EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 10, No. 5, 2017, 1078-1091
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

The study of multidimensional mixed problem for one
class of third order semilinear psevdohyperbolic

equations

Samed J.Aliyev1∗, Arzu G.Aliyeva2

1 Department of Mechanics and Mathematics, Baku State University, AZ1148
Baku, Azerbaijan
2 Department of Differential Equations, Institute of Mathematics and Mechanics, Azerbaijan
National Academy of Sciences, AZ1141, Baku, Azerbaijan.

Abstract. This work is dedicated to the study of existence in small of classical solution of mul-
tidimensional mixed problem for one class of third order semilinear psevdohyperbolic equations.
Conception of classical solution for mixed problem under consideration is introduced. After apply-
ing Fourier method, the solution of original problem is reduced to the solution of some countable
system of nonlinear inteqro-differential equations in unknown Fourier coefficients of the sought
solution. Besides, existence theorem in small of classical solution of the mixed problem is proved
by contracted mappings principle.

2010 Mathematics Subject Classifications: 35L76, 35L82.

Key Words and Phrases: semilinear, psevdohyperbolic equations, mixed problem, classical
solution.

1. Introduction

This work is dedicated to the study existence of classical solution for the following
multidimensional mixed problem:

∂2u(t, x)

∂t2
− ∂

∂t
(L(u(t, x))) =

= F (t, x, u(t, x), ut(t, x),Ou(t, x),Out(t, x),O2u(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)

∗Corresponding author.

Email addresses: samed59@bk.ru (Samed J.Aliyev), arzu.aliyeva@bk.ru (A. G. Aliyeva)

http://www.ejpam.com 1078 c© 2017 EJPAM All rights reserved.



S. J.Aliyev, A. G.Aliyeva / Eur. J. Pure Appl. Math, 10 (5) (2017), 1078-1091 1079

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)

∂

∂xj

)
− a(x) · u(t, x), (4)

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

aijξiξj ≥ α ·
n∑
i=1

ξ2
i (α = const > 0),

where ξi (i = 1, ..., n) are arbitrary real numbers; F, ϕ, ψ are the given functions, and
u(t, x) is a sought function; we define the classical solution of problem (1)-(3) as a func-
tion u(t, x) if it is continuous on domain QT = [0, T ] × Ω together with the 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), for which satisfies the equation (1) on the closed domain QT ,

initial conditions (2) on Ω and boundary condition (3) in the usual sense.
It must note that, many problems in elasticity theory, in particular the problems of

longitudinal vibration of the viscoelastic non-homogeneous bar, some wave problems for
elastic-viscidal liquid and etc. lead to the problems type (1)-(3).

In the works [4-6] considered a special case of the equation (1), when L = ∆, F (t, x, 0,
..., 0) = 0 and proven a theorems of existence and uniqueness of the classical solution for
initial functions ϕ(x), ψ(x) with sufficiently small (in a certain metric) norms.

We mention the work [1] in which is introduced the definition of almost everywhere
solution of the problem (1)-(3) for arbitrary dimension n (i.e. for arbitrary number of the
variables) and proven the local existence and global uniqueness theorems for the almost
everywhere solution.

We mention also the results of [2] which is complete progression of the results of [1].
In particular in the work [2] the apriori estimates for the almost everywhere solution of
the considered mixed problem are esthablished in three steps, which are getting stronger
from step to step.

In the work [10] is proven the existence and uniqueness of the strong global solution
of the one special case of the problem (1)-(3), when L = ∆, n ≤ 3 and F = ∆u+ f(u).

Finally we mention the work [8] in which proven theorems about existence and unique-
ness of the generalized, almost everywhere and classical solution for one special one-
dimensional case of the problem (1)-(3), when n = 1, Ω = (0, 1), Lu = α · uxx.

2. Auxiliaries

In this section, we introduce a number of concepts, notations and facts to be used
later.



S. J.Aliyev, A. G.Aliyeva / Eur. J. Pure Appl. Math, 10 (5) (2017), 1078-1091 1080

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).
For investigation of the problem (1)-(3) we recall one property of the operator L, gener-

ating 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 υs(x) which are complete and orthonor-

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

L, if it is not identically zero and

∫
Ω


n∑

i,j=1

aij(x)
∂υs(x)

∂xi
· ∂Φ(x)

∂xj
+ a(x)υs(x)Φ(x)

 dx = λ2
s ·
∫
Ω

υs(x)Φ(x)dx (5)

for any function Φ(x) ∈
◦
D(Ω).

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

a totality of all the functions of the from

u(t, x) =
∞∑
s=1

us(t)υs(x)

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.

3. Let ∀t ∈ [0, T ] Ai(t, x) (i = 0, 1, ..., n), B(t, x) ∈ L2(Ω). Then the following
inequality hold ([7, p. 135]):

∞∑
s=1


∫
Ω

 n∑
i,j=1

aij(x)Ai(t, x)
∂

∂xj

(
υs(x)

λs

)
+ a(x)B(t, x)

υs(x)

λs

 dx


2

≤
∫
Ω


n∑

i,j=1

aij(x)Ai(t, x)Aj(t, x) + a(x)B2(t, x)

 dx. (6)

4. We will also often use the following lemma from [9, p.88, lemma 1] by O.A. La-
dyzhenskaya:



S. J.Aliyev, A. G.Aliyeva / Eur. J. Pure Appl. Math, 10 (5) (2017), 1078-1091 1081

Lemma 1. Let the following conditions be satisfied for a natural number m ≥ 2:
a) aij(x) (i, j = 1, 2, ..., n) ∈ Cm−1(Ω̄)), a(x) ∈ Cm−2(Ω̄), ∂Ω ≡ S ∈ Cm, υ(x) ∈

Cm(Ω̄);

b) υ(x)|s = L(υ(x))|s = ... = L[m−1
2 ](υ(x))

∣∣∣
s

= 0.

Then the following inequalities hold:

Hm(υ) ≡
m∑
k=0

Jk(υ) ≤ C
{

r∑
s=0

J0(Lsυ) +
r−1∑
s=0

J1(Lsυ)

}
for m = 2r,

Hm(υ) ≤ C
{

r∑
s=0

J1(Lsυ) +
r∑
s=0

J0(Lsυ)

}
for m = 2r + 1, where

Jk(u, υ) =

∫
Ω

∑
1 ≤ α1, ..., αk ≤ n
1 ≤ β1, ..., βk ≤ n

aα1β1 ...aαkβk
∂ku(x)

∂xα1 ...∂xαk
· ∂kυ(x)

∂xβ1 ...∂xβk
dx

Jk(u) = Jk(u, u)(k ≥ 1); J0(u) = J0(u, u), J0(u, υ) =

∫
Ω

u(x)υ(x)dx,

C > 0 is some constant not depending on υ(x).

It is evident that for the function of the form u(t, x) =
∞∑
s=1

us(t)υs(x)

J0(u(t, x)) =

∫
Ω

u2(t, x)dx =

∞∑
s=1

u2
s(t), (7)

hold and, due to the equality (5) we have

J1(u(t, x)) =

∫
Ω


n∑

i,j=1

aij(x)
∂u(t, x)

∂xi
· ∂u(t, x)

∂xj

 dx

≤
∫
Ω


n∑

i,j=1

aij(x)
∂u(t, x)

∂xi
· ∂u(t, x)

∂xj
+ a(x)u2(t, x)

 dx =

∞∑
s=1

λ2
su

2
s(t). (8)

Besides, for any function u(x) ∈W k
2 (Ω), the following inequality hold [9, p.84, inequal-

ity (18)]:

Ak ·
∫
Ω

∑
1≤α1,...,αk≤n

(
∂ku(x)

∂xα1 ...∂xαk

)2

dx ≤ Jk(u)

≤ Bk ·
∫
Ω

∑
1≤α1,...,αk≤n

(
∂ku(x)

∂xα1 ...∂xαk

)2

dx (k ≥ 1) (9)



S. J.Aliyev, A. G.Aliyeva / Eur. J. Pure Appl. Math, 10 (5) (2017), 1078-1091 1082

where Ak > 0, Bk > 0 are some constants not depending on u(x) ∈W k
2 (Ω).

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

u(t, x) =
∞∑
s=1

us(t)υs(x),

where

us(t) =

∫
Ω

u(t, x)υs(x)dx (s = 1, 2, ...).

Then, after applying the Fourier method, finding the unknown Fourier coefficients
us(t) (s = 1, 2, ...) for the classical 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

=(u(τ, x))
[
1− eλ2s(t−τ)

]
υs(x)dxdτ (s = 1, 2, ...; t ∈ [0, T ]), (10)

where

ϕs =

∫
Ω

ϕ(x)υs(x)dx, ψs =

∫
Ω

ψ(x)υs(x)dx (s = 1, 2, ...),

=(u(τ, x)) ≡ F (τ, x, u(τ, x), uτ (τ, x), ux(τ, x), uτ x(τ, x), uxx(τ, x)). (11)

Proceeding from the definition of classical solution of problem (1)-(3), it is easy to
prove the following

Lemma 2. If u(t, x) =
∞∑
s=1

us(t)υs(x) is any classical 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 (10).

Proof. Let u(t, x) =
∞∑
s=1

us(t)υs(x) be any classical solution of problem (1)-(3). Then

it is evident that

T∫
0

∫
Ω

{
utt(t, x)− ∂

∂t
(L(u(t, x)))−=(u(t, x))

}
Φ(t, x)dxdt = 0 (12)

for each Φ(t, x) ∈ L2(QT ), and = is defined by (11).



S. J.Aliyev, A. G.Aliyeva / Eur. J. Pure Appl. Math, 10 (5) (2017), 1078-1091 1083

If, in particular, we take

Φ(t, x) =

{
(t− τ)2υs(x) for 0 ≤ t ≤ τ, x ∈ Ω,
0 for τ < t ≤ T, x ∈ Ω,

where s = 1, 2, ... and τ ∈ [0, T ], then with the help of integration by parts with respect
to t twice in the first term and once in the second term of (12) and taking the initial
conditions (2) into consideration we easily get

2

τ∫
0

us(t)dt− 2λ2
s

τ∫
0

(t− τ)us(t)dt−
τ∫

0

(t− τ)2=s(u, t)dt

−2τϕs − τ2ψs − λ2
sτ

2ϕs = 0, (13)

where

=s(u, t) ≡
∫
Ω

=(u(t, x))υs(x)dx.

Differentiating (13) three times with respect to τ we have the next problem{
u′′s(τ) + λ2

su
′
s(τ) = =s(u, τ) (s = 1, 2, ...; t ∈ [0, T ]),

us(0) = ϕs, u′s(0) = ψs,

which is obviously equivalent to system (10). Lemma is proved.

6. We agree to assume that all the quantities throughout this work are real, all the
functions are real –valued, and all the integrals are understood in the sense of Lebesgue.

3. Main Result

In this section, using contracted mappings principle, the following existence in small
(i.e. for sufficiently small values of T ) theorem for the classical solution of problem (1)-(3)
is proved for n:

Theorem 1. Let

1. aij(x) ∈ C[n2 ]+3(Ω̄) (i, j = 1, 2, ..., n); a(x) ∈ C[n2 ]+2(Ω̄); S ∈ C[n2 ]+4; the eigenfunc-
tions υs(x) of the operator L under boundary condition υs(x)|s = 0 be

[
n
2

]
+ 4 times

continuously differentiable on Ω̄.

2. ϕ(x) ∈ W
[n2 ]+4

2 (Ω), ϕ(x), Lϕ(x), ..., L

[
n+2

4

]
+1
ϕ(x) ∈

◦
D(Ω); ψ(x) ∈ W

[n2 ]+3

2 (Ω),

ψ(x), Lψ(x), ..., L[n4 ]+1ψ(x) ∈
◦
D(Ω).



S. J.Aliyev, A. G.Aliyeva / Eur. J. Pure Appl. Math, 10 (5) (2017), 1078-1091 1084

3. a) ∂kF (t, ξ1, ..., ξÑ )
/
∂ξα1

1 ...∂ξ
αÑ
Ñ
∈ C(Q̄T × (−∞,∞)N (k = 0, 1, ...,

[
n
2

]
+ 2);

b) ∂kF (t, ξ1, ..., ξn, 0, 0, ξn+3, ..., ξÑ )
/
∂ξα1

1 ...∂ξ
αÑ
Ñ
≡ 0 (k = 0, 1, ..., 2

[
n+2

4

]
)

∀t ∈ [0, T ] , (ξ1..., ξn) ∈ S, ξn+3, ..., ξN ∈ (−∞,∞), N = 2 + 2n+ n2, Ñ = n+N .

4. ∀R > 0 in Q̄T × (−∞,∞)N

∣∣∣∣∂2
[
n+2

4

]
F (t, x, u1, ... uN )

/
∂xα∂uγ11 ...∂u

γN
N −

−∂2
[
n+2

4

]
F (t, x, ũ1, ..., ũN )

/
∂xα∂uγ11 ...∂u

γN
N

∣∣∣∣ ≤ CR N∑
i=1

|ui − ũi| ,

where α = (α1, ..., αn), |α|+
N∑
i=1

γi = 2
[
n+2

4

]
, and CR > 0 is a constant.

Then there exists in small (i.e. for sufficiently small values of T ) a unique in large
(i.e. for any finite value of T ) classical solution of problem (1)-(3).

Proof. We consider the following operator Q in space B
[n2 ]+4,[n2 ]+3

2,2,T :

Q(u(t, x)) = W (t, x) + P(u(t, x)) (14)

where

W (t, x) =
∞∑
s=1

{
ϕs +

1

λ2
s

[
1− e−λ2st

]
ψ(s)

}
υs(x),

P(u(t, x)) =
∞∑
s=1

1

λ2
s

t∫
0

∫
Ω

=(u(τ, x))
[
1− e−λ2s(t−τ)

]
υs(x)dxdτ · υs(x),

and operator = is defined by (11). It is easy to see that under condition 3 of this theorem

∀u(t, x) ∈ B[n2 ]+4,[n2 ]+3

2,2,T :

P(u(t, x)) =

∞∑
s=1

(−1)r+1

λ
[n2 ]+4
s

t∫
0

∫
Ω

Lr+1(=(u(τ, x)))
[
1− e−λ2s(t−τ)

]
×υs(x)dxdτ · υs(x) for n = 4r, 4r + 1, (15)

P(u(t, x)) =

∞∑
s=1

(−1)r+1

λ
[n2 ]+4
s

t∫
0

∫
Ω


n∑

i,j=1

aij(x)
∂

∂xi
Lr+1(=(u(τ, x)))


× ∂

∂xj

(
υs(x)

λs

)
+ a(x)Lr+1(=(u(τ, x)))× υs(x)

λs

}



S. J.Aliyev, A. G.Aliyeva / Eur. J. Pure Appl. Math, 10 (5) (2017), 1078-1091 1085

×
[
1− e−λ2s(t−τ)

]
dxdτ · υs(x) for n = 4r + 2, 4r + 3. (16)

To ease our writings, we will use the following notations:

‖u‖
B

[
n
2

]
+4,

[
n
2

]
+3

2, 2, t

= ‖u‖Et (t ∈ [0, T ]).

Next, using inequalities (9), lemma 1, and conditions 1,2 of this theorem, we obtain
that W (t, x) ∈ ET , because

‖W (t, x)‖ET ≤

{
2
∞∑
s=1

(
λ

[n2 ]+4
s · ϕs

)2

+ 2
∞∑
s=1

(
λ

[n2 ]+2
s · ψs

)2
} 1

2

+

{ ∞∑
s=1

(
λ

[n2 ]+3
s · ψs

)2
} 1

2

≤

 C ·
(∥∥Lr+2ϕ

∥∥
L2(Ω)

+
∥∥Lr+1ψ

∥∥
L2(Ω)

+
√
D (Lr+1ψ)

)
< +∞ for n = 4r, 4r + 1,

C ·
(√

D (Lr+2ϕ) +
√
D (Lr+1ψ) +

∥∥Lr+2ψ
∥∥
L2(Ω)

)
< +∞ for n = 4r + 2, 4r + 3,

where C > 0 is some constant,

D(z(t, x)) ≡
∫
Ω


n∑

i,j=1

aij(x)
∂z(t, x)

∂xi
· ∂z(t, x)

∂xj
+ a(x) · z2(t, x)

 dx.

Now we consider operator Q defined by (14) in a closed ball Kr of space ET with
a center in zero and a radius r > ‖W (t, x)‖ET . Then, using Bessel’s inequality (for
n = 4r, 4r + 1) and inequality (6) (for n = 4r + 2, 4r + 3), where functions Ai(t, x)
(i = 1, 2, ..., n) and B(t, x) must be replaced by ∂

∂xi
Lr+1 (=(u(t, x)))(i = 1, 2, ..., n) and

Lr+1(=(u(t, x))), respectively, we obtain from (14), (15) and (16) that ∀u ∈ ET :

‖Q(u)‖ET ≤ ‖W (t, x)‖ET +

(2T + 1) ·
T∫

0

∫
Ω

[
Lr+1 (=(u(t, x)))

]2
dxdt


1
2

= ‖W (t, x)‖ET +
√

2T + 1 ·
∥∥Lr+1(=(u(t, x)))

∥∥
L2(QT )

≤ ‖W (t, x)‖ET + C̃ ·
√

2T + 1 · ‖=(u(t, x))‖
W

0,2(r+1)
t,x,2 (QT )

for n = 4r, 4r + 1, (17)

‖Q(u)‖ET ≤ ‖W (t, x)‖ET +

(2T + 1) ·
T∫

0

∫
Ω


n∑

i,j=1

aij(x)
∂

∂xi
Lr+1 (=(u(t, x)))



S. J.Aliyev, A. G.Aliyeva / Eur. J. Pure Appl. Math, 10 (5) (2017), 1078-1091 1086

× ∂

∂xj
Lr+1 (=(u(t, x))) + a(x)

[
Lr+1(u(t, x))

]2}
dxdt

} 1
2

= ‖W (t, x)‖ET

+
√

2T + 1 ·
{∥∥D (Lr+1(=(u(t, x)))

)∥∥
L(0,T )

} 1
2

≤ ‖W (t, x)‖ET + ˜̃C ·
√

2T + 1 · ‖=(u(t, x))‖
W

0,2(r+1)+1
t,x,2 (QT )

for n = 4r + 2, 4r + 3, (18)

where C̃ > 0 and ˜̃C > 0 are some constants.
Due to estimates (17) and (18), we have for every n and ∀u ∈ ET :

‖Q(u)‖ET ≤ ‖W (t, x)‖ET + C ·
√

2T + 1 · ‖=(u(t, x))‖
W

0,

[
n
2

]
+2

t,x,2 (QT )

. (19)

Using Sobolev’s imbedding theorems, inequalities (9), lemma 1, estimates (7), (8), and
the structure of space ET , we have for every u ∈ ET and t ∈ [0, T ]:

∥∥∥Dk
tD

αu(t, x)
∥∥∥
L2(Ω)

≤ C · ‖u‖Et ≤ C · ‖u‖ET (k = 0, 1; 0 ≤ k + |α| ≤
[n

2

]
+ 4), (20)

∥∥∥Dk
tD

αu(t, x)
∥∥∥
C(Q̄T )

≤ C · ‖u‖ET (k = 0, 1; 0 ≤ k + |α| ≤ 3), (21)

∥∥∥Dk
tD

αu(t, x)
∥∥∥
Lq(Ω)

≤ Cq · ‖u‖Et ≤ Cq · ‖u‖ET
(
k = 0, 1; 0 ≤ k + |α| ≤

[n
2

]
+ 4
)
, (22)

where C > 0, Cq > 0 are some constants not depending on u and t, and

1 ≤ q ≤ 2n

n− 2
([
n
2

]
+ 4− k − |α|

) =
2n

n− 2
([
n
2

]
+ 4
)

+ 2 (k + |α|)
, q < +∞. (23)

Due to estimates (21) and condition 3a of this theorem, ∀u ∈ Kr:

∥∥∂sF (t, x, u(t, x), ut(t, x), ux(t, x), utx(t, x), uxx(t, x)
/
∂xα∂uγ11 ...∂u

γn
N

∥∥
C(Q̄T )

≤ Ãr

(
0 ≤ |α|+

N∑
i=1

γi = s ≤
[n

2

]
+ 2

)
, (24)

where Ar > 0 is some constant depending on the radius of closed sphere Kr in space ET
with a center in zero and a radius r. Next, due to estimates (24) and (21), to have an
estimate for



S. J.Aliyev, A. G.Aliyeva / Eur. J. Pure Appl. Math, 10 (5) (2017), 1078-1091 1087

∂[n2 ]+2

∂xα1
1 ...∂xαnn

{=(u(t, x))}

with u ∈ Kr, it suffices to estimate in L2(QT ) the products of the following form:

l∏
i=1

Dkil
t Dαilu(t, x)

(
l = 1, 2, ...,

[n
2

]
+ 2; 4 ≤ kil + |αil| ;

l∑
i=1

(kil + |αil|) ≤ l · 3 +
([n

2

]
+ 2− l

)
=
[n

2

]
+ 2l + 2; kil ≤ 1

)
. (25)

Using estimate (22) and taking (23) into account, for products of the form (25) with
any u ∈ Kr and t ∈ [0, T ] we have:∥∥∥∥∥

l∏
i=1

Dkil
t Dαilu(t, x)

∥∥∥∥∥
L2(Ω)

≤
l∏

i=1

∥∥∥Dkil
t Dαilu(t, x)

∥∥∥
L2pi(Ω)

≤ C · ‖u‖lEt ≤ C · ‖u‖
l
ET
≤ C · rl, (26)

where C > 0 is some constant and

1 ≤ pi ≤
n

n− 2
([
n
2

]
+ 4
)

+ 2 (kil + |αil|)
, pi < +∞,

l∑
i=1

1

pi
= 1, (27)

the possibility of last equality following from the relation below:

l∑
i=1

n− 2
([
n
2

]
+ 4
)

+ 2 (kil + |αil|)
n

=
1

n

{(
n− 2

([n
2

]
+ 4
))
· l + 2

l∑
i=1

(kil + |αil|)

}

≤ 1

n

{(
n− 2

([n
2

]
+ 4
))
· l + 2

([n
2

]
+ 2l + 2

)}
=

1

n

{
n+ (l − 1)

(
n− 2

[n
2

]
− 4
)}
≤ 1, (28)

when l ≥ 2, the latter part of (28) becomes a strict inequality because n− 2
[
n
2

]
− 4 < 0.

Now, using estimates (24), (21) and (26), from (19) we obtain that ∀u ∈ Kr:

‖Q(u)‖ET ≤ ‖W‖ET +
√
T ·
√

2T + 1 · Cr, (29)

where Cr > 0 is some number depending on r.
Next, similar to (19), ∀u, ũ ∈ Kr we have:

‖Q(u)−Q(ũ)‖ET ≤ C ·
√

2T + 1 · ‖=(u(t, x))−=(ũ(t, x))‖
W

0,

[
n
2

]
+2

t,x,2 (QT )

. (30)



S. J.Aliyev, A. G.Aliyeva / Eur. J. Pure Appl. Math, 10 (5) (2017), 1078-1091 1088

Due to estimates (21) and condition 4 of this theorem, ∀u, ũ ∈ Kr:∥∥∂sF (t, x, u(t, x)ut(t, x), ux(t, x), utx(t, x), uxx(t, x)
/
∂xα∂uγ11 ...∂u

γN
N

−∂sF (t, x, ũ(t, x)ũt(t, x), ũx(t, x), ũtx(t, x), ũxx(t, x)
/
∂xα∂uγ11 ...∂u

γN
N

∥∥
C(Q̄T )

≤ Ãr · ‖u− ũ‖ET

(
0 ≤ |α|+

N∑
i=1

γi = s ≤
[n

2

]
+ 2

)
, (31)

where Ãr > 0 is some constant depending on r.
Next, using estimate (22) and taking (23) into account, similar to (26) we have for

every t ∈ [0, T ]:∥∥∥∥∥
l∏

i=1

Dkil
t Dαilui(t, x)

∥∥∥∥∥
L2(Ω)

≤
l∏

i=1

∥∥∥Dkil
t Dαilui(t, x)

∥∥∥
L2pi(Ω)

≤ C ·
l∏

i=1

‖ui‖Et ≤ C ·
l∏

i=1

‖ui‖ET ≤ C · r
l−1 · ‖u− ũ‖ET , (32)

where the conditions (27) are satisfied, one of functions ui (i = 1, 2, ..., l) is equal to u− ũ
while the other ui’s are equal to u and ũ, with u, ũ ∈ Kr.

Now, using estimates (31), (21) and (32) (in (21) u must be replaced by u− ũ), from
(30) we obtain that ∀u, ũ ∈ Kr:

‖Q(u)−Q(ũ)‖ET ≤
√
T ·
√

2T + 1 · C̃r · ‖u− ũ‖ET , (33)

where C̃r > 0 is some number depending on r.
It can be seen from the inequalities (29) and (33) that for sufficiently small values of T

the operator Q is a contraction in sphere Kr, and, consequently, has a unique fixed point
u(t, x) in Kr. Then it is evident that

u(t, x) = Q(u(t, x)) = W (t, x) + P(u(t, x)) = W (t, x)

+
∞∑
s=1

1

λ2
s

t∫
0

∫
Ω

=(u(t, x)) ·
[
1− e−λ2s(t−τ)

]
υs(x)dxdτ · υs(x).

Thus, Fourier coefficients us(t) of the found function

u(t, x) ∈ Kr ⊂ ET = B
[n2 ]+4,[n2 ]+3

2,2,T

satisfy system (10).



S. J.Aliyev, A. G.Aliyeva / Eur. J. Pure Appl. Math, 10 (5) (2017), 1078-1091 1089

And now let’s show that the function u(t, x) is a classical solution of problem (1)-(3).
As

∞∑
s=1

us(t)υs(x) = u(t, x) ∈ B[n2 ]+4,[n2 ]+3

2,2,T ,

then it is evident that the functions upq(t, x) =
q∑
s=p

us(t)υs(x) (1 ≤ p ≤ q) satisfy conditions

of lemma 1. Then, from (9) (for u = upq(t, x) =
q∑
s=p

us(t)υs(x) and k ≤
[
n
2

]
+ 4), due to

lemma 1 (for m =
[
n
2

]
+ 4, υ = upq(t, x)), we obtain ∀t ∈ [0, T ]:

‖upq(t, x)‖2

W

[
n
2

]
+4

2 (Ω)

≤ C ·
q∑
s=p

[
λ

[n2 ]+4
s us(t)

]2

≤ C ·
q∑
s=p

(
λ

[n2 ]+4
s max

0≤t≤T
|us(t)|

)2

, (34)

where C > 0 is some constant.
From (34), due to convergence of number series

∞∑
s=1

(
λ

[n2 ]+4
s max

0≤t≤T
|us(t)|

)2

it follows that ‖upq(t, x)‖
W

[
n
2

]
+4

2 (Ω)

→ 0 uniformly with regard to t ∈ [0, T ] as p, q →∞.

Similar to the foregoing discussions, it is easy to show that
∥∥∥∂upq(t,x)

∂t

∥∥∥
W

[
n
2

]
+3

2 (Ω)

→ 0

uniformly with regard to t ∈ [0, T ] as p, q →∞, because numerical series

∞∑
s=1

(
λ

[n2 ]+3
s max

0≤t≤T

∣∣u′s(t)∣∣)2

is convergent.

So we got that the series
∞∑
s=1

us(t)υs(x),
∞∑
s=1

u′s(t)υs(x) and the ones obtained from

them by differentiating them with regard to x1, ..., xn up to
[
n
2

]
+ 4 and

[
n
2

]
+ 3 times,

respectively, converge in L2(Ω) uniformly with regard to t ∈ [0, T ]. Then it is evident that

u(t, x) ∈ C
(

[0, T ] ; W
[n2 ]+4

2 (Ω)

)
, ut(t, x) ∈ C

(
[0, T ] ; W

[n2 ]+3

2 (Ω)

)
. (35)

From (35), due to S.L. Sobolev’s imbedding theorems, we obtain that each of the
functions

u(t, x), ut(t, x), uxi(t, x) (i = 1, 2, ...n), ut xi(t, x) (i = 1, 2, ...n), uxixj (t, x) (i, j = 1, 2, ...n),



S. J.Aliyev, A. G.Aliyeva / Eur. J. Pure Appl. Math, 10 (5) (2017), 1078-1091 1090

utxixj (t, x) (i, j = 1, 2, ...n), uxixjxk(t, x) (i, j, k = 1, 2, ...n)

is continuous in the closed domain Q̄T .
It is easy to show that for each fixed t ∈ [0, T ] and for all x ∈ Ω̄:

=(u(t, x)) =
∞∑
s=1

∫
Ω

=(u(t, x))υs(x)dx

 υs(x),

∂2u(t, x)

∂t2
− ∂

∂t
(L(u(t, x))) = =(u(t, x)). (36)

From (36), it follows that utt(t, x) ∈ C(Q̄T ) and the function u(t, x) satisfies the
equation (1) on the closed domain Q̄T , initial conditions (2) on Ω̄ and boundary condition
(3) in the usual sense.

Moreover,

‖u(t, x)− ϕ(x)‖
W

[
n
2

]
+4

2 (Ω)

→ 0, ‖ut(t, x)− ψ(x)‖
W

[
n
2

]
+3

2 (Ω)

→ 0 as t→ +0.

Thus, function u(t, x) is a classical solution of problem (1)-(3)
And now we prove the uniqueness (in large) of the classical solution of problem (1)-(3).

Let u(t, x) =
∞∑
s=1

us(t)υs(x) and ũ(t, x) =
∞∑
s=1

ũs(t)υs(x) be two arbitrary classical

solutions of problem (1)-(3). Then, due to lemma 2, from system (10) we obtain that

‖u− ũ‖2
B2,1

2,2,T
≤ (2T + 1)

T∫
0

‖=(u(t, x))−=(ũ(t, x))‖2L2(Ω) dt, (37)

where operator = is defined by (11). Then it is evident that u(t, x) − ũ(t, x) ∈ B2,1
2,2,T ,

because =(u(t, x)), =(ũ(t, x)) ∈ C(Q̄T ).
Next, similar to (37), from system (10) we have for every t ∈ [0, T ]:

‖u− ũ‖2
B2,1

2,2,T
≤ (2T + 1)

t∫
0

‖=(u(τ, x))−=(ũ(τ, x))‖2L2(Ω) dτ. (38)

From (38), due to condition 4 of this theorem and using the structure of spase B2,1
2,2,T ,

for every t ∈ [0, T ] we have:

‖u− ũ‖2
B2,1

2,2,t
≤ C

t∫
0

‖u− ũ‖2
B2,1

2,2,τ
dτ,

where C > 0 is some constant.
From here, on applying Bellman’s inequality ([3], pp. 188-189), we obtain that ∀t ∈

[0, T ] ‖u− ũ‖2
B2,1

2,2,t
= 0. Hence, u = ũ. Theorem is proved.



REFERENCES 1091

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