IDENTIFICATIONOFTHERIGHTHANDSIDEOFAQUASILINEAR - kaznu.kz

IRSTI 27.31.17; 27.35.38; 27.35.41 DOI: https://doi.org/10.26577/JMMCS.2021.v110.i2.05

S.E. Aitzhanov^1∗ , G.R. Ashurova¹ , K.A. Zhalgassova²

1

Аl-Farabi Kazakh National University, Kazakhstan, Almaty

2

M.Auezov South Kazakhstan University, Kazakhstan, Shymkent

∗

e-mail: Aitzhanov.Serik81@gmail.com

IDENTIFICATION OF THE RIGHT HAND SIDE OF A QUASILINEAR PSEUDOPARABOLIC EQUATION WITH MEMORY TERM

The study of equations of mathematical physics, including inverse problems, is relevant today.

This work is devoted to the fundamental problem of studying the solvability and qualitative properties of the solution of the inverse problem for a quasilinear pseudoparabolic equation (also called Sobolev-type equations) with memory term. To date, studies of direct and inverse problems for a pseudoparabolic equations are rapidly developing in connection with the needs of modeling and control of processes in thermal physics, hydrodynamics, and mechanics of a continuous medium. The pseudoparabolic equations similar to those considered in this work arise in the description of heat and mass transfer processes, processes of non-Newtonian fluids motion, wave processes, and in many other areas.

The main types of the inverse problems are: boundary, retrospective, coefficient and geometric.

The boundary and retrospective inverse problems lead to the study of linear problems. In turn, the statements related to the study of coefficient and geometric types bring to the nonlinear problems.

Coefficient inverse problems are divided into two main groups: coefficient inverse problems, where the unknown is a function of one or several variables, and finite-dimensional coefficient inverse problems.

In this article the existence and uniqueness of a weak and strong solution of the inverse problem in a bounded domain are proved by the Galerkin method. Also we used Sobolev’s embedding theorems, and obtained a priori estimates for the solution. Moreover, we get local and global theorems on the existence of the solution.

Key words: Pseudoparabolic equation, inverse problem, existence, uniqueness, local solvability, global solvability, non-local condition.

C.E. Айтжанов

^1∗

, Г.Р. Ашурова

¹

, К.Ә. Жалғасова

²

1

Әл-Фараби атындағы Қазақ Ұлттық университетi, Қазақстан, Алматы қ.

2

М.Әуезов атындағы Оңтүстiк Қазақстан мемлекеттiк универитетi, Қазақстан, Шымкент қ.

∗

e-mail: Aitzhanov.Serik81@gmail.com

Жады бар квазисызықты псевдопараболалық теңдеудiң оң жағын анықтау Математикалық физика теңдеулерiн, оның iшiнде керi есептердi зерттеу бүгiнгi күнi өзектi болып табылады. Бұл жұмыс квазисызықтық псевдопараболалық теңдеу үшiн керi есептiң шешiмдiлiгi мен сапалық қасиеттерiн зерттеудiң iргелi мәселесiне арналған (Соболев типтi теңдеулер деп те аталады). Бүгiнгi таңда псевдопараболалық теңдеулер үшiн тура және керi есептердi зерттеу жылу физикасы, гидродинамика және үздiксiз орта механикасындағы процестердi модельдеу және басқару қажеттiлiктерiне байланысты тез дамып келедi. Осы жұмыста қарастырылған псевдопараболалық теңдеулер жылу-масса алмасу процестерiн, ньютондық емес сұйықтықтардың қозғалыс процестерiн, толқындық процестердi және басқа да көптеген салаларды сипаттау кезiнде пайда болады.

c 2021 Al-Farabi Kazakh National University

Керi есептердiң негiзгi түрлерiне мыналар жатады: шекаралық, ретроспективтi, коэффици- енттiк және геометриялық. Шекаралық және ретроспективтi керi есептер -сызықтық есептер- дi зерттеуге, ал коэффициенттiк және геометриялық есептер -сызықтық емес есептердi зерт- теуге алып келедi. Коэффициенттiк керi есептер екi негiзгi түрге бөлiнедi-коэффициенттiк керi есептер, онда бiр немесе бiрнеше айнымалылардан тәуелдi функциясы белгiсiз және шек- тi өлшемдi коэффициенттiк керi есептер. Мақалада Галеркин әдiсiмен шенелген облыстағы керi есептiң әлсiз және әлдi шешiмiнiң бар және жалғыздығы дәлелденедi. Соболевтiң енгi- зу теоремалары қолданылып, шешiмнiң априорлық бағалаулары алынды. Шешiмнiң локалдi және глобалдi шешiмдiлiгi туралы теоремалар алынды.

Түйiн сөздер: Псевдопараболалық теңдеу, керi есеп, шешiмнiң бар болуы, шешiмнiң жалғы- здығы, локалдi шешiмдiлiк, глобалдi шешiмдiлiк, локалдi емес шарт.

C.E. Айтжанов

^1∗

, Г.Р. Ашурова

¹

, К.Ә. Жалғасова

²

1

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

2

Южно-Казахстанский государственный университет имени М. О. Ауэзова, Казахстан, г. Шымкент

∗

e-mail: Aitzhanov.Serik81@gmail.com

Идентификация правой части квазилинейного псевдопараболического уравнения с памятью

Исследование уравнений математической физики, в том числе обратных задач на сегодняш- ний день является актуальной. Эта работа посвящена фундаментальной проблеме исследо- ванию разрешимости и качественных свойств решения обратной задачи для квазилинейно- го псевдопараболическогоуравнения (называемых также уравнениями соболевского типа) с памятью. На сегодняшний день исследования прямых и обратных задач для псевдопарабо- лических уравнений бурно развиваются в связи с потребностями моделирования и управле- ния процессами в теплофизике, гидродинамике и механике сплошной среды. Псевдопарабо- лические уравнения подобные рассматриваемым в данной работе возникают при описании процессов тепломассопереноса, процессов движение неньютоновских жидкостей, волновых процессов и во многих других областях. К основным типам обратных задач относятся: гра- ничные, ретроспективные, коэффициентные и геометрические. Граничные и ретроспектив- ные обратные задачи приводят к исследованию линейных задач. В свою очередь, постанов- ки, к которым приводит исследование коэффициентных и геометрических задач, являют- ся нелинейными. Коэффициентные обратные задачи подразделяются на два основных вида

— коэффициентные обратные задачи, в которых неизвестной является функция одной или нескольких переменных, и конечномерные коэффициентные обратные задачи. В статье ме- тодом Галеркина доказывается существование и единственность слабого и сильного решения обратной задачи в ограниченной области. Использование теорем вложения Соболева, получе- ны априорные оценки решения. Получены локальная и глобальная теорема о существовании решения.

Ключевые слова: Псевдопараболическое уравнение, обратная задача, существования, един- ственность, локальная разрешимость, глобальная разрешимость, нелокальное условие.

1 Introduction

LetΩis a bounded area of a spaceR^N, N≥1 with a sufficiently smooth boundaryΓ,QT is a cylinder Ω×(0, T)of finite height T, S= Γ×(0, T). Let b(x, t), h(x, t), u0(x), ϕ(t), ω(x) are given functions,χ, a andβ are positive constants. Consider an inverse problem in the cylinderQT for a pseudoparabolic equation with a nonlocal overdetermination condition. Find a pair of functions{u(x, t), f(t)}that satisfy:

ut−χ∆ut−a∆u−

t

Z

0

g(t−τ)∆u(τ)dτ =b(x, t)|u|^β−2u+f(t)h(x, t), (1)

u(x,0) =u0(x), (2)

u|_S= 0, (3) Z

Ω

u(x, t)(ω(x)−χ∆ω(x))dx=ϕ(t). (4)

The monograph ([1] and see its references) considers a wide class of direct problems for nonlinear Sobolev-type equations. In papers [2-12] some inverse problems similar to our problem statement were studied.

Let us note some papers on inverse problems for Sobolev-type equations with the integral overdetermination condition. Yaman M. [7] obtained sufficient conditions for both destruction and stability of the solution. The work [8] establishes an existence theorem for regular solutions of the inverse problem of recovering coefficients in equations of composite type. A.I. Kozhanov and L.A. Teleshova [9] have proved the existence theorem for regular solutions of a nonlinear inverse problem for nonstationary differential equations of higher order is proved. Kozhanov A.I. and Namsaraeva G.V. [10] showed the existence and uniqueness of the regular solutions of linear inverse problem for equation of Sobolev type. The work [11] is devoted to the investigation of the inverse problem for the equation

ut−χ∆ut−∆u=b(x, t)|u|^β−2u+f(t)h(x),

with the conditions (2)-(4). In this paper the existence of a weak solution is proved, the asymptotic behavior of the solutions is shown att→ ∞. Moreover, sufficient conditions of finite time "blow up"of the solution are obtained. In the work [12] is dedicated to the inverse problem for a pseudoparabolic equation with p-Laplacian.

In the present work, we proved the existence of a weak solution and showed the asymptotic behavior of the solutions att→ ∞. We obtained sufficient conditions of finite time "blow up"of the solution, furthermore we get sufficient conditions for the disappearance (vanishing) of the solution in a finite time.

From a physical point of view, the considered initial-boundary value problem is a mathematical model of quasi-stationary processes in semiconductors and magnets allowing for a wide variety of physical factors.

Let the functionsh(x, t), ω(x), ϕ(t),u0(x)satisfy the following conditions:

h₁(t)≡R

Ω

h(x, t)ω(x)dx6= 0, ∀t∈[0, T],

h(x, t)∈L_∞(0, T; L2(Ω))∩L2(QT)∩L_β∗(QT), β∗= _β−1^β , β≥2.

(5)

ω∈L2(Ω)∩Lβ(Ω)∩

0

W₂²(Ω), β≥2. (6)

Z

Ω

u₀(x)ω(x)dx=ϕ(0), ϕ(t)∈W₂¹[0, T], |ϕ⁰(t)| ≤C, u₀∈

0

W₂¹(Ω)∩L_β(Ω). (7)

2 Main results

2.1 Reducing the inverse problem (1)-(4) to a direct problem

Lemma 1 The problem (1) - (4) is equivalent to the following problem for a nonlinear pseudoparabolic equation containing a nonlinear nonlocal operator of the functionu(x, t)

ut−χ∆ut−a∆u−

t

Z

0

g(t−τ)∆u(τ)dτ =b(x, t)|u|^β−2u+F(t, u)h(x, t), x∈Ω, t >0, (8)

u(x,0) =u₀(x), x∈Ω, u|_S = 0. (9)

Here

F(t, u) = 1 h₁(t)



ϕ⁰(t) +a Z

Ω

∇u∇ωdx+

t

Z

0

g(t−τ) Z

Ω

∇u(τ)∇ωdxdτ − Z

Ω

b(x, t)|u|^β−2uωdx



. (10)

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

Ω

u_tωdx−χR

Ω

∆u_tωdx−aR

Ω

∆uωdx−R

Ω t

R

0

g(t−τ)∆u(τ)dτ ωdx=

=R

Ω

b(x, t)|u|^β−2uωdx+R

Ω

f(t)h(x, t)ωdx,

(11)

next if conditions (4) and (5) are satisfied, then F(t, u) = 1

h1(t)



ϕ⁰(t) +a Z

Ω

∇u∇ωdx+

t

Z

0

g(t−τ) Z

Ω

∇u(τ)∇ωdxdτ− Z

Ω

b(x, t)|u|^β−2uωdx



. (12) Therefore the relation (10) is fulfilled. Now consider the problem (8)-(9). If the relation (10) is performed, then it obviously leads to the equality (12) . Then

F(t, u) = _h¹

1

ϕ⁰(t) +aR

Ω

∇u∇ωdx+

t

R

0

g(t−τ)R

Ω

∇u(τ)∇ωdxdτ−R

Ω

b(x, t)|u|^β−2uωdx

=

= _h¹

1

ϕ⁰(t)−aR

Ω

∆uωdx−

t

R

0

g(t−τ)R

Ω

∆u(τ)ωdxdτ−R

Ω

b(x, t)|u|^β−2uωdx

. By virtue of (11) we obtain that

F(t, u) = _h¹

1

ϕ⁰(t) +aR

Ω

∇u∇ωdx+

t

R

0

g(t−τ)R

Ω

∇u(τ)∇ωdxdτ−R

Ω

b(x, t)|u|^β−2uωdx

=

= _h¹

1

ϕ⁰(t)−R

Ω

(ut−χ∆ut)ωdx+R

Ω

b(x, t)|u|^β−2uωdx+

+R

Ω

f(t)h(x, t)ωdx+R

Ω

b(x, t)|u|^β−2uωdx

,

ϕ⁰(t)− Z

Ω

ut(ω−χ∆ω)dx= 0.

In this way, _dt^d

ϕ(t)−R

Ω

u(ω−χ∆ω)dx

= 0. We denote byv(t) =ϕ(t)−R

Ω

u(ω−χ∆ω)dx. Then the functionv(t)can be found as a solution of the Cauchy problem:v⁰(t) = 0, v(0) = 0.(v(0) = 0follows from the agreement condition (7)). The unique solution of the problem is the function v(t) = 0, consequently, R

Ωu(ω−χ∆ω)dx=ϕ(t).

Definition 1 A function u(x, t) from the space W₂¹(0, T;

0

W₂¹(Ω)) is called a weak solution of the problem (8)-(9) which satisfies the integral identity

T

R

0

R

Ω

∂u

∂tv+χ∇u_t∇v+a∇u∇v dxdt+

T

R

0 t

R

0

g(t−τ)R

Ω

∇u(τ)∇v(t)dxdτ dt−

−

T

R

0

R

Ω

b(x, t)|u|^β−2uvdxdt=

T

R

0

R

Ω

F(t, u)hvdxdt,

(13)

for allv(x, t)∈L₂(0, T;

0

W₂¹(Ω)).

Definition 2 A function u(x, t) from the space ut,4u,4ut,∈ L2(QT), is called a strong solution of the problem (8)-(9) which satisfies the integral identity

T

R

0

R

Ω

∂u

∂tv−χ4utv−a4uv dxdt+

T

R

0 t

R

0

g(t−τ)R

Ω

4u(τ)v(t)dxdτ dt−

−

T

R

0

R

Ω

b(x, t)|u|^β−2uvdxdt=

T

R

0

R

Ω

F(t, u)hvdxdt,

(14)

for allv(x, t)∈L₂(Q_t).

2.2 Existence of a weak solution

Theorem 1 Let the conditions (5)-(7) are performed, and also 2 < β < _N−2^2N , N ≥ 3. Then there is a weak solution u(x, t)of the problem (8)-(9) on the interval(0, T), T < T₀, and besides accepts the following inclusions:

u∈L∞(0, T;

0

W₂¹(Ω)),∇u∈L2(QT), QT = Ω×(0, T), u_t∈L₂(0, T;

0

W₂¹(Ω)),|u|^β−2u∈L_∞(0, T;L β β−1

(Γ)).

Proof. Let us choose in

0

W₂¹(Ω) some system of functions {Ψj(x)} forming a basis in a given space (∆Ψ +λΨ = 0, Ψ|_Γ = 0). We will look for an approximate solution of the problem (8)-(9) in the form

um(x, t) =

m

X

k=1

Cmk(t)Ψk(x) (15)

where the coefficientsCmk(t)are searched out from the conditions

m

P

k=1

C_mk⁰ (t)R

Ω

ΨkΨj+χ

m

P

i=1

∂Ψ_k

∂xi ·^∂Ψ_∂x^j

i

dx+a

m

P

k=1

Cmk(t)R

Ω

∇Ψk∇Ψjdx+

+

m

P

k=1 t

R

0

g(t−τ)C_mk(τ)R

Ω

∇Ψ_k∇Ψ_jdxdτ−

−

m

P

k=1

Cmk(t)R

Ω

b(x, t)|um|^β−2ΨkΨjdx=R

Ω

F(t, um)h(x, t)Ψjdx.

(16)

um0=um(0) =

m

X

k=1

Cmk(0)Ψk =

m

X

k=1

αkΨk (17)

and besides

u_m0→u₀ strongly in

0

W₂¹(Ω) at m→ ∞ (18)

We introduce a notation

C~_m≡ {C1m(t), ..., C_mm(t)}^T, ~α≡ {α1, ..., α_m}^T, a_kj=

Z

Ω

[Ψ_kΨ_j+χ(∇Ψ_k,∇Ψ_j)]dx, b_kj= Z

Ω

∇Ψ_k∇Ψ_jdx,

fkj =−a Z

Ω

∇Ψk∇Ψjdx+ Z

Ω

b(x, t)|um|^β−2ΨkΨjdx+ Z

Ω

F(t, um)h(x, t)Ψjdx,

Am

C~m

≡n ajk

C~m

o , Bm

C~m

≡n bjk

C~m

o , ~Fm

C~m

≡n fjk

C~m

oC~m. Then the system of equations (16) and condition (17) take the matrix form.

A_mC~_m⁰ +

t

Z

0

g(t−τ)B_mC_m(τ)dτ ≡F~_m C~_m

, C~_m(0) =~α. (19)

According to the Cauchy theorem, the problem (19) has at least one solution C~_m on a certain time interval t∈(0, T_m), T_m >0. Below we obtain a priori estimates foru_m which is independent ofm and, in some cases, valid for any finite t.

2.3 A priori estimates

To get the first estimate, we multiply both sides of the equality (16) byC_mj(t)and summarize both sides of the obtained equality overj= 1, m. As a result, we get the equality

1 2 d dt

R

Ω

[|um|²+χ|∇um|²]dx+aR

Ω

|∇um|²dx+

+

t

R

0

g(t−τ)R

Ω

∇u_m(τ)∇u_m(t)dxdτ =R

Ω

b(x, t)|u_m|^βdx+R

Ω

F(t, u_m)hu_mdx.

(20)

Lemma 2 If u∈

0

W₂¹(Ω), 2< β < _N^2N₋₂, N ≥3, then the next 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.

From the lemma 2 follows the inequality kuk^β_β,Ω≤C1

kuk²_2,Ω+χk∇uk²_2,Ω^β₂ ,

whereC₁= max (

1; ^(1−α)α

α 1−αC

1−α2 0

χ^1−αα

)!^β₂ . We estimate the right hand side of (20)

Z

Ω

b(x, t)|um|^βdx

≤b0kumk^β_β,Ω≤C1b0

kumk²_2,Ω+χk∇umk²_2,Ω^β₂

, (21)

ϕ⁰(t) h1(t)

Z

Ω

humdx

≤ |ϕ⁰(t)|

|h1(t)|khk_2,Ωkumk_2,Ω≤1 4 sup

0≤t≤T

|ϕ⁰(t)|²

|h1(t)|²khk²_2,Ω+kumk²_2,Ω, (22)

a h1(t)

R

Ω

hu_mdx

t

R

0

g(t−τ)R

Ω

∇um(τ)∇ωdxdτ

≤

≤ _|h^a

1(t)|kh(x, t)k_2,Ωkumk_2,Ω

t

R

0

g(t−τ)k∇um(τ)k_2,Ωk∇ωk_2,Ωdτ ≤

≤ ^a₄²

k∇ωk_2,Ω sup

0≤t≤T 1

|h1(t)|kh(x, t)k_2,Ω ² t

R

0

k∇um(τ)k²_2,Ωdτ+kumk²_2,Ω.

(23)

t

R

0

g(t−τ)R

Ω

∇um(τ)∇um(t)dxdτ

≤

t

R

0

g(t−τ)k∇um(τ)k_2,Ωk∇um(t)k_2,Ωdτ ≤

≤ ^g_2a⁰

t

R

0

k∇um(τ)k²_2,Ωdτ +^a₂k∇um(t)k²_2,Ω.

1 h₁(t)

R

Ω

humdxR

Ω

b(x, t)|um|^β−2um·ωdx

≤ _|h^b0

1(t)|kumk^β_β,Ωkωk_β,Ωkhk β β−1,Ω≤

≤C1b0kωk_β,Ω sup

0≤t≤T 1

h²₁(t)kh(x, t)k β β−1,Ω

k∇umk²_2,Ω+C1kumk²_2,Ω^β₂ .

(24)

d dt

R

Ω

[|um|²+χ|∇um|²]dx+aR

Ω

|∇um|²dx≤

≤C2

kumk²_2,Ω+χk∇umk²_2,Ω +C3

kumk²_2,Ω+χk∇umk²_2,Ω^β₂ + +C₄

t

R

0

k∇u_m(τ)k²_2,Ωdτ+¹₄ sup

0≤t≤T

|ϕ⁰(t)|²

|h1(t)|²khk²_2,Ω.

(25)

We denote byy(t)≡χk∇uk²_2,Ω+kuk²_2,Ω,then (25) takes the form dy(t)

dt ≤C4 t

Z

0

y(τ)dτ +C3[y(t)]^β2 +C2y(t) +1 4 sup

0≤t≤T

|ϕ⁰(t)|²

|h1(t)|²khk²_2,Ω. By integrating from0 tot, we get

y(t)≤y(0) +1 4 sup

0≤t≤T

|ϕ⁰(t)|²

|h1(t)|²

t

Z

0

khk²_2,Ωdτ+C₅(t^{2(β−1)β−2} +t) +C₆

t

Z

0

[y(τ)]^β2dτ,

Applying the Bihari lemma to (25), if C4

C3

e^{C3p−22 t}−1

< 1

z(0) + ^C_C²

3 +_2(C^C0

3−γ)

^p−2₂

, 0≤t < T,

then the next inequality is true

z(t)≤ z(0) +^C_C²

3 +_2(C^C0

3−γ)

1−

z(0) +^C_C²

3 +_2(C^C0

3−γ)

^p−2_{2 C}

4

C3

e^{C3p−22 t}−1p−2²

.

χk∇umk²_2,Ω+kumk²_2,Ω≤

≤ (χk∇um(x,0)k²_2,Ω+kum(x,0)k²_2,Ω+C)^eC3t

1−(χk∇um(x,0)k²_2,Ω+kum(x,0)k²_2,Ω+C)^{β−22 C}_C⁴₃

e^C3β−22 t

−1

_β−2² . (26)

From this estimate, we can conclude that there isT0>0 such that

ku_mk²_2,Ω+χk∇u_mk²_2,Ω≤C₅, f or all t∈[0, T], T < T₀, (27) where the constantC5 does not depend onm∈N.

Returning to (25) and taking into account (27), we obtain one more inequality:

t

Z

0

Z

Ω

|∇um|²dxdt≤C5. (28)

Now we multiply equality (16) byC_mj⁰ (t)and summarize overj= 1, m. As a result, we get k∂_τu_mk²_2,Ω+χk∇∂_τu_mk²_2,Ω+^a_2dt^d R

Ω

|∇u_m|²dx+

+

t

R

0

g(t−τ)R

Ω

∇um(τ)∇∂tum(t)dxdτ =_β¹_dt^d R

Ω

b(x, t)|um|^βdx−

−_β¹R

Ω

b_t(x, t)|um|^βdx+R

Ω

F(t, u_m)h(x, t)∂_tu_mdx.

(29)

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

t

R

0

k∂_τu_mk²_2,Ω+χk∇∂_τu_mk²_2,Ω

dτ+^a₂R

Ω

|∇u_m|²dx=¹₂R

Ω

|∇u_m(x,0)|²dx−

−¹_βR

Ω

b(x,0)|um(x,0)|^βdx+_β¹R

Ω

b(x, t)|um|^βdx−

−¹_β

t

R

0

R

Ω

b_τ(x, τ)|um|^βdxdτ−

t

R

0 τ

R

0

g(τ−s)R

Γ∇um(s)∇∂τu_m(τ)dΓdsdτ+ +

t

R

0

R

Ω

F(t, um)h(x, t)∂τumdxdτ.

(30)

We estimate the right hand side of (30):

1 β

R

Ω

b(x, t)|um|^βdx

≤ ^b_β⁰kumk^β_β,Ω≤

≤ ^C1_β^b0

kumk²_2,Ω+χk∇umk²_2,Ω^β₂

≤^C1_β^b0C

β 2

5 ,

(31)

1 β

t

R

0

R

Ω

bτ(x, τ)|um|^βdxdτ

≤ ^b_β⁰

t

R

0

kumk^β_β,Ωdτ ≤

≤ ^C1_β^b0

t

R

0

kumk²_2,Ω+χk∇umk²_2,Ω^β₂

dτ ≤^C1_β^b0C

β 2

5 t,

(32)

t

R

0

R

Ω ϕ⁰(τ)

h1(τ)h(x, τ)∂_τu_mdxdτ

≤

≤ ³₂

t

R

0

R

Ω

ϕ⁰(τ)h(x,τ) h₁(τ)

2

dxdτ+¹₆

t

R

0

R

Ω

|∂τum|²dxdτ.

(33)

t

R

0

R

Ω h(x,τ)

h₁(τ)∂τumdxR

Ω

b(x, τ)|u|^β−2uωdxdτ

≤

≤ ²₃C

2β−2 β

1

b₀kωk_β,Ω sup

0≤t≤T

kh(x,t)k_2,Ω h1(t)

² t

R

0

ku_mk²_2,Ω+χk∇u_mk²_2,Ω^2β−2_β dτ+ +¹₆

t

R

0

k∂_τu_mk²_2,Ωdτ ≤¹₆

t

R

0

k∂_τu_mk²_2,Ωdτ+ +²₃C

2β−2 β

1

b₀kωk_β,Ω sup

0≤t≤T

kh(x,t)k_2,Ω h1(t)

² C

2β−2 β

5 t.

(34)

t

R

0 τ

R

0

g(τ−s)R

Γ∇um(s)∇∂τum(τ)dΓdsdτ

≤

t

R

0 τ

R

0

g(τ−s)k∇um(s)k_2,Ωk∇∂τum(τ)k_2,Ωdτ ≤

≤ ^g₂⁰

t

R

0

(t−τ)k∇um(τ)k²_2,Ωdτ+¹₂

t

R

0

k∇∂τum(τ)k²_2,Ωdτ.

(35)

a

t

R

0

R

Ω h(x,τ)

h₁(τ)∂τumdx

τ

R

0

g(τ−s)R

Ω

∇um(s)∇ωdxdsdτ

≤

≤ ¹₆

t

R

0

k∂τumk²_2,Ωdτ+^2g₃⁰

ak∇ωk_2,Ω sup

0≤t≤T

kh(x,t)k_2,Ω

|h1(t)|

2 t

R

0

(t−τ)k∇um(τ)k²_2,Ωdτ.

(36)

We substitute the obtained inequalities into the identity (30), and from the estimate (27), we get the second estimate

t

Z

0

k∂τumk²_2,Ω+χk∇∂τumk²_2,Ω

dτ ≤C6. (37)

2.4 Passage to the limit

From the obtained estimates (27), (28), (31) imply the next assertions respectively:

u_m is bounded in L_∞(0, T;

0

W₂¹(Ω)), (38)

∇um is bounded in L2(QT), QT = Ω×(0, T), (39)

∂tum is bounded in L2(0, T;

0

W₂¹(Ω)), (40)

In addition, by virtue of the conditions which set forβ:

|um|^β−2um is bounded in L∞(0, T;L β

β−1(Ω)),2< β < 2N

N−2, N ≥3. (41)

From (38) follows that there exists a subsequenceum_k of the sequenceum, *-weakly converging to some element u∈L_∞(0, T;

0

W₂¹(Ω)), that isum_k → u*-weakly inL_∞(0, T;

0

W₂¹(Ω)). Similarly, from (39)-(41) follows that there exists a sequence such that {um_k} ⊂ {um}, that um_k → uweakly in L2(0, T;

0

W₂¹(Ω)).

By virtue of the Rellich-Kondrashov theorem, the embeddingW₂¹(Q_T)intoL₂(Q_T)is a compact. This means that we can choose the sequence u_mk in such way that u_mk → u in the norm of L₂(Q_T), therefore it converges almost everywhere [13].

The reasoning above allows us to pass to the limit in (16). But first, we multiply each of the equality (16) by d_j(t)∈C[0, T]and summarize both sides of the obtained equality overj= 1, m. Then we integrate with respect to tfrom0 toT, and get

t

R

0

R

Ω

∂um

∂t µ+χ∇umt∇µ+a∇um∇µ dxdt+

t

R

0 t

R

0

g(t−τ)R

Ω

∇um(τ)∇µ(t)dxdτ dt−

−

t

R

0

R

Ω

b(x, t)|um|^β−2u_mµdxdt=

t

R

0

R

Ω

F(t, u_m)hµdxdt,

(42)

whereµ(x, t) =

m

P

j=1

dj(t)Ψj(x).

Considering the obtained inclusions and convergences we pass to the limit in (42) atm → ∞and get (14) for v =µ. Since the set of all functionsµ(x, t) is dense in W₂¹(0, T;

0

W₂¹(Ω)), then the limit relation is performed for allv(x, t)∈L2(0, T;

0

W₂¹(Ω)).

3 Uniqueness of the weak solution

Theorem 2 Let u₀(x) ∈

0

W₂¹(Ω), 2 < β ≤ ^2(N−1)_N−2 , N ≥ 3 are fulfilled .Then the weak solution of the problem (8)-(9)on the interval(0, T)is unique (in the sense of Definition 1).

Proof. Suppose that the problem (8)-(9) has two solutions:u1(x, t)and u2(x, t). Then their difference u(x, t) =u₁(x, t)−u₂(x, t)satisfies the conditionu(x,0) = 0and the identity

t

R

0

R

Ω

(uτv+χ∇uτ∇v+a∇u∇v)dxdτ+

t

R

0 τ

R

0

g(τ−s)R

Ω

∇u(s)∇v(τ)dxdsdτ =

=

t

R

0

R

Ω

b(x, τ) |u1|^β−2u1− |u1|^β−2u1

vdxdτ+ +

t

R

0

R

Ω

(F(τ, u₁)−F(τ, u₂))hvdxdτ,

By virtue ofv(x, t)∈L2(0, T;

0

W₂¹(Ω)), then asv(x, t)we may takeu(x, t), that is we putv(x, t) =u(x, t)

t

R

0

R

Ω

uτu+χ∇uτ∇u+a|∇u|² dxdτ+

t

R

0 τ

R

0

g(τ−s)R

Ω

∇u(s)∇u(τ)dxdsdτ =

=

t

R

0

R

Ω

b(x, τ) |u1|^β−2u1− |u1|^β−2u1

udxdτ+

+

t

R

0

R

Ω

(F(τ, u₁)−F(τ, u₂))hudxdτ.

(43)

We estimate the right hand side of the inequality (43), using the following inequality||u₁|^qu₁− |u₂|^qu₂| ≤ (q+ 1) (|u₁|^q+|u₂|^q)|u₁−u₂|,q >0.

R

Ω

b(x, τ) |u₁|^β−2u₁− |u₂|^β−2u₂ udx

≤b₁(β−1)R

Ω

|u₁|^β−2+|u₂|^β−2 u²dx≤

≤b1(β−1)

R

Ω

|u1|^β−2+|u2|^β−22

u²dx ¹₂

R

Ω

u²dx ¹₂

≤

≤b1(β−1)

R

Ω

|u1|^β−2+|u2|^β−2r−2^2r dx ^r−2_2r

R

Ω

u^rdx ¹_r

R

Ω

u²dx ¹₂

≤

≤b₁(β−1)

R

Ω

|u1|^{2r(β−2)r−2} dx ^r−2_2r

+

R

Ω

|u2|^{2r(β−2)r−2} dx ^r−2_2r !

×

×

R

Ω

u^rdx ¹_r

R

Ω

u²dx ¹₂

.

We putr= _N^2N₋₂, 2< β ≤ ^2(N_N−2⁻¹⁾, N ≥3. Then by the Sobolev embedding theorem

0

W₂¹(Ω)⊂Lr(Ω) and

0

W₂¹(Ω)⊂L2r(β−2)/(r−2)(Ω).

In this case, taking into account the smoothness class of the solutionsu1(x, t)andu2(x, t), we arrive at the estimate

R

Ω

b(x, τ) |u₁|^β−2u₁− |u₂|^β−2u₂ udx

≤

≤ ^a₈k∇uk²_2,Ω+C₁₆⁰ kuk²_2,Ω≤^a₈k∇uk²_2,Ω+C16

kuk²_2,Ω+χk∇uk²_2,Ω .

(44)

Analogically,

1 h1

R

Ω

b(x, τ) |u1|^β−2u1− |u2|^β−2u2

ωdxR

Ω

hudx

≤

≤ ^a₈k∇uk²_2,Ω+C₁₇

kuk²_2,Ω+χk∇uk²_2,Ω .

1 h₁aR

Ω

∇u∇ωdxR

Ω

hudx

≤_|h¹

1|k∇uk_2,Ωk∇ωk_2,Ωkuk_2,Ωkhk_2,Ω≤

≤ ^a₈k∇uk²_2,Ω+C18

kuk²_2,Ω+χk∇uk²_2,Ω .

a h₁(t)

R

Ω

humdx

t

R

0

g(t−τ)R

Ω

∇um(τ)∇ωdxdτ

≤

≤C19

kuk²_2,Ω+χk∇uk²_2,Ω +C20

t

R

0

kuk²_2,Ω+χk∇uk²_2,Ω dτ.

t

R

0

g(t−τ)R

Ω

∇um(τ)∇um(t)dxdτ

≤

t

R

0

g(t−τ)k∇um(τ)k_2,Ωk∇um(t)k_2,Ωdτ ≤

≤ k∇um(t)k_2,Ω

t

R

0

g(t−τ)k∇um(τ)k_2,Ωdτ ≤

≤²_a

t

R

0

g²(t−τ)dτ

t

R

0

k∇um(τ)k²_2,Ωdτ+^a₈k∇um(t)k²_2,Ω≤

≤^2g_a⁰

t

R

0

k∇u_m(τ)k²_2,Ωdτ+^a₈k∇u_m(t)k²_2,Ω≤

≤C21

kuk²_2,Ω+χk∇uk²_2,Ω +C22

t

R

0

kuk²_2,Ω+χk∇uk²_2,Ω dτ.

(45)

By virtue of (44)-(45), we obtain R

Ω

|u|²dx+χR

Ω

|∇u|²dx+^a₂

t

R

0

R

Ω

|∇u|²dxdτ≤

≤C23 t

R

0

kuk²_2,Ω+χk∇uk²_2,Ω

dτ+C24 t

R

0 τ

R

0

ku(s)k²_2,Ω+χk∇u(s)k²_2,Ω dsdτ.

(46)

R

Ω

|u|²dx+χR

Ω

|∇u|²dx≤

≤C25 t

R

0

(t−τ+ 1)

kuk²_2,Ω+χk∇uk²_2,Ω dτ.

(47)

By virtue of Gronwall’s lemma from the inequality (47), we get R

Ω

|u|²dx+χR

Ω

|∇u|²dx = 0 almost everywhere on the time interval (0, T), which shows the uniqueness of the weak solution.From Lemma 1 we can establish the solvability of the inverse problem (1)-(4). Let u(x, t)be a solution of the initial-boundary value problem (8)-(9) from the space (Theorem 1) u∈L_∞(0, T;

0

W₂¹(Ω)),∇u∈L₂(Q_T), Q_T = Ω×(0, T), ut ∈ L2(0, T;

0

W₂¹(Ω)),|u|^β−2u ∈ L_∞(0, T;L β

β−1(Γ)). Obviously, the function f(t) from the relation (10) belongs to the spaceL_∞(0, T). What was proved above means that the found functionsu(x, t)andf(t)give a weak solution of the inverse problem.

4 Global solvability of the problem (8)-(9).

Consider the case when1< β≤2. Let the conditions h1(t)≡R

Ω

h(x, t)ω(x)dx6= 0, ∀t∈[0, T],

h(x, t)∈L_∞(0, T; L2(Ω))∩L2(QT), 1< β≤2, ω∈L2(Ω)∩Lβ(Ω)∩

0

W₂²(Ω), 1< β ≤2, (48)

Z

Ω

u₀(x)ω(x)dx=ϕ(0), ϕ(t)∈W₂¹[0, T], |ϕ⁰(t)| ≤C, u₀∈

0

W₂¹(Ω)∩L_β(Ω).

are fulfilled.

Lemma 3 Ifu∈

0

W₂¹(Ω), 1< β ≤2, then the next inequality is performed Z

Ω

|um|^βdx≤



 Z

Ω

|u|²dx





β 2

|Ω|^2−β2 ≤C1



1 + Z

Ω

|u|²dx+χ Z

Ω

|∇u|²dx



.