EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 12, No. 2, 2019, 577-589
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

On the existence of solution to multidimensional third
order nonlinear equations

Samed J.Aliyev1,∗, Arzu Q.Aliyeva2, Goncha Z. Abdullayeva1

1 Department of Mathematics and Methodology of its Teaching,
Baku State University, Z.Khalilov str.23, AZ1148, Baku, Azerbaijan
2 Department of Differential Equations, Institute of Mathematics and Mechanics,
Azerbaijan National Academy of Sciences, B.Vahabzade str.9, AZ1141, Baku, Azerbaijan

Abstract. In this paper, we prove existence of an almost everywhere solution to mixed problem
for a class of third order differential equations by non-zero rotation principle. Also studied the
correctness of the formulation of the considered problem.

2010 Mathematics Subject Classifications: 35L76, 35L82.

Key Words and Phrases: Nonlinear operator, continuously differentiable, correct formulation.

1. Introduction

The paper is devoted to the problem of existence of almost everywhere solution and
correctness of the formulation to the following multidimensional mixed problem for the
third order nonlinear equation:

∂2u(t, x)

∂t2
− ∂

∂t
L(u(t, x)) = F(u(t, x)) (t ∈ [0, T ], x ∈ Ω), (1)

u(0, x) = ϕ(x) (x ∈ Ω), ut(0, x) = ψ(x) (x ∈ Ω), (2)

u(t, x)|Γ = 0, (3)

where 0 < T < +∞; x = (x1, ..., xn), Ω is a bounded n dimensional domain with an
enough smooth boundary S; Γ = [0, T ]× S;

L(u(t, x)) =
n∑

i,j=1

∂

∂xi

(
aij(x)

∂u(t, x)

∂xj

)
− a(x)u(t, x), (4)

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v12i2.3382

Email addresses: samed59@bk.ru (Samed J.Aliyev),
arzu.aliyeva@bk.ru (Arzu Q.Aliyeva), a.q.z.41@mail.ru (Goncha Z. Abdullayeva).

http://www.ejpam.com 577 c© 2019 EJPAM All rights reserved.



S. J.Aliyev, A. Q.Aliyeva, G. Z. Abdullayeva / Eur. J. Pure Appl. Math, 12 (2) (2019), 577-589 578

functions aij(x) (i, j = 1, n) and a(x) are measurable, and bounded in Ω and satisfy in Ω
the following conditions:

aij(x) = aji(x), a(x) ≥ 0,
n∑

i,j=1

aij(x)ξiξj ≥ α ·
n∑
i=1

ξ2
i (α = const > 0)

where ξi (i = 1, 2, . . . , n) are arbitrary real numbers; ϕ, ψ are the given functions; F is
some, generally speaking, nonlinear operator, and u(t, x) is a sought function.

Note that the results of this work improve the results of our article [1]. There have
been many works devoted to the study of mixed problems for nonlinear third order equa-
tions (see [2, 3, 5, 8, 10, 12] and references therein), where the problem of existence and
uniqueness in appropriate spaces, the problem of blow up of solutions and the problems
of asymptotic behavior of solutions are studied.

As well as we know the equations considered in previous publications do not cover the
class of equations we study.

Considered by us equations appear in modeling dynamical processes, in elasticity the-
ory and in modeling dynamics of shallow water waves (see [7,11]).

2. Auxiliaries

In what follows we are using the following notations and facts.
1. We denote by Ḋ(Ω) the class of all continuously differentiable functions on Ω which

vanished near the boundary of Ω. The closure of Ḋ(Ω) with respect to the norm of W 1
2 (Ω)

we denote by
◦
D(Ω). Hence

◦
D(Ω) ⊂W 1

2 (Ω).
Denote Ḋ1(QT ) (QT ≡ [0, T ]×Ω) the class of all continuously differentiable functions

on the cylinder QT are equal to zero in the δ neighborhood of the lateral surface on the
cylinder QT , having the form: QT,δ ≡ [0, T ] × Ωδ where Ωδ is a δ neighborhood of the
boundary of Ω. The closure of Ḋ1(QT ) with respect to the norm of W 1

2 (QT ) we denote

by
◦
D1(QT ). Hence

◦
D1(QT ) ⊂W 1

2 (QT ).

Definition. The function u(t, x) ∈
◦
D1(QT ) belonging to the space L2(QT ) together

with all its derivatives ut(t, x),uxi(t, x) (i = 1, n), utxi(t, x) (i = 1, n), uxixj (t, x) (i, j =
1, n), utt(t, x), utxixj (t, x) (i, j = 1, n) satisfying equation (1) almost everywhere in QT
and taking initial values (2) almost everywhere in Ω is called an almost everywhere solution
of the problem (1)-(3).

2. For investigation of the problem (1)-(3) we recall one property of the operator
L, generating by the differential expression (4) and boundary condition (3): there are
denumerable number of negative eigenvalues

0 > −λ2
1 ≥ −λ2

2 ≥ ... ≥ −λ2
s ≥ ..., (0 < λs → +∞ as s→∞)

with the corresponding generalized eigenfunctions vs(x) which are complete and orthonor-

mal in L2(Ω). We call function vs(x) ∈
◦
D(Ω) a generalized eigenfunction of the operator



S. J.Aliyev, A. Q.Aliyeva, G. Z. Abdullayeva / Eur. J. Pure Appl. Math, 12 (2) (2019), 577-589 579

L, if it is not identically zero and

∫
Ω


n∑

i,j=1

aij(x)
∂vs(x)

∂xi
· ∂Φ(x)

∂xj
+ a(x)vs(x)Φ(x)

 dx = λ2
s

∫
Ω

vs(x)Φ(x)dx

for any function Φ(x) ∈
◦
D(Ω).

As the system {vs(x)}∞s=1 is complete orthonormal in L2(Ω), then it is evident that
every almost everywhere solution of problem (1)-(3) has the following form:

u(t, x) =

∞∑
s=1

us(t)vs(x),

where

us(t) =

∫
Ω

u(t, x)vs(x)dx (s = 1, 2, ...).

Then, after applying the Fourier method, finding the unknown Fourier coefficients
us(t) (s = 1, 2, ...) for the almost everywhere solution u(t, x) of the problem (1)-(3) is
reduced to the solution of the following countable system of nonlinear integro-differential
equations:

us(t) = ϕs +
1

λ2
s

(1− e−λ2st)ψs

+
1

λ2
s

t∫
0

∫
Ω

F(u(τ, x)) ·
[
1− e−λ2s(t−τ)

]
vs(x)dxdτ (s = 1, 2, ...; t ∈ [0, T ]), (5)

where

ϕs =

∫
Ω

ϕ(x)vs(x)dx, ψs =

∫
Ω

ψ(x)vs(x)dx (s = 1, 2, ...).

Proceeding from the definition of almost every where solution of problem (1)-(3), it is
easy to prove (see [1]) the following

Lemma. If u(t, x) =
∞∑
s=1

us(t)vs(x) is any almost everywhere solution of problem

(1)-(3) and the generalized derivatives ∂
∂xk

aij(x) (i, j, k = 1, 2, . . . , n) are bounded on Ω,
then functions us(t) (s = 1, 2, ...) satisfy system (5).

3. We denote by Bα0,...,αl
β0,...,βl,T

a totality of all the functions of the from

u(t, x) =
∞∑
s=1

us(t)vs(x)



S. J.Aliyev, A. Q.Aliyeva, G. Z. Abdullayeva / Eur. J. Pure Appl. Math, 12 (2) (2019), 577-589 580

considered in QT = [0, T ]× Ω, where us(t) ∈ C(l)([0, T ]) for all s and

NT (u) ≡
l∑

i=0

{ ∞∑
s=1

(
λαis · max

0≤t≤T

∣∣∣u(i)
s (t)

∣∣∣)βi} 1
βi

< +∞,

with αi ≥ 0, 1 ≤ βi ≤ 2 (i = 0, 1, . . . , n). We define the norm in this set as ‖u‖ = NT (u).
It is evident that all these spaces are Banach spaces ([6, p.50]).

4. Let G be class all functions u(t, x) which have the properties

u(t, x), ut(t, x), uxi(t, x) (i = 1, n), utxi(t, x) (i = 1, n), uxixj (t, x) (i, j = 1, n), utt(t, x),

utxixj (t, x) (i, j = 1, n) ∈ L2(QT ).

3. On the existence of almost everywhere solution

In this section, using non-zero rotation principle, the following existence theorem for
the almost everywhere solution of problem (1)-(3) is proved for n:

Theorem 1. Let

(i) aij(x) ∈ C(2)(Ω̄) (i, j = 1, n); a(x) ∈ C(1)(Ω̄); S ∈ C(3); the eigenfunctions vs(x)
of the operator L under boundary condition vs(x)|S = 0 be three times continuously

differentiable on Ω̄; ϕ(x) ∈W 3
2 (Ω); ϕ(x), Lϕ(x) ∈

◦
D(Ω); ψ(x) ∈W 2

2 (Ω)
⋂ ◦
D(Ω).

2. F = F1 + F2 + F3, where
a) the operator F1 acts from the B2

2,T into the space W 0,1
t,x,2(QT ) continuously and for

all u ∈ B2
2,T , t ∈ [0, T ] :

‖F1(u(t, x))‖W 1
2 (Ω) ≤ a1(t) + a2(t) · ‖u‖γ

B2
2,T

+ a3(t) · ‖u‖B2
2,t
, (6)

where ai(t) ∈ L2(0, T ) (i = 1, 2, 3) and 0 < γ < 1;

b) the operator F2 acts from the closed ballK∗
(
‖u‖B2

2,T
≤ 1

λ1
a
)

into the spaceW 0,1
t,x,2(QT )

continuously, where

a > a0 ≡ max{y : y2 ≤ (A1 +A2 |y|2γ) ·A3}, (7)

A1 ≡ 2 ‖w(t, x)‖2
B3,2

2,2,T
+ 6(2T + 1) · C2

0 · ‖a1(t)‖2L2(0,T ) , (8)

A2 ≡ 6(2T + 1) · C2
0 ·

1

λ2γ
1

· ‖a2(t)‖2L2(0,T ) , (9)



S. J.Aliyev, A. Q.Aliyeva, G. Z. Abdullayeva / Eur. J. Pure Appl. Math, 12 (2) (2019), 577-589 581

A3 ≡ exp

{
6(2T + 1) · C2

0 ·
1

λ2
1

· ‖a3(t)‖2L2(0,T )

}
, (10)

w(t, x) =
∞∑
s=1

{
ϕs +

1

λ2
s

[
1− e−λ2st

]
ψs

}
· vs(x), (11)

C0 ≡ max

{
n · max

i,j=1,n
{‖aij(x)‖C(Ω̄)}, ‖a(x)‖C(Ω̄)

} 1
2

; (12)

c)

inf
u∈M
{‖u−Q1(u)‖

B3,2
2,2,T
− ‖Q2(u)‖

B3,2
2,2,T
} > 0, (13)

where M the boundary of the ball K(‖u‖
B3,2

2,2,T
≤ a), Q1(u) = w + P (F1(u)),

Q2(u) = P (F2(u)) and

P (u(t, x)) ≡
∞∑
s=1

1

λ2
s

t∫
0

∫
Ω

u(τ, ξ)vs(ξ) ·
[
1− e−λ2s(t−τ)

]
dξdτ · vs(x); (14)

d) the operator F3 acts from the closed ball Kρ(‖u‖B3,2
2,2,T
≤ ρ) into the space W 0,1

t,x,2(QT )

and for all u, v ∈ Kρ :

‖F3(u)− F3(v)‖
W 0,1
t,x,2(QT )

≤ q · ‖u− v‖
B3,2

2,2,T
, (15)

where

ρ ≥ a, ρ ≥ ρ0 ≡ sup
u∈K
{‖Q1(u) +Q2(u)‖

B3,2
2,2,T
},

(√
T +

1√
2

)
· C0 · q ≡ q0 ≤ 1− ρ0

ρ
, q0 < 1; (16)

e)

inf
u∈M
{‖u−Q1(u)−Q2(u)‖

B3,2
2,2,T
− ‖Q3(u)‖

B3,2
2,2,T
} > 0, Q3(u) = P (F3(u)); (17)

f) F3(0) = 0.

3. a) For any u ∈ B3,2
2,2,T for almost all t ∈ [0, T ], F1(u(t, x)) ∈

◦
D(Ω);

b) For any u ∈ K for almost all t ∈ [0, T ], F2(u(t, x)) ∈
◦
D(Ω);

c) For any u ∈ Kρ for almost all t ∈ [0, T ], F3(u(t, x)) ∈
◦
D(Ω).

Then problem (1)-(3) has an almost everywhere solution.



S. J.Aliyev, A. Q.Aliyeva, G. Z. Abdullayeva / Eur. J. Pure Appl. Math, 12 (2) (2019), 577-589 582

Proof. Using condition 3 of this theorem, we have

Q1(u(t, x)) = w(t, x) +

∞∑
s=1

1

λ3
s

t∫
0

∫
Ω

 n∑
i,j=1

aij(ξ)
∂

∂ξi
F1(u(τ, ξ)) · ∂

∂ξj

(
vs(ξ)

λs

)

+a(ξ)F1(u(τ, ξ)) · vs(ξ)
λs

]
·
[
1− e−λ2s(t−τ)

]
dξdτ · vs(x) ∀u ∈ B3,2

2,2,T , (18)

Q2(u(t, x)) =
∞∑
s=1

1

λ3
s

t∫
0

∫
Ω

 n∑
i,j=1

aij(ξ)
∂

∂ξi
F2(u(τ, ξ)) · ∂

∂ξj

(
vs(ξ)

λs

)

+a(ξ)F2(u(τ, ξ)) · vs(ξ)
λs

]
·
[
1− e−λ2s(t−τ)

]
dξdτ · vs(x) ∀u ∈ K, (19)

Q3(u(t, x)) =
∞∑
s=1

1

λ3
s

t∫
0

∫
Ω

 n∑
i,j=1

aij(ξ)
∂

∂ξi
F3(u(τ, ξ)) · ∂

∂ξj

(
vs(ξ)

λs

)

+a(ξ)F3(u(τ, ξ)) · vs(ξ)
λs

]
·
[
1− e−λ2s(t−τ)

]
dξdτ · vs(x) ∀u ∈ Kρ. (20)

It is easy to obtain that, for any u, v ∈ B3,2
2,2,T

‖Q1(u)−Q1(v)‖
B3,2

2,2,T
≤
(√

T +
1√
2

)
T∫

0

∫
Ω

 n∑
i,j=1

aij(ξ)
∂

∂ξi
(F1(u(τ, ξ))

−F1(v(τ, ξ))) · ∂
∂ξj

(F1(u(τ, ξ))− F1(v(τ, ξ))) + a(ξ)(F1(u(τ, ξ))− F1(v(τ, ξ)))2

]
dξdτ

} 1
2

≤
(√

T +
1√
2

)
· C0 · ‖F1(u(t, x))− F1(v(t, x))‖

W 0,1
t,x,2(QT )

. (21)

From (21) by virtue of the condition 2a this theorem it follows that the operator Q1

acts continuously from the B2
2,T into B3,2

2,2,T . Since, the space B3,2
2,2,T imbedded into the

space B2
2,T compactly ([6, Theorem 1.1, p.51]), then the operator Q1 acts in the B3,2

2,2,T

compactly.
We consider in B3,2

2,2,T the equations

u = µQ1(u) µ ∈ [0, 1], (22)

and a priori estimate their all the possible solutions uµ(t, x). Then, using inequality (6)
∀µ ∈ [0, 1] and t ∈ [0, T ] we have



S. J.Aliyev, A. Q.Aliyeva, G. Z. Abdullayeva / Eur. J. Pure Appl. Math, 12 (2) (2019), 577-589 583

‖uµ‖2B3,2
2,2,t

= ‖µQ1(uµ)‖2
B3,2

2,2,t
≤ ‖Q1(uµ)‖2

B3,2
2,2,t
≡ ‖w + P (F1(uµ(t, x)))‖2

B3,2
2,2,t

≤ 2 ‖w‖2
B3,2

2,2,T
+ 2(2T + 1) · C2

0 ·
t∫

0

‖F1(uµ(τ, x))‖2W 1
2 (Ω) dτ

≤ 2 ‖w‖2
B3,2

2,2,T
+ 6(2T + 1) · C2

0 ·

{
‖a1(t)‖2L2(0,T ) + ‖a2(t)‖2L2(0,T ) ·

1

λ2γ
1

· ‖uµ‖2γ
B3,2

2,2,T

}

+6(2T + 1) · C2
0 ·

1

λ2
1

t∫
0

a2
3(τ) · ‖uµ‖2B3,2

2,2,τ
dτ, (23)

where C0 is defined by (12).
From (23), on applying Bellman’s inequality [4, pp. 188,189] and using notations

(8)-(10), we obtain that ∀µ ∈ [0, 1] :

‖uµ‖2B3,2
2,2,T
≤
(
A1 +A2 · ‖uµ‖2γ

B3,2
2,2,T

)
·A3.

From here, using notation (7), we have

‖uµ(t, x)‖2
B3,2

2,2,T
≤ a0 ∀µ ∈ [0, 1], (24)

that is, all the possible solutions uµ of equations (22) are a priori bounded in B3,2
2,2,T and

belong to the ball K0(‖u‖
B3,2

2,2,T
≤ a0).

From (22) and (24) we obtain that ∀µ ∈ [0, 1] completely continuous vector field
Tµ = J − µQ1 has no zeros on the boundary M of the ball K(‖u‖

B3,2
2,2,T

≤ a), where

J is a unit vector field and a is a number appearing in the condition 2b this theorem.
Consequently, completely continuous vector fields T0 = J and T1 = J −Q1 are homotopic
on the sphere M . Then their rotation δ on M are the same, namely:

δ(J −Q1;M) = δ(J ;M) = 1.

Now, we consider the operator Q2 in the closed ball K. Just as the completely conti-
nuity of the operator Q1 in B3,2

2,2,T , was shown it is easy to show that the operator Q2 acts

compactly from K into B3,2
2,2,T . Further, on the boundary M of the ball K we consider

completely continuous vector fields Fλ = J−Q1−λQ2, λ ∈ [0, 1]. Due to of the condition
2b this theorem ∀λ ∈ [0, 1] and u ∈M we have

‖u−Q1(u)− λQ2(u)‖
B3,2

2,2,T
≥ ‖u−Q1(u)‖

B3,2
2,2,T
− ‖Q2(u)‖

B3,2
2,2,T

> 0.



S. J.Aliyev, A. Q.Aliyeva, G. Z. Abdullayeva / Eur. J. Pure Appl. Math, 12 (2) (2019), 577-589 584

Hence, in particular, it follows that completely continuous vector fields F0 = J − Q1

and F1 = J−Q1−Q2 are homotopic on the sphere M . Consequently, on M their rotations
are equal to:

δ(J −Q1 −Q2;M) = δ(J −Q1;M) = δ(J ;M) = 1. (25)

And now in the ball Kρ(‖u‖B3,2
2,2,T
≤ ρ) we consider the operator Q3. Similar to (21),

∀u, v ∈ Kρ we have

‖Q3(u)−Q3(v)‖
B3,2

2,2,T
≤
(√

T +
1√
2

)
· C0 · ‖F3(u)− F3(v)‖

W 0,1
t,x,2(QT )

≤
(√

T +
1√
2

)
· C0 · q · ‖u− V ‖B3,2

2,2,T
= q0 · ‖u− V ‖B3,2

2,2,T
.

For each fixed V0 ∈ Kρ0(‖u‖
B3,2

2,2,T
≤ ρ0) (where the number ρ0 is defined by (16)) and

ε ∈ [0, 1] in the ball Kρ we consider the following equation

u = ε ·Q3(u) + V0. (26)

Taking advantage fact that

Q3(0) = P (F3(0)) = P (0) = 0,

for any u, u1, u2 ∈ Kρ we have

‖ε ·Q3(u) + V0‖B3,2
2,2,T
≤ ‖Q3(u)‖

B3,2
2,2,T

+ ‖V0‖B3,2
2,2,T

= ‖Q3(u)−Q3(0)‖
B3,2

2,2,T
+ ‖V0‖B3,2

2,2,T
≤
(√

T +
1√
2

)
· C0 · q · ‖u‖B3,2

2,2,T
+ ‖V0‖B3,2

2,2,T

= q0 · ‖u‖B3,2
2,2,T

+ ‖V0‖B3,2
2,2,T
≤ q0 · ρ+ ρ0 ≤ ρ,

‖ε ·Q3(u1) + V0 − [ε ·Q3(u2) + V0]‖
B3,2

2,2,T

≤ ‖Q3(u1)−Q3(u2)‖
B3,2

2,2,T
≤ q0 ‖u1 − u2‖B3,2

2,2,T
.

As q0 < 1, then by virtue of the contracted mappings principle, the operator A(u) =
ε ·Q3(u) + V0 has a unique fixed point u0 in Kρ. Comparing to each V0 ∈ Kρ0 the unique
in Kρ solution u0 ∈ Kρ0 of equation (26), we generate some operator Rε, acting from Kρ0

into Kρ. Next, for any ε ∈ [0, 1] and V ∈ Kρ0 using notation

Rε(V ) ≡ ε ·Q3(Rε(V )) + V,



S. J.Aliyev, A. Q.Aliyeva, G. Z. Abdullayeva / Eur. J. Pure Appl. Math, 12 (2) (2019), 577-589 585

it is easy to get for any V1, V2 ∈ Kρ0

‖Rε(V1)−Rε(V2)‖
B3,2

2,2,T
≤ 1

1− ε · q0
· ‖V1 − V2‖B3,2

2,2,T
. (27)

Due to notation (16) we have (Q1 + Q2)K ⊂ Kρ0 . Consequently, for each ε ∈ [0, 1]
the operator Rε is defined, in particular, and on (Q1 + Q2)K. Due to (27) the operator
Rε satisfies a Lipschitz condition (and therefore continuous) on (Q1 +Q2)K. And as, the
operators Q1 and Q2 are completely continuous on K, then for each ε ∈ [0, 1] the operator
Rε(Q1 +Q2) compactly on K. Further, due to (17), for any u ∈M and ε ∈ [0, 1] we have

‖u−Q1(u)−Q2(u)− ε ·Q3(u)‖
B3,2

2,2,T

≥ ‖u−Q1(u)−Q2(u)‖
B3,2

2,2,T
− ‖Q3(u)‖

B3,2
2,2,T

> 0.

Consequently, the completely continuous vector field J−Q1−Q2−ε·Q3 for any ε ∈ [0, 1]
does not have zeros on M . Then does not have zeros on M same way the completely
continuous vector field J − Rε(Q1 +Q2), because each zero on M field J − Rε(Q1 +Q2)
is zero field J −Q1 −Q2 − ε ·Q3.

Thus, the completely continuous vector fields J − R0(Q1 + Q2) = J − Q1 − Q2 and
J −R1(Q1 +Q2) are homotopic on M . Consequently, due to (25) we have

δ(J −R1(Q1 +Q2);M) = δ(J −Q1 −Q2;M) = 1.

Hence, by virtue of the non-zero rotation principle [9, p. 207], the completely contin-
uous vector field J − R1(Q1 + Q2) has at least one zero inside the ball of K. Since each
such zero is a zero of the field J −Q1 −Q2 −Q3, it is thus proved that there exists in K
at least one fixed point u(t, x) an operator Q1 + Q2 + Q3 = Q. Further, it easy to verify
(in absolutely the same way as in the proof of Theorem of [2]), that the function u(t, x) is
an almost everywhere solution of problem (1)-(3). The theorem is proved.

4. Correct formulation of the problem

In this section, using Bellman’s inequality ([4, p. 188-189]), the following theorem on
continuous dependence (in a certain sense) on initial functions ϕ(x), ψ(x), and nonlinear
operator F for the almost everywhere solution of problem (1)-(3). Problem (1)-(3) with
the data ϕ̃, ψ̃, F̃ we let’s name problem Ã.

Theorem 2. Let:

(i) Condition 1 of Theorem 1 be satisfied.

(ii) ϕ̃(x) ∈W 3
2 (Ω); ϕ̃(x), Lϕ̃(x) ∈

◦
D(Ω); ψ̃(x) ∈W 2

2 (Ω)
⋂ ◦
D(Ω).

(iii) For each u ∈ B3,2
2,2,T

⋃
(G
⋂
B2,1

2,2,T ) for almost all t ∈ [0, T ], F(u(t, x)) ∈
◦
D(Ω).



S. J.Aliyev, A. Q.Aliyeva, G. Z. Abdullayeva / Eur. J. Pure Appl. Math, 12 (2) (2019), 577-589 586

(iv) For each u ∈ B3,2
2,2,T for almost all t ∈ [0, T ], F̃(u(t, x)) ∈

◦
D(Ω).

(v) The operators F and F̃ acts from B3,2
2,2,T

⋃
(G
⋂
B2,1

2,2,T ) into W 0,1
t,x,2(QT ) so that, for

all u, v ∈ B3,2
2,2,T and t ∈ [0, T ]

‖F(u(t, x))‖W 1
2 (Ω) ≤ a(t) + b(t) · ‖u‖

B3,2
2,2,T

, a(t), b(t) ∈ L2(0, T ),

‖F(u(t, x))− F(v(t, x))‖W 1
2 (Ω) ≤ c(t) · ‖u− v‖B3,2

2,2,T
, c(t) ∈ L2(0, T ),

∥∥∥F̃(u(t, x))− F̃(v(t, x))
∥∥∥
W 1

2 (Ω)
≤ c̃(t) · ‖u− v‖

B3,2
2,2,T

, c̃(t) ∈ L2(0, T ),

sup
u∈K1

{∥∥∥F(u(t, x))− F̃(u(t, x))
∥∥∥
W 0,1
t,x,2(QT )

}
≡ ε < +∞,

where

K1 = K1(‖u‖
B3,2

2,2,T
≤ a1),

a1 ≡ {[2 ‖w(t, x)‖2
B3,2

2,2,T
+ 4(2T + 1) ·C2

0 · ‖a(t)‖2L2(0,T )] · exp[4(2T + 1) ·C2
0 · ‖b(t)‖

2
L2(0,T )]}

1
2 ,

the function w(t, x) is defined by (11) and the number C0 is defined by (12).
Then for the unique almost everywhere solutions u(t, x) and ũ(t, x) of problems (1)-(3)

and Ã, respectively, we have

‖u(t, x)− ũ(t, x)‖
B3,2

2,2,T
≤ {
√

3 · C0 · ‖L(ϕ(x)− ϕ̃(x))‖W 1
2 (Ω)

+
√

6 · C0 ·
∥∥∥ψ(x)− ψ̃(x)

∥∥∥
W 1

2 (Ω)
+
√

6 ·
∥∥∥L(ψ(x)− ψ̃(x))

∥∥∥
L2(Ω)

+
√

6(2T + 1) · C0 · ε} · exp{3(2T + 1) · C2
0 · ‖c̃(t)‖

2
L2(0,T )}, (28)

where the operator L is defined by (4) and the number C0 is defined by (12).
Proof: By Theorem 2 from work [1] each of the problems (1)-(3) and à has a unique

almost everywhere solution

u(t, x) =
∞∑
s=1

us(t)vs(x) and ũ(t, x) =
∞∑
s=1

ũs(t)vs(x),

respectively, so that u ∈ K1 ⊂ B3,2
2,2,T , ũ ∈ B3,2

2,2,T . Then, by virtue of the lemma in section
2, the functions us(t) (s = 1, 2, . . .) and ũs(t) (s = 1, 2, . . .) satisfy system (5), so that for



S. J.Aliyev, A. Q.Aliyeva, G. Z. Abdullayeva / Eur. J. Pure Appl. Math, 12 (2) (2019), 577-589 587

ũs(t) (s = 1, 2, . . .) in the system (5) instead of ϕs, ψs and F(u) need to take ϕ̃s, ψ̃s and
F̃(u), respectively.

Using this fact, from system (5) it is easy to obtain that ∀t ∈ [0, T ]:

‖u− ũ‖2
B3,2

2,2,t
≤ 3

∥∥∥∥∥
∞∑
s=1

(ϕs − ϕ̃s)vs(x)

∥∥∥∥∥
2

B3,2
2,2,t

+ 3

∥∥∥∥∥
∞∑
s=1

1

λ2
s

(1− e−λ2st)(ψs − ψ̃s)vs(x)

∥∥∥∥∥
2

B3,2
2,2,t

+3

∥∥∥∥∥∥
∞∑
s=1

1

λ2
s

t∫
0

∫
Ω

[F(u(τ, ξ))− F̃(ũ(τ, ξ))] · [1− e−λ2s(t−τ)]vs(ξ)dξdτ · vs(x)

∥∥∥∥∥∥
2

B3,2
2,2,t

≤ 3
∞∑
s=1

[λ3
s(ϕs − ϕ̃s)]2 + 6

∞∑
s=1

[λs(ψs − ψ̃s)]2 + 6
∞∑
s=1

[λ2
s(ψs − ψ̃s)]2

+3(2T + 1)C2
0 ·

t∫
0

∥∥∥F(u(τ, x))− F̃(ũ(τ, x))
∥∥∥2

W 1
2 (Ω)

dτ

≤ 3
∞∑
s=1

[λ3
s(ϕs − ϕ̃s)]2 + 6

∞∑
s=1

[λs(ψs − ψ̃s)]2 + 6
∞∑
s=1

[λ2
s(ψs − ψ̃s)]2

+6(2T + 1) · C2
0 ·

 t∫
0

∥∥∥F(u(τ, x))− F̃(u(τ, x))
∥∥∥2

W 1
2 (Ω)

dτ

+

t∫
0

∥∥∥F̃(u(τ, x))− F̃(ũ(τ, x))
∥∥∥2

W 1
2 (Ω)

dτ


≤ 3 · C2

0 · ‖L(ϕ(x)− ϕ̃(x))‖2W 1
2 (Ω) + 6 · C2

0 ·
∥∥∥ψ(x)− ψ̃(x)

∥∥∥2

W 1
2 (Ω)

+6 ·
∥∥∥L(ψ(x)− ψ̃(x))

∥∥∥2

L2(Ω)
+ 6(2T + 1) · C2

0 ·
∥∥∥F(u(τ, x))− F̃(u(τ, x))

∥∥∥2

W 0,1
t,x,2(QT )

+6(2T + 1) · C2
0 ·

t∫
0

c̃2(τ) · ‖u− ũ‖
B3,2

2,2,τ
dτ ≤ 3 · C2

0 · ‖L(ϕ(x)− ϕ̃(x))‖2W 1
2 (Ω)

+6 · C2
0 ·
∥∥∥ψ(x)− ψ̃(x)

∥∥∥2

W 1
2 (Ω)

+ 6 ·
∥∥∥L(ψ(x)− ψ̃(x))

∥∥∥2

L2(Ω)
+ 6(2T + 1) · C2

0 · ε2



REFERENCES 588

+6(2T + 1) · C2
0 ·

t∫
0

c̃2(τ) · ‖u− ũ‖2
B3,2

2,2,τ
dτ.

From here, on applying Bellman’s inequality ([4, p.188, 189]), we obtain the estimate
(28).

Acknowledgements

The authors would like to express their deep gratitude to the editorial team and the
anonymous referees of European Journal of Pure and Applied Mathematics, for the careful
reading of the manuscript as well as their valuable comments and suggestions which helped
to improve the present paper.

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