Existence and uniqueness of weak solutions for nonlocal parabolic problems via the Galerkin method

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J. Math. Anal. Appl. ••• (••••) •••–•••

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Existence and uniqueness of weak solutions for nonlocal parabolic problems via the Galerkin method Miguel Yangari 1 Research Center on Mathematical Modelling (MODEMAT) & Departamento de Matemática, Escuela Politécnica Nacional, Ladrón de Guevara E11-253, P.O. Box 17-01-2759, Quito, Ecuador

a r t i c l e

i n f o

Article history: Received 26 January 2018 Available online xxxx Submitted by M. Musso Keywords: Nonlocal operator Nonlocal vector calculus Weak solution Galerkin method

a b s t r a c t Fractional differential equations are becoming increasingly popular as a modeling tool to describe a wide range of non-classical phenomena with spatial heterogeneities throughout the applied science and engineering. A recently developed nonlocal vector calculus is exploited to provide a variational analysis for a general class of nonlocal operators which include fractional Laplacians on bounded domains in Rn . We develop the Galerkin method to prove existence and uniqueness of weak solutions to nonlocal parabolic problems. Moreover, we study the existence of orthonormal basis of eigenvectors associated to these nonlocal operators. © 2018 Elsevier Inc. All rights reserved.

1. Introduction In this paper, we consider the nonlocal operator L defined by  Lu(x) := 2 (u(y) − u(x))γ(x, y)dy for all x ∈ Rn

(1.1)

Rn

with γ(x, y) := α(x, y)·α(x, y), the Euclidean inner product of an antisymmetric function α : Rn ×Rn −→ Rk . Thus, if T > 0, f, g are given functions and ∂t represents the partial derivative respect to the time variable, we are interested in the well-posedness of the nonlocal parabolic problem ⎧ ⎪ in ΩT ⎨ ∂t u − Lu = f, (1.2) u = 0, on ΩI × [0, T ] ⎪ ⎩ u = g, on Ω × {t = 0} where Ω is open and bounded subset of Rn with piecewise smooth boundary and satisfies the interior cone condition, ΩT := Ω × (0, T ] and the interaction set ΩI := {y ∈ Rn \Ω : α(x, y) = 0 for every x ∈ Ω}.

1

E-mail address: [email protected]. The author was supported by Escuela Politécnica Nacional, Proyecto PIJ 15-22.

https://doi.org/10.1016/j.jmaa.2018.03.058 0022-247X/© 2018 Elsevier Inc. All rights reserved.

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Nonlocal problems have attracted the attention of the scientific community in recent years, its application to models in nonlocal Dirichlet forms are studied in [3], kinetic equations [4], [18], nonlocal heat conduction [6], phase transitions [5], [13], image processing [7], [14], [15], [19] and machine learning [22]. Also, nonlocal operators appear in physical models when the diffusive phenomena are better described by Lévy processes allowing long jumps than by Brownian processes, see, for example, [20] for a description of some of these models. The Lévy processes occur widely in physics, chemistry and biology. The aim of this paper is to state a variational formulation to the nonlocal parabolic problem (1.2), moreover, we prove the existence and uniqueness of weak solutions developing a nonlocal version of the Galerkin method in function spaces close related to the nonlocal operator L. Following the standard procedure described in section 7.1.2 in [12], we approximate the solution of (1.2) by the projection of the solution and the equation into finite dimensional subspaces formed through an orthonormal basis of eigenvectors associated to the nonlocal operator L. In problems with classical diffusion, elements of the vector calculus such as the divergence theorem and the Green’s identities play a crucial role in several aspects such as defining weak formulations of boundary-value problems. In the nonlocal case, a nonlocal calculus developed in [17], [9] equally serves to state variational formulations to the nonlocal equations, introducing nonlocal divergence and gradient operators that mimic their local counterparts. We can also define function spaces which are equivalents to fractional Sobolev spaces. The central difference between the local and nonlocal models is that, in the former case, interactions between two domains occur due to contact, whereas in the latter case, interactions can occur at a distance, i.e., we have nonlocal interactions, this is the reason to define the interaction set ΩI which is associated to the operator L. Finally, we notice that the Galerkin method has proved to be a powerful method to find solutions to problems involving different type of nonlocal operators in the time and space variable, see for example [1], [2], [11], [16], [21] and references therein. Some of these works also include numerical simulations to either corroborate new findings or to compare convergence of different methods. 2. Weak formulation and main result In order to state a variational formulation to (1.2), following the notation given in [9], the action of the nonlocal divergence operator D : Rn −→ R is defined as  D(ν)(x) =

(ν(x, y) + ν(y, x)) · α(x, y)dy

(2.1)

Rn

with ν : Rn × Rn −→ Rk . Given the mapping u : Rn −→ R, the action of the nonlocal gradient operator D∗ : Rn × Rn −→ Rn on u is defined as D∗ (u)(x, y) = −(u(y) − u(x)) · α(x, y).

(2.2)

It is shown in [9] that D∗ and D are adjoint operators, also, Lu(x) := −D(D∗ u)(x).

(2.3)

Moreover, given positive constants γ0 and ε, we first assume that the symmetric kernel γ that appears in the definition of L in (1.1) satisfies: for all x ∈ Ω ∪ ΩI a) γ(x, y) ≥ 0 ∀ y ∈ Bε (x) with γ(x, y) ≥ γ0 > 0 ∀ y ∈ Bε/2 (x). b) γ(x, y) = 0 ∀ y ∈ (Ω ∪ ΩI ) \ Bε (x),

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where Bε (x) := {y ∈ Ω ∪ ΩI : |y − x| ≤ ε}. Obviously, the previous hypothesis implies that although interactions are nonlocal, they are limited to a ball of radius ε. For what follows, we assume that there exist s ∈ (0, 1) and positive constants γ∗ and γ ∗ such that, for all γ∗ γ∗ ≤ γ(x, y) ≤ , |y − x|n+2s |y − x|n+2s

∀ y ∈ Bε (x).

(2.4)

Now, we respectively define the nonlocal energy seminorm, nonlocal energy space, and nonlocal volumeconstrained energy space by ⎛ 1 |||u||| := ⎝ 2



⎞ 12



D∗ (u)(x, y) · D∗ (u)(x, y)⎠

Ω∪ΩI Ω∪ΩI

V (Ω ∪ ΩI ) := {u ∈ L2 (Ω ∪ ΩI ) : |||u||| < +∞} Vc (Ω ∪ ΩI ) := {u ∈ V (Ω ∪ ΩI ) : u = 0 on ΩI }. In [10], it is shown that, the nonlocal energy space V (Ω ∪ ΩI ) is equivalent to the fractional-order Sobolev space H s (Ω ∪ ΩI ) for s ∈ (0, 1) defined in (2.4). These equivalences imply that the volume constrained space Vc (Ω ∪ ΩI ) is a Hilbert space equipped with the norm |||u|||. Moreover, we denote by Vc (Ω) the dual space of Vc (Ω ∪ ΩI ) with respect to the standard L2 (Ω) duality pairing. A norm on Vc (Ω) can be defined by

h Vc (Ω) =

sup v∈Vc (Ω∪ΩI ),v=0

|



hvdx| . |||v|||

Ω

In what follows, for simplicity of notation we use u to represent the partial derivative in time ∂t u. Hence, the Green’s identities of the nonlocal vector calculus [9] enables to define the weak formulation of the value problem (1.2) as follows: Definition 2.1. If f ∈ L2 ((0, T ; L2 (Ω)) and g ∈ L2 (Ω), a weak solution of (1.2) is a function u ∈ L2 ((0, T ; Vc (Ω ∪ ΩI ))

with u ∈ L2 ((0, T ; Vc (Ω ∪ ΩI ))

such that 1. (u , v)L2 (Ω) + B[u, v] = (f, v)L2 (Ω) , for all v ∈ Vc (Ω ∪ ΩI ), a.e. 0 ≤ t ≤ T , 2. u(0) = g on Ω where





B[u, v] :=

D∗ (u) · D∗ (v)dydx.

Ω∪ΩI Ω∪ΩI

Defining L2c (Ω ∪ ΩI ) := {u ∈ L2 (Ω ∪ ΩI ) : u = 0 on ΩI }, we are in a position to state our main theorem. Theorem 2.2. There exists a unique weak solution of (1.2) in the sense of the Definition 2.1. Also, the solution belongs to C([0, T ]; L2c (Ω ∪ ΩI )). The plan to prove Theorem 2.2 is organized as follows. In Section 3, we state crucial results concerning the existence of an orthonormal basis of eigenvectors associated to the nonlocal operator L. Section 4 is devoted to the proof of Theorem 2.2, in which we develop the Galerkin method for nonlocal parabolic problems.

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3. Existence of Hilbert basis Firstly, we consider the nonlocal elliptic problem −Lu = h, u = 0,

in Ω on ΩI

(3.1)

The Green’s identities of the nonlocal vector calculus [9] enables to define the weak formulation of the value problem (3.1) as follows: Definition 3.1. If h ∈ L2 (Ω), a weak solution of (3.1) is a function u ∈ Vc (Ω ∪ ΩI ) such that  B[u, v] = hvdx, ∀ v ∈ Vc (Ω ∪ ΩI ). Ω

Theorem 3.2. The problem (3.1) has a unique weak solution in the above sense. Moreover, the solution satisfies |||u||| ≤ c h L2 (Ω) .

(3.2)

for some constant c > 0. Proof. We recall that Vc (Ω ∪ ΩI ) is Hilbert space with norm ||| · |||. Moreover, by definition of ||| · ||| we easily get the coercivity of B[·, ·]. Now, by the Cauchy–Schwarz inequality applied to the inner product associated to ||| · |||, we directly have





∗ ∗

D v · D u dydx

≤ |||u||||||v|||

Ω∪ΩI Ω∪ΩI

for all u, v ∈ Vc (Ω ∪ ΩI ). Moreover,





hv dx ≤ h L2 (Ω) v L2 (Ω∪Ω ) I



Ω

≤ c h L2 (Ω) |||v||| the last two inequalities follows since v ∈ Vc (Ω ∪ ΩI ), thus v = 0 on ΩI and v L2 (Ω) = v L2 (Ω∪ΩI ) , also by Lemma 4.3 in [10], there exists a constant c > 0 such that

v L2 (Ω∪ΩI ) ≤ c|||v|||. Hence, the problem (3.1) satisfies all the hypotheses of the Lax–Milgram theorem, so that the problem has a unique solution u ∈ Vc (Ω ∪ ΩI ) and this solution satisfies (3.2), indeed,    2 ∗ ∗ |||u||| = D u · D u dydx = hu dx ≤ c h L2 (Ω) |||u||| Ω∪ΩI Ω∪ΩI

Ω

then, |||u||| ≤ c h L2 (Ω) .

2

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2 Theorem 3.3. There exists an orthonormal basis (wk )∞ k=1 of Lc (Ω ∪ ΩI ) and a sequence of positive numbers ∞ (λk )k=1 such that λk → +∞ and



wk ∈ Vc (Ω ∪ ΩI ) in Ω −Lwk = λk wk ,

in the weak sense, for all k ∈ N. Proof. By Theorem 3.2, for each h ∈ L2 (Ω), there exists a unique weak solution u ∈ Vc (Ω ∪ ΩI ) of (3.1). Since L2c (Ω ∪ ΩI ) ⊂ L2 (Ω) and Vc (Ω ∪ ΩI ) ⊂ L2c (Ω ∪ ΩI ), we can define the operator S := (−L)−1 : L2c (Ω ∪ ΩI ) −→ L2c (Ω ∪ ΩI ) h −→ Sh = u with u the unique weak solution of (3.1) associated to h. We have that S is a linear bounded operator, indeed, by Lemma 4.3 in [10] and (3.2) c−1 u L2 (Ω∪ΩI ) ≤ |||u||| ≤ h L2 (Ω) ≤ h L2 (Ω∪ΩI ) since Sh = u, then

Sh L2 (Ω∪ΩI ) ≤ c h L2 (Ω∪ΩI ) . Moreover, S is a compact operator, since, given h ∈ L2c (Ω ∪ ΩI ), u = Sh verifies |||u|||2 = B[u, u] = (h, u)L2c (Ω) ≤ c h L2 (Ω∪ΩI ) |||u||| we have |||Sh||| ≤ c h L2 (Ω∪ΩI ) .

(3.3)

Now, taking (hk ) a bounded sequence in L2c (Ω ∪ ΩI ), by (3.3), (Shk ) is bounded in Vc (Ω ∪ ΩI ). Proceeding exactly as in the proof of Lemma 4.3 in [10], we have that Vc(Ω ∪ ΩI ) is compactly embedded in L2c (Ω ∪ ΩI ), thus, there exists a convergent subsequence (Shkn ) in L2c (Ω ∪ΩI ), therefore, we conclude that S is a compact operator. Also, let us note that S is a self-adjoint operator since B[u, v] = B[v, u]. Finally, (Sh, h)L2c (Ω∪ΩI ) = (u, h)L2c (Ω∪ΩI ) = (u, h)L2c (Ω) = B[u, u] = |||u|||2 ≥ 0 i.e., S is a positive operator. Now, since L2c (Ω ∪ ΩI ) is a separable Hilbert space, Theorem 9.4 in [8] implies the existence of a sequence of real positive eigenvalues (μk )∞ k=1 of S such that μk → +∞ and the eigenvectors 2 (wk )∞ make up an orthonormal basis of L (Ω ∪Ω ). Moreover, by definition of S and since wk ∈ L2c (Ω ∪ΩI ), I c k=1 we have Swk = μk wu ∈ vc (Ω ∪ ΩI ).

(3.4)

Since μk = 0, we have wu ∈ vc (Ω ∪ ΩI ) for all k ∈ N. Also, taking λk = 1/μk , we conclude that wk is an eigenvector associated to λk of −L in the weak sense. 2 Lemma 3.4. There exists a sequence (wk )∞ k=1 in Vc (Ω ∪ ΩI ) that verifies

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2 a) (wk )∞ k=1 is an orthonormal basis of Lc (Ω ∪ ΩI ), ∞ b) (wk )k=1 is an orthogonal basis of Vc (Ω ∪ ΩI ). 2 Proof. By Theorem 3.3, there exists an orthonormal basis (wk )∞ k=1 of Lc (Ω ∪ ΩI ) with wk ∈ Vc (Ω ∪ ΩI ) an ∞ eigenvector of −L for all k ∈ N. We claim that (wk )k=1 is an orthogonal basis of Vc (Ω ∪ ΩI ). Indeed, since

−Lwk = λk wk ,

∀k∈N

in the weak sense, we get B[wk , wj ] = (λk wk , wj )L2 (Ω) = λk (wk , wj )L2 (Ω∪ΩI ) for all j, k ∈ N. Since B[·, ·] is by definition the inner product (·, ·)Vc (Ω∪ΩI ) and (wk )∞ k=1 is an orthonormal in L2c (Ω ∪ ΩI ), we conclude that (wk )∞ is orthogonal in V (Ω ∪ Ω ). Finally, let us prove that Vc (Ω ∪ ΩI ) = c I k=1 ∞

(wk )k=1 . To see this, since Vc (Ω ∪ ΩI ) is a Hilbert space, we have that ⊥

∞ Vc (Ω ∪ ΩI ) = (wk )∞ k=1  ⊕ (wk )k=1  . ⊥

⊥

∞ Hence, we need to prove that (wk )∞ k=1  = {0}. Indeed, if h ∈ (wk )k=1  , then (h, wk )Vc (Ω∪ΩI ) = 0 for all k ∈ N, but

0 = (h, wk )Vc (Ω∪ΩI ) = B[h, wk ] = λk (h, wk )L2 (Ω) = λk (h, wk )L2 (Ω∪ΩI ) . Since λk = 0, we get (h, wk )L2 (Ω∪ΩI ) = 0 for all k ∈ N, then, using the fact that (wk )∞ k=1 is an orthonormal basis of L2c (Ω ∪ ΩI ), we have h=

∞ 

(h, wk )L2 (Ω∪ΩI ) wk = 0,

on Ω ∪ ΩI .

2

k=1

Lemma 3.5. With continuous and dense embedding, we have Vc (Ω ∪ ΩI ) → L2c (Ω ∪ ΩI ) → Vc (Ω ∪ ΩI ). Proof. By definition of Hilbert triple (for instance, we refer to chapter 23.4 of [23]), we need to show that Vc (Ω ∪ ΩI ) and L2c (Ω ∪ ΩI ) are real separable Hilbert spaces and the embedding Vc (Ω ∪ ΩI ) → L2c (Ω ∪ ΩI ) is dense and continuous. Indeed, L2c (Ω ∪ ΩI ) is a separable Hilbert space since it is a closed subspace of L2 (Ω ∪ ΩI ). As described in section 4.3 in [10], Vc (Ω ∪ ΩI ) is a Hilbert space which is a subset of L2c (Ω ∪ ΩI ) with continuous embedding, hence Vc (Ω ∪ ΩI ) is separable. Finally, to prove density, we need to state that Vc (Ω ∪ ΩI ) = L2c (Ω ∪ ΩI ). The first inclusion is direct, hence, we focus in L2c (Ω ∪ ΩI ) ⊂ Vc (Ω ∪ ΩI ). Indeed, 2 let u ∈ L2c (Ω ∪ ΩI ), since (wk )∞ k=1 is an orthonormal basis of Lc (Ω ∪ ΩI ), we have that u=

∞  k=1

(u, wk )L2 (Ω∪ΩI ) wk = lim Un n→∞

n with Un := k=1 (u, wk )L2 (Ω∪ΩI ) wk ∈ Vc (Ω ∪ ΩI ) and the convergence in the norm of L2c (Ω ∪ ΩI ), thus, we conclude the proof. 2

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4. Galerkin approximation In order to develop the Galerkin method, by Lemma 3.3, we consider sequence (wk )∞ k=1 in Vc (Ω ∪ ΩI ) 2 which is an orthonormal basis of Lc (Ω ∪ ΩI ) and orthogonal basis in Vc (Ω ∪ ΩI ). Hence, for a fixed m ∈ N, since B[·, ·] is bilinear and according to standard existence theory for ODE, there exist functions um : [0, T ] −→ Vc (Ω ∪ ΩI ) of the form um (t) =

m 

dkm (t)wk

(4.1)

k=1

where the coefficients dkm (t) with 0 ≤ t ≤ T verify ∀ k = 1, ..., m

dkm (0) = (g, wk )L2 (Ω) ,

(4.2)

and (um , wk )L2 (Ω) + B[um , wk ] = (f, wk )L2 (Ω) .

(4.3)

In order to prove the sequence (um )∞ m=1 converge to the weak solution of (1.2), we first state some uniform bounds for the approximate solutions (um )∞ m=1 . Theorem 4.1. There exists a positive constant C depending on Ω and T such that max um (t) L2 (Ω∪ΩI ) + um L2 (0,T ;Vc (Ω∪ΩI )) + um L2 (0,T ;Vc (Ω∪ΩI ))

0≤t≤T

≤ C( f L2 (0,T ;L2 (Ω)) + g L2 (Ω) ) for all m ∈ N. Proof. Multiplying (4.3) by dkm and adding in k = 1, ..., m, by the definition of (4.1), we have (um , um )L2 (Ω) + B[um , um ] = (f, um )L2 (Ω)

(4.4)

for 0 ≤ t ≤ T . Noting that B[um , um ] = |um | 2 , we have  1  ∂t um 2L2 (Ω∪ΩI ) + |um | 2 2 1 1 ≤ f 2L2 (Ω) + um 2L2 (Ω∪ΩI ) . 2 2

(4.5)

Furthermore,   ∂t um 2L2 (Ω∪ΩI ) ≤ f 2L2 (Ω) + um 2L2 (Ω∪ΩI )

(4.6)

2 by the Gronwall inequality and since (wk )∞ k=1 is orthonormal in L (Ω ∪ ΩI ), we have

  max um (t) 2L2 (Ω∪ΩI ) ≤ c g 2L2 (Ω) + f 2L2 (0,T ;L2 (Ω)) .

0≤t≤T

(4.7)

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Now, by (4.5) and (4.7), we get  

um 2L2 (0,T ;Vc (Ω∪ΩI )) ≤ c g 2L2 (Ω) + f 2L2 (0,T ;L2 (Ω)) .

(4.8)

2 To conclude, fix any v ∈ Vc (Ω ∪ ΩI ) with v = 0. Since (wk )∞ k=1 is an orthonormal basis of Lc (Ω ∪ ΩI ), we have

v=

m 

(v, wk )L2c (Ω∪ΩI ) wk +

k=1

∞ 

(v, wk )L2c (Ω∪ΩI ) wk := v 1 + v 2

k=m+1

with the convergences in the norm of L2 (Ω ∪ ΩI ). Also, since wk ∈ Vc (Ω ∪ ΩI ) for all k ∈ N, we have v 1 ∈ Vc (Ω ∪ ΩI ). To see that v 2 ∈ Vc (Ω ∪ ΩI ), notice that n 

(v, wk )L2c (Ω∪ΩI ) wk ∈ (wk )∞ k=1 ,

∀ n≥m+1

k=m+1

hence, w :=

∞ 

(v, wk )L2c (Ω∪ΩI ) wk ∈ (wk )∞ k=1 ,

∀ n≥m+1

k=m+1

with the convergence in the norm of Vc (Ω ∪ ΩI ), but, since (wk )∞ k=1 is a orthogonal basis in Vc (Ω ∪ ΩI ), we have Vc (Ω ∪ ΩI ) = (wk )∞ , thus w ∈ V (Ω ∪ Ω ). Now, by Lemma 4.3 in [10] c I k=1 c−1 w −

∞ 

(v, wk )L2c (Ω∪ΩI ) wk L2 (Ω∪ΩI ) ≤ |||w −

k=m+1

∞ 

(v, wk )L2c (Ω∪ΩI ) wk ||| −→ 0.

k=m+1

Then, the convergence of w is also in the norm of L2 (Ω ∪ ΩI ), thus, by uniqueness of the limit we conclude that w = v 2 , therefore, v 2 ∈ Vc (Ω ∪ ΩI ). 1 1 Also, since (wk )∞ k=1 is orthogonal in Vc (Ω ∪ ΩI ), we have that |||v ||| ≤ |||v|||. Moreover, since v is a linear combination of (wk )m k=1 , by (4.3) (um , v 1 )L2 (Ω) + B[um , v 1 ] = (f, v 1 )L2 (Ω) . Hence, since (wk , v 2 )L2 (Ω) = 0 for k = 1, ..., m, we have (um , v)L2 (Ω∪ΩI ) = (um , v)L2 (Ω) = (um , v 1 )L2 (Ω) = (f, v 1 )L2 (Ω) − B[um , v 1 ]. Therefore, |(um , v)L2 (Ω∪ΩI ) | ≤ f L2 (Ω) v 1 L2 (Ω) + |||um |||.|||v 1 ||| ≤ c( f L2 (Ω) + |||um |||)|||v|||. By definition of the norm in Vc (Ω ∪ ΩI ), we get

um Vc (Ω∪ΩI ) ≤ c( f L2 (Ω) + |||um |||). Finally, from (4.8), we conclude

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um 2L2 (0,T ;Vc (Ω∪ΩI ))

9

T ≤c

f (s) 2L2 (Ω) + |||um (s)|||2 ds 0

≤ c( f 2L2 (0,T ;L2 (Ω)) + g 2L2 (Ω) ).

2

Proof of Theorem 2.2. Let us note that L2 (0, T ; Vc (Ω ∪ ΩI )) and L2 (0, T ; Vc (Ω ∪ ΩI )) are reflexive Banach spaces, since Vc (Ω ∪ ΩI ) and Vc (Ω ∪ ΩI ) are also reflexive Banach spaces. By Theorem 4.1, the sequences 2  ∞ 2  (um )∞ k=1 is bounded in L (0, T ; Vc (Ω ∪ ΩI )) and (um )k=1 is bounded in L (0, T ; Vc (Ω ∪ ΩI )), hence, by ∞ 2 Proposition 23.19 in [23], there exists a subsequence (uml )l=1 and u ∈ L (0, T ; Vc (Ω ∪ ΩI )) with u ∈ L2 (0, T ; Vc (Ω ∪ ΩI )) such that uml u in L2 (0, T ; Vc (Ω ∪ ΩI )), uml



2

u in L

(0, T ; Vc (Ω

∪ ΩI )).

(4.9) (4.10)

Next fix an integer N > 0 and consider the polynomial function

v(t) =

N 

t k wk .

(4.11)

k=1

We choose m ≥ N , multiply (4.3) by tk , sum k = 1, ..., N and integrate with respect to t, since (um , v)L2 (Ω) = (um , v)L2 (Ω∪ΩI ) , we get T

(um , v)L2 (Ω∪ΩI )

T + B[um , v]dt =

0

(f, v)L2 (Ω) dt.

(4.12)

0

Now, since Vc (Ω ∪ ΩI ) is the dual space of Vc (Ω ∪ ΩI ) with respect to the standard L2 (Ω ∪ ΩI ) dual pairing, taking by simplicity m = ml , by (4.9), we have T

(um , v)L2 (Ω∪ΩI ) dt −→

0

T

(u , v)L2 (Ω∪ΩI ) dt.

(4.13)

B[u , v]dt.

(4.14)

0

Otherwise, by (4.10) and since B[·, ·] = (·, ·)Vc (Ω∪ΩI ) T 0

B[um , v]dt

T −→ 0

Thus, by (4.12), (4.13) and (4.14) T 0

(u , v)L2 (Ω∪ΩI ) + B[u, v]dt =

T (f, v)L2 (Ω) dt.

(4.15)

0

By part (iii.) in Proposition 23.23 from [23], the functions of the form (4.11) are dense in L2 (0, T ; Vc (Ω ∪ΩI )), thus, we have that (4.15) holds for all v ∈ L2 (0, T ; Vc (Ω ∪ΩI )). In particular, taking w ∈ Vc (Ω ∪ΩI ), t ∈ [0, T ] and defining

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10

φ(t, ·) : [0, T ] −→ [0, 1] s −→ φ(t, s) = χ[0,t] (s) the characteristic function in [0, t], we have φ(t, ·)w : [0, T ] −→ Vc (Ω ∪ ΩI ) s −→ φ(t, s)w moreover, φ(t, ·)w ∈ L2 (0, T ; Vc (Ω ∪ ΩI )). Hence, by (4.15), we have T



T

 φ(t, s) (u , w)L2 (Ω∪ΩI ) + B[u, w] ds = 

φ(t, s)(f, w)L2 (Ω) ds

0

0

and by definition of φ(t, ·), we get t

   (u , w)L2 (Ω∪ΩI ) + B[u, w] ds =

0

t (f, w)L2 (Ω) ds. 0

Applying a derivative with respect to t and since w = 0 on ΩI (u , w)L2 (Ω) + B[u, w] = (f, w)L2 (Ω)

(4.16)

for all w ∈ Vc (Ω ∪ ΩI ) and a.e. 0 ≤ t ≤ T . Moreover, since u ∈ L2 (0, T ; Vc (Ω ∪ ΩI )), u ∈ L2 (0, T ; Vc (Ω ∪ ΩI )) and Lemma 3.5, by Proposition 23.23 in [23], we conclude that u ∈ C([0, T ]; L2c (Ω ∪ ΩI )). Now, to show that the limit satisfies the initial condition u(0) = g, we consider the test function ψw 1 with ψ ∈ C 1 ([0, T ]) such that ψ(0) = −1, ψ(T ) = 0 and w ∈ (wk )N k=1 . Since C ([0, T ]; Vc (Ω ∪ ΩI )) is 2 2 dense in the space of functions L (0, T ; Vc (Ω ∪ ΩI )) with time weak derivative in L (0, T ; Vc (Ω ∪ ΩI )) and by Lemma 3.5, we use the integration by parts formula Theorem 23.23 in [23] to get T

T



(u , ψw)L2 (Ω∪ΩI ) dt = (u(0), ψ)L2 (Ω∪ΩI ) − 0

ψ  (u, w)L2 (Ω∪ΩI ) dt.

0

Therefore, by (4.15), we obtain T (u(0), w)L2 (Ω∪ΩI ) =



T

ψ (u, w)L2 (Ω∪ΩI ) dt + 0

  ψ (f, w)L2 (Ω) − B[u, w] dt.

0

Now, similarly to (4.12) with m ≥ N , we get T (um (0), w)L2 (Ω∪ΩI ) =



T

ψ (um , w)L2 (Ω∪ΩI ) dt + 0

  ψ (f, w)L2 (Ω) − B[um , w] dt.

0

Moreover, we have (um (0), wk )L2 (Ω∪ΩI ) = (g, wk )L2 (Ω) for all k = 1, ..., N , and since w ∈ (wk )N k=1 , we have that

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M. Yangari / J. Math. Anal. Appl. ••• (••••) •••–•••

(um (0), w)L2 (Ω∪ΩI ) = (g, w)L2 (Ω) ,

11

∀ m ≥ N.

Taking m = ml , from (4.9) and the above equations, we get (u(0), w)L2 (Ω∪ΩI ) = (g, w)L2 (Ω) ,

∀ w ∈ (wk )N k=1 .

Now, since N ∈ N is arbitrary and wk = 0 in ΩI , we have (u(0) − g, wk )L2 (Ω) = 0,

∀ k ∈ N.

2 2 Since, (wk )∞ k=1 is an orthonormal basis of Lc (Ω ∪ΩI ), then, its restrictions are an orthonormal basis of Lc (Ω), therefore,

u(0) − g =

∞ 

(u(0) − g, wk )L2 (Ω) wk = 0,

on Ω

k=1

thus, we conclude u(0) = g in Ω. Finally, we claim that u is the unique weak solution of (1.2). Indeed, it suffices to check that the only weak solution of (1.2) with f = g = 0 is u = 0. To prove this, taking u = w in (4.16) with f = 0, we have   ∂t u 2L2 (Ω∪ΩI ) = −2B[u, u] = −2|||u||| ≤ 0. Finally, by Gronwall inequality, we conclude

u 2L2 (Ω∪ΩI ) ≤ u(0) 2L2 (Ω∪ΩI ) = g 2L2 (Ω) = 0 therefore, u = 0 on Ω ∪ ΩI .

2

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