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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

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 редактор–Б.Е. Кангужин, д.ф.-м.н., профессор, КазНУ им. аль-Фараби,

заместитель научного редактора–Д.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

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. Aitzhanov^1∗ , 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: [email protected]

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. Айтжанов^1∗, 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: [email protected]

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

4 Solvability of the inverse problem for . . .

С.E. Айтжанов^1∗, 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: [email protected]

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

reС´Гovering the right-hand side of the Klein-Gordon equation u_tt−χ∆u_t−

a₀ +a₁k∇uk^2r_2,Ω

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

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

u|_S = 0, (3)

and the overdetermination condition Z

Ω

u(x, t)ω(x)dx=ϕ(t), t∈(0, T). (4)

Here Ω ⊂ R^N, 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(Q_T), h₁(t)≡R

Ω

h(x, t)ω(x)dx6= 0, ∀t∈[0, T], (5)

ϕ(t)∈W₂²(0, T), R

Ω

u₀(x)ω(x)dx =ϕ(0), R

Ω

u₁(x)ω(x)dx=ϕ⁰(0), u₀ ∈

0

W₂²(Ω), u₁ ∈

0

W₂¹(Ω).

(6)

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)

u_tt−χ∆u_t−

a₀+a₁k∇uk^2r_2,Ω

∆u+|u_t|^q−2u_t =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

Ω

∇u_t∇ωdx+

a₀+a₁k∇uk^2r_2,Ω R

Ω

∇u∇ωdx+

+R

Ω

|ut|^q−2utωdx−R

Ω

b(x, t)|u|^p−2uωdx

.

(9)

6 Solvability of the inverse problem for . . .

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

Ω

(utt−χ∆ut)ωdx−

a0+a1k∇uk^2r_2,Ω R

Ω

∆uωdx−

−R

Ωb(x, t)|u|^p−2uωdx=R

Ω

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

Ω

∇ut∇ωdx+

a0+a1k∇uk^2r_2,Ω R

Ω

∇u∇ωdx+

+R

Ω

|u_t|^q−2u_tωdx−R

Ω

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

Ω

∇ut∇ωdx+

a0+a1k∇uk^2r_2,Ω R

Ω

∇u∇ωdx+

+R

Ω

|u_t|^q−2u_tωdx−R

Ω

b(x, t)|u|^p−2uωdx

=

= _h¹

1(t)

ϕ⁰⁰(t)−χR

Ω

∆u_tωdx−

a₀+a₁k∇uk^2r_2,Ω R

Ω

∆uωdx+

+R

Ω

|u_t|^q−2u_tωdx−R

Ω

b(x, t)|u|^p−2uωdx

. By virtue of (10), we obtain that

F(t, u) = _h¹

1(t)

ϕ⁰⁰(t) +χR

Ω

∇u_t∇ωdx+

a₀ +a₁k∇uk^2r_2,Ω R

Ω

∇u∇ωdx−R

Ω

b(x, t)|u|^p−2uωdx

=

= _h¹

1(t)

ϕ⁰⁰(t)−χR

Ω

∆u_tωdx−

a₀+a₁k∇uk^2r_2,Ω R

Ω

∆uωdx−R

Ω

b(x, t)|u|^p−2uωdx

= _h¹

1(t)

ϕ⁰⁰(t)−R

Ω

u_ttωdx+R

Ω

b(x, t)|u|^p−2uωdx+R

Ω

f(t)h(x, t)ωdx−R

Ω

b(x, t)|u|^p−2uωdx

.

ϕ⁰⁰(t)− Z

Ω

u_ttωdx= 0.

In this way, _dt^d22

ϕ(t)−R

Ω

uωdx

= 0. Denote by v(t) = ϕ(t)−R

Ω

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

Ω

u(x, t)ω(x)dx=ϕ(t).

3 Existence of the solution. Galerkin approximations

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

Proof. Let us choose in

0

W₂¹(Ω) some system of functions {Ψ_j(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 u_m(x, t) =

m

X

k=1

C_mk(t)Ψ_k(x) (12)

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

m

P

k=1

C_mk⁰⁰ (t)R

Ω

Ψ_kΨ_jdx+χ

m

P

k=1

C_mk⁰ (t)R

Ω

∇Ψ_k∇Ψ_jdx+

a₀ +a₁k∇u_mk^2r_2,Ω R

Ω

∇u_m∇Ψ_jdx+

+

m

P

k=1

C_mk⁰ (t)R

Ω

|∂_tu_m|^p−2Ψ_kΨ_jdx−R

Ω

b(x, t)|u_m|^p−2u_mΨ_jdx=R

Ω

F(t, u_m)Ψ_jdx.

(13)

u_m0 =u_m(0) =

m

P

k=1

C_mk(0)Ψ_k =

m

P

k=1

α_0kΨ_k, u_m1 =u⁰_m(0) =

m

P

k=1

C_mk⁰ (0)Ψ_k =

m

P

k=1

α_1kΨ_k

(14)

and besides

u_m0 →u₀ strongly in

0

W₂²(Ω) at m→ ∞ um1 →u1 strongly in

0

W₂¹(Ω) at m→ ∞

(15)

Let us introduce denotations

C~_m≡ {C_1m(t), ..., C_mm(t)}^T , ~α≡ {α₁, ..., α_m}^T , a_kj = Z

Ω

Ψ_kΨ_jdx, b_kj =χ Z

Ω

(∇Ψ_k,∇Ψ_j)dx,

f_kj =χR

Ω

(∇Ψ_k,∇Ψ_j)dx+

a₀+a₁k∇u_mk^2r_2,Ω R

Ω

∇Ψ_k∇Ψ_jdx+

+

m

P

k=1

C_mk⁰ (t)R

Ω

|∂tum|^p−2ΨkΨjdx−R

Ω

b(x, t)|um|^p−2ΨkΨjdx+R

Ω

F(t, um)Ψjdx, A_m

C~_m

≡n a_jk

C~_mo , ~F_m

C~_m, ~C_m⁰

≡n f_jk

C~_m, ~C_m⁰ o C~_m.

8 Solvability of the inverse problem for . . .

Then the system of equations (13) and condition (14) take the matrix form A_mC~_m⁰⁰ ≡F~_m

C~_m, ~C_m⁰ , C~_m(0) =~α₀, ~C_m⁰ (0) =α~₁.

(16)

Relations (16) represent the Cauchy problem for the system of ordinary differential equations, which is solvable on the segment [0, T_m]. 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

Ω

|um|^σdx≤



 Z

Ω

|u|²dx





σ 2

|Ω|^2−σ2 ≤C0



 Z

Ω

|u|²dx+χ Z

Ω

|∇u|²dx



.

Lemma 3. If u ∈

0

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

kuk²_β,Ω ≤C₀²k∇uk^2α_2,Ωkuk^2(1−α)_2,Ω ≤χk∇uk²_2,Ω+ (1−α)α^1−αα C

2 1−α

0

χ^1−αα kuk²_2,Ω, whereC₀ =_2(N−1)

N−2

α

, α = ^(β−2)N_2β , 0< α <1.

We multiply the equality (13) by C_mj⁰ (t) and summarize over j = 1, m. As a result, we take

1 2

d dt

R

Ω

|∂_tu_m(t)|²dx+χR

Ω

|∂_t∇u_m|²dx+^a₂⁰_dt^d k∇u_mk²_2,Ω+ +_2r+2^a1 _dt^d k∇u_mk^2r+2_2,Ω +R

Ω

|∂_tu_m(t)|^qdx=

=R

Ω

b(x, t)|um|^p−2um∂tumdx+R

Ω

F(t, um)h∂tumdx.

(17)

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

1 2

R

Ω

|∂_tu_m(t)|²dx+χ

t

R

0

R

Ω

|∂_τ∇u_m|²dxdτ + ^a₂⁰ k∇u_mk²_2,Ω+ +_2r+2^a1 k∇u_mk^2r+2_2,Ω +

t

R

0

R

Ω

|∂_τu_m|^qdxdτ =

= ¹₂R

Ω

|∂_tu_m(x,0)|²dx+^a₂⁰ k∇u_m(x,0)k²_2,Ω+_2r+2^a1 k∇u_m(x,0)k^2r+2_2,Ω +

+

t

R

0

R

Ω

b(x, τ)|um|^p−2um∂τumdxdτ+

t

R

0

R

Ω

F(t, um)h∂τumdxdτ.

(18)

Denote by y(t) = 1

2 Z

Ω

|∂_tu_m(t)|²dx+a₀

2 k∇u_mk²_2,Ω+ a₁

2r+ 2k∇u_mk^2r+2_2,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

Ω

b(x, τ)|u_m|^p−2u_m∂_τu_mdxdτ

≤b₀

t

R

0

R

Ω

|u_m|^p−1∂_τu_mdxdτ ≤

≤ k∂_τu_mk_2,Q

t

_t R

0

R

Ω

|u_m|ⁿ⁻²²ⁿ dxdτ

ⁿ⁻²_{2n t} R

0

R

Ω

|u_m|^(p−2)ndxdτ ¹_n

≤

≤ k∂_τu_mk²_2,Q

t +C₁k∇u_mk

2n n−2

2,Qt.

(19)

χ

t

R

0

R

Ω 1 h1(τ)

R

Ω

∂_τ∇u_m∇ωdx h∂_τu_mdxdτ

≤

≤χ

t

R

0 1

h1(τ)k∂_τ∇u_mk_2,Ωk∇ωk_2,Ωkhk_2,Ωk∂_τu_mk_2,Ωdτ ≤

≤ ^χ₂

t

R

0

k∂_τ∇u_mk²_2,Ωdτ+ ^χ₂k∇ωk²_2,Ω sup

0≤t≤T

kh(x,t)k²_2,Ω

|h1(t)|² t

R

0

k∂_τu_mk²_2,Ωdτ.

t

R

0

R

Ω 1 h1(τ)

a₀ +a₁k∇u_mk^2r_2,Ω R

Ω∇u_m∇ωdx h∂_τu_mdxdτ

≤

≤

t

R

0 1 h1(τ)

a₀ +a₁k∇u_mk^2r_2,Ω

k∇u_mk_2,Ωk∇ωk_2,Ωkhk_2,Ωk∂_τu_mk_2,Ωdτ ≤

≤a₀

t

R

0

k∇u_mk²_2,Ωdτ +a₀k∇ωk²_2,Ω

t

R

0 1

h²₁(τ)khk²_2,Ωk∂_τu_mk²_2,Ωdτ+ +a₁

t

R

0

k∇u_mk^2r+2_2,Ω dτ +C₂k∇ωk^2r+2_2,Ω sup

0≤t≤T

kh(x,t)k^2r+2_2,Ω

|h1(t)|^2r+2 t

R

0

k∂_τu_mk^2r+2_2,Ω dτ,

C₂ = a₁(2r+ 1)^2r+1 (2r+ 2)^2r+2 .

t

R

0

R

Ω 1 h1(τ)

R

Ω

|∂_τu_m|^q−2∂_τu_mωdx h∂_τu_mdxdτ

≤

≤

t

R

0 1

h1(τ)k∂τumk^q−1_q,Ω kωk_q,Ωkhk_2,Ωk∂τumk_2,Ωdτ ≤

≤ ¹₂

t

R

0

k∂_τu_mk^q_q,Ωdτ +C₃kωk^q_q,Ω sup

0≤t≤T

kh(x,t)k^q_2,Ω

|h₁(t)|^q t

R

0

k∂_τu_mk^q_2,Ωdτ, C3 = q−1

q ^q_2q−1¹ .

10 Solvability of the inverse problem for . . .

t

R

0

R

Ω 1 h1(τ)

R

Ω

b(x, t)|u_m|^p−2u_mωdx h∂_τu_mdxdτ

≤

≤b₀kωk_p,Ω sup

0≤t≤T

kh(x,t)k_2,Ω h1(t)

t

R

0

k∂_τu_mk_2,Ωku_mk^p−1_p,Ω dτ ≤

≤b₀kωk_p,Ω sup

0≤t≤T

kh(x,t)k_2,Ω h1(t)

_t R

0

ku_mk^p_p,Ωdτ +

t

R

0

k∂_τu_mk^p_2,Ωdτ

≤

≤b0kωk_p,Ω sup

0≤t≤T

kh(x,t)k_2,Ω h1(t)

t

R

0

kumk²_2,Ω+χk∇umk²_2,Ω^p₂ dτ+

+b₀kωk_p,Ω sup

0≤t≤T

kh(x,t)k_2,Ω h1(t)

t

R

0

k∂_τu_mk^p_2,Ωdτ.

Denote by y(t) = 1

2 Z

Ω

|∂_tu_m(t)|²dx+ a₀

4C(Ω)ku_mk²_2,Ω+a₀

4 k∇u_mk²_2,Ω+ a₁

2r+ 2k∇u_mk^2r+2_2,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)C₅C₄^d−1t_d−1¹ ,

i.Рµ. ¹2

R

Ω

|∂_tu_m(t)|²dx+^a₄⁰C(Ω)ku_mk²_2,Ω+^a₄⁰ k∇u_mk²_2,Ω+_2r+2^a1 k∇u_mk^2r+2_2,Ω ≤ ^!4

[^{1−(d−1)C5C}4^d−1t]^d−11 .

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

Ω

|∂_tu_m(t)|²dx+ku_mk²_2,Ω+k∇u_mk²_2,Ω+ +k∇u_mk^2r+2_2,Ω +

T

R

0

R

Ω

|∂_τ∇u_m|²dxdτ+

T

R

0

R

Ω

|∂_τu_m|^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) byλ_jC_mj(t) andC_mj⁰⁰ (t), then summarize overj = 1, m. As a result, we get the next relations

−_dt^d R

Ω

∂_tu_m(t)∆u_m(t)dx− k∂_t∇u_mk²_2,Ω+ ^χ_2dt^d k∆u_mk²_2,Ω+ +

a₀+a₁k∇u_mk^2r_2,Ω

k∆u_mk²_2,Ω−R

Ω

|∂_tu_m(t)|^q−2∂_tu_m(t)∆u_mdx=

=−R

Ω

b(x, t)|um|^p−2um∆umdx−R

Ω

F(t, um)h∆umdx.

R

Ω

|∂_t²u_m(t)|²dx+^χ_2dt^d R

Ω

|∂_t∇u_m|²dx−

a₀+a₁k∇u_mk^2r_2,Ω R

Ω

∆u_m∂²_tu_mdx+

+¹_qdt^d R

Ω

|∂tum|^qdx=R

Ω

b(x, t)|um|^p−2um∂_t²umdx+R

Ω

F(t, um)∂_t²umdx.

By integrating these relations from 0 to t, we get

χ

2 k∆umk²_2,Ω+

t

R

0

a0+a1k∇umk^2r_2,Ω

k∆umk²_2,Ωdτ = ^χ₂ k∆um(0)k²_2,Ω+R

Ω

∂tum(t)∆um(t)dx−

−R

Ω

∂_tu_m(0)∆u_m(0)dx+

t

R

0

k∂_τ∇u_mk²_2,Ωdτ +

t

R

0

R

Ω

|∂_τu_m(τ)|^q−2∂_τu_m(τ)∆u_mdxdτ−

−

t

R

0

R

Ω

b(x, τ)|u_m|^p−2u_m∆u_mdxdτ −

t

R

0

R

Ω

F(τ, u_m)h∆u_mdxdτ.

(21)

χ 2

R

Ω

|∂_t∇u_m(t)|²dx+¹_qR

Ω

|∂_tu_m(t)|^qdx+

t

R

0

R

Ω

|∂_τ²u_m(x, τ)|²dxdτ =

=

t

R

0

a₀+a₁k∇u_mk^2r_2,Ω R

Ω

∆u_m∂_τ²u_mdxdτ+ +Rt

0

R

Ω

b(x, τ)|um|^p−2um∂_τ²umdxdτ +

t

R

0

R

Ω

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∆u_mk²_2,Ω+

T

Z

0

a₀+a₁k∇u_mk^2r_2,Ω

k∆u_mk²_2,Ωdt≤C₇, for all t∈[0, T], T < T₀, (23)

Z

Ω

|∂t∇um(t)|²dx+ Z

Ω

|∂tum(t)|^qdx+

T

Z

0

Z

Ω

|∂_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∆u_mk²_2,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 u_xx ∈L₂(Q_T). In this way, ∆u, ∆u_t, u_tt ∈L₂(Q_T).

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

u_tt−χ∆u_t−a₀∆u−a₁

k∇u₁k^2r_2,Ω∆u₁− k∇u₂k^2r_2,Ω∆u₂ + +|u_1t|^q−2u_1t− |u_2t|^q−2u_2t=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

Ω

[uτ τ −χ∆uτ −a0∆u −a1

k∇u1k^2r_2,Ω∆u1− k∇u2k^2r_2,Ω∆u2

+ +|u_1τ|^q−2u_1τ − |u_2τ|^q−2u_2τ]u_τdxdτ =

t

R

0

R

Ω

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

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

Ω

u²_t(t)dx+χ

t

R

0

R

Ω

|∇u_τ|²dxdτ + ^a₂⁰ R

Ω

|∇u|²dx+

t

R

0

R

Ω

|u_τ|^qdxdτ ≤

≤ −a₁

t

R

0

k∇u₁k^2r_2,ΩR

Ω

∇u∇u_τdx−

k∇u₁k^2r_2,Ω− k∇u₂k^2r_2,Ω R

Ω

∇u₂∇u_τdx

dτ+

+

t

R

0

R

Ω

b(x, τ) (|u₁|^p−2+|u₂|^p−2)u_τdxdτ +

t

R

0

R

Ω

h(x, τ) (F(τ, u₁)−F(τ, u₂))u_τdxdτ.

(28)

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

t

R

0

R

Ω

b(x, τ) (|u₁|^p−2u₁− |u₂|^p−2u₂)u_τdxdτ

≤b₁(p−1)

t

R

0

R

Ω

(|u₁|^p−2+|u₂|^p−2)uu_τdxdτ ≤

≤b₁(p−1) _t

R

0

R

Ω

|u₁|^{2r(p−2)r−2} dxdτ ^r−2_2r

+ _t

R

0

R

Ω

|u₂|^{2r(p−2)r−2} dxdτ

^r−2_2r !

×

× _t

R

0

R

Ω

u^rdxdτ

¹_{r t} R

0

R

Ω

u²_τdxdτ ¹₂

.

Let us put r = _N^2N₋₂, p ≤ 2 + _N−2¹ , N ≥ 3. Then by the Sobolev embedding theorem H¹(Ω) →→ L_r(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

Ω

b(x, τ) |u₁|^p−2u₁− |u₂|^p−2u₂

u_τdxdτ

≤C₁

t

Z

0

ku_τk²_2,Ω+k∇uk²_2,Ω+kuk²_2,Ω

dτ. (29) Let us estimate the first term

a₁

t

R

0

k∇u₁k^2r_2,ΩR

Ω

∇u∇u_τdx−

k∇u₁k^2r_2,Ω− k∇u₂k^2r_2,Ω R

Ω

∇u₂∇u_τdx

dτ

≤

≤a₁

t

R

0

k∇u₁k^2r_2,Ωk∇uk_2,Ωk∇u_τk_2,Ωdτ+a₁

t

R

0

k∇u₁k^2r−2_2,Ω +k∇u₂k^2r−2_2,Ω

k∇u₂k_2,Ωk∇u_τk_2,Ω×

×R

Ω

(|∇u₁|² − |∇u₂|²)dxdτ ≤a₁

t

R

0

k∇u₁k^2r_2,Ωk∇uk_2,Ωk∇u_τk_2,Ωdτ+ +a1C₂⁰

t

R

0

k|∇u1|+|∇u2|k_2,Ωk∇uk_2,Ωk∇uτk_2,Ω ≤

≤ ^χ₄

t

R

0

k∇u_τk²_2,Ωdτ +C₂

t

R

0

ku_τk²_2,Ω+k∇uk²_2,Ω+kuk²_2,Ω dτ.

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

Ω

u²_t(t)dx+C₀R

Ω

|u|²dx+a₀R

Ω

|∇u|²dx+χ

t

R

0

R

Ω

|∇u_τ|²dxdτ +

t

R

0

R

Ω

|u_τ|^qdxdτ ≤

≤C₄

t

R

0

ku_τk²_2,Ω+a₀k∇uk²_2,Ω+C₀kuk²_2,Ω

dτ+C₅

t

R

0

ku_τk²_2,Ω+a₀k∇uk²_2,Ω+C₀kuk²_2,Ωd

dτ, where d >1.

From the last inequality follows that R

Ω

u²_t(t)dx+C₀R

Ω

|u|²dx+a₀R

Ω

|∇u|²dx≤C₄

t

R

0

ku_τk²_2,Ω+a₀k∇uk²_2,Ω+C₀kuk²_2,Ω dτ+ +C5

t

R

0

kuτk²_2,Ω+a0k∇uk²_2,Ω+C0kuk²_2,Ωd

dτ, where by Bihari’s lemma, impliesR

Ω

u²_t(t)dx+C₀R

Ω

|u|²dx+a₀R

Ω

|∇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.