Analysis of the nonlocal wave propagation problem with volume constraints

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Analysis of the nonlocal wave propagation problem with volume constraints

Fatim-Zahra Aït-Bella, Mohammed El Rhabi, Abdelilah Hakim, Amine Laghrib

To cite this version:

Fatim-Zahra Aït-Bella, Mohammed El Rhabi, Abdelilah Hakim, Amine Laghrib. Analysis of the

nonlocal wave propagation problem with volume constraints. Mathematical Modeling and Computing,

2020, 7 (2), pp.334-344. �10.23939/mmc2020.02.334�. �hal-02985244�

AIMS’ Journals

VolumeX, Number0X, XX200X pp.X–XX

ANALYSIS OF THE NONLOCAL WAVE PROPAGATION PROBLEM WITH VOLUME CONSTRAINTS

Fatim Zahra Ait Bella

^∗

, Abdelilah Hakim

Cadi Ayyad University, LAMAI FST Marrakech, Morocco

Mohammed El Rhabi

Ecole des Ponts ParisTech (ENPC) Paris, France

Amine Laghrib

Sultan Moulay Slimane University, LMA FST B´eni-Mellal, Morocco

Abstract. In the current paper, we investigate a nonlocal hyperbolic problem with volume constraints. The main motivation of this work is to apply the nonlocal vector calculus, introduced and developed by DU et al. [3] to such problem. Moreover, based on some density arguments, some a priori estimates and using the Galerkin approach, we prove existence and uniqueness of a weak solution to the nonlocal wave equation.

1. Introduction. The study of nonlocal problems has gained great attention over the last two decades. Nonlocal models involve integral equations and fractional derivatives allowing nonlocal interactions, that is to say, the interaction may occur even when the closures of two domains have an empty intersection. Such models are effective in modeling material singularities and are widely considered in a variety of applications, including image analyses [6]-[10], phase transition [4][11], machine learning [12] and obstacle problem [5]...

In a major advance in 2013, Du et al. [3] introduced nonlocal vector calculus as a nonlocal framework to understand and analyze nonlocal problems. It defines non- local fluxes , nonlocal analogues of the gradient, divergence, and curl operators, and presentes some basic nonlocal integral theorems that mimic the classical in- tegral theorems of the vector calculus for differential operators, the authors have also provided connection between the nonlocal operators and their usual differential counterparts in a distributional sense then in a weak sense by introducing nonlocal weighted operators.

The present paper was motivated by [2], where the authors threw light on the anal- ogy between nonlocal and local diffusion problems with a convincing explanation of the usefulness, in the nonlocal case, of volume constraints which represent the nonlocal analogue of the boundary conditions of the classical theory. Our purpose

2010Mathematics Subject Classification. Primary: 65M60; Secondary: 35L05 .

Key words and phrases. Galerkin approximation, nonlocal operators, nonlocal vector calculus, volume constrained problems, wave equation.

∗Corresponding author: [email protected].

1

is to discuss the well posedeness of a hyperbolic problem considering a nonlocal diffusion operator instead of the Laplacian operator. Furthermore, the study of the eigenvalue problem corresponding to the nonlocal Dirichlet problem is carried out.

The paper is divided into two main sections. The first part gives a brief overview of the basic concepts of the nonlocal vector calculus and emphasises the existence of an orthogonal basis of eigenfunctions associated to the considered nonlocal oper- ator. In the second part, we formulate the nonlocal wave equation and exploit the Galerkin method to prove existence and uniqueness of weak solution to the nonlocal hyperbolic problem.

2. Statement of the elliptic nonlocal problem. In the present section we give the position of the elliptic volume constrained problem and present the energy spaces needed to study the nonlocal problem:

D(ξ.D

^∗

(u)) = f on Ω

u = 0 on Ω

_I

(1)

Where Ω is an open and bounded subset of R

ⁿ

with piecewise smooth boundary and satisfies the interior cone condition, ξ is a symmetric second-order tensor, and f ∈ L

²

(Ω) is a given function.

Recall the definition of some nonlocal operators, see [3]. Given a vector function ν(x, y) : R

ⁿ

× R

ⁿ

→ R

^k

and an antisymmetric vector function α(x, y) : R

ⁿ

× R

ⁿ

→ R

^k

, the action of the nonlocal divergence operator D on ν is defined as

D(ν)(x) :=

Z

Rⁿ

(ν(x, y) + ν (y, x)).α(x, y)dy f or x ∈ R

ⁿ

(2) Given a scalar function u(x) : R

ⁿ

→ R , the adjoint of D is the operator D

^∗

whose action on u is given by

D

^∗

(u)(x, y) = −(u(y) − u(x))α(x, y) f or x, y ∈ R

ⁿ

(3) The operator −D

^∗

is considered as a nonlocal gradient, also,

D(ξ.D

^∗

u)(x) = −2 Z

Rⁿ

(u(y) − u(x))α(x, y).(ξ(x, y).α(x, y))dy

Given positive constants γ

₀

and ε, we first assume that the symmetric kernel γ(x, y) = α(x, y).(ξ(x, y).α(x, y))

satisfies, for all x ∈ Ω ∪ Ω

I

1) γ(x, y) ≥ 0 ∀y ∈ B

ε

(x) 2) γ(x, y) ≥ γ

0

> 0 ∀y ∈ B

_ε/2

(x) 3) γ(x, y) = 0 ∀y ∈ (Ω ∪ Ω

I

) \ B

ε

(x) where B

ε

(x) := {y ∈ Ω ∪ Ω

I

: |y − x| ≤ ε}

4) There exist s ∈ (0, 1) and positive constants γ

_∗

and γ

^∗

such that, for all x ∈ Ω γ

_∗

|y − x|

^n+2s

≤ γ(x, y) ≤ γ

^∗

|y − x|

^n+2s

f or y ∈ B

ε

(x)

let us also recall the definition of the interaction domain corresponding to Ω:

Ω

I

:= {y ∈ R

ⁿ

\Ω : α(x, y) 6= 0 for some x ∈ Ω} (4) To investigate the problem (1), the following nonlocal energy space will be used constantly [2]. We adopt:

V (Ω ∪ Ω

_I

) := {u ∈ L

²

(Ω ∪ Ω

_I

) : |||u||| < ∞} (5)

equipped with the nonlocal energy norm

|||u||| :=

1 2

Z

Ω∪Ω_I

Z

Ω∪Ω_I

D

^∗

(u).(ξ.D

^∗

(u))dydx

¹₂

(6) we then introduce the nonlocal volume constrained energy space [2]:

V

c

(Ω ∪ Ω

I

) := {u ∈ V (Ω ∪ Ω

I

) : u = 0 on Ω

I

} the norm

|||f |||

_V^∗

c(Ω∪ΩI)

:= sup

ϕ∈V_c(Ω∪Ω_I)

|||ϕ|||≤1

hf, ϕi

_V^∗

c(Ω∪ΩI),Vc(Ω∪ΩI)

denotes the norm for the dual space V

_c^∗

(Ω ∪ Ω

I

) of V

c

(Ω ∪ Ω

I

).

Next, using the nonlocal Green’s first identities [3], we state the following definition:

Definition 2.1. We say that u ∈ V

_c

(Ω ∪ Ω

_I

) is a weak solution of the nonlocal elliptic problem (1) if

Z

Ω∪ΩI

Z

Ω∪ΩI

D

^∗

(u).(ξ.D

^∗

(ϕ))dydx = Z

Ω

f ϕdx ∀ϕ ∈ V

c

(Ω ∪ Ω

I

). (7) Then, according to the definition of the nonlocal energy norm (6), we immediately announce the following theorem:

Theorem 2.2. There exist two constants M

₁

, M

₂

> 0 such that

| Z

Ω∪ΩI

Z

Ω∪ΩI

D

^∗

(u).(ξ.D

^∗

(ϕ))dydx |≤ M

₁

|||u||||||ϕ||| (8) and

M

2

|||u|||

²

≤ Z

Ω∪ΩI

Z

Ω∪ΩI

D

^∗

(u).(ξ.D

^∗

(u))dydx (9) for all u, ϕ ∈ V

c

(Ω ∪ Ω

I

)

Theorem 2.3. For each f ∈ L

²

(Ω), there exists a unique weak solution u ∈ V

_c

(Ω ∪ Ω

_I

) of the nonlocal elliptic problem (1).

Proof. Using the previous theorem (2.2), we obtain the result of existence and uniqueness via a direct application of the Lax-Milgram theorem.

2.1. The nonlocal Dirichlet eigenvalue problem. In this subsection, we focus our attention on seeking the set of numbers µ such that the following eigenvalues problem (10) corresponding to the Dirichlet nonlocal problem (1):

Z

Ω∪ΩI

Z

Ω∪ΩI

D

^∗

(u).(ξ.D

^∗

(ϕ))dydx = µ Z

Ω

uϕdx ∀ϕ ∈ V

c

(Ω ∪ Ω

I

), (10) has a solution u ∈ V

c

(Ω ∪ Ω

I

).

We state the following result:

Theorem 2.4. 1) Each eigenvalue of the problem (10) is real.

2) If we repeat each eigenvalue according to its multiplicity, we have that the set Σ of the eigenvalues of the operator D(ξ.D

^∗

(.)) is as follows:

Σ = (µ

_j

)

_j≥1

(11)

where 0 < µ

1

≤ µ

2

≤ ... ≤ µ

n

≤ ... and µ

j

→

j→∞

∞.

3) There exists an orthonormal basis (v

_j

)

_j≥1

of L

²

(Ω ∪ Ω

_I

), where v

_j

∈ V

_c

(Ω ∪ Ω

_I

)

is an eigenvector corresponding to µ

_j

for j ≥ 1.

Proof. Let K be the mapping

K : L

²

(Ω ∪ Ω

I

) → V

c

(Ω ∪ Ω

I

) f 7→ u

f

where u is the unique solution of (1) given by Theorem (2.3).

We claim that the operator K is bounded, indeed:

|||u|||

²

= Z

Ω∪Ω_I

Z

Ω∪Ω_I

D

^∗

(u).(ξ.D

^∗

(u))dydx

≤ k f k

L²(Ω)

k u k

L²(Ω)

according to the nonlocal Poincar´ e inequality [2], there exists a positive constant C such that:

|||Kf ||| ≤k f k

_L2(Ω∪ΩI)

since the embedding

I : V

_c

(Ω ∪ Ω

_I

) → L

²

(Ω ∪ Ω

_I

) u 7→ u

is compact, we directly deduce that the mapping

I ◦ K : L

²

(Ω ∪ Ω

_I

) → L

²

(Ω ∪ Ω

_I

) f 7→ u

f

is linear and compact.

On the other hand, if w is the unique solution of the problem:

D(ξ.D

^∗

(w)) = f on Ω

w = 0 on Ω

_I

and v if the solution of:

D(ξ.D

^∗

(v)) = g on Ω

v = 0 on Ω

_I

where f, g ∈ L

²

(Ω ∪ Ω

_I

). We have:

((I ◦ K)f, g)

L²(Ω∪ΩI)

= Z

Ω

wgdx = Z

Ω∪ΩI

Z

Ω∪ΩI

D

^∗

(v).(ξ.D

^∗

(w))dydx ((I ◦ K)g, f)

_L2(Ω∪ΩI)

=

Z

Ω

vf dx = Z

Ω∪ΩI

Z

Ω∪ΩI

D

^∗

(w).(ξ.D

^∗

(v))dydx which prove that the operator I ◦ K is symmetric. In addition:

((I ◦ K)f, f)

_L2(Ω∪ΩI)

= Z

Ω∪Ω_I

Z

Ω∪Ω_I

D

^∗

(w).(ξ.D

^∗

(w))dydx ≥ 0

We apply the theory of compact and symmetric operators from [13] to conclude the existence of real eigenvalues of I ◦ K, and that the corresponding eigenvectors (v

j

)

_j≥1

form a complete orthonormal system in L

²

(Ω ∪ Ω

I

).

To conclude the proof, notice that:

(I ◦ K)v = λv is equivalent to D(ξ.D

^∗

(v)) = 1

λ v = µv

Theorem 2.5. Let (v

j

)

_j≥1

be the eigenvectors corresponding to (µ

j

)

_j≥1

given by Theorem (2.4), then (v

j

)

j≥1

forms an orthogonal basis of V

c

(Ω ∪ Ω

I

).

Proof. The orthogonality of the eigenvectors follows from:

Z

Ω∪ΩI

Z

Ω∪ΩI

D

^∗

(v

_j

).(ξ.D

^∗

(v

_k

))dydx = µ

_j

(v

_j

, v

_k

)

_L2(Ω)

= µ

j

δ

i,j

On the other hand, for each u ∈ V

_c

(Ω ∪ Ω

_I

) we have:

u = X

j≥¹

(u, v

j

)

L²(Ω∪ΩI)

v

j

= X

j≥¹

R

Ω∪Ω_I

R

Ω∪Ω_I

D

^∗

(u).(ξ.D

^∗

(v

j

))dydx µ

j

v

_j

= X

j≥¹

R

Ω∪ΩI

R

Ω∪ΩI

D

^∗

(u).(ξ.D

^∗

(v

_j

))dydx

|||v

_j

|||

²

v

j

which concludes the proof.

Since (v

j

)

j≥1

is an orthogonal basis of V

c

(Ω ∪ Ω

I

) for any j ∈ N , we can define the orthogonal projection on the j-dimensional subspace of V

_c

(Ω ∪ Ω

_I

) spanned by v

₁

, v

₂

, ..., v

_j

.

Proposition 1. Let P

n

, Q

n

be the orthogonal projections defined, for all n ∈ N , by:

P

_n

(u) :=

n

X

j=1

(u, v

_j

)

_L2(Ω∪Ω_I)

v

_j

∀u ∈ L

²

(Ω ∪ Ω

_I

) (12)

Q

n

(u) :=

n

X

j=1

R

Ω∪ΩI

R

Ω∪ΩI

D

^∗

(u).(ξ.D

^∗

(v

_j

))dydx

|||v

j

|||

²

v

j

∀u ∈ V

c

(Ω ∪ Ω

I

) (13) Then

P

_n

u

^L

2(Ω∪ΩI)

−−−−−−→ u ∀u ∈ L

²

(Ω ∪ Ω

_I

) Then

Q

n

u −−−−−−→

^{Vc(Ω∪ΩI)}

u ∀u ∈ V

c

(Ω ∪ Ω

I

) These convergences come simply from the following result:

Proposition 2. Let P

_n

, Q

_n

be the orthogonal projections defined by definitions (12) and (13), then:

k P

n

k

_L(L2(Ω∪ΩI),L²(Ω∪ΩI))

=k Q

n

k

_{L(Vc(Ω∪ΩI}),V_c(Ω∪ΩI))

= 1 If we set

P

_n

(u) =

n

X

j=1

< u, v

_j

>

_V∗

c(Ω∪ΩI),V_c(Ω∪ΩI)

v

_j

∀u ∈ V

_c^∗

(Ω ∪ Ω

_I

) then

k P

n

k

_L(V^∗

c(Ω∪ΩI),V_c^∗(Ω∪ΩI))

= 1

Proof. Let u ∈ L

²

(Ω ∪ Ω

I

), then k u k

²_L2(Ω∪ΩI)

= lim

n→∞

n

X

j=1

(u, v

j

)

²_L2(Ω∪ΩI)

= lim

n→∞

k P

n

(u) k

²_L2(Ω∪ΩI)

subsequently

k P

n

k

_L(L²_(Ω∪ΩI),L²(Ω∪ΩI))

= sup

u∈L²(Ω∪ΩI)

u6=0

k P

_n

(u) k

L²(Ω∪ΩI)

k u k

_L2(Ω∪ΩI)

≤ 1

to conclude the proof, note that: P

n

(v

j

) = v

j

Secondly, we have, for u ∈ V

c

(Ω ∪ Ω

I

):

Q

n

(u) =

n

X

j=1

R

Ω∪ΩI

R

Ω∪ΩI

D

^∗

(u).(ξ.D

^∗

(v

_j

))dydx

|||v

_j

|||

²

v

j

=

n

X

j=1

(u, v

j

)

_L2(Ω∪ΩI)

v

j

therefore

|||Q

n

(u)||| =

n

X

j=1

(u, v

j

)

²_L2(Ω∪Ω_I)

R

Ω∪ΩI

R

Ω∪ΩI

D

^∗

(v

j

).(ξ.D

^∗

(v

j

))dydx

|||v

j

|||

²

v

j

≤

∞

X

j=1

(u, v

j

)

²_L2(Ω∪ΩI)

|||v

j

|||

²

≤ |||u|||

²

and as Q

n

(v

j

) = v

j

, we claim that k Q

n

k

_{L(Vc(Ω∪ΩI),Vc(Ω∪ΩI))}

= 1.

Furthermore, if we extend the projection P

n

to V

_c^∗

(Ω ∪ Ω

I

) we obtain:

| < P

_n

(u), ϕ >

_V∗

c(Ω∪Ω_I),V_c(Ω∪Ω_I)

| = |

n

X

j=1

< u, v

_j

>

_V∗

c(Ω∪Ω_I),V_c(Ω∪Ω_I)

(ϕ, v

_j

)

_L2(Ω∪Ω_I)

|

= | < u, P

n

(ϕ) >

V_c^∗(Ω∪ΩI),V_c(Ω∪ΩI)

|

≤ k u k

V_c^∗(Ω∪ΩI)

|||ϕ|||

hence

k P

n

(u) k

_V^∗

c(Ω∪ΩI)

≤k u k

_V^∗

c(Ω∪ΩI)

3. The nonlocal wave equation.

3.1. statement of the problem. We denote by Ω an open set of R

ⁿ

and Ω

_I

its corresponding interaction domain. We will always assume that Ω and Ω

_I

are bounded with piecewise smooth boundary and satisfy the interior cone condition.

The example of nonlocal hyperbolic equation that we consider is the following: we seek a real valued function u = u(x, t), x ∈ Ω, t ∈]0, T ], solution to



 



 



u

_tt

+ D(ξ.D

^∗

(u)) = f in Ω×]0, T ]

u = 0 on Ω

_I

×]0, T ]

u(x, 0) = g(x) on Ω

u

t

(x, 0) = h(x) on Ω

(14)

Where D, D

^∗

are, respectively, the nonlocal divergence (2) and the nonlocal gradi- ent (3), ξ(x, y) denotes a symmetric, positive definite second order tensor having el- ements that are symmetric functions of x and y and f : Ω×]0, T [→ R , g, h : Ω → R are given.

First, we specify in which sense we want to solve the problem. (14)

Definition 3.1. If f ∈ L

²

(0, T ; L

²

(Ω)), g ∈ V

_c

(Ω ∪ Ω

_I

) and h ∈ L

²

(Ω ∪ Ω

_I

) we say a function u ∈ L

²

(0, T ; V

c

(Ω ∪ Ω

I

)) with u

⁰

∈ L

²

(0, T ; L

²

(Ω ∪ Ω

I

)) and u

⁰⁰

∈ L

²

(0, T ; V

_c^∗

(Ω ∪ Ω

I

)) is a weak solution of the nonlocal constrained problem (14) if

hu

⁰⁰

, ϕi

_V^∗

c(Ω∪ΩI),Vc(Ω∪ΩI)

+ Z

Ω∪ΩI

Z

Ω∪ΩI

D

^∗

(u).(ξ.D

^∗

(ϕ))dydx = Z

Ω

f ϕdx

∀ϕ ∈ V

_c

(Ω ∪ Ω

_I

) and a.e 0 ≤ t ≤ T , with u(0) = g, u

⁰

(0) = h

3.2. Galerkin approximation. Let (v

_j

)

_j≥1

be the eigenvectors corresponding to the eigenvalues (λ

_j

)

_j≥1

of the problem (10), given by Theorem (2.4).

For a fixed n ≥ 1, we are looking for a function u

n

: [0, T ] → V

c

(Ω ∪ Ω

I

) of the form u

_n

(t) :=

n

X

j=1

p

_jn

(t)v

_j

(15)

such that



 

 

 

 

 



 

 

 

 

 



j = 1, ..., n

hu

⁰⁰_n

, v

j

i

V_c^∗(Ω∪ΩI),V_c(Ω∪ΩI)

+ Z

Ω∪ΩI

Z

Ω∪ΩI

D

^∗

(u

n

).(ξ.D

^∗

(v

j

))dydx=

Z

Ω

f

n

v

j

dx u

_n

(0) =

n

X

j=1

(g, v

_j

)

_L2(Ω)

v

_j

u

⁰_n

(0) =

n

X

j=1

(h, v

_j

)

_L2(Ω)

v

_j

(16) where (f

n

)

n

∈ D(Ω × (0, T )) such that f

n

L²(Ω×(0,T))

−−−−−−−−→ f with k f

n

k

_L2(Ω×(0,T))

≤k f k

_L2(Ω×(0,T))

Theorem 3.2. For each integer n ≥ 1 there exists a unique function u

n

of the form (15) satisfying (16).

Proof. To solve the problem (16), we shall find

p

n

(t) = (p

1n

(t), p

2n

(t), ..., p

nn

(t)) ∈ R

ⁿ

solution to



 

 

 

 

 

 

j = 1, ..., n p

⁰⁰_jn

+

n

X

k=1

Z

Ω∪ΩI

Z

Ω∪ΩI

D

^∗

(v

_j

).(ξ.D

^∗

(v

_k

))p

_kn

(t)dydx=

Z

Ω

f

_n

v

_j

dx p

_jn

(0) = (g, v

_j

)

_L2(Ω)

p

⁰_jn

(0) = (h, v

_j

)

_L2(Ω)

(17)

According to standard existence theory for ODE, there exists a unique function

p

n

(t) = (p

1,n

(t), p

2,n

(t), ..., p

n,n

(t)) (18)

satisfying (17) for a.e 0 ≤ t ≤ T .

3.3. Energy estimates. In order to show that (u

n

)

_n≥1

converges to a weak solu- tion of (14) we will need some uniform estimates.

Theorem 3.3. There exists a positive constant M such that

0≤t≤T

max

|||u

n

(t)|||+ k u

⁰_n

(t) k

L²(Ω∪ΩI)

+ k u

⁰⁰_n

k

L²(0,T;V_c^∗(Ω∪ΩI))

≤ M k f k

_L2(0,T;L²(Ω))

+|||g|||+ k h k

_L2(Ω)

f or n ≥ 1 Proof. We multiply equation (16) by p

⁰_jn

(t), sum for j = 1, ..., n, we find

hu

⁰⁰_n

, u

⁰_n

i

_V^∗

c(Ω∪ΩI),Vc(Ω∪ΩI)

+ Z

Ω∪ΩI

Z

Ω∪ΩI

D

^∗

(u

n

).(ξ.D

^∗

(u

⁰_n

))dydx = Z

Ω

f

n

u

⁰_n

dx for a.e 0 ≤ t ≤ T .

which give d

dt

k u

⁰_n

k

²_L2(Ω∪Ω_I)

+ Z

Ω∪ΩI

Z

Ω∪ΩI

D

^∗

(u

n

).(ξ.D

^∗

(u

n

))dydx

≤ 2 k f

n

k

_L2(Ω)

k u

⁰_n

k

_L2(Ω)

≤ k f

n

k

²_L2(Ω)

+ k u

⁰_n

k

²_L2(Ω∪Ω_I)

+|||u

n

|||

²

Gronwall’s inequality and the proposition (2) imply

k u

⁰_n

k

²_L2(Ω∪ΩI)

+|||u

n

|||

²

≤ M

|||P

n

(g)|||

²

+ k P

n

(h) k

²_L2(Ω∪ΩI)

+ k f

n

k

_L2(0,T;L²(Ω))

≤ M

|||g|||

²

+ k h k

²_L2(Ω∪ΩI)

+ k f

n

k

²_L2(0,T;L²(Ω))

as 0 ≤ t ≤ T was chosen arbitrarily, we obtain:

0≤t≤T

max

k u

⁰_n

(t) k

²_L2(Ω∪Ω_I)

+|||u

n

|||

²

≤ M

|||g|||

²

+ k h k

²_L2(Ω∪Ω_I)

+ k f k

²_L2(0,T;L²(Ω))

(19) To conclude, we fix any ϕ ∈ V

c

(Ω ∪ Ω

I

) with

|||ϕ||| ≤ 1 and ϕ = P

n

(ϕ) + ψ

where (ψ, v

_j

)

_L2(Ω∪ΩI)

= 0 f or 1 ≤ j ≤ n, we get:

hu

⁰⁰_n

, ϕi

V_c^∗(Ω∪ΩI),V_c(Ω∪ΩI)

= Z

Ω

f

n

P

n

(ϕ)dx−

Z

Ω∪ΩI

Z

Ω∪ΩI

D

^∗

(u

n

).(ξ.D

^∗

(P

n

(ϕ)))dydx Consequently, since |||P

n

(ϕ)||| ≤ 1

|hu

⁰⁰_n

, ϕi|

V_c^∗(Ω∪ΩI),V_c(Ω∪ΩI)

≤ M (k f

n

k

²_L2(Ω)

+|||u

n

|||) finally, using (19) we get

u

⁰⁰_n

_L2(0,T;V∗

c(Ω∪Ω_I))

≤ M (k f k

²_L2(0,T;L²(Ω))

+|||g|||

²

+ ||h||

²_L2(Ω)

)

3.4. Existence and uniqueness result.

Theorem 3.4. The nonlocal parabolic problem (14) admits a unique weak solution.

Proof. Using the previous Theorem (3.3), we conclude that (u

_n

)

_n≥1

is bounded in L

²

(0, T ; V

_c

(Ω ∪ Ω

_I

)), with (u

⁰_n

)

_n≥1

is bounded in L

²

(0, T ; L

²

(Ω ∪ Ω

_I

)), and (u

⁰⁰_n

)

_n≥1

is bounded in L

²

(0, T ; V

_c^∗

(Ω ∪ Ω

I

)).

Consequently, there exists a subsequence still denoted (u

n

)

_n≥1

and a function u ∈ L

²

(0, T ; V

c

(Ω ∪ Ω

I

)) with u

⁰

∈ L

²

(0, T ; L

²

(Ω ∪ Ω

I

)) and

u

⁰⁰

∈ L

²

(0, T ; V

_c^∗

(Ω ∪ Ω

I

)), such that:







u

_n

* u in L

²

(0, T ; V

_c

(Ω ∪ Ω

_I

) u

⁰_n

* u

⁰

in L

²

(0, T ; L

²

(Ω ∪ Ω

_I

) u

⁰⁰_n

* u

⁰⁰

in L

²

(0, T ; V

_c^∗

(Ω ∪ Ω

I

)

(20) next, fix an integer N and select n ≥ N , choose a function ψ ∈ L

²

(0, T ) and ϕ ∈ V

c

(Ω ∪ Ω

I

). We multiply (16) by P

n

(ϕ)ψ, sum j = 1, ..., N and integrate with respect to t to discover:

Z

T 0

hu

⁰⁰_n

, P

n

(ϕ)ψ(t)i

_V^∗

c(Ω∪ΩI),Vc(Ω∪ΩI)

+ Z

Ω∪ΩI

Z

Ω∪ΩI

D

^∗

(u

n

).(ξ.D

^∗

(P

n

(ϕ)ψ))dydx

dt (21)

= Z

T

0

Z

Ω

f

_n

P

_n

(ϕ)ψdxdt

by passing to weak limits, together with the fact that P

_n

(ϕ) −−−−−−→

^{Vc(Ω∪ΩI)}

ϕ we obtain:

Z

T 0

hu

⁰⁰

, ϕi

V_c^∗(Ω∪ΩI),V_c(Ω∪ΩI)

ψ(t) + Z

Ω∪ΩI

Z

Ω∪ΩI

D

^∗

(u).(ξ.D

^∗

(ϕψ(t)))dydx

dt (22)

= Z

T

0

Z

Ω

f ϕψ(t)dxdt

for all ψ ∈ L

²

(0, T ) and ϕ ∈ V

_c

(Ω ∪ Ω

_I

). This terminates the proof.

It remains to prove that u(0) = g and u

⁰

(0) = h, For this purpose, we choose any function ϕ ∈ V

c

(Ω∪Ω

I

) and ψ ∈ C

¹

([0, T ]) such that ψ(T ) = ψ

⁰

(T ) = 0. Integrating by parts twice with respect to t in (22) yields:

Z

T 0

Z

Ω∪ΩI

uϕψ

⁰⁰

(t)dx + Z

Ω∪ΩI

Z

Ω∪ΩI

D

^∗

(u).(ξ.D

^∗

(ϕψ(t)))dydx

dt (23)

= Z

T

0

Z

Ω

f ϕψ(t)dxdt − Z

Ω

u(0)ϕψ

⁰

(0)dx + Z

Ω

u

⁰

(0)ϕψ(0)dx Similarly from (21) we get:

Z

T 0

Z

Ω∪ΩI

u

_n

P

_n

(ϕ)ψ

⁰⁰

(t)dx + Z

Ω∪ΩI

Z

Ω∪ΩI

D

^∗

(u

_n

).(ξ.D

^∗

(P

_n

(ϕ)ψ(t)))dydx

dt (24)

= Z

T

0

Z

Ω

f

n

P

n

(ϕ)ψ(t)dxdt − Z

Ω

u

n

(0)P

n

(ϕ)ψ

⁰

(0)dx + Z

Ω

u

⁰_n

(0)P

n

(ϕ)ψ(0)dx by passing to the limit, we obtain:

Z

T 0

Z

Ω∪Ω_I

uϕψ

⁰⁰

(t)dx + Z

Ω∪Ω_I

Z

Ω∪Ω_I

D

^∗

(u).(ξ.D

^∗

(ϕψ(t)))dydx

dt (25)

= Z

T

0

Z

Ω

f ϕψ(t)dxdt − Z

Ω

gϕψ

⁰

(0)dx + Z

Ω

hϕψ(0)dx comparing those results, we conclude that u(0) = g and u

⁰

= h.

Finally, we announce the uniqueness of the weak solution to (14).

Theorem 3.5. A weak solution of (14) is unique.

Proof. Since the equation is linear, to show uniqueness it is sufficient to show that the only solution u of (14) with zero data f = g = h = 0 is u = 0.

To verify this, fix 0 ≤ s ≤ T and set ϕ(t) =

R

s

t

u(τ)dτ if 0 ≤ t < s

0 if s ≤ t ≤ T. (26)

Then ϕ(t) ∈ V

_c

(Ω ∪ Ω

_I

) for each 0 ≤ t ≤ T , which allows to write Z

s

0

hu

⁰⁰

, ϕi

V_c^∗(Ω∪ΩI),V_c(Ω∪ΩI)

+ Z

Ω∪ΩI

Z

Ω∪ΩI

D

^∗

(u).(ξ.D

^∗

(ϕ))dydx

dt = 0 since u

⁰

= 0 and ϕ(s) = 0 by integrating by parts, we obtain:

Z

s 0

− Z

Ω∪ΩI

u

⁰

ϕ

⁰

dx + Z

Ω∪ΩI

Z

Ω∪ΩI

D

^∗

(u).(ξ.D

^∗

(ϕ))dydx

dt = 0 now as ϕ

⁰

= −u(0 ≤ t ≤ s), we acquire:

Z

s 0

Z

Ω∪ΩI

u

⁰

udx − Z

Ω∪ΩI

Z

Ω∪ΩI

D

^∗

(ϕ

⁰

).(ξ.D

^∗

(ϕ))dydx

dt = 0 then

Z

s 0

d dt

k u k

²_L2(Ω∪Ω_I)

− Z

Ω∪ΩI

Z

Ω∪ΩI

D

^∗

(ϕ).(ξ.D

^∗

(ϕ))dydx

dt = 0 here

k u(s) k

²_L2(Ω∪ΩI)

+|||ϕ(0)||| = 0 Consequently u = 0 on [0, T ].

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Received xxxx 20xx; revised xxxx 20xx.

E-mail address:[email protected]

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