u\prime n(0) = \psi n, we obtain (3.1).
Lemma 3.1 is proved.
When obtaining the main result we need the following lemma.
Lemma 3.2. Let the operator P be defined in the space Lp,p - 2
\bigl(
[0, T ], Lp(0, \pi )
\bigr)
by the formula
P (f)(t, x) =
\infty \sum
n=1
1
\alpha n2
t\int
0
fn(\tau )
\bigl(
1 - e - \alpha n2(t - \tau )
\bigr)
d\tau \mathrm{s}\mathrm{i}\mathrm{n}nx, f \in Lp,p - 2
\bigl(
[0, T ], Lp(0, \pi )
\bigr)
,
where fn(t) =
2
\pi
\int \pi
0
f(t, x) \mathrm{s}\mathrm{i}\mathrm{n}nxdx. Then P : Lp,p - 2
\bigl(
[0, T ], Lp(0, \pi )
\bigr)
\rightarrow B
1+ 2
q
, 2
q
p,p,T and
\bigm\| \bigm\| P (f)\bigm\| \bigm\|
B
1+2
q , 2q
p,p,T
\leq L\| f\|
Lp,p - 2
\bigl(
[0,T ],Lp(0,\pi )
\bigr) , (3.2)
where L =
q
1
q T
1
q + \alpha
1
p
\alpha q
1
q
.
Proof. For every f \in Lp,p - 2
\bigl(
[0, T ], Lp(0, \pi )
\bigr)
, we have
\bigm\| \bigm\| P (f)\bigm\| \bigm\|
B
1+2
q , 2q
p,p,T
=
\left( \infty \sum
n=1
\left\{ n1+ 2
q \mathrm{m}\mathrm{a}\mathrm{x}
[0,T ]
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| 1
\alpha n2
t\int
0
fn(\tau )(1 - e - \alpha n2(t - \tau ))d\tau
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\right\}
p\right)
1
p
+
+
\left( \infty \sum
n=1
\left\{ n 2
q \mathrm{m}\mathrm{a}\mathrm{x}
[0,T ]
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
t\int
0
fn(\tau )e
- \alpha n2(t - \tau )d\tau
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\right\}
p\right)
1
p
.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 3
320 B. T. BILALOV, M. I. ISMAILOV, Z. A. KASUMOV
Applying the Hölder inequality, we obtain
\| P (f)\|
B
1+2
q , 2q
p,p,T
\leq 1
\alpha
\left( \infty \sum
n=1
np - 2
T\int
0
| fn(\tau )| pd\tau \mathrm{m}\mathrm{a}\mathrm{x}
[0,T ]
\left( t\int
0
(1 - e - \alpha n2(t - \tau ))qd\tau
\right)
p
q
\right)
1
p
+
+
\left( \infty \sum
n=1
n
2p
q
T\int
0
| fn(\tau )| pdt\mathrm{m}\mathrm{a}\mathrm{x}
[0,T ]
\left( t\int
0
e - \alpha n2(t - \tau )d\tau
\right)
p
q
\right)
1
p
\leq
\leq T
1
q
\alpha
\left( \infty \sum
n=1
np - 2
T\int
0
| fn(\tau )| pd\tau
\right)
1
p
+
+
\left( \infty \sum
n=1
n
2p
q
T\int
0
| fn(\tau )| pdt
\biggl(
1
\alpha n2q
\biggr) p
q
\mathrm{m}\mathrm{a}\mathrm{x}
[0,T ]
\bigl(
1 - e - \alpha n2t
\bigr) p
q
\right)
1
p
.
Taking into account that
\mathrm{m}\mathrm{a}\mathrm{x}
[0,T ]
t\int
0
(1 - e - \alpha n2(t - \tau ))qd\tau \leq T and \mathrm{m}\mathrm{a}\mathrm{x}
[0,T ]
t\int
0
e - \alpha n2(t - \tau )d\tau \leq 1
\alpha n2q
,
we have
\| P (f)\|
B
1+2
q , 2q
p,p,T
\leq T
1
q
\alpha
\left( \infty \sum
n=1
np - 2
T\int
0
| fn(\tau )| pd\tau
\right)
1
p
+
1
\alpha
1
q q
1
q
\left( \infty \sum
n=1
T\int
0
| fn(\tau )| pdt
\right)
1
p
\leq
\leq q
1
q T
1
q + \alpha
1
p
\alpha q
1
q
\left( \infty \sum
n=1
np - 2
T\int
0
| fn(\tau )| pd\tau
\right)
1
p
= L
\left( T\int
0
\Biggl( \infty \sum
n=1
np - 2 | fn(\tau )| p
\Biggr)
d\tau
\right)
1
p
.
Lemma 3.2 is proved.
3.1. Uniqueness of solution. Now let us state the main uniqueness result for the generalized
solution of the problem (1.1) – (1.3).
Theorem 3.1. Let the following conditions be satisfied:
1) F : B
1+ 2
q
, 2
q
p,p,T \rightarrow Lp,p - 2
\bigl(
[0, T ], Lp,p - 2(0, \pi )
\bigr)
;
2) \forall u(t, x), v(t, x) \in B
1+ 2
q
, 2
q
p,p,T and t \in [0, T ]:\bigm\| \bigm\| F (u)(t, \cdot ) - F (v)(t, \cdot )
\bigm\| \bigm\|
Lp,p - 2(0,\pi )
\leq c(t)\| u - v\|
B
1+2
q , 2q
p,p,t
, (3.3)
where c(t) \in Lp(0, T ) is some positive function.
Then the problem (1.1) – (1.3) can not have more than one generalized solution.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 3
ON SOLVABILITY OF ONE CLASS OF THIRD ORDER DIFFERENTIAL EQUATIONS 321
Proof. Assume the contrary, i.e., assume that the problem (1.1) – (1.3) has at least two different
generalized solutions u(t, x) and v(t, x). Let \{ un(t)\} n\in N and \{ vn(t)\} n\in N be the sequences of
coefficients of the functions u(t, x) and v(t, x), respectively. From Lemma 3.1, we obtain
u(t, x) - v(t, x) =
\infty \sum
n=1
(un(t) - vn(t)) \mathrm{s}\mathrm{i}\mathrm{n}nx =
=
\infty \sum
n=1
1
\alpha n2
t\int
0
\bigl(
Fn(u, \tau ) - Fn(v, \tau )
\bigr) \bigl(
1 - e - \alpha n2(t - \tau )
\bigr)
d\tau \mathrm{s}\mathrm{i}\mathrm{n}nx =
= P
\bigl(
F (u)(t, x) - F (v)(t, x)
\bigr)
.
Then, for all t \in [0, T ] by virtue of (3.2), using (3.3), we have
\bigm\| \bigm\| u - v
\bigm\| \bigm\| p
B
1+2
q , 2q
p,p,t
=
\bigm\| \bigm\| P (F (u) - F (v))
\bigm\| \bigm\| p
B
1+2
q , 2q
p,p,t
\leq Lp
t\int
0
cp(\tau )
\bigm\| \bigm\| u - v
\bigm\| \bigm\| p
B
1+2
q , 2q
p,p,\tau
d\tau .
Hence, by Lemma 2.1, we obtain \| u - v\| p
B
1+2
q , 2q
p,p,t
= 0. Consequently, u(t, x) = v(t, x).
Theorem 3.1 is proved.
Lemma 3.3. Assume \varphi (x) \in W
(2)
p (0, \pi ), \{ n2\varphi n\} n\in N \in lp,p - 2, \varphi (0) = \varphi (\pi ) = 0, \psi (x) \in
\in W
(1)
p (0, \pi ), \{ n\psi n\} n\in N \in lp,p - 2, \psi (0) = \psi (\pi ) = 0 and let wn(t) = \varphi n +
1
\alpha n2
\bigl(
1 - e - \alpha n2t
\bigr)
\psi n.
Then the function w(t, x) =
\sum \infty
n=1
wn(t) \mathrm{s}\mathrm{i}\mathrm{n}nx belongs to the space B
1+ 2
q
, 2
q
p,p,T .
Proof. It is clear that the series
\sum \infty
n=1
wn(t) \mathrm{s}\mathrm{i}\mathrm{n}nx is convergent. Let us show that w(t, x) \in
\in B
1+ 2
q
, 2
q
p,p,T , i.e., let us show the convergence of the series
\sum \infty
n=1
\Bigl(
n
1+ 2
q \| wn\| C[0,T ]
\Bigr) p
and\sum \infty
n=1
\Bigl(
n
2
q
\bigm\| \bigm\| w\prime
n
\bigm\| \bigm\|
C[0,T ]
\Bigr) p
. We have
\| wn\| C[0,T ] \leq | \varphi n| +
1
\alpha n2
\bigm| \bigm| \psi n
\bigm| \bigm| \bigm\| \bigm\| w\prime
n
\bigm\| \bigm\|
C[0,T ]
=
\bigm| \bigm| \psi n
\bigm| \bigm| .
Then, taking into account that \{ n2\varphi n\} n\in N , \{ n\psi n\} n\in N \in lp,p - 2, we obtain\Biggl( \infty \sum
n=1
\Bigl(
n
1+ 2
q \| wn\| C[0,T ]
\Bigr) p\Biggr) 1
p
\leq
\Biggl( \infty \sum
n=1
\Biggl(
n
1+ 2
q | \varphi n| +
n
2
q
\alpha
\bigm| \bigm| \psi n
\bigm| \bigm| \Biggr) p\Biggr) 1
p
\leq
\leq
\Biggl( \infty \sum
n=1
\Bigl(
n
1+ 2
q | \varphi n|
\Bigr) p\Biggr) 1
p
+
1
\alpha
\Biggl( \infty \sum
n=1
\Bigl(
n
2
q
\bigm| \bigm| \psi n
\bigm| \bigm| \Bigr) p\Biggr) 1
p
=
=
\Biggl( \infty \sum
n=1
\bigl(
n2| \varphi n|
\bigr) p
np - 2
\Biggr) 1
p
+
1
\alpha
\Biggl( \infty \sum
n=1
\bigl(
n
\bigm| \bigm| \psi n
\bigm| \bigm| \bigr) pnp - 2
\Biggr) 1
p
< +\infty .
Also we get
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 3
322 B. T. BILALOV, M. I. ISMAILOV, Z. A. KASUMOV
\infty \sum
n=1
\Bigl(
n
2
q
\bigm\| \bigm\| w\prime
n
\bigm\| \bigm\|
C[0,T ]
\Bigr) p
=
\infty \sum
n=1
\Bigl(
n
2
q | \psi n|
\Bigr) p
=
\infty \sum
n=1
\bigl(
n| \psi n|
\bigr) p
np - 2 < +\infty .
Consequently, w(t, x) \in B
1+ 2
q
, 2
q
p,p,T .
Lemma 3.3 is proved.
3.2. Existence of solution. Now let us consider solvability of the generalized solution of the
problem.
Theorem 3.2. Let the following conditions be satisfied:
1) \varphi (x) \in W
(2)
p (0, \pi ), \{ n2\varphi n\} n\in N \in lp,p - 2, \varphi (0) = \varphi (\pi ) = 0, \psi (x) \in W
(1)
p (0, \pi ),
\{ n\psi n\} n\in N \in lp,p - 2, \psi (0) = \psi (\pi ) = 0;
2) F : B
1+ 2
q
, 2
q
p,p,T \rightarrow Lp,p - 2
\bigl(
[0, T ], Lp,p - 2(0, \pi )
\bigr)
\forall u \in B
1+ 2
q
, 2
q
p,p,T , t \in [0, T ]:\bigm\| \bigm\| F (u)(t, \cdot )\bigm\| \bigm\|
Lp,p - 2(0,\pi )
\leq a(t) + b(t)\| u\|
B
1+2
q , 2q
p,p,t
, (3.4)
where a(t), b(t) \in Lp(0, T ) are some positive functions;
3) \forall u(t, x), v(t, x) \in K
\Bigl(
\| u\|
B
1+2
q , 2q
p,p,T
\leq R
\Bigr)
t \in [0, T ]:
\bigm\| \bigm\| F (u)(t, \cdot ) - F (v)(t, \cdot )
\bigm\| \bigm\|
Lp,p - 2(0,\pi )
\leq c(t) \| u - v\|
B
1+2
q , 2q
p,p,t
, (3.5)
where c(t) \in Lp(0, T ), R
p = A \mathrm{e}\mathrm{x}\mathrm{p}
\int T
0
Bp(t)dt, A = 2p - 1\| w\| p
B
1+2
q , 2q
p,p,T
+ Lp
0\| a\|
p
Lp(0,T ), B(t) =
= L0b(t), L0 = 2
2
qL.
Then the problem (1.1) – (1.3) has a unique generalized solution.
Proof. Consider the operator Q in the space B
1+ 2
q
, 2
q
p,p,T defined by the formula
Q(u)(t, x) = w(t, x) + P (F (u)(t, x)).
By using (3.2), (3.3), we obtain
\bigm\| \bigm\| Q(u)
\bigm\| \bigm\| p
B
1+2
q , 2q
p,p,t
\leq 2p - 1
\Biggl(
\| w\| p
B
1+2
q , 2q
p,p,t
+ \| P (F (u))\| p
B
1+2
q , 2q
p,p,t
\Biggr)
\leq
\leq 2p - 1
\left( \| w\| p
B
1+2
q , 2q
p,p,t
+ Lp
t\int
0
\| F (u)\| pLp,p - 2(0,\pi )
dt
\right) \leq
\leq 2p - 1
\left( \| w\| p
B
1+2
q , 2q
p,p,t
+ 2p - 1Lp
t\int
0
(ap(\tau ) + bp(\tau )\| u\| p
B
1+2
q , 2q
p,p,t
)d\tau
\right) =
= 2p - 1 \| w\| pBp,p,t
+ Lp
0
t\int
0
ap(\tau )d\tau +Lp
0
t\int
0
bp(\tau )\| u\| p
B
1+2
q , 2q
p,p,t
d\tau \leq
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 3
ON SOLVABILITY OF ONE CLASS OF THIRD ORDER DIFFERENTIAL EQUATIONS 323
\leq A+
t\int
0
Bp(\tau )\| u\| p
B
1+2
q , 2q
p,p,t
d\tau . (3.6)
Let us build the sequence
\bigl\{
uk(t, x)
\bigr\} \infty
k=0
\subset B
1+ 2
q
, 2
q
p,p,T as follows:
u0(t, x) = 0, uk(t, x) = Q(uk - 1)(t, x), k = 1, 2, . . . .
According to (3.6), for every t \in [0, T ], we obtain
\| u1\| p
B
1+2
q , 2q
p,p,t
= \| Q(u0)\| p
B
1+2
q , 2q
p,p,t
\leq A \leq A+A
t\int
0
Bp(\tau )d\tau ,
\| u2\| p
B
1+2
q , 2q
p,p,t
= \| Q(u1)\| p
B
1+2
q , 2q
p,p,t
\leq A+
t\int
0
Bp(\tau ) \| u1\| p
B
1+2
q , 2q
p,p,t
d\tau \leq
\leq A+
t\int
0
Bp(\tau )(A+A
\tau \int
0
Bp(s)ds)d\tau =
= A+A
t\int
0
Bp(\tau )d\tau +A
t\int
0
Bp(\tau )
\tau \int
0
Bp(s)dsd\tau =
= A
\left( 1 +
t\int
0
Bp(\tau )d\tau +
t\int
0
d
dt
\biggl( \int \tau
0
Bp(s)ds
\biggr) 2
2
d\tau
\right) =
= A
\left( 1 +
t\int
0
Bp(\tau )d\tau +
\biggl( \int t
0
Bp(\tau )d\tau
\biggr) 2
2
\right) .
Continuing this process in a similar way, we get
\| uk\| p
B
1+2
q , 2q
p,p,t
\leq A
\left( 1 +
t\int
0
Bp(\tau )d\tau + . . .+
\biggl( \int t
0
Bp(\tau )d\tau
\biggr) k
k!
\right) , k = 0, 1, . . . .
Hence it follows
\| uk\| p
B
1+2
q , 2q
p,p,T
\leq A \mathrm{e}\mathrm{x}\mathrm{p}
T\int
0
Bp(t)dt = Rp, uk(t, x) \in K, k = 0, 1, . . . .
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 3
324 B. T. BILALOV, M. I. ISMAILOV, Z. A. KASUMOV
Let us estimate
\bigm\| \bigm\| un+k - uk
\bigm\| \bigm\| p
B
1+2
q , 2q
p,p,T
for any n, k = 1, 2, . . . . Taking into account (3.2), (3.4)
and (3.5), we have
\| un+k - uk\| p
B
1+2
q , 2q
p,p,t
=
\bigm\| \bigm\| \bigm\| Q(un+k - 1) - Q(uk - 1)
\bigm\| \bigm\| \bigm\| p
B
1+2
q , 2q
p,p,t
=
=
\bigm\| \bigm\| \bigm\| P \bigl( F (un+k - 1) - F (uk - 1)
\bigr) \bigm\| \bigm\| \bigm\| p
B
1+2
q , 2q
p,p,t
\leq
\leq Lp
t\int
0
\bigm\| \bigm\| \bigm\| F (un+k - 1) - F (uk - 1)
\bigm\| \bigm\| \bigm\| p
Lp,p - 2(0,\pi )
dt \leq
\leq Lp
t\int
0
cp(\tau )
\bigm\| \bigm\| un+k - 1 - uk - 1
\bigm\| \bigm\| p
B
1+2
q , 2q
p,p,\tau
d\tau .
Then
\| un+k - uk\| p
B
1+2
q , 2q
p,p,t
\leq Lp
t\int
0
cp(\tau ) \| un+k - 1 - uk - 1\| p
B
1+2
q , 2q
p,p,\tau
d\tau \leq
\leq Lp
t\int
0
cp(\tau )
\left( Lp
\tau \int
0
cp(s) \| un+k - 2 - uk - 2\| p
B
1+2
q , 2q
p,p,s
ds
\right) d\tau =
= L2p
t\int
0
cp(\tau )
\tau \int
0
cp(s)ds \| un+k - 2 - uk - 2\| p
B
1+2
q , 2q
p,p,\tau
d\tau \leq . . .
. . . \leq L2p
t\int
0
d
d\tau
\biggl( \int \tau
0
cp(s)ds
\biggr) 2
2
\| un+k - 2 - uk - 2\| p
B
1+2
q , 2q
p,p,\tau
d\tau \leq
\leq Lpk
t\int
0
d
d\tau
\biggl( \int \tau
0
cp(s)ds
\biggr) k
k!
\| un - u0\| p
B
1+2
q , 2q
p,p,\tau
d\tau \leq
\leq Lpk\| un\| p
B
1+2
q , 2q
p,p,t
t\int
0
d
d\tau
\biggl( \int \tau
0
cp(s)ds
\biggr) k
k!
d\tau =
= Lpk \| un\| p
B
1+2
q , 2q
p,p,t
\biggl( \int t
0
cp(\tau )d\tau
\biggr) k
k!
.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 3
ON SOLVABILITY OF ONE CLASS OF THIRD ORDER DIFFERENTIAL EQUATIONS 325
Thus, the following inequality is true:
\| un+k - uk\| p
B
1+2
q , 2q
p,p,T
\leq
LpkRp \| c\| pkLp(0,T )
k!
.
Consequently, the sequence
\bigl\{
uk(t, x)
\bigr\} \infty
k=1
is fundamental in B
1+ 2
q
, 2
q
p,p,T , and therefore it converges
to some u(t, x) \in K. Further, we have\bigm\| \bigm\| Q(uk) - Q(u)
\bigm\| \bigm\|
B
1+2
q , 2q
p,p,T
=
\bigm\| \bigm\| P (F (uk) - F (u))
\bigm\| \bigm\|
B
1+2
q , 2q
p,p,T
\leq
\leq L \| F (uk) - F (u)\| Lp([0,T ],Lp,p - 2(0,\pi ))
\leq L \| c\| Lp(0,T ) \| uk - u\|
B
1+2
q , 2q
p,p,T
,
and, therefore, Q(uk) converges in B
1+ 2
q
, 2
q
p,p,T to Q(u) as k \rightarrow \infty . Then
u(t, x) = \mathrm{l}\mathrm{i}\mathrm{m}
k\rightarrow \infty
uk(t, x) = \mathrm{l}\mathrm{i}\mathrm{m}
k\rightarrow \infty
Q(uk - 1)(t, x) = Q(u)(t, x) =
\infty \sum
n=1
un(t) \mathrm{s}\mathrm{i}\mathrm{n}nx,
where
un(t) = \varphi n +
1
\alpha n2
\bigl(
1 - e - \alpha n2t
\bigr)
\psi n +
1
\alpha n2
t\int
0
Fn(u, \tau )
\bigl(
1 - e - \alpha n2(t - \tau )
\bigr)
d\tau .
Let us show that u(t, x) is a generalized solution of the problem (1.1) – (1.3). Obviously,
u(0, x) =
\infty \sum
n=1
un(0) \mathrm{s}\mathrm{i}\mathrm{n}nx =
\infty \sum
n=1
\varphi n \mathrm{s}\mathrm{i}\mathrm{n}nx = \varphi (x),
ut(0, x) =
\infty \sum
n=1
u\prime n(0) \mathrm{s}\mathrm{i}\mathrm{n}nx =
\infty \sum
n=1
\psi n \mathrm{s}\mathrm{i}\mathrm{n}nx = \psi (x).
It remains to show the validity of the identity (2.1). Assume that
um(t, x) =
m\sum
n=1
un(t) \mathrm{s}\mathrm{i}\mathrm{n}nx,
Jm =
T\int
0
\pi \int
0
\Bigl\{
um,t(t, x)vt(t, x) - \alpha um,xx(t, x)vt(t, x) + F (u)(t, x)v(t, x)
\Bigr\}
dxdt -
- \alpha
\pi \int
0
\varphi \prime \prime (x)v(0, x)dx+
\pi \int
0
\psi (x)v(0, x)dx. (3.7)
We have
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 3
326 B. T. BILALOV, M. I. ISMAILOV, Z. A. KASUMOV
T\int
0
\pi \int
0
um,t(t, x)vt(t, x)dxdt =
=
T\int
0
\pi \int
0
m\sum
n=1
u\prime n(t) \mathrm{s}\mathrm{i}\mathrm{n}nxvt(x, t)dxdt =
m\sum
n=1
\pi \int
0
\left( T\int
0
u\prime n(t)vt(t, x)dt
\right) \mathrm{s}\mathrm{i}\mathrm{n}nxdx =
=
m\sum
n=1
\pi \int
0
\left( u\prime n(t)v(t, x)| T0 -
T\int
0
u\prime \prime n(t)v(t, x)dt
\right) \mathrm{s}\mathrm{i}\mathrm{n}nxdx =
= -
\pi \int
0
m\sum
n=1
\psi n \mathrm{s}\mathrm{i}\mathrm{n}nxv(0, x)dx -
T\int
0
\pi \int
0
m\sum
n=1
u\prime \prime n(t) \mathrm{s}\mathrm{i}\mathrm{n}nxv(x, t)dxdt,
T\int
0
\pi \int
0
um,xx(t, x)vt(t, x)dxdt = -
T\int
0
\pi \int
0
m\sum
n=1
n2un(t)vt(t, x) \mathrm{s}\mathrm{i}\mathrm{n}nxdxdt =
= -
m\sum
n=1
n2
\pi \int
0
\left( un(t)v(t, x)| T0 -
T\int
0
u\prime n(t)v(t, x)dt
\right) \mathrm{s}\mathrm{i}\mathrm{n}nxdx =
=
\pi \int
0
m\sum
n=1
n2\varphi n \mathrm{s}\mathrm{i}\mathrm{n}nxv(0, x)dx+
T\int
0
\pi \int
0
m\sum
n=1
n2u\prime n(t)v(t, x) \mathrm{s}\mathrm{i}\mathrm{n}nxdxdt.
Consequently, taking into account u\prime \prime n(t) + \alpha n2u\prime n(t) = Fn(u, t), we obtain
Jm =
T\int
0
\pi \int
0
\bigl(
F (u)(t, x) -
m\sum
n=1
Fn(u, t) \mathrm{s}\mathrm{i}\mathrm{n}nx
\bigr)
v(t, x)dxdt+
+
\pi \int
0
\Biggl(
\psi (x) -
m\sum
n=1
\psi n \mathrm{s}\mathrm{i}\mathrm{n}nx
\Biggr)
v(0, x)dx+
+\alpha
\pi \int
0
\Biggl(
\varphi \prime \prime (x) +
m\sum
n=1
n2\varphi n \mathrm{s}\mathrm{i}\mathrm{n}nx
\Biggr)
v(0, x)dx.
As a result, we have
| Jm| \leq
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| F (u)(t, x) -
m\sum
n=1
Fn(u, t) \mathrm{s}\mathrm{i}\mathrm{n}
n\pi
\ell
x
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
Lp(D)
\bigm\| \bigm\| v(t, x)\bigm\| \bigm\|
Lq(D)
+
+
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \psi (x) -
m\sum
n=1
\psi n \mathrm{s}\mathrm{i}\mathrm{n}nx
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
Lp(0,\pi )
\bigm\| \bigm\| v(0, x)\bigm\| \bigm\|
Lq(0,\pi )
+
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 3
ON SOLVABILITY OF ONE CLASS OF THIRD ORDER DIFFERENTIAL EQUATIONS 327
+\alpha
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \varphi \prime \prime (x) +
m\sum
n=1
n2\varphi n \mathrm{s}\mathrm{i}\mathrm{n}nx
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
Lp(0,\pi )
\bigm\| \bigm\| v(0, x)\bigm\| \bigm\|
Lq(0,\pi )
\rightarrow 0
as m \rightarrow \infty . Passing to the limit in (3.7) as m \rightarrow \infty , we obtain the validity of the integral iden-
tity (2.1). The uniqueness of generalized solution follows from Theorem 3.1.
Theorem 3.2 is proved.
4. Conclusion. For the initial boundary-value problem (1.1) – (1.3) with nonlinear right-hand
side and zero boundary conditions, the concept of generalized solution belonging to Banach space is
introduced. Under some conditions, the existence and uniqueness of generalized solution is proved.
In particular, for we obtain the previously known results in this field. Note that, using the known
technique, we can obtain similar results for the same problem with nonzero boundary data. Moreover,
the same technique is also applicable to the multidimensional analog of this problem. Of course, this
problem can be treated by other methods too, for example, by operator method. To do so, you need
to define the corresponding mapping operators and use the methods of this theory.
References
1. A. Ashyralyev, K. Belakroum, On the stability of nonlocal boundary value problem for a third order PDE, AIP Conf.
Proc., 2183, Article 070012 (2019), https://doi.ord/10.1063/1.5136174.
2. Sh. Amirov, A. I. Kozhanov, A mixed problem for a class of strongly nonlinear higher-order equations of Sobolev
type, Dokl. Math., 88, № 1, 446 – 448 (2013) (in Russian).
3. Yu. P. Apakov, B. Yu. Irgashev, Boundary-value problem for a degenerate high-odd-order equation, Ukr. Math. J.,
66, № 10, 1475 – 1490 (2015).
4. C. Latrous, A. Memou, A three-point boundary value problem with an integral condition for a third-order partial
differential equation, Abstr. and Appl. Anal., 2005, № 1, 33 – 43 (2005).
5. Y. Apakov, S. Rutkauskas, On a boundary value problem to third order PDE with multiple characteristics, Nonlinear
Anal. Model. and Control, 16, № 3, 255 – 269 (2011).
6. M. Kudu, I. Amirali, Method of lines for third order partial differential equations, J. Appl. Math. and Phys., 2, № 2,
33 – 36 (2014).
7. A. Ashyralyev, D. Arjmand, M. Koksal, Taylor’s decomposition on four points for solving third-order linear time-
varying systems, J. Franklin Inst. Eng. and Appl. Math., 346, 651 – 662 (2009).
8. A. Ashyralyev, D. Arjmand, A note on the Taylor’s decomposition on four points for a third-order differential
equation, Appl. Math. and Comput., 188, № 2, 1483 – 1490 (2007).
9. Kh. Belakroum, A. Ashyralyev, A. Guezane-Lakoud, A note on the nonlocal boundary value problem for a third order
partial differential equation, AIP Conf. Proc., 1759, Article 020021 (2016), http://dx.doi.org/10.1063/1.4959635.
10. A. Ashyralyev, P. E. Sobolevskii, New difference schemes for partial differential equations, Birkhäuser, Basel etc.
(2004).
11. Kh. Belakroum, A.Ashyralyev, A. Guezane-Lakoud, A note on the nonlocal boundary value problem for a third order
partial differential equation, Filomat, 32, № 3, 801 – 808 (2018).
12. A. Ashyralyev, Kh. Belakroum, A. Guezane-Lakoud, Stability of boundary-value problems for third-order partial
differential equations, Electron. J. Different. Equat., 53, 1 – 11 (2017).
13. J. Nagumo, S. Arimoto, S. Yoshizawa, An active pulse transmission line simulating nerve axon, Proc. JRE, 50, № 10,
2061 – 2070 (1962).
14. A. Reiko, H. Yojiro, On global solutions for mixed problem of a semilinear differential equation, Proc. Japan Acad.,
39, № 10, 721 – 725 (1963).
15. M. Jamaguti, The asymptotic behavior of the solution of a semilinear partial differential equation related to an active
pulse transmission line, Proc. Japan Acad., 39, № 10, 726 – 730 (1963).
16. J. M. Greenberg, R. C. MacCamy, V. J. Mizel, On the existence, uniqueness and stability of solutions of the equation
\sigma \prime (ux)uxx + \lambda uxtx = \rho 0utt , J. Math. and Mech., 17, № 7, 707 – 728 (1968).
17. J. M. Greenberg, On the existence, uniqueness and stability of solutions of the equation \rho 0utt = E(ux) \cdot uxx+\lambda uxxt ,
J. Math. Anal. and Appl., 25, 575 – 591 (1969).
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 3
328 B. T. BILALOV, M. I. ISMAILOV, Z. A. KASUMOV
18. J. M. Greenberg, R. C. MacCamy, On the exponential stability solutions of the equation E(ux) \cdot uxx+\lambda uxtx = \rho utt ,
J. Math. Anal. and Appl., 31, № 2, 406 – 417 (1970).
19. P. L. Davis, On the existence, uniqueness and stability of solutions of a nonlinear functional defferential equation, J.
Math. Anal. and Appl., 34, № 1, 128 – 140 (1971).
20. A. I. Kozhanov, Mixed problem for some classes of third order nonlinear equations, Mat. Sb., 118, № 4, 504 – 522
(1982) (in Russian).
21. S. A. Gabov, G. Y. Malysheva, A. G. Sveshnikov, On one equation of composite type related to the oscillations of
stratified fluid, Differents. Uravneniya, 19, № 7, 1171 – 1180 (1983) (in Russian).
22. B. A. Iskenderov, V. Q. Sardarov, Mixed problem for Boussinesq equation in cylindrical dominain and behavior of
its solution at t \rightarrow +\infty , Trans. Nat. Akad. Sci. Azerb., 22, № 4, 117 – 134 (2002).
23. A. B. Aliev, A. A. Kazimov, The asymptotic behavior of weak solution of Cauchy problem for a class Sobolev type
semilinear equation, Trans. Nat. Akad. Sci. Azerb. Ser. Phys.-Techn. and Math. Sci., 26, № 4, 23 – 30 (2006).
24. T. K. Yuldashev, On an optimal control problem for a nonlinear pseudohyperbolic equation, Model. Anal. Inform.
Sist., 20, № 5, 78 – 89 (2013).
25. G. A. Rasulova, Study of mixed problem for one class of third order quasilinear differential equations, Differents.
Uravneniya, 3, № 9, 1578 – 1591 (1967) (in Russian).
26. A. I. Kozhanov, Mixed problem for one class of nonclassical equations, Differents. Uravneniya, 15, № 2, 272 – 280
(1979) (in Russian).
27. O. A. Ladyzhenskaya, Mixed problem for hyperbolic equation, Gostekhizdat, Moscow (1953) (in Russian).
28. G. I. Laptev, On one third order quasilinear partial differential equation, Differents. Uravneniya, 24, № 7, 1270 – 1272
(1988) (in Russian).
29. P. L. Davis, A quasilinear hyperbolic and related third-order equations, J. Math. Anal. and Appl., 51, № 3, 596 – 606
(1975).
30. R. S. Zhamalov, Directional derivative problem for one third order equation, Kraevyye Zadachi dlya Neklassicheskikh
Uravnenii Matematicheskoi Fiziki, 17, 115 – 116 (1989) (in Russian).
31. K. I. Khudaverdiyev, A. A. Veliyev, Study of one-dimensional mixed problem for one class of third order
pseudohyperbolic equations with nonlinear operator right-hand side, Chashioglu, Baku (2010) (in Russian).
32. Z. I. Khalilov, A new method for solving the equations of vibrations of elastic system, Izv. Azerb. Branch AS USSR,
№ 4, 168 – 169 (1942).
33. V. A. Ilyin, On solvability of mixed problem for hyperbolic and parabolic equations, Uspekhi Mat. Nauk, 15, № 2,
97 – 154 (1960) (in Russian).
34. K. I. Khudaverdiyev, On generalized solutions of one-dimensional mixed problem for a class of quasilinear differential
equations, Uch. Zap. Artsakh. Gos. Univ., № 4, 29 – 42 (1965) (in Russian).
35. V. A. Chernyatin, Justification of Fourier method for mixed problem for partial differential equations, Izd-vo Mosk.
Un-ta (1991) (in Russian).
36. B. T. Bilalov, Z. G. Guseynov, K -Bessel and K -Hilbert systems. K -bases, Dokl. RAN, 429, № 3, 298 – 300 (2009).
37. B. T. Bilalov, Bases and tensor product, Trans. Nat. Acad. Sci. Azerb., 25, № 4, 15 – 20 (2005).
38. B. T. Bilalov, Some questions of approximation, Elm, Baku (2016).
39. A. Zygmund, Trigonometric series, vols. 1, 2, Mir, Moscow (1965).
40. S. Kachmazh, G. Steinhous, Theory of orthogonal series, GIFML, Moscow (1958).
Received 20.07.18,
after revision — 14.02.20
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 3
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| id | umjimathkievua-article-195 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:02:03Z |
| publishDate | 2021 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/b5/6b3170ad42915117a169b060357929b5.pdf |
| spelling | umjimathkievua-article-1952025-03-31T08:48:21Z On solvability of one class of third order differential equations On solvability of one class of third order differential equations Bilalov, B. T. Ismailov, M. I. Kasumov, Z. A. Bilalov, B. T. Ismailov, M. I. Kasumov, Z. A. mixed problem Fourier method generalized solution Paley theorem mixed problem Fourier method generalized solution Paley theorem UDC 517.9 One-dimensional mixed problem for one class of third order partial differential equation with nonlinear right-hand side is considered. The concept of generalized solution for this problem is introduced. By the Fourier method, the problem of existence and uniqueness of generalized solution for this problem is reduced to the problem of solvability of the countable system of nonlinear integro-differential equations. Using Bellman's inequality, the uniqueness of generalized solution is proved. Under some conditions on initial functions and the right-hand side of the equation, the existence theorem for the generalized solution is proved using the method of successive approximations. УДК 517.9 Про розв’язнiсть одного класу диференцiальних рiвнянь третього порядку Розглянуто одновимірну мішану задачу для одного класу диференціальних рівнянь третього порядку з частинними похідними з нелінійною правою частиною. Введено поняття узагальненого розв'язку для цієї задачі. За допомогою методу Фур'є задачу існування та єдиності узагальненого розв'язку зведено до задачі розв'язності зліченної системи нелінійних інтегро-диференціальних рівнянь. З використанням нерівності Беллмана доведено єдиність узагальненого розв'язку. При деяких умовах на початкові функції та праву частину рівняння на основі методу послідовних ітерацій доведено теорему про існування узагальненого розв'язку. Institute of Mathematics, NAS of Ukraine 2021-03-22 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/195 10.37863/umzh.v73i3.195 Ukrains’kyi Matematychnyi Zhurnal; Vol. 73 No. 3 (2021); 314 - 328 Український математичний журнал; Том 73 № 3 (2021); 314 - 328 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/195/8977 Copyright (c) 2021 Migdad I. Ismailov, B. T. Bilalov, Z. A. Kasumov |
| spellingShingle | Bilalov, B. T. Ismailov, M. I. Kasumov, Z. A. Bilalov, B. T. Ismailov, M. I. Kasumov, Z. A. On solvability of one class of third order differential equations |
| title | On solvability of one class of third order differential equations |
| title_alt | On solvability of one class of third order differential equations |
| title_full | On solvability of one class of third order differential equations |
| title_fullStr | On solvability of one class of third order differential equations |
| title_full_unstemmed | On solvability of one class of third order differential equations |
| title_short | On solvability of one class of third order differential equations |
| title_sort | on solvability of one class of third order differential equations |
| topic_facet | mixed problem Fourier method generalized solution Paley theorem mixed problem Fourier method generalized solution Paley theorem |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/195 |
| work_keys_str_mv | AT bilalovbt onsolvabilityofoneclassofthirdorderdifferentialequations AT ismailovmi onsolvabilityofoneclassofthirdorderdifferentialequations AT kasumovza onsolvabilityofoneclassofthirdorderdifferentialequations AT bilalovbt onsolvabilityofoneclassofthirdorderdifferentialequations AT ismailovmi onsolvabilityofoneclassofthirdorderdifferentialequations AT kasumovza onsolvabilityofoneclassofthirdorderdifferentialequations |