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ӘЛ-ФАРАБИ атындағы ҚАЗАҚ ҰЛТТЫҚ УНИВЕРСИТЕТI

ХАБАРШЫ

Математика, механика, информатика сериясы

КАЗАХСКИЙ НАЦИОНАЛЬНЫЙ УНИВЕРСИТЕТ имени АЛЬ-ФАРАБИ

ВЕСТНИК

Серия математика, механика, информатика

AL-FARABI KAZAKH NATIONAL UNIVERSITY

Journal of Mathematics, Mechanics and Computer Science

№3 (115)

Алматы

«Қазақ университетi»

2022

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Зарегистрирован в Министерстве информации и коммуникаций Республики Казахстан, свидетельство №16508-Ж от 04.05.2017 г. (Время и номер первичной постановки на учет

№766 от 22.04.1992 г.). Язык издания: казахский, русский, английский. Выходит 4 раза в год.

Тематическая направленность: теоретическая и прикладная математика, механика, информатика.

Редакционная коллегия

научный редакторБ.Е. Кангужин, д.ф.-м.н., профессор, КазНУ им. аль-Фараби,

заместитель научного редактораД.И. Борисов, д.ф.-м.н., профессор, Институт математики с вычислительным центром Уфимского научного центра РАН, Башкирский государственный педагогический

университет им. М. Акмуллы, Россия

ответственный секретарь С.Е. Айтжанов, к.ф.-м.н., доцент, КазНУ им. аль-Фараби Гиви Пэйман – профессор, Университет Питтсбурга, США

Моханрадж Муругесан – профессор, Инженерный и технологический колледж Хиндустана, Индия Эльбаз Ибрагим Мохаммед Абу Эльмагд – профессор, Национальный исследовательский

институт астрономии и геофизики, Египет

Д.Б. Жакебаев – PhD доктор, профессор, КазНУ им.аль-Фараби, Казахстан С.И. Кабанихин – д.ф.-м.н., профессор, чл.-корр. РАН, Институт вычислительной математики и математической геофизики СО РАН, Россия

М. Майнке – профессор, Департамент Вычислительной гидродинамики Института аэродинамики, Германия

В.Э. Малышкин – д.т.н., профессор, Новосибирский государственный технический университет, Россия

З.Б. Ракишева – к.ф.-м.н., доцент, КазНУ им.аль-Фараби, Казахстан

М. Ружанский – д.ф.-м.н., профессор, Имперский колледж Лондона, Великобритания С.М. Сагитов – д.ф.-м.н., профессор, Университет Гетеборга, Швеция

Ф.А. Сукочев – профессор, академик АН Австралии, Университет Нового Южного Уэльса, Австралия И.А. Тайманов – д.ф.-м.н., профессор, академик РАН, Институт математики

им. С.Л. Соболева СО РАН, Россия

В.Н. Темляков – д.ф.-м.н., профессор, Университет Южной Каролины, США Шиничи Накасука – PhD доктор, профессор, Университет Токио, Япония Гиви Пейман – профессор, Университет Питтсбурга, США

Мохаммед Отман – PhD доктор, профессор, Университет Путра, Малайзия

Альберто Кабада – PhD доктор, профессор, Университет Сантьяго де Компостела, Испания Хорхе Феррейра – PhD доктор, профессор, Федеральный университет Флуминенс, Бразилия Кантони Вирджинио – PhD доктор, профессор, Университет Павии, Италия

Эжилчелван Пол Девадосс – PhD доктор, профессор, Ньюкаслский университет, Великобритания

Минсу Хан – PhD доктор, профессор, Корейский передовой институт науки и технологий, Южная Корея Триго Пауло – PhD доктор, профессор, Высший инженерный институт Лиссабона, Португалия

Б.С. Дарибаев – PhD доктор, КазНУ им.аль-Фараби, Казахстан Т.С. Иманкулов – PhD доктор, КазНУ им.аль-Фараби, Казахстан

Научное издание

Вестник КазНУ. Серия “Математика, механика, информатика”, № 3 (115) 2022.

Редактор – С.Е. Айтжанов. Компьютерная верстка – С.Е. Айтжанов

ИБ N 15229

Формат60×84 1/8. Бумага офсетная. Печать цифровая. Объем 11,18 п.л.

Заказ N 13463. Издательский дом “Қазақ университетi”

Казахского национального университета им. аль-Фараби. 050040, г. Алматы, пр.аль-Фараби, 71, КазНУ.

Отпечатано в типографии издательского дома “Қазақ университетi”.

c КазНУ им. аль-Фараби, 2022

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1-бөлiм Раздел 1 Section 1

Математика Математика Mathematics

IRSTI 27.31.21 DOI: https://doi.org/10.26577/JMMCS.2022.v115.i3.01

S.E. Aitzhanov1∗ , J. Ferreira2 , K.A. Zhalgassova3

1Al-Farabi Kazakh National University, Kazakhstan, Almaty

2Federal University of Fluminense, Brazil, Volta Redonda

3 M.Auezov South Kazakhstan University, Kazakhstan, Shymkent

e-mail: aitzhanovserik81@gmail.com

SOLVABILITY OF THE INVERSE PROBLEM FOR THE PSEUDOHYPERBOLIC EQUATION

This paper investigates the solvability of the inverse problem of finding a solution and an unknown coefficient in a pseudohyperbolic equation known as the Klein-Gordon equation. A distinctive feature of the given problem is that the unknown coefficient is a function that depends only on the time variable. The problem is considered in the cylinder, the conditions of the usual initial-boundary value problem are set. The integral overdetermination condition is used as an additional condition. In this paper, the inverse problem is reduced to an equivalent problem for the loaded nonlinear pseudohyperbolic equation. Such equations belong to the class of partial differential equations, not resolved with respect to the highest time derivative, and they are also called composite type equations. The proof uses the Galerkin method and the compactness method (using the obtained a priori estimates). For the problem under study, the authors prove existence and uniqueness theorems for the solution in appropriate classes.

Key words: Pseudohyperbolic equation, inverse problem, Klein-Gordon equation, Galerkin method, compactness method, existence, uniqueness.

С.E. Айтжанов1∗, Ж. Феррейра2, K.A. Жалғасова3

1Әл-Фараби атындағы Қазақ ұлттық университетi, Қазақстан, Алматы қ.

2Флюминенс Федерал университетi, Бразилия, Вольта-Редонда қ.

3M.Әуезов атындағы Оңтүстiк Қазақстан университетi, Қазақстан, Шымкент қ.

e-mail: aitzhanovserik81@gmail.com

Псевдогиперболалық теңдеу үшiн керi есептiң шешiмдiлiгi

Мақалада Клейн-Гордон теңдеуi деген атпен белгiлi псевдогиперболалық теңдеудiң шешiмiн және оң жақ коэффициентiн табу керi есебi зерттеледi. Бұл есеп iзделiндi коэффициенттiң тек уақыттан тәуелдi функция болуымен ерекшеленедi. Есеп цилиндрлiк аймақта қарасты- рылады, әдеттегiдей бастапқы-шеттiк есептiң шарттары қойылады. Қосымша шарт ретiнде интегралдық түрдегi артық анықталған шарт берiлген. Бұл жұмыста керi есеп жүктелген сызықтық емес псевдогиперболалық теңдеу үшiн қойылған эквиваленттi есепке келтiрiледi.

Мұндай теңдеулер уақыт бойынша ең жоғары туындыға қатысты шешiлмеген дербес туын- дылы дифференциалдық теңдеулер класына жатады және оларды құрама типтi теңдеулер деп те атайды. Дәлелдеуде Галеркин әдiсi және компакт әдiсi (априорлық бағалаулар алу арқылы) қолданылады. Жұмыста зерттелiп отырған есептiң сәйкес кластардағы шешiмнiң бар болу және жалғыздық теоремалары дәлелденедi.

Түйiн сөздер: Псевдогиперболалық теңдеу, керi есеп, Клейн-Гордон теңдеуi, Галеркин әдiсi, компакт әдiсi, шешiмнiң бар болуы және жалғыздығы.

c 2022 Al-Farabi Kazakh National University

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4 Solvability of the inverse problem for . . .

С.E. Айтжанов1∗, Ж. Феррейра2, K.A. Жалгасова3

1Казахский национальный университет имени аль-Фараби, Казахстан, г. Алматы

2Федеральный университет Флуминенсе, Бразилия, г. Вольта-Редонда

3Южно-Казахстанский университет имени M.Ауезова, Казахстан, г. Чимкент

e-mail: aitzhanovserik81@gmail.com

Разрешимость обратной задачи для псевдогиперболического уравнения

Исследуется разрешимость обратной задачи нахождения решения и неизвестного коэффи- циента в псевдогиперболическом уравнении, известного как уравнение Клейна-Гордона. От- личительной особенностью изучаемой задачи является то, что неизвестный коэффициент является функцией, зависящей лишь от временной переменной. Задача рассматривается в цилиндрической области, задаются условия обычной начально-краевой задачи. В качестве дополнительного условия используется условие интегрального переопределения. В работе обратная задача сводится к эквивалентной задаче для нагруженного нелинейного псевдо- гиперболического уравнения. Подобные уравнения относятся к классу дифференциальных уравнений в частных производных, не разрешенные относительно старшей производной по времени и они также называются уравнениями составного типа. При доказательстве приме- няются метод Галеркина и метод компактности (с использованием полученных априорных оценок). Для изучаемой задачи авторы доказывают теоремы существования и единственно- сти решения в рассматриваемых классах.

Ключевые слова: Псевдогиперболическое уравнение, обратная задача, уравнение Клейна- Гордона, метод Галеркина, метод компактности, существование, единственность.

1 Introduction

The work is devoted to the study of the solvability of the inverse problem of reС´Гovering an external influence in the pseudohyperbolic equation known as the Klein-Gordon equation.

Nowadays, inverse problems have become a powerful and rapidly developing field of knowledge, penetrating almost all areas of mathematics. Similar inverse problems arise naturally in the mathematical modeling of certain processes occurring in the media with unknown characteristics. Since it is the characteristics of the medium that determine the coefficients of the corresponding differential equation or the coefficients of the external influence. The Klein-Gordon equation plays an important role in mathematical physics. This equation is used in modeling various phenomena of relativistic quantum mechanics [1] and nonlinear optics, in studying the behavior of elementary particles and dislocation propagation in crystals, as well as in studying nonlinear wave equations [2]. For such equations, many problems have been investigated in different formulations by various methods [3]-[14].

In this paper, the inverse problem under study is reduced to an equivalent problem for the loaded nonlinear pseudohyperbolic equation. Pseudohyperbolic equations belong to the class of partial differential equations, not solved with respect to the highest time derivative, and they are also known as composite type equations. Initial-boundary value problems for linear and nonlinear pseudohyperbolic equations were studied in various works [15]-[20].

Moreover, it is necessary to note the works [21]-[25], where studied the qualitative properties of solutions of inverse problems for hyperbolic type equations.

In the cylinder QT = {(x, t) : x ∈ Ω, 0 < t < T} we consider the inverse problem of

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reС´Гovering the right-hand side of the Klein-Gordon equation utt−χ∆ut

a0 +a1k∇uk2r2,Ω

∆u+|ut|q−2ut=b(x, t)|u|p−2u+f(t)h(x, t), (x, t)∈QT, (1) with initial conditions

u(x,0) =u0(x), ut(x,0) =u1(x), x∈Ω, (2) the boundary condition

u|S = 0, (3)

and the overdetermination condition Z

u(x, t)ω(x)dx=ϕ(t), t∈(0, T). (4)

Here Ω ⊂ RN, N ≥ 1 is bounded area, ∂Ω is sufficiently smooth boundary, b(x, t), h(x, t), u0(x),u1(x),ω(x),ϕ(t)are the given functions,χ, a0, a1, p, q and r are positive constants.

Let the given functions of the problem (1)-(4) satisfy the conditions ω∈L2(Ω)T

0

W22(Ω),

h(x, t)∈C1(QT), h1(t)≡R

h(x, t)ω(x)dx6= 0, ∀t∈[0, T], (5)

ϕ(t)∈W22(0, T), R

u0(x)ω(x)dx =ϕ(0), R

u1(x)ω(x)dx=ϕ0(0), u0

0

W22(Ω), u1

0

W21(Ω).

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2 Materials and methods 2.1 The Equivalent Problem

Lemma 1. The problem (1)-(4) is equivalent to the next problem for nonlinear pseudoparabolic equation containing nonlinear nonlocal operator from function u(x, t)

utt−χ∆ut

a0+a1k∇uk2r2,Ω

∆u+|ut|q−2ut =b(x, t)|u|p−2u+F(t, u)h(x, t), x∈Ω, t >0, (7)

u(x,0) =u0(x), x∈Ω, u|S = 0. (8)

Here

F(t, u) = h1

1(t)

ϕ00(t) +χR

∇ut∇ωdx+

a0+a1k∇uk2r2,Ω R

∇u∇ωdx+

+R

|ut|q−2utωdx−R

b(x, t)|u|p−2uωdx

.

(9)

(6)

6 Solvability of the inverse problem for . . .

Proof. Indeed, it follows from equation (1) that R

(utt−χ∆ut)ωdx−

a0+a1k∇uk2r2,Ω R

∆uωdx−

−R

b(x, t)|u|p−2uωdx=R

f(t)h(x, t)ωdx, (10)

next, if conditions (4) and (5) are performed, then F(t, u) = h1

1(t)

ϕ00(t) +χR

∇ut∇ωdx+

a0+a1k∇uk2r2,Ω R

∇u∇ωdx+

+R

|ut|q−2utωdx−R

b(x, t)|u|p−2uωdx

.

(11)

Therefore, the relation (9) is satisfied.

Now let us consider the problem (7)-(8). If the relation (9) is satisfied, then equality (11) obviously follows from it. Then

F(t, u) = h1

1(t)

ϕ00(t) +χR

∇ut∇ωdx+

a0+a1k∇uk2r2,Ω R

∇u∇ωdx+

+R

|ut|q−2utωdx−R

b(x, t)|u|p−2uωdx

=

= h1

1(t)

ϕ00(t)−χR

∆utωdx−

a0+a1k∇uk2r2,Ω R

∆uωdx+

+R

|ut|q−2utωdx−R

b(x, t)|u|p−2uωdx

. By virtue of (10), we obtain that

F(t, u) = h1

1(t)

ϕ00(t) +χR

∇ut∇ωdx+

a0 +a1k∇uk2r2,Ω R

∇u∇ωdx−R

b(x, t)|u|p−2uωdx

=

= h1

1(t)

ϕ00(t)−χR

∆utωdx−

a0+a1k∇uk2r2,Ω R

∆uωdx−R

b(x, t)|u|p−2uωdx

= h1

1(t)

ϕ00(t)−R

uttωdx+R

b(x, t)|u|p−2uωdx+R

f(t)h(x, t)ωdx−R

b(x, t)|u|p−2uωdx

.

ϕ00(t)− Z

uttωdx= 0.

In this way, dtd22

ϕ(t)−R

uωdx

= 0. Denote by v(t) = ϕ(t)−R

uωdx. Then the function v(t) can be found as a solution of the Cauchy problem: v00(t) = 0, v(0) = 0, v0(0) = 0.

(v(0) = 0, v0(0) = 0 follows from the matching condition (5)). The unique solution of the problem is the function v(t) = 0, consequently, R

u(x, t)ω(x)dx=ϕ(t).

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3 Existence of the solution. Galerkin approximations

Theorem 1. Let the conditions (5), (6) and 2 ≤ p < 2n−2n−2, n ≥ 3, q ≥ 2, r > 1 are performed. Then there exists the generalized solution ∆u, ∆ut, utt ∈L2(QT) of the problem (7)-(8).

Proof. Let us choose in

0

W21(Ω) some system of functions {Ψj(x)} forming a basis in the given space. As a basis, we can take the eigenfunctions of the Sturm-Liouville problem

∆Ψ +λΨ = 0, Ψ|∂Ω = 0.

We will look for an approximate solution of the problem (7)-(8) in the form um(x, t) =

m

X

k=1

Cmk(t)Ψk(x) (12)

where coefficients Cmk(t) are searched out from the relations

m

P

k=1

Cmk00 (t)R

ΨkΨjdx+χ

m

P

k=1

Cmk0 (t)R

∇Ψk∇Ψjdx+

a0 +a1k∇umk2r2,Ω R

∇um∇Ψjdx+

+

m

P

k=1

Cmk0 (t)R

|∂tum|p−2ΨkΨjdx−R

b(x, t)|um|p−2umΨjdx=R

F(t, umjdx.

(13)

um0 =um(0) =

m

P

k=1

Cmk(0)Ψk =

m

P

k=1

α0kΨk, um1 =u0m(0) =

m

P

k=1

Cmk0 (0)Ψk =

m

P

k=1

α1kΨk

(14)

and besides

um0 →u0 strongly in

0

W22(Ω) at m→ ∞ um1 →u1 strongly in

0

W21(Ω) at m→ ∞

(15)

Let us introduce denotations

C~m≡ {C1m(t), ..., Cmm(t)}T , ~α≡ {α1, ..., αm}T , akj = Z

ΨkΨjdx, bkj =χ Z

(∇Ψk,∇Ψj)dx,

fkj =χR

(∇Ψk,∇Ψj)dx+

a0+a1k∇umk2r2,Ω R

∇Ψk∇Ψjdx+

+

m

P

k=1

Cmk0 (t)R

|∂tum|p−2ΨkΨjdx−R

b(x, t)|um|p−2ΨkΨjdx+R

F(t, umjdx, Am

C~m

≡n ajk

C~mo , ~Fm

C~m, ~Cm0

≡n fjk

C~m, ~Cm0 o C~m.

(8)

8 Solvability of the inverse problem for . . .

Then the system of equations (13) and condition (14) take the matrix form AmC~m00 ≡F~m

C~m, ~Cm0 , C~m(0) =~α0, ~Cm0 (0) =α~1.

(16)

Relations (16) represent the Cauchy problem for the system of ordinary differential equations, which is solvable on the segment [0, Tm]. In order to verify the existence of the solution on [0, T], we obtain a priori estimates.

Lemma 2. Ifu∈

0

W21(Ω),1< σ ≤2, then the following inequality is performed Z

|um|σdx≤

 Z

|u|2dx

σ 2

|Ω|2−σ2 ≤C0

 Z

|u|2dx+χ Z

|∇u|2dx

.

Lemma 3. If u ∈

0

W21(Ω),2 < β < N2N−2, N ≥ 3,, then the following inequality is performed

kuk2β,Ω ≤C02k∇uk2,Ωkuk2(1−α)2,Ω ≤χk∇uk22,Ω+ (1−α)α1−αα C

2 1−α

0

χ1−αα kuk22,Ω, whereC0 =2(N−1)

N−2

α

, α = (β−2)N , 0< α <1.

We multiply the equality (13) by Cmj0 (t) and summarize over j = 1, m. As a result, we take

1 2

d dt

R

|∂tum(t)|2dx+χR

|∂t∇um|2dx+a20dtd k∇umk22,Ω+ +2r+2a1 dtd k∇umk2r+22,Ω +R

|∂tum(t)|qdx=

=R

b(x, t)|um|p−2umtumdx+R

F(t, um)h∂tumdx.

(17)

We integrate with respect toτ from 0 tot, then we get the relation

1 2

R

|∂tum(t)|2dx+χ

t

R

0

R

|∂τ∇um|2dxdτ + a20 k∇umk22,Ω+ +2r+2a1 k∇umk2r+22,Ω +

t

R

0

R

|∂τum|qdxdτ =

= 12R

|∂tum(x,0)|2dx+a20 k∇um(x,0)k22,Ω+2r+2a1 k∇um(x,0)k2r+22,Ω +

+

t

R

0

R

b(x, τ)|um|p−2umτumdxdτ+

t

R

0

R

F(t, um)h∂τumdxdτ.

(18)

(9)

Denote by y(t) = 1

2 Z

|∂tum(t)|2dx+a0

2 k∇umk22,Ω+ a1

2r+ 2k∇umk2r+22,Ω .

Estimating the right-hand side of (18) using Lemma 2 and 3, as well as the H¨older and Young inequality, we obtain

t

R

0

R

b(x, τ)|um|p−2umτumdxdτ

≤b0

t

R

0

R

|um|p−1τumdxdτ ≤

≤ k∂τumk2,Q

t

t R

0

R

|um|n−22n dxdτ

n−22n t R

0

R

|um|(p−2)ndxdτ 1n

≤ k∂τumk22,Q

t +C1k∇umk

2n n−2

2,Qt.

(19)

χ

t

R

0

R

1 h1(τ)

R

τ∇um∇ωdx h∂τumdxdτ

≤χ

t

R

0 1

h1(τ)k∂τ∇umk2,Ωk∇ωk2,Ωkhk2,Ωk∂τumk2,Ωdτ ≤

χ2

t

R

0

k∂τ∇umk22,Ωdτ+ χ2k∇ωk22,Ω sup

0≤t≤T

kh(x,t)k22,Ω

|h1(t)|2 t

R

0

k∂τumk22,Ωdτ.

t

R

0

R

1 h1(τ)

a0 +a1k∇umk2r2,Ω R

∇um∇ωdx h∂τumdxdτ

t

R

0 1 h1(τ)

a0 +a1k∇umk2r2,Ω

k∇umk2,Ωk∇ωk2,Ωkhk2,Ωk∂τumk2,Ωdτ ≤

≤a0

t

R

0

k∇umk22,Ωdτ +a0k∇ωk22,Ω

t

R

0 1

h21(τ)khk22,Ωk∂τumk22,Ωdτ+ +a1

t

R

0

k∇umk2r+22,Ω dτ +C2k∇ωk2r+22,Ω sup

0≤t≤T

kh(x,t)k2r+22,Ω

|h1(t)|2r+2 t

R

0

k∂τumk2r+22,Ω dτ,

C2 = a1(2r+ 1)2r+1 (2r+ 2)2r+2 .

t

R

0

R

1 h1(τ)

R

|∂τum|q−2τumωdx h∂τumdxdτ

t

R

0 1

h1(τ)k∂τumkq−1q,Ω kωkq,Ωkhk2,Ωk∂τumk2,Ωdτ ≤

12

t

R

0

k∂τumkqq,Ωdτ +C3kωkqq,Ω sup

0≤t≤T

kh(x,t)kq2,Ω

|h1(t)|q t

R

0

k∂τumkq2,Ωdτ, C3 = q−1

q q2q−11 .

(10)

10 Solvability of the inverse problem for . . .

t

R

0

R

1 h1(τ)

R

b(x, t)|um|p−2umωdx h∂τumdxdτ

≤b0kωkp,Ω sup

0≤t≤T

kh(x,t)k2,Ω h1(t)

t

R

0

k∂τumk2,Ωkumkp−1p,Ω dτ ≤

≤b0kωkp,Ω sup

0≤t≤T

kh(x,t)k2,Ω h1(t)

t R

0

kumkpp,Ωdτ +

t

R

0

k∂τumkp2,Ω

≤b0kωkp,Ω sup

0≤t≤T

kh(x,t)k2,Ω h1(t)

t

R

0

kumk22,Ω+χk∇umk22,Ωp2 dτ+

+b0kωkp,Ω sup

0≤t≤T

kh(x,t)k2,Ω h1(t)

t

R

0

k∂τumkp2,Ωdτ.

Denote by y(t) = 1

2 Z

|∂tum(t)|2dx+ a0

4C(Ω)kumk22,Ω+a0

4 k∇umk22,Ω+ a1

2r+ 2k∇umk2r+22,Ω . d= max

n n−2, p

2, q

2, r+ 1

. Then from the relation (18), we get

y(t)≤C4+C5

t

Z

0

[y(τ)]ddτ.

Applying for this the generalized Bihari lemma, then the next inequality is true

y(t)≤ C4

1−(d−1)C5C4d−1td−11 ,

i.Рµ. 12

R

|∂tum(t)|2dx+a40C(Ω)kumk22,Ω+a40 k∇umk22,Ω+2r+2a1 k∇umk2r+22,Ω!4

[1−(d−1)C5C4d−1t]d−11 .

From this estimate we can conclude that there exists T0 >0 such that R

|∂tum(t)|2dx+kumk22,Ω+k∇umk22,Ω+ +k∇umk2r+22,Ω +

T

R

0

R

|∂τ∇um|2dxdτ+

T

R

0

R

|∂τum|qdxdτ ≤C6,

(20)

for all t∈[0, T], T < T0, whereC6 is constant which does not depend on m∈N.

We multiply the relation (13) byλjCmj(t) andCmj00 (t), then summarize overj = 1, m. As a result, we get the next relations

dtd R

tum(t)∆um(t)dx− k∂t∇umk22,Ω+ χ2dtd k∆umk22,Ω+ +

a0+a1k∇umk2r2,Ω

k∆umk22,Ω−R

|∂tum(t)|q−2tum(t)∆umdx=

=−R

b(x, t)|um|p−2um∆umdx−R

F(t, um)h∆umdx.

(11)

R

|∂t2um(t)|2dx+χ2dtd R

|∂t∇um|2dx−

a0+a1k∇umk2r2,Ω R

∆um2tumdx+

+1qdtd R

|∂tum|qdx=R

b(x, t)|um|p−2umt2umdx+R

F(t, um)∂t2umdx.

By integrating these relations from 0 to t, we get

χ

2 k∆umk22,Ω+

t

R

0

a0+a1k∇umk2r2,Ω

k∆umk22,Ωdτ = χ2 k∆um(0)k22,Ω+R

tum(t)∆um(t)dx−

−R

tum(0)∆um(0)dx+

t

R

0

k∂τ∇umk22,Ωdτ +

t

R

0

R

|∂τum(τ)|q−2τum(τ)∆umdxdτ−

t

R

0

R

b(x, τ)|um|p−2um∆umdxdτ −

t

R

0

R

F(τ, um)h∆umdxdτ.

(21)

χ 2

R

|∂t∇um(t)|2dx+1qR

|∂tum(t)|qdx+

t

R

0

R

|∂τ2um(x, τ)|2dxdτ =

=

t

R

0

a0+a1k∇umk2r2,Ω R

∆umτ2umdxdτ+ +Rt

0

R

b(x, τ)|um|p−2umτ2umdxdτ +

t

R

0

R

F(τ, um)∂τ2umdxdτ.

(22)

Analogically, we estimate the right-hand side of (21) and (22), applying lemmas 2 and 3, H¨older and Young inequalities, Bihari’s lemma and a priori estimate (20), as a result we obtain

k∆umk22,Ω+

T

Z

0

a0+a1k∇umk2r2,Ω

k∆umk22,Ωdt≤C7, for all t∈[0, T], T < T0, (23)

Z

|∂t∇um(t)|2dx+ Z

|∂tum(t)|qdx+

T

Z

0

Z

|∂τ2um(x, τ)|2dxdt≤C8, for all t ∈[0, T], T < T0, (24) where C7 and C8 are constants which does not depend onm ∈N.

From the obtained estimates (20), (23) and (3) follows the estimate

T

Z

0

k∂t∆umk22,Ωdt≤C9, for all t∈[0, T], T < T0, m∈N. (25) Then by using (20), (23), (3) and (25), considering the conditions of the theorem, we can show the existence of the derivative uxx ∈L2(QT). In this way, ∆u, ∆ut, utt ∈L2(QT).

(12)

12 Solvability of the inverse problem for . . .

4 Uniqueness of the generalized solution

Theorem 2. Let the conditions (5),r >2, q >2, 2< p ≤2 +N−21 , N ≥3,are performed.

Then the generalized solution of the problem (1)-(3) on the segment (0, T) is unique.

Proof. Assume that the problem (7)-(8) has two generalized solutions: u1(x, t) and u2(x, t). Let us put u(x, t) = u1(x, t)− u2(x, t) . Then there are the following equalities

utt−χ∆ut−a0∆u−a1

k∇u1k2r2,Ω∆u1− k∇u2k2r2,Ω∆u2 + +|u1t|q−2u1t− |u2t|q−2u2t=b(x, t) (|u1|p−2u1− |u2|p−2u2) + +h(x, t) (F(t, u1)−F(t, u2)), x∈Ω, t >0,

(26)

u(x,0) = 0, ut(x,0) = 0, x∈Ω, u|S = 0. (27) We consider the equality

t

R

0

R

[uτ τ −χ∆uτ −a0∆u −a1

k∇u1k2r2,Ω∆u1− k∇u2k2r2,Ω∆u2

+ +|u|q−2u − |u|q−2u]uτdxdτ =

t

R

0

R

[b(x, τ) (|u1|p−2u1− |u2|p−2u2) + +h(x, τ) (F(τ, u1)−F(τ, u2))]uτdxdτ.

By applying the next inequalities

||u1|qu1− |u2|qu2| ≤(q+ 1) (|u1|q+|u2|q)|u1−u2| atq >0,

|(|u1|qu1− |u2|qu2) (u1−u2)| ≥ |u1−u2|q+2 atq >0.

As a result, we obtain the inequality

1 2

R

u2t(t)dx+χ

t

R

0

R

|∇uτ|2dxdτ + a20 R

|∇u|2dx+

t

R

0

R

|uτ|qdxdτ ≤

≤ −a1

t

R

0

k∇u1k2r2,ΩR

∇u∇uτdx−

k∇u1k2r2,Ω− k∇u2k2r2,Ω R

∇u2∇uτdx

dτ+

+

t

R

0

R

b(x, τ) (|u1|p−2+|u2|p−2)uτdxdτ +

t

R

0

R

h(x, τ) (F(τ, u1)−F(τ, u2))uτdxdτ.

(28)

We estimate the right-hand side of the inequality (28), applying the H¨older’s inequality

t

R

0

R

b(x, τ) (|u1|p−2u1− |u2|p−2u2)uτdxdτ

≤b1(p−1)

t

R

0

R

(|u1|p−2+|u2|p−2)uuτdxdτ ≤

≤b1(p−1) t

R

0

R

|u1|2r(p−2)r−2 dxdτ r−22r

+ t

R

0

R

|u2|2r(p−2)r−2 dxdτ

r−22r !

×

× t

R

0

R

urdxdτ

1r t R

0

R

u2τdxdτ 12

.

(13)

Let us put r = N2N−2, p ≤ 2 + N−21 , N ≥ 3. Then by the Sobolev embedding theorem H1(Ω) →→ Lr(Ω) and H1(Ω) →→ L2r(p−2)/(r−2)(Ω). In this case, taking into account the smoothness class of solutions u1(x, t) and u2(x, t), we come to the estimate

t

Z

0

Z

b(x, τ) |u1|p−2u1− |u2|p−2u2

uτdxdτ

≤C1

t

Z

0

kuτk22,Ω+k∇uk22,Ω+kuk22,Ω

dτ. (29) Let us estimate the first term

a1

t

R

0

k∇u1k2r2,ΩR

∇u∇uτdx−

k∇u1k2r2,Ω− k∇u2k2r2,Ω R

∇u2∇uτdx

≤a1

t

R

0

k∇u1k2r2,Ωk∇uk2,Ωk∇uτk2,Ωdτ+a1

t

R

0

k∇u1k2r−22,Ω +k∇u2k2r−22,Ω

k∇u2k2,Ωk∇uτk2,Ω×

×R

(|∇u1|2 − |∇u2|2)dxdτ ≤a1

t

R

0

k∇u1k2r2,Ωk∇uk2,Ωk∇uτk2,Ωdτ+ +a1C20

t

R

0

k|∇u1|+|∇u2|k2,Ωk∇uk2,Ωk∇uτk2,Ω

χ4

t

R

0

k∇uτk22,Ωdτ +C2

t

R

0

kuτk22,Ω+k∇uk22,Ω+kuk22,Ω dτ.

The third term is estimated in a similar way. From the obtained estimates, we get R

u2t(t)dx+C0R

|u|2dx+a0R

|∇u|2dx+χ

t

R

0

R

|∇uτ|2dxdτ +

t

R

0

R

|uτ|qdxdτ ≤

≤C4

t

R

0

kuτk22,Ω+a0k∇uk22,Ω+C0kuk22,Ω

dτ+C5

t

R

0

kuτk22,Ω+a0k∇uk22,Ω+C0kuk22,Ωd

dτ, where d >1.

From the last inequality follows that R

u2t(t)dx+C0R

|u|2dx+a0R

|∇u|2dx≤C4

t

R

0

kuτk22,Ω+a0k∇uk22,Ω+C0kuk22,Ω dτ+ +C5

t

R

0

kuτk22,Ω+a0k∇uk22,Ω+C0kuk22,Ωd

dτ, where by Bihari’s lemma, impliesR

u2t(t)dx+C0R

|u|2dx+a0R

|∇u|2dx= 0almost everywhere on the time interval (0, T), which means that the generalized solution is unique.

5 Conclusion

In the paper, we investigated the solvability of the inverse problem of determining the solution of the pseudohyperbolic equation, also an unknown coefficient of a special form which identifies the external source. The methods used are based on the transition from the original problem to the equivalent problem for the loaded nonlinear pseudohyperbolic equation. For this problem we use Galerkin’s method to prove the existence of a strong generalized solution. The obtained results on the solvability of the inverse problem are new and can be useful to study another problems in the given area.

Сурет

Figure 1: Lattice T
Figure 4: Lattice L n , n ≥ 2
Figure 1: Verification of the numerical calculation algorithm
Figure 2: BHE geometrical configurations
+7

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