Yavar Kian
email: yavar.kian@univ-amu.fr

Masahiro Yamamoto

ON EXISTENCE AND UNIQUENESS OF SOLUTIONS FOR SEMILINEAR FRACTIONAL WAVE EQUATIONS

Let Ω be a C 2 -bounded domain of R d , d = 2, 3, and fix Q = (0, T ) × Ω with T ∈ (0, +∞].

In the present paper we consider a Dirichlet initial-boundary value problem associated to the semilinear fractional wave equation

to the Caputo fractional derivative of order α, A is an elliptic operator and the nonlinearity

We first provide a definition of local weak solutions of this problem by applying some properties of the associated linear equation

Then, we prove existence of local solutions of the semilinear fractional wave equation for some suitable values of b > 1. Moreover, we obtain an explicit dependence of the time of existence of solutions with respect to the initial data that allows longer time of existence for small initial data.

∂ xi a ij (x)∂ xj u + V (x)u(x), x ∈ Ω,
where a ij = a ji ∈ C 1 (Ω) and V ∈ L κ (Ω), for some κ > d, satisfy d i,j=1 a ij (x)ξ i ξ j c|ξ| 2 , x ∈ Ω, ξ = (ξ 1 , . . . , ξ d ) ∈ R d and V 0 a.e. in Ω.

We set T ∈ (0, +∞], Σ = (0, T ) × ∂Ω and Q = (0, T ) × Ω. We consider the following initialboundary value problem (IBVP in short) for the fractional semilinear wave equation

           ∂ α t u + Au = f b (u), (t, x) ∈ Q, u(t, x) = 0, (t, x) ∈ Σ, u(0, x) = u 0 (x), ∂ t u(0, x) = u 1 (x), x ∈ Ω, (1.1)
where 1 < α < 2, ∂ α t denotes the Caputo fractional derivative with respect to t,

∂ α t u(t, x) := 1 Γ(2 -α) t 0 (t -s) 1-α ∂ 2 s u(s, x)ds, (t, x) ∈ Q, b > 1 and f b ∈ C 1 (R) satisfies f b (0) = 0 and |f b (u)| C |u| b-1 , u ∈ R.
The main purpose of this paper is to give a suitable definition of solutions of (1.1) and to study the well-posedeness of this problem.

1.2. Physical motivations and known results. Recall that equation (1.1) is associated to anomalous diffusion phenomenon. More precisely, for 1 < α < 2, the linear part of equation (1.1) is frequently used for super-diffusive model of anomalous diffusion such as diffusion in heterogeneous media. In particular, in the linear case (i.e., f b ≡ 0), some physical background is found in Sokolov, Klafter and Blumen [START_REF] Sokolov | Fractional kinetics[END_REF]. As for analytical results in the case of 1 < α < 2, we refer to Mainardi [START_REF] Mainardi | On the initial value problem for the fractional diffusion-wave equation[END_REF] as one early work, and also to §6.1 in Kilbas, Srivastava and Trujillo [START_REF] Kilbas | Theory and applications of fractional differential equations[END_REF], §10.10 in Podlubny [START_REF] Podlubny | Fractional differential equations[END_REF]. For 0 < α < 1, we define ∂ α t u by ∂ α t u(t, x) := 1 Γ(1-α) t 0 (t -s) -α ∂ s u(s, x)ds, and there are works in view of the theory of partial differential equations (e.g., Beckers and Yamamoto [START_REF] Beckers | Regularity and uniqueness of solution to linear diffusion equation with multiple time-fractional derivatives[END_REF], Luchko [START_REF] Luchko | Some uniqueness and existence results for the initial-boundary-value problems for the generalized timefractional diffusion equation[END_REF], Sakamoto and Yamamoto [START_REF] Sakamoto | Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems[END_REF]). Such researches are rapidly developing and here we do not intend to give any comprehensive lists of references.

In contrast to the wave equation, even linear fractional wave equations are not well studied. In fact, few authors treated the well-posedness of the linear IBVP associated to (1.1) and to our best knowledge even the definition of weak solutions does not allow source term with low regularity. For a general study of the linear fractional wave equation and the regularity of solutions we refer to [START_REF] Sakamoto | Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems[END_REF].

When we consider e.g., reaction effects in a super-diffusive model, we have to introduce a semilinear term.

To the best knowledge of the authors, there are no publications on fractional semilinear wave equations by the Strichartz estimate which is a common technique for semilinear wave and Schrödinger equations. In fact, for the wave equation (α = 2), the well-posedness of problem (1.1) has been studied by various authors. In the case Ω = R k with k 3 and A = -∆, the global well-posedness has been proved both in the subcritical case 1 < b < 1 + 4 k-2 by Ginibre and Velo [START_REF] Ginibre | The global Cauchy problem for nonlinear Klein-Gordon equation[END_REF], and in the critical case b = 1+ 4 k-2 by Grillarkis [START_REF] Grillakis | Regularity and asymptotic behavior of the wave equation with a critical non-linearity[END_REF] and, Shatah and Struwe [START_REF] Shatah | Regularity results for nonlinear wave equations[END_REF][START_REF] Shatah | Well-Posedness in the energy space for semilinear wave equation with critical growth[END_REF]. For Ω = R 2 , Nakamura and Ozawa [START_REF] Nakamura | Global solutions in the critical Sobolev space for the wave e quations with nonlinearity of exponential growth[END_REF][START_REF] Nakamura | The Cauchy problem for nonlinear wave equations in the Sobolev space of critical order[END_REF] proved global well-posedness with exponentially growing nonlinearity. Without being exhaustive, for other results related to regularity of solutions or existence of solutions for more general semilinear hyperbolic equations we refer to [START_REF] Grillakis | Regularity and asymptotic behavior of the wave equation with a critical non-linearity[END_REF][START_REF] Ginibre | Regularity of solutions of critical and subcritical nonlinear wave equations[END_REF][START_REF] Kapitanski | Weak and yet weaker solutions of semi-linear wave equations[END_REF][START_REF] Kian | Cauchy problem for semilinear wave equation with time-dependent metrics[END_REF][START_REF] Lindelbag | On existence and scattering with minimal regularity for the semilinear wave equation[END_REF]. In the case of Ω a smooth bounded domain of R 3 , [START_REF] Burq | Global existence for energy critical waves in 3-D domains[END_REF] proved the global well-posedness in the critical case b = 5. In addition, following the strategy set by [START_REF] Burq | Global existence for energy critical waves in 3-D domains[END_REF], [START_REF] Ibrahim | Strichartz type estimates and the well-posedness of an energy critical 2D wave equation in a bounded domain[END_REF] treated the case of Ω a smooth bounded domain of R 2 with exponentially growing nonlinearity. 1.3. Main results. In order to give a suitable definition of solutions of (1.1) we first need to consider the IBVP associated to the linear fractional wave equation

           ∂ α t u + Au = f (t, x), (t, x) ∈ Q, u(t, x) = 0, (t, x) ∈ Σ, u(0, x) = u 0 (x), ∂ t u(0, x) = u 1 (x), x ∈ Ω.
(1.

2)

The present paper contains three main results. Our two first main results are related to properties of solutions of (1.2), while our last result concerns the nonlinear problem (1.1).

Let us first remark that in contrast to usual derivatives, there is no exact integration by parts formula for fractional derivatives. Therefore, it is difficult to introduce the definition of weak solutions of (1.2) in the sense of distributions. To overcome this gap we give the following definition of weak solutions of (1.2). Let 1 (0,T ) (t) be the characteristic function of (0, T ).

Definition 1.1. Let u 0 ∈ L 2 (Ω), u 1 ∈ H -1 (Ω) and f ∈ L 1 (0, T ; L 2 (Ω)). We say that problem (1.2) admits a weak solution if there exists v ∈ L ∞ loc (R + ; L 2 (Ω)) such that: 1) v |Q = u and inf{ε > 0 : e -εt v ∈ L 1 (R + ; L 2 (Ω))} = 0, 2) for all p > 0 the Laplace transform V (p) = +∞ 0 e -pt v(t, .)dt with respect to t of v solves    (A + p α )V (p) = F (p) + p α-1 u 0 + p α-2 u 1 , in Ω, V (p) = 0, on ∂Ω, where F (p) = L[f (t, .)1 (0,T ) (t)](p) = T 0 e -pt f (t, .)dt.
Remark 1. Recall (e.g. formula (2.140) page 80 of [START_REF] Podlubny | Fractional differential equations[END_REF]) that for h ∈ C 2 (R + ) satisfying inf{ε > 0 :

e -εt h (k) ∈ L 1 (R + ), k = 0, 1, 2} = ε 0 we have L[∂ α h](p) = p α H(p) -p α-1 h(0) -p α-2 h (0), p > ε 0 , where H(p) = L[h](p) = +∞ 0
e -pt h(t)dt. Therefore, for sufficiently smooth data u 0 , u 1 , f (e.g. [START_REF] Sakamoto | Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems[END_REF]) one can check that problem (1.2) admits a unique strong solution which is also a weak solution of (1.2).

Consider the operator A acting on L 2 (Ω) with domain

D(A) = {g ∈ H 1 0 (Ω) : Ag ∈ L 2 (Ω)} defined by Au = Au, u ∈ D(A).
Recall that in view of the Sobolev embedding theorem (e.g. [START_REF] Grisvard | Elliptic problems in nonsmooth domains[END_REF]Theorem 1.4.4.1]) the multiplication operator u → V u is bounded from H 1 (Ω) to L 2 (Ω). Thus, we have D(A) = H 2 (Ω) ∩ H 1 0 (Ω). Moreover, by V ≥ 0 in Ω, the operator A is a strictly positive selfadjoint operator with a compact resolvent. Therefore, the spectrum of A consists of a non-decreasing sequence of strictly positive eigenvalues (λ n ) n 1 . Let us also introduce an orthonormal basis in the Hilbert space L 2 (Ω) of eigenfunctions (ϕ n ) n 1 of A associated to the non-decreasing sequence of eigenvalues (λ n ) n 1 .

From now on, by ., . , we denote the scalar product of L 2 (Ω). For all s 0, we denote by A s the operator defined by

A s h = +∞ n=1 h, ϕ n λ s n ϕ n , h ∈ D(A s ) = h ∈ L 2 (Ω) : +∞ n=1 | h, ϕ n | 2 λ 2s n < ∞
and consider on D(A s ) the norm

h D(A s ) = +∞ n=1 | h, ϕ n | 2 λ 2s n 1 2
, h ∈ D(A s ).

By duality, we can also set D(A -s ) = D(A s ) by identifying L 2 (Ω) = L 2 (Ω) which is a Hilbert space with the norm

h D(A -s ) = ∞ n=1 h, ϕ n -2s λ -2s n 1 2
.

Here ., . -2s denotes the duality bracket between D(A -s ) and D(A s ). Since D(A 1/2 ) = H 1 0 (Ω), we identify H -1 (Ω) with D(A -1/2 ).

Using eigenfunction expansions we show our first main result where we state existence and uniqueness of weak solutions of (1.2).

Theorem 1.2. Let u 0 ∈ L 2 (Ω), u 1 ∈ H -1 (Ω) = D(A -1 2 ), f ∈ L 1 (0, T ; L 2 (Ω)). Then, problem (1.2)
admits a unique weak solution u ∈ C([0, T ]; L 2 (Ω)) satisfying

u C([0,T ];L 2 (Ω)) C( u 0 L 2 (Ω) + u 1 H -1 (Ω) + f L 1 (0,T ;L 2 (Ω)) ). (1.3)
Moreover, assuming that there exists 0 < r < 1 4 such that u 0 ∈ H 2r (Ω), we have u ∈ W 1,1 (0, T ; L 2 (Ω)) and

u W 1,1 (0,T ;L 2 (Ω)) C( u 0 H 2r (Ω) + u 1 H -1 (Ω) + f L 1 (0,T ;L 2 (Ω)) ).
(1.4)

Recall that for γ, r, s 0, 1 p, q, p, q ∞, Strichartz estimates for solutions u of (1.2) denotes estimates of the form

u C([0,T ];H 2r (Ω)) + u L p (0,T ;L q (Ω)) C( u 0 H 2γ (Ω) + u 1 H 2s (Ω) + f L p (0,T ;L q (Ω)) ).
It is well known that these estimates, introduced by [START_REF] Strichartz | A priori estimates for the wave equation and some applications[END_REF] and extended to the endpoints by [START_REF] Keel | Endpoint Strichartz estimates[END_REF] for both wave and Schrödinger equations, are important tools in the study of well-posedness of nonlinear equations (e.g. [START_REF] Burq | Global existence for energy critical waves in 3-D domains[END_REF][START_REF] Grillakis | Regularity and asymptotic behavior of the wave equation with a critical non-linearity[END_REF][START_REF] Ginibre | Regularity of solutions of critical and subcritical nonlinear wave equations[END_REF][START_REF] Ibrahim | Strichartz type estimates and the well-posedness of an energy critical 2D wave equation in a bounded domain[END_REF][START_REF] Kian | Cauchy problem for semilinear wave equation with time-dependent metrics[END_REF]). In the present paper we prove these estimates for solutions of (1.2). For this purpose, we consider 1 p, q ∞ and 0 < γ < 1 satisfying:

1) q = ∞, for d 4 < γ < 1, 2) 2 < q < ∞, for γ = d 4 , 3) q = 2d d-4γ , for 0 < γ < d 4 .
(1.5)

1) p < 1 1-α(1-γ) , for γ > 1 -1 α , 2) p = ∞, for γ 1 -1 α .
(1.6)

Then, our second main result can be stated as follows.

Theorem 1.3. (Strichartz estimates) Assume that 1 p, q ∞ and 0 < γ < 1 fulfill (1.5), (1.6) and set

s = max 0, γ - 1 α , r = min 1 - 1 α , γ . Let u 0 ∈ D(A γ ), u 1 ∈ D(A s ), f ∈ L 1 (0, T ; L 2 (Ω)).
Then, the unique weak solution u of problem (1.2)

is lying in L p (0, T ; L q (Ω)) ∩ C([0, T ]; H 2r (Ω)) and fulfills estimate u C([0,T ];H 2r (Ω)) + u L p (0,T ;L q (Ω)) C( u 0 H 2γ (Ω) + u 1 H 2s (Ω) + f L 1 (0,T ;L 2 (Ω)) ). (1.7)
Here the constant C takes the form

C = C 0 (1 + T ) δ , (1.8)
where

δ =    max (α(1 -γ) -1, 1 -α(γ -s), 1 -α(r -s), α(1 -r) -1) , for p = ∞, max 1 p , 1 -α(γ -s) + 1 p , 1 -α(r -s), α(1 -r) -1, α(1 -γ) -1 + 1 p , for p < ∞ (1.9)
and C 0 depends only on Ω, γ, d, α, p.

In the last section we apply estimates (1.7) to prove our last result which is related to the existence and uniqueness of local solutions of (1.1). For this purpose, we first need to define local solutions of (1.1). In section 2 (see also [START_REF] Sakamoto | Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems[END_REF]), using the eigenfunction expansions we introduce the operators

S 1 (t)h = ∞ k=1 E α,1 (-λ k t α ) h, ϕ k ϕ n , h ∈ L 2 (Ω), S 2 (t)h = ∞ k=1 tE α,2 (-λ k t α ) h, ϕ k ϕ n , h ∈ L 2 (Ω), S 3 (t)h = ∞ k=1 t α-1 E α,α (-λ k t α ) h, ϕ k ϕ k , h ∈ L 2 (Ω),
where for all α > 0, β ∈ R, E α,β denotes the Mittag-Leffler function given by

E α,β (z) = ∞ k=0 z k Γ(αk + β) .
It is well known (e.g. [START_REF] Beckers | Regularity and uniqueness of solution to linear diffusion equation with multiple time-fractional derivatives[END_REF][START_REF] Podlubny | Fractional differential equations[END_REF][START_REF] Luchko | Some uniqueness and existence results for the initial-boundary-value problems for the generalized timefractional diffusion equation[END_REF][START_REF] Sakamoto | Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems[END_REF]) that for all t > 0 we have

S j (t) ∈ B(L 2 (Ω)), j = 1, 2, 3. Moreover, in view of Theorem 1.2, for u 0 , u 1 ∈ L 2 (Ω) and f ∈ L 1 (0, T ; L 2 (Ω)), the unique weak solution of (1.2)
is given by

u(t) = S 1 (t)u 0 + S 2 (t)u 1 + t 0 S 3 (t -s)f (s)ds. (1.10)
For all T > 0, we introduce the space

X T = C([0, T ]; L 2 (Ω)) ∩ L b (0, T ; L 2b (Ω))
with the norm

v X T = v C([0,T ];L 2 (Ω)) + v L b (0,T ;L 2b (Ω)) .
Recall that, by applying the Hölder inequality, one can check that for all u, v ∈ X T we have

f b (u), f b (v) ∈ L 1 (0, T ; L 2 (Ω)) with f b (u) L 1 (0,T ;L 2 (Ω)) C b u b L b (0,T ;L 2b (Ω)) C b u b X T (1.11)
and

f b (u) -f b (v) L 1 (0,T ;L 2 (Ω)) C b u -v X T ( u b-1 X T + v b-1 X T ), (1.12)
where the constant C b > 0 depends only on b, f b . Therefore, in view of Theorem 1.2, the map H b defined by

H b u(t) = t 0 S 3 (t -s)f b (u(s))ds, u ∈ X T is locally Lipschitz from X T to C([0, T ]; L 2 (Ω)).
Definition 1.4. Let u 0 , u 1 ∈ L 2 (Ω) and T > 0. We say that (1.1) admits a weak solution on (0,

T ) if the map G b : X T → C([0, T ]; L 2 (Ω)) defined by G b u(t) = S 1 (t)u 0 + S 2 (t)v 2 + t 0 S 3 (t -s)f b (u(s))ds admits a fixed point u ∈ X T . Such a fixed point u ∈ X T is called a weak solution to (1.1) on (0, T ).
We say that problem (1.1) admits a local weak solution if there exists T > 0, depending on u 0 , u 1 , such that problem (1.1) admits a weak solution on (0, T ).

Now we can state our result of existence and uniqueness of local solutions for (1.1). We recall that δ > 0 is given in (1.9).

Theorem 1.5. Let b > 1 satisfy

dα dα + 4(1 -α) < b < dα + 4 dα + 4(1 -α) (1.13)
and let

γ = d(b -1) 4b , q = 2b, s = max(0, γ - 1 α ), r = min(1 - 1 α , γ), 1 < 1 2 -α . (1.14)
Then, we can choose p ∈ b,

1 1-α(1-γ) such that for u 0 ∈ D(A γ ), u 1 ∈ D(A s ), T 0 > 0, problem (1.1)
admits a local weak solution u lying in L p (0, T ; L q (Ω)) ∩ C([0, T ]; H 2r (Ω)) ∩ W 1, (0, T ; L 2 (Ω)) for some T T 0 that takes the form

T = min C( u 0 H 2γ (Ω) + u 1 H 2s (Ω) ) - p(b-1) p-b , T 0 , (1.15)
where we set

C = C0 (1 + T 0 ) δ b-1 , (1.16)
and C0 depends only on f b , Ω, α, b, p and d. Moreover, this local weak solution u is a unique local weak solution of (1.1) lying in L p (0, T ; L q (Ω)) and satisfies

u C([0,T ];H 2r (Ω)) + u L p (0,T ;L q (Ω)) + u W 1, (0,T ;L 2 (Ω)) C( u 0 H 2γ (Ω) + u 1 H 2s (Ω) ).
(1.17)

Here the constant C > 0 depends on d, Ω, f b , b, T 0 , p, α.

A direct consequence of Theorem 1.5 is the following.

Corollary 1.6. Assume that conditions (1.13) and (1.14

) are fulfilled. Let u 0 ∈ D(A γ ), u 1 ∈ D(A s ) satisfy C0 ( u 0 H 2γ (Ω) + u 1 H 2s (Ω) ) - p(b-1) p(1+δ)-b > 1 for some b < p < 1 1-α(1-γ)
, where the constant C0 is introduced in (1.16). Then, for any T > 0 satisfying

T < C0 ( u 0 H 2γ (Ω) + u 1 H 2s (Ω) ) - p(b-1) p(1+δ)-b , (1.18)
problem (1.1) admits a unique weak solution u on (0, T ) lying in L p (0, T ;

L q (Ω)) ∩ C([0, T ]; H 2r (Ω)) ∩ W 1, (0, T ; L 2 (Ω)).
This last result means that for smaller initial data we obtain longer time of existence of weak solutions.

Let us remark that, this paper seems to be the first where the Definition 1.1 of weak solutions of (1.2) is considered. The main contribution of Definition 1.1 comes from the fact that it allows wellposedness of (1.2) with weak conditions. Indeed, in contrast to other definitions of weak solutions for (1.2) (e.g. [21, Definition 2.1] used by [START_REF] Sakamoto | Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems[END_REF] to prove existence of weak solutions of (1.2) with f ∈ L 2 (Q),

u 0 ∈ L 2 (Ω), u 1 = 0 in [21, Corollary 2.5, 2.6]
), applying Definition 1.1 we can show well-posedness of (1.2) with f ∈ L 1 (0, T ; L 2 (Ω)), u 0 ∈ L 2 (Ω) and u 1 ∈ H -1 (Ω). The choice of Definition 1.1 is inspired both by the analysis of [START_REF] Podlubny | Fractional differential equations[END_REF] and the connection between elliptic equations and fractional diffusion equations used by [START_REF] Li | Uniqueness in inverse boundary value problems for fractional diffusion equations[END_REF]. Note also that Definition 1.1 plays an important role in the Definition 1.4 of weak solutions of (1.1).

Let us observe that in contrast to the wave equation the solution of (1.2) are not described by a semigroup. Therefore, we can not apply many arguments that allow to improve the Strichartz estimates

(1.7) such as the T T * method of [START_REF] Keel | Endpoint Strichartz estimates[END_REF]. Nevertheless, we prove local existence of solution of (1.1) with estimates (1.7). Note also that estimates (1.7) are derived from suitable estimates of Mittag-Leffler functions.

To our best knowledge this paper is the first treating well-posedness for semilinear fractional wave equations. Contrary to semilinear wave equations, it seems difficult to give a suitable definition of the energy for (1.1). This is mainly due to that fact that, once again, there is no exact integration by parts formula for fractional derivatives as well as properties of composition and conjugation of the fractional Caputo derivative ∂ α t (e.g. [START_REF] Podlubny | Fractional differential equations[END_REF]Section 2]). For this reason, it seems complicate to derive global well-posedness from local well-posedness. However, using the explicit dependence with respect to T of the constant in (1.7) we can establish an explicit dependence of the time of existence T of (1.1) with respect to the initial conditions u 0 , u 1 . From this result, we prove long time of existence for small initial data (see Corollary 1.6).

1.4. Outline. The paper is composed of four sections. In Section 2, we treat the well-posedness of the linear problem (1.2) and we show Theorem 1.2. Then, in Section 3 we prove the Strichartz estimates associated to these solutions and given by Theorem 1.3. Finally, in Section 4 we prove the local existence of solutions stated in Theorem 1.5 and Corollary 1.6.

The linear equation

The goal of this section is to prove Theorem 1.2. For this purpose, for k 1 we introduce

u k ∈ C(R + ) defined by u k (t) = E α,1 (-t α λ k )u 0,k + tE α,2 (-t α λ k )u 1,k + t 0 (t -s) α-1 E α,α (-(t -s) α λ k )f k (s)ds, t > 0, (2.1)
where

u 0,k = u 0 , ϕ k , u 1,k = u 1 , ϕ k -1 , f k (s) = f (s), ϕ k 1 (0,T ) (s). We will show that k 1 u k (t)ϕ k (x)
converge to a weak solution of (1.2) and this weak solution is unique. Let us first recall the following estimates of the behavior of the Mittag-Leffler function.

Lemma 2.1. (Theorem 1.6, [START_REF] Podlubny | Fractional differential equations[END_REF]) If 0 < α < 2, β ∈ R, πα/2 < µ < min(π, πα), then

|E α,β (z)| C 1 + |z| , z ∈ C, µ |argz| π,
where the constant C > 0 depends only on α, β, µ.

Applying Lemma 2.1, one can check that, for all t > 0 and all m, n ∈ N * , we have

n k=m u k (t)ϕ k L 2 (Ω) C n k=m u 0,k ϕ k L 2 (Ω) + Ct 1-α 2 n k=m (λ k t α ) 1 2 1 + λ k t α λ -1 2 k u 1,k ϕ k L 2 (Ω) + Ct α-1 T 0 n k=m f k (s)ϕ k L 2 (Ω)
ds.

Thus, for all T 1 > 0 we obtain

sup t∈(0,T1) n k=m u k (t)ϕ k L 2 (Ω) C n k=m u 0,k ϕ k L 2 (Ω) + C(T 1 ) 1-α 2 n k=m λ -1 2 k u 1,k ϕ k L 2 (Ω) + C(T 1 ) α-1 T 0 n k=m f k (s)ϕ k L 2 (Ω)
ds and it follows that

lim m,n→∞ sup t∈(0,T1) n k=m u k (t)ϕ k L 2 (Ω) = 0.
Therefore, for any T 1 > 0 the serie

k 1 u k (t)ϕ k converge uniformly in t ∈ (0, T 1 ) to v ∈ C(R + , L 2 (Ω)).
In addition, for all N ∈ N * and t > 0, we have

N k=1 u k (t)ϕ k L 2 (Ω) C u 0 L 2 (Ω) + t 1-α 2 u 1 D(A - 1
2 )

+ Ct α-1 f L 1 (0,T ;L 2 (Ω)) . (2.2)
Here and henceforth N * denotes the set of all the natural number > 0. Therefore, we deduce

inf{ε > 0 : e -εt v ∈ L 1 (R + ; L 2 (Ω))} = 0
and (2.2) implies that, for all N ∈ N * , t > 0 and p > 0, we obtain

N k=1 e -pt u k (t)ϕ k L 2 (Ω) C e -pt u 0 L 2 (Ω) + e -pt t 1-α 2 u 1 D(A -1 2 ) + Ce -pt t α-1 f L 1 (0,T ;L 2 (Ω)) .
Then, an application of Lebesgue's dominated convergence for functions taking values in L 2 (Ω) yields

V (p, .) = L[v(t, .)](p) = ∞ k=1 L[u k ](p)ϕ k = ∞ k=1 U k (p, .) with U k (p, .) = L[u k ](p)ϕ k .
Moreover, the properties of the Lalpace transform of the Mittag-Leffler function (e.g. formula (1.80) pp 21 of [START_REF] Podlubny | Fractional differential equations[END_REF]) imply

U k (p) = p α-1 u 0,k + p α-2 u 1,k + F k (p) p α + λ k ϕ k = (A + p α ) -1 ( p α-1 u 0 + F (p), ϕ k + p α-2 u 1 , ϕ k -1 )ϕ k with F k (p) = L[f k ](p) = F (p), ϕ k . Thus U k (p, .) solves      (A + p α )U k (p) = p α-1 u 0 + F (p), ϕ k + p α-2 u 1 , ϕ k -1 ϕ k , in Ω, U k (p) = 0, on ∂Ω.
Combining this with the fact that u 0 , u 1 , F (p, .) ∈ H -1 (Ω) = D(A -1 2 ), we deduce that

k 1 U k (p, .) converge in H 1 0 (Ω) to V (p, .) and k 1 (A + p α )U k (p) converge in H -1 (Ω) to F (p) + p α-1 u 0 + p α-2 u 1 .
Therefore, V (p, .) solves

   (A + p α )V (p) = F (p) + p α-1 u 0 + p α-2 u 1 , in Ω, V (p) = 0, on ∂Ω.
(2.3) Thus, u = v |Q is a weak solution of (1.2). This proves the existence of weak solutions lying in C([0, T ]; L 2 (Ω)) and by the same way we obtain estimate (1.3). It remains to show that this solution is unique and, when u 0 ∈ H 2r (Ω), that it is lying in W 1,1 (0, T ; L 2 (Ω)) and that it fulfills (1.4).

We first prove the uniqueness of solutions. Let v 1 , v 2 be two weak solutions of (1.2). Then, for j = 1, 2, there exist w j ∈ L ∞ loc (R + ; L 2 (Ω)) such that:

w j |Q = u j , inf{ε > 0 : e -εt w j ∈ L 1 (R + ; L 2 (Ω))} = 0
and, for all p > 0, the Laplace transform W j (p) with respect to t of w j solves (2.3). Let p > 0 and set

W (p) = W 1 (p) -W 2 (p) ∈ L 2 (Ω) and note that W (p) solves    (A + p α )W (p) = 0, in Ω, W (p) = 0, on ∂Ω.
The uniqueness of the solution of this elliptic problem implies that W (p) = 0. Therefore, for all p > 0 we have W 1 (p) = W 2 (p) which implies that w 1 = w 2 and by the same way

v 1 = w 1|Q = w 2|Q = v 2 .
This proves the uniqueness.

From now on we assume that u 0 ∈ H 2r (Ω), for r ∈ (0, 1/4), and we will show that u ∈ W 1,1 (0, T ; L 2 (Ω)) and that it fulfills (1.4). For this purpose, we establish the following lemmata.

Here we recall that u k , f k , u 0,k , u 1,k appear in (2.1).

Lemma 2.2. For λ > 0, α > 0 and positive integer m ∈ N * , we have

d m dt m E α,1 (-λt α ) = -λt α-m E α,α-m+1 (-λt α ), t > 0 and d dt (tE α,2 (-λt α )) = E α,1 (-λt α ), t > 0.
Proof. The power series defining E α,1 (-λt α ) and tE α,2 (-λt α ) for t > 0 admit the termwise differentiation any times, and the termwise differentiation yields the conclusions.

Lemma 2.3. For all k 1 and 1 < 1 2-α , we have u k ∈ W 1, (0, T ) and

∂ t u k (t) = -λ k t α-1 E α,α (-t α λ k )u 0,k + E α,1 (-t α λ k )u 1,k + t 0 (t -s) α-2 E α,α-1 (-(t -s) α λ k )f k (s)ds, (2.4)
for a.e. t ∈ (0, T ).

Proof. First we consider the case f k = 0. Then, we have

u k (t) = E α,1 (-t α λ k )u 0,k + tE α,2 (-t α λ k )u 1,k , t > 0.
In view of Lemma 2.2, we see that u k ∈ C 1 ([0, T ]) and (2.4) is fulfilled.

Second we consider the the case u 0,k = u 1,k = 0. Introduce, for all ε > 0 the function

u ε k (t) = t-ε 0 (t -s) α-1 E α,α (-(t -s) α λ k )f k (s)ds, 0 < t < T.
In view of Lemma 2.2, we have u ε k ∈ W 1, (0, T ) and

∂ t u ε k (t) = ε α-1 E α,α (-λ k ε α )f k (t -ε) + t-ε 0 (t -s) α-2 E α,α-1 (-(t -s) α λ k )f k (s)ds, a.a. t ∈ (0, T ).
On the other hand, one can easily check that (u ε k ) ε>0 converge to u k as ε → 0 in D (0, T ) and

(∂ t u ε k ) ε>0 converge to t → t 0 (t -s) α-2 E α,α-1 (-(t -s) α λ k )f k (s)ds
as ε → 0 in D (0, T ), where D (0, T ) is the space of distributions in (0, T ). Therefore, in the sense of D (0, T ) we have

∂ t u k (t) = t 0 (t -s) α-2 E α,α-1 (-(t -s) α λ k )f k (s)ds, 0 < t < T,
which implies (2.4). In addition, applying (2.1), we obtain

|∂ t u k (t)| C t 0 (t -s) α-2 |f k (s)| ds.
Then, according to the Young inequality, we deduce that ∂ t u k ∈ L l (0, T ). Therefore, we have u k , ∂ t u k ∈ L l (0, T ), which means that u k ∈ W 1, (0, T ). Combining these two cases, we complete the proof of Lemma 2.3.

Let us remark that, using the fact that 0 < 2r < 1 2 and D(A

1 2 ) = H 1 0 (Ω), one can check by interpolation that u 0 ∈ H 2r (Ω) = H 2r 0 (Ω) = D(A r
) (e.g. [14, Chapter 1, Theorems 11.1 and 11.6]) and

∞ k=1 λ 2r k |u 0,k | 2 C u 0 2 H 2r (Ω) . (2.5)
In view of (2.4), applying our previous arguments, for all m, n ∈ N * , we obtain

n k=m ∂ t u k ϕ k L 1 (0,T ;L 2 (Ω)) C n k=m (λ k t α ) 1-r 1 + (λ 1 α k t) α t αr-1 λ r k u 0,k ϕ k L 1 (0,T ;L 2 (Ω)) + C n k=m (λ k t α ) 1 2 1 + (λ 1 α k t) α t -α 2 λ -1 2 k u 1,k ϕ k L 1 (0,T ;L 2 (Ω)) + C T 0 t 0 (t -s) α-2 n k=m f k (s)ϕ k L 2 (Ω)
dsdt.

The Young inequality implies

n k=m ∂ t u k ϕ k L 1 (0,T ;L 2 (Ω)) C T αr αr n k=m λ r k u 0,k ϕ k L 2 (Ω) + C T 1-α 2 1 -α 2 n k=m λ -1 2 k u 1,k ϕ k L 2 (Ω) + C T α-1 α -1 n k=m f k (s)ϕ k L 1 (0,T ;L 2 (Ω))
.

Thus, we have

lim m,n→+∞ n k=m ∂ t u k ϕ k L 1 (0,T ;L 2 (Ω))
= 0, which means that n k=1 ∂ t u k (t)ϕ k (x) is a Cauchy sequence and a convergent sequence in L 1 (0, T ; L 2 (Ω)). Since n k=1 u k (t)ϕ k (x) converge to u in C([0, T ]; L 2 (Ω)), combining this with (2.2), we deduce that Second step: Let u 0 = u 1 = 0. In view of Lemma 2.1, for all 0 < t < T , we have

λ 2γ k t α-1 E α,α (-λ k t α ) f, ϕ k 2 t 2(α(1-γ)-1) | f, ϕ k | 2 (λ k t α ) γ 1 + λ k t α 2 .
Thus, for all 0 < t < T and h ∈ L 2 (Ω), we deduce that S 3 (t)h ∈ D(A γ ) ⊂ H 2γ (Ω) with

S 3 (t)h H 2γ (Ω) Ct α(1-γ)-1 h L 2 (Ω) , 0 < t < T.
By the Sobolev embedding theorem, for all 0 < t < T , we have S 3 (t)h ∈ H 2γ (Ω) ⊂ L q (Ω) with

S 3 (t)h L q (Ω) C S 3 (t)h H 2γ (Ω) Ct α(1-γ)-1 h L 2 (Ω) .
Applying this estimate, we obtain u(t, .) L q (Ω)

t 0 S 3 (t -s)f (s) L q (Ω) ds C t 0 (t -s) α(1-γ)-1 f (s) L 2 (Ω) ds.
By t → t α(1-γ)-1 ∈ L p (0, T ), the Young inequality yields

u L p (0,T ;L q (Ω)) C T α(1-γ)-1+ 1 p (p(α(1 -γ) -1) + 1) 1/p f L 1 (0,T ;L 2 (Ω)) . (3.5)
Repeating the above arguments, we deduce that

u(t, .) H 2r (Ω) C t 0 (t -s) α(1-r)-1 f (s) L 2 (Ω) ds. Then, since α(1 -r) -1 α(1 -(1 -α -1
)) -1 = 0, we deduce from the Young inequality that u(t, .) H 2r (Ω) CT α(1-r)-1 f L 1 (0,T ;L 2 (Ω)) .

Combining this estimate with (3.3) -(3.5), we deduce (1.7) for u 0 = u 1 = 0. This completes the proof of Theorem 1.3.

4.

Local solutions of (1.1)

In this section we will apply the results of the previous section to prove Theorem 1. On the other hand, for 1 -1 α < γ < d 4 , one can check that

γ < dα 4 + dα ⇐⇒ d(b -1) 4b < dα 4 + dα ⇐⇒ b < dα + 4 dα + 4(1 -α) . (4.2)
Therefore, γ given by (1.14) fulfills

1 - 1 α < γ < dα 4 + dα , which yields 1 1 -α 1 -dα 4+dα < 1 1 -α(1 -γ)
.

Therefore, we can choose p satisfying

b < dα + 4 dα + 4(1 -α) = 1 1 -α(1 -dα 4+dα ) < p < 1 1 -α(1 -γ) .
Moreover, for q given by (1.14) we have q = 2d d-4γ . Thus, for q, γ given by (1.14) and b < p < 1 1-α(1-γ) , p, q, γ fulfill conditions (1.5) and (1.6) with p > b. Provided that 0 < T ≤ T 0 and M > 0 will be chosen suitably later, we set

Y T = L p (0, T ; L q (Ω)) ∩ C([0, T ]; H 2r (Ω)) and B M = {u ∈ Y T : u Y T M }.
Moreover, we set

u Y T = u L p (0,T ;L q (Ω)) + u C([0,T ];H 2r (Ω)) .
We fix the constant C b > 0 which appears in estimates (1.11) and (1.12). We note that C b is independent of T . We put C = C 0 (1 + T 0 ) δ , where the constants C 0 , δ are introduced in (1.8), (1.9) and are independent of T . Finally we fix

C = C (1 + C b ) + 1. Since p > b, for all u ∈ Y T we have u ∈ L b (0, T ; L 2b (Ω)).
Therefore, in view of Theorem 1.3 and estimates (1.7), (1.8) and (1.11), we have

G b (u) ∈ Y T and G b (u) Y T C ( u 0 H 2γ (Ω) + u 1 H 2s (Ω) + f b (u) L 1 (0,T ;L 2 (Ω)) ) C ( u 0 H 2γ (Ω) + u 1 H 2s (Ω) + C b u b L b (0,T ;L 2b (Ω)) ) C( u 0 H 2γ (Ω) + u 1 H 2s (Ω) + u b L b (0,T ;L 2b (Ω)) ). (4.3)
On the other hand, by the Hölder inequality one can check that

T 0 u(t, .) b L q (Ω) dt T 0 u(t, .) p L q (Ω) dt b p T 1-b p , which implies u L b (0,T ;L 2b (Ω)) T p-b bp u L p (0,T ;L q (Ω)) . (4.4)
Applying this estimate to (4.3), we obtain

G b (u) Y T C( u 0 H 2γ (Ω) + u 1 H 2s (Ω) + T p-b p u b Y T ). (4.5)
We set M = 2C( u 0 H 2γ (Ω) + u 1 H 2s (Ω) ) and T = min (3CM b-1 ) -p p-b , T 0 . With these values of M and T , one can easily verify that (4.5) implies

G b u Y T M, u ∈ B M .
In the same way, applying estimates (1.12) and (1.7) in

G b u -G b v = t 0 S 3 (t -s)[f b (u(s)) -f b (v(s))]ds, we obtain G b u -G b v Y T C u -v L b (0,T ;L 2b (Ω)) ( u b-1 L b (0,T ;L 2b (Ω)) + v b-1 L b (0,T ;L 2b (Ω)) ).
Then, (4.4) and the choice of T imply that for every u, v ∈ B M , we have

G b u -G b v Y T CT p-b p u -v L p (0,T ;L q (Ω)) ( u b-1 L p (0,T ;L q (Ω)) + v b-1 L p (0,T ;L q (Ω)) ) 2CM b-1 T p-b p u -v Y T 2 3 u -v Y T . Therefore, G b is a contraction from B M to B M . Consequently G b admits a unique fixed point u ∈ B M
which is a local weak solution of (1.1). Moreover, from our choice of M and T we deduce (1.15) and

(1.16).

Now we show that this solution is unique in L p (0, T ; L q (Ω)). For this purpose, consider the space

Z T = C([0, T ]; L 2 (Ω)) ∩ L p (0, T ; L q (Ω)) with the norm v Z T = v C([0,T ];L 2 (Ω)) + v L p (0,T ;L q (Ω)) , v ∈ Z T .
Repeating our previous arguments we can show that G b is a contraction from B M to B M with B M = {u ∈ Z T : u Z T M }. Therefore, the fixed point u ∈ B M of G b is a unique local weak solution of (1.1) lying in L p (0, T ; L q (Ω)). Now let us show that the unique weak solution of (1.1) lying in L p (0, T ; L q (Ω)) is also lying in W 1, (0, T ; L 2 (Ω)) and it fulfills (1.17 for all m, n ∈ N * . In view of (4.1), we have

(αγ -1) > α -2 2 -α > -1.
Therefore, the Young inequality yields Combining this with the fact that Combining this estimate with (4.6) and letting N → ∞, we deduce (1.17). Thus, the proof of Theorem Since T 0 1 we can replace T 0 by T 0 + 1 in condition (1.15). Therefore, with this value of T 0 , condition (1.15) holds. Thus, according to Theorem 1.5, problem (1.1) admits a unique weak solution u on (0, T ) lying in L p (0, T ; L q (Ω)) ∩ C([0, T ]; H 2r (Ω)) ∩ W 1, (0, T ; L 2 (Ω)).

1 . Introduction 1 . 1 .

 111 Statement of the problem. Let Ω be a C 2 -bounded domain of R d with d = 2, 3. In what follows, we define A by the differential operator Au(x) = -

5 and Corollary 1 . 6 .

 16 Proof of Theorem 1.5 Note first that for γ and b given by (1.13) and (1.14), we have γ < d 4 and d d -4γ = b > dα dα + 4(1 -α) , which implies by 1 < α < 2 and d = 2

  ). Since u Z T M and T = min (3CM b-1 ) -p p-b , T 0 , by(1.11), we obtain that f b (u) ∈ L 1 (0, T ; L 2 (Ω)) satisfiesf b (u) L 1 (0,T ;L 2 (Ω)) C( u 0 H 2γ (Ω) + u 1 H 2s (Ω) ).(4.6)Now let us setf k (t) = f b (u(t)), ϕ k , u 0,k = u 0 , ϕ k , u 1,k = u 1 , ϕ k . Then, in view of Lemma 2.3, u k (t) = u(t), ϕ k ∈ W 1, (0, T ) fulfills (2.4). Repeating the arguments used in the last part of the proof of Theorem 1.2, we obtain n k=m (∂ t u k )ϕ k L (0,T ;L 2 (Ω))

k 1 (

 1 ∂ t u k )(t)ϕ k (x)is a Cauchy sequence and is a convergent sequence in L (0, T ; L 2 (Ω)).

k 1 u

 1 k (t)ϕ k (x) converge to u in C([0, T ]; L 2 (Ω)), we deduce that

k 1 u

 1 k (t)ϕ k (x) converge to u in W 1, (0, T ; L 2 (Ω)). Finally, for all N ∈ N * , we findN k=1 u k ϕ k W 1, (0,T ;L 2 (Ω)) C( u 0 H 2γ (Ω) + u 1 H 2s (Ω) + f b (u) L 1 (0,T ;L 2 (Ω)) ).

1. 5 (

 5 is completed.Proof of Corollary 1.6. Let T > 0 fulfill (1.18) and set T 0 = T . Without lost of generality we can assume that T 1. Then, we have u 0 H 2γ (Ω) + u 1 H 2s (Ω) )

Acknolwedgements. The second author is partially supported by Grant-in-Aid for Scientific Research (S) 15H05740 of Japan Society for the Promotion of Science.

u k (t)ϕ k (x) converge to u in W 1,1 (0, T ; L 2 (Ω)). Finally, repeating our previous arguments and applying (2.5), for all N ∈ N * , we find

Then, combining this estimate with (2.2) and taking the limit N → ∞, we deduce (1.3). Thus, the proof of Theorem 1.2 is completed.

Strichartz estimates

The goal of this section is to show Theorem 1.3. We divide the proof of Theorem 1.3 into two steps.

First we prove estimates (1.7) for the weak solution u of (1.2) with f = 0 and then for u 0 = u 1 = 0.

Henceforth C > 0 denotes generic constants which are dependent only on Ω, d, α, γ.

First step: Let f = 0 and let 1 p, q ∞, 0 < γ < 1 fulfill (1.5) and (1.6). Then, (1.10) implies that

Applying estimate Lemma 2.1, we deduce that for

We have 0 ≤ γ -s < 1 by the definition of γ, s. Therefore, in the same way, Lemma 2.1 yields that, for all 0 < t < T , we have

Thus, for all 0 < t < T , we deduce that S 2 (t)

By the Sobolev embedding theorem, for all 0 < t < T , we have u(t, .) ∈ H 2γ (Ω) ⊂ L q (Ω) and

On the other hand, we have 1 -(γ -s)α 0 and so u ∈ L ∞ (0, T ; L q (Ω)) and

In the same way, we have


David Guerrero

Patrick Nierat

Laurent Proulhac

Supplying cities from abroad

Keywords: freight transport, city supply, urban hierarchy, France

This article aims to identify the role of urban areas in freight systems. To achieve this goal, the article examines the nature and intensity of inter-urban freight flows in France. It draws on the results of a French survey that describes shipments sent by firms. It paints a simplified picture of the geography of freight flows with regard to the light they shed on economic interactions between urban centers. It shows that the pattern of inter-urban freight flows in France is hierarchical: large urban areas are supplied by smaller ones.

INTRODUCTION

Urban areas rely on freight transportation, which is essential to supply them with goods and to enhance their economic development. The scale and nature of freight flows depend on the size and shape of individual urban centers and also to a large extent on how their activities are structured. This is the purpose of most urban freight transport surveys [START_REF] Gonzalez-Feliu | Sustainable urban logistics: Concepts, methods and information systems[END_REF][START_REF] Taniguchi | Predicting the effects of city logistics schemes[END_REF], which, however, deal with only a small number of urban areas and do not extend beyond their boundaries. Because urban areas are just very small parts of the global freight system, accurate forecasting of urban freight should integrate larger areas, at least at a national level. Some of these shortcomings can be overcome by exploiting a shipper survey, which provides an insight into national interests and provides a fuller description of the transport flows among urban areas. The aim of this article then is to use the results of this survey to explore the organization of inter-urban freight flowsand to determine if a clear pattern can be eventually identified.

Pursuing these two avenues requires conceptualizing inter-urban freight flows. Toward this end, we use the concept of urban hierarchy, which refers to a pyramid-shaped network dominated by very large urban centers whose economies are largely diversified. Formalized in the first half of the 20th century by [START_REF] Christaller | Die zentralen Orte in Süddeutschland: eine ökonomischgeographische Untersuchung über die Gesetzmässigkeit der Verbreitung und Entwicklung der Siedlungen mit städtischen Funktionen[END_REF], it was mainly developed in the 1950s and 1960s through a number of national studies. In these studies, urban hierarchy is analyzed using two main information sources: stocks and networks. In the analysis from stocks, population and activities in urban centers are studied to identify thresholds that form the boundaries between the different levels of urban hierarchy [START_REF] Berry | The functional bases of the central place hierarchy[END_REF]. In the analysis from networks, urban centers are distinguished by the size of their catchment areas and/or their position within systems of different kinds. For this article, we narrow our focus to the works about urban hierarchy that are based on networks and flows.

Transport Networks Studies: A Partitioned Approach of the Urban Hierarchy

Transport is a recurrent theme of urban hierarchy. In their quest to investigate the entire urban hierarchy, researchers tackle different means of transport separately. The work by [START_REF] Godlund | The function and growth of bus traffic within the sphere of urban influence[END_REF] in Sweden and by [START_REF] Green | Community of interest areas: Notes on the hierarchy of central places and their hinterlands[END_REF] in the United Kingdom shows how urban centers are hierarchically structured by a bus network. Godlund also shows that the position of individual Swedish urban centers within a bus network is strongly correlated with diversity of economic activities present in those urban centers. These two studies, focused on medium-sized and small urban areas, are rich in information about the organization of the lowest levels of the urban hierarchy because they can help to identify the boundaries of regional catchment areas. However, this type of approach has limitations when it comes to analyzing the catchment areas of large urban centers insofar as medium-and long-distance transport increasingly takes place by air. In this respect, the pioneering study by [START_REF] Taaffe | The urban hierarchy: An air passenger definition[END_REF] on air links between US urban centers provides an excellent counterpart to the work on bus connections. It shows the growth in direct air links among four big airports and all the other urban centers and demonstrates the dominant role of New York, Chicago, Los Angeles, and San Francisco in air links. This leads to a marginalization of many regional airports, which experience a weakening in their role as intermediaries between large and small airports. The decline of the intermediate level and the growth of the four main airports reveal the changing hierarchical organization of US urban centers.

Global City Studies: A Selective Approach of the Urban Hierarchy

This same hierarchical model of interurban relations reemerged more recently in the research field of "global cities," focusing less on transport-based approaches and more on the higher levels of hierarchy. [START_REF] Friedmann | The world city hypothesis[END_REF], for example, showed how the development of multinational firms, with their international division of labor, has increased the gap between urban centers. In his approach, based on the number of headquarters located in each urban center, global cities are those urban centers that concentrate the control functions of multinational firms. Urban centers are then organized in categories depending on their ranking on the different sectors of world economy, mainly on advanced producers and financial services.

The hierarchical approach of global cities has been widely criticized by scholars since the 1990s. [START_REF] Castells | The information age: Economy, society and culture (Vol. 1: The rise of the network society)[END_REF] argues that the global city phenomenon cannot be reduced to competition between large urban centers. According to him, non-hierarchical relationships of cooperation between urban centers have become more and more important in the age of cyberspace and cannot be neglected. More globally, hierarchical approaches are criticized by the lack of empirical work specifying how hierarchical ties between urban centers are organized [START_REF] Taylor | Regionality in the world city network[END_REF].

Contemporary studies of urban systems focus mainly on the higher levels of urban hierarchy (i.e., global cities) and on control functions of multinational firms (i.e., advanced producers and financial services). Small and medium-sized urban centers and other dimensions of economy, such as manufacturing and logistics, remain clearly underrepresented, except in a few cases that are essentially focused on port-city relationships [START_REF] Ducruet | Activité portuaire et villes[END_REF].

Freight Flows: Toward a More Inclusive and Integrated Approach to Urban Hierarchy

The purpose of this article is to help to fill this void by analyzing the physical links between the higher and lower levels of urban hierarchy in France. To achieve this goal, it examines the nature and intensity of interurban freight flows in France. It paints a simplified picture of the geography of freight flows with regard to the light they shed on economic interactions between urban centers. However, beyond simply deploying a new indicator to analyze the relations between urban areas, the article seeks to elucidate the consequences and implications of the location of the system of production on the transport system.

The rest of this article is structured as follows. The next section sets out our hypotheses and outlines the data and methods used. Section 3 describes the distribution of freight flows between urban areas as it relates to their size. It proposes a macroscopic approach to examining the spatial organization of productive systems. The final section offers some concluding remarks.

Measuring Interurban City Supply: Data and Method

In this section, we describe the research methodology employed. Our purpose to analyze freight flows between urban areas requires multiple methodological choiceswhich are discussed here. To ensure reliability and robustness of the findings, the research approach must be adapted to available data.

The Urban Area as the Unit of Observation of Interurban City Supply

Because this article looks at the question of inter-urban freight flows, its aim is significantly different from that of other articles on urban logistics. What we analyze here are not intra-urban movements between urban centers and the outskirts, but the relationship between different urban areas. We therefore felt it necessary to use a wider definition of urban entities to include the urban outskirts, where many of the entities that generate flows are located. The definition of the urban perimeter in this article is that used by INSEE (the French National Institute of Statistics and Economic Studies) and refers to the geographical definition of urban areas established in 1999. The urban area is a unit with more than 10,000 jobs, and a suburb is a unit in which at least 40% of the resident working population works within the central zone or within its catchment area. The principle of definition of the 354 French urban areas is similar to other functional urban regions defined abroad, as for example functional urban areas for Western Europe, metropolitan statistical areas in the United States, or urban employment areas in Japan.

Data on Flows

In this work we use the 2004 national shipper survey (ECHO), conducted by IFSTTAR (French Institute of Science and Technology for Transport, Development and Networks) [START_REF] Guilbault | Transport and logistics demand: New inputs from large shipper's surveys in France[END_REF][START_REF] Guilbault | Apports des enquêtes chargeurs: Connaissance des chaînes de transport de marchandises et de leurs déterminants logistiques[END_REF]. It provides a tool for observing demand by shippers and by flows leading from the shipper to the end consignee. The sample survey in the study consists of 2,935 plants and 10,462 shipments. The scope of the survey covers all firms in mainland France with ten employees or more, operating in "freight-related activities": wholesale trade (retailing is not included), manufacturing (excluding the extractive industries and public works), mail-order sales, agricultural cooperatives, warehousing services, and industrial waste processing centers. The ECHO survey is then focused mainly on business-to-business freight flows. In France, these freight-related activities account for 20% of total jobs. It should be noted that that the initial aim of the ECHO survey was not to be statistically representative of the different levels of urban hierarchy. The sampling plan used in this survey is based mainly on the type of activity and size (number of jobs) of firms.

This study is focused exclusively on inter-urban flows between French urban areas, so the data consist of 6,303 shipments and 114,000 tons (56% and 60% of the total sample survey). Intra-urban shipments (1,406) and shipments to foreign countries (2,753) have been excluded. So in all our sample covers some 70,000 plants and estimated transport volumes after adjustment of 641 million metric tons and 490 million shipments.

Choice of the Shipment as a Transport Measurement Unit

The use of the shipment as a transport measurement unit in addition to the conventional units of tonnes or tonne kilometres is an original feature of shipper surveys. The shipment is defined as a quantity of freight that is made available at a given time in order to be transported during a single transport operation from a given shipper to a given consignee and is the natural observation unit for reconstructing transport chains. However, this unit also has conceptual advantages. The shipment is the link between the shipper and the consignee. Likewise, it is a representation of transport that is closely integrated with the production and distribution process that informs us about the economic context of the exchange of goods. In addition, it provides us with a better perception of shipper practices and the logistical constraints that affect them. Shippers develop their transport policy not only on the basis of the tonnages they produce but also on the number and spatial dispersion of their clients and the frequency at which they send out shipments. There is a great difference between sending 1,000 tonnes of freight to a single consignee or 1,000 shipments of 1 tonne at different times to different consignees. The number of tonnes is the same, but the economic rationale and the type of transport differ in every respect. Likewise, by using the shipment as an observation unit, we are able to observe the activity of carriers in terms of the number of operations they perform. Doing so provides a picture of the carrier population, which is close to that based on the total turnover of transport firms.

The results from shipper surveys thus enable us to use a variety of measurement units, not only the number of tonnes or tonne kilometres but also the number of shipments or the number of shippers responsible for the flows in question. Passing from one to the other of these units is highly instructive and provides an understanding of the market that is markedly different depending on the preferred unit: the relative weight of the different sectors of activity in the generation of traffic changes completely, as do the transport modes.

Methodological Choices

Identifying Size: Classes of Urban Areas Based on Population

The fine stratification of the shipper survey by firm activity and size means that its traffic data are statistically representative at the national level. It lends to economic breakdown into major production types. But, the sampling plan does not include a regional breakdown or origin-destination analysis. This means that the shipper survey does not represent shipments at the level of each urban area. In order to overcome this deficiency, the urban areas where the firms' shipping and receiving freight flows are located were aggregated into five categories (see the appendix): a rural area and four urban area classes by population size: the Paris urban area; 18 large urban areas (400,000 to 1.6 million inhabitants), which include a dozen regional capitals outside the Paris area (i.e., Lyon, Marseille, Nice, Lille, Bordeaux, Toulouse, Strasbourg, etc.); 58 medium-sized urban areas (100,000-399,000 inhabitants); and 274 small urban areas (under 100,000). The total weight of each class was comparable in terms of population and jobs (Table 1). The average distance of urban areas has been calculated for every level of urban area, except Paris and rural areas, for which it cannot logically be calculated. The results show that the average distances within each level of urban area are very similar (Table 1).

Measuring Inter-Urban Freight Flows

In the ECHO shipper survey, the results can be presented with two complementary units: tonnage and shipment. Used alongside tonnage, the shipment provides an enhanced level of observation for reconstituting traffic chains and as a logistical indicator of shipper practices. Both these different levels of information are used in the article. We restricted our analysis of freight flows only to the two ends of the shipping chain, in other words, to consider only the urban area of origin, where the sender is located, and the urban area of destination, where the consignee is located. Intermediate logistics platforms in urban areas through which these shipments may transit are not included because this is a question of logistics rather than production or consumption. Finally, when looking at the results, it should always be remembered that internal flows within urban areas are never included. These intraurban flows represent 29% of shipments and 17% of tons and are logically more important in larger urban areas. Paris is a special case and calls for some explanations: (1) the surface of the urban area of Paris is very large (much more than the second largest one), so its intra-urban flows, which are not taken into account by our analysis, are important; (2) we find inside Paris several urban entities of different sizes, so the hypothesis of an internal urban hierarchy shouldn't be excluded; (3) because intra-urban flows have been excluded, Paris is therefore underrepresented in the matrix. Furthermore, freight transport flows are analyzed by the direction: from the origin to the destination of the flows (Table 2). Our research approach consists of a matrix that differentiates the five urban areas of origin and the five urban areas of destination. Thus the survey sample of flows is divided across the 25 classes that are in fact 24 because intra-urban flows of Paris are outside the scope of this research. To assess the shipper survey sample, we grouped the flows into three categories:

-Upward flows regroup flows that are directed to greater urban areas. For example, a flow from a small urban area to a large urban area is an upward flow. Within the upward flows, the methodology used to distinguish flows puts them immediately upward, which means that flows are directed to the level just above. -Downward flows regroup flows that are directed to smaller urban areas. For example, a flow from a medium urban area to a rural area is a downward flow. Downward and upward flows are referred as the hierarchic flows. -Horizontal flows aggregate flows between urban areas of the same size of the urban hierarchy.

It should not be confused with intra-urban flows, because horizontal flows are not within the same urban area. For this reason, the Paris urban area is excluded from the horizontal flows. The method used here provides a representative sample of the inter-urban freight flows to ensure reliability and robustness of the findings.

The Organization of Freight Flows among Size Classes of Urban Areas

This third part explores the interrelations, hierarchical or horizontal, between urban areas according to their size class. We try to determine if the jobs and the flows exhibit similar location patterns. The location pattern of economic activities should theoretically be reproduced in the pattern of freight flows by urban areas forming either horizontal or hierarchical networks.

Distributions of Jobs and Flows Are Similar

One can expect that the volume of flows generated by the different levels of the urban hierarchy is proportional to their economic potential. To examine this relationship, shipments, tonnes, and jobs are compared (Figure 1). Inbound and outbound flows are underrepresented in Paris and the large urban areas: their shares of shipments (33%) and tonnes (28%) are less important than their share of jobs (40%). By way of a corollary, this first finding confirms the importance of medium-sized and small urban areas and rural areas in structuring the geography of freight flows. In particular, the findings emphasize the structural role of medium-sized urban areas in terms of shipments and tonnage, and the rural areas, in tonnage but not in shipments.

The relations between flows and jobs vary with the direction of flows: inbound and outbound (Figure 1). Inbound flows seem to be more related to jobs than outbound ones (Figure 1a). The inbound flows of Paris are exceptionally limited (10% of shipments and tonnes) in view of its importance in jobs (19% of jobs). In addition to the underrepresentation of inbound flows in Paris, rural and small urban areas receive more flows than expected because of their jobs. Moreover rural inbound shipments are rather heavy (1.6 t). Outbound flows are less linked with jobs (Figure 1b) and exhibit a different pattern depending on the unit used: shipments or tonnes. Again, Paris flows are limited, but less limited in shipments than in tonnes, meaning that their average weight is rather light (0.9 t). Rural areas also generate relatively few shipments but their average weight is very heavy (2.5 t). This exceptional average weight is partly because of the importance of agricultural activities (production, wholesale trade), which tends to generate large shipments. The intensity of freight flows also varies depending on the level involved, reflecting the different roles of urban centers inside the freight system. Shipments are to be equally distributed between the levels of urban hierarchy (Table 3). Outbound shipments are very concentrated: level 3 (medium-sized urban areas) alone accounts for one-third of shipments and the share of the other levels generates 14% to 18%. Comparatively, inbound shipments are better distributed between the different levels of hierarchy (21% to 23%), with the exception of Paris (10%), where those are considerably underrepresented.

The distribution of tonnages between the levels of urban hierarchy varies less depending on the direction of the flows. Again, flows are noticeably underrepresented in Paris (10% of outbound and inbound tonnages). The main imbalances between inbound and outbound tonnages can be found at levels 2 (16% of outbound and 20% of inbound tonnage) and 3 (25% of outbound tonnage and 19% of inbound tonnage).

Shipments involving rural areas (level 5) are the heaviest, regardless to the direction of flows (2.5 t inbound, 1.6 t outbound). The deviation between rural and the other flows is higher for outbound (twice the average weight) than for inbound (1.2 the average weight). The light weight of the shipments shipped from levels 1 (0.9 t) and 3 (1 t) deserves to be highlighted.

The average distance of shipments also varies significantly depending on the level of urban hierarchy, and that is true for both directions. The shortest distance of shipments is 210 km (level 3) and the maximum distance is 309 km (level 1). The average distance of the shipments generated by the other levels of the urban hierarchy is close to the average value (between 231 km and 255 km). For inbound flows the distance varies regularly with the level of hierarchy: it goes from 306 km and 340 km in levels 1 and 2 to 165 for level 5. As showed in Table 1, the differences of scope of flows between levels cannot be attributed to the distances between urban areas, because the latter are rather the same for every level.

In order to provide a standardized view of our results, a normalized ratio between flows and employment has been calculated for each level of the urban hierarchy (Table 4). Values under 100 reflect situations in which the flows generated by the urban level are less important than expected owing to its economic potential (measured by the number of jobs). Conversely, values above 100 characterize situations in which flows are more important than expected. The strikingly low values of Paris (53.7 to 82.1) reflect its weakness as a generator of freight flows. On the contrary, the high values of medium-sized urban areas for shipments (112.7 to 170.4) reflect their strong influence on the French freight system. Rural areas are characterized by the underrepresentation of outbound shipments and overrepresentation of tonnes (both with regard to the inbound and outbound flows). All in all, the distribution of flows between the different levels of the urban hierarchy cannot be solely be attributed to the distribution of employment.

A System Dominated by Vertical Flows

The hierarchical analysis shows that vertical flows prevail (85% of shipments and 90% of tons). Inside this hierarchy, one in two shipments goes downward (51% of shipments, 42% of tons) and one in three to a higher-ranked entity (35% of shipments, 47% of tons) (Table 5). By contrast, urban and rural areas have few exchanges with same-ranked entities (horizontal flows, representing only 15% of shipments and 10% of tons). The matrix also highlights the nested pattern of flows: 39% of shipments and tons go to neighboring levels of urban hierarchy. The average weight of shipments is 1.3 tons and varies considerably depending on its hierarchical pattern. Horizontal shipments are lighter (0.9 t) than the average. Among vertical flows, upward shipments (1.8 t) are significantly heavier than downward ones (1.1 t). Shipments are conveyed, on average, over 245 km. This average distance varies little regardless to the hierarchical pattern. Upward flows are conveyed over longer distances (272 km) than downward ones (226 km). The length of horizontal flows is close to the average (250 km). All in all, these global results (which cover very different activities) show that freight flows are essentially hierarchical and confirm our main hypothesis. Flows take place mainly between urban areas of (very) different population sizes. Comparatively, ahierarchical flows are marginal. Inside the hierarchical flows, the share of shipments going immediately to the next level (directed to the level n-1 or n+1) is particularly high. It represents about a half of upward shipments and tonnages and slightly less for downward flows. This means that freight flows between urban areas are often organized in interlocking subsystems.

Regarding the weight, upward shipments (1.8 t) are considerably heavier than downward and horizontal ones (1.1 t and 0.9 t, respectively). Globally, the share of Paris (level 1) on outbound flows is twice as little as expected, except outbound shipments, in which it is slightly higher (16%) but still above the average (20%). Medium-sized urban areas (level 3) seem to play an important role in the generation of outbound shipments (33%), contrasting with the marginal role of rural areas (level 5, 14% of outbound shipments). The average distance of shipments varies with the urban hierarchy, more regularly for inbound than for outbound. Shipments departing from rural areas (2.5 t) are twice as heavy as the average (1.3 t). This overall analysis provides empirical evidence of hierarchical organization of freight flows between urban areas. It doesn't explain the causes, but simply the consequences in the spatial organization of freight flows.

DISCUSSION AND CONCLUSION

This article provides a new analysis of the organization of freight flows between French urban areas based on a shipper survey. It brings empirical evidence that confirms theoretical assumptions on the hierarchical pattern of inter-urban freight flows. Conclusions may be grouped in the three following ways.

The supply of urban areas is hierarchically organized: inter-urban flows take place mainly between urban areas of different levels. The characteristics of supplying vary a lot depending on the hierarchical level of urban area. Paris and the large urban areas are essentially supplied from smaller urban areas, and these shipments are lighter than the average. Conversely, small urban and rural areas are supplied from higher levels of urban hierarchy, and these shipments are heavier than the average. This result is consistent with the simultaneous decline of manufacturing and the development of distribution activities observed in large urban areas of developed countries since the 1980s. In this context, the main challenge for decision makers is to manage the rapid expansion of distribution activities, which have become increasingly concentrated in large urban areas (Guerrero & Proulhac, forthcoming).

Inside hierarchical flows, upward and downward flows are very different. On average upward shipments are almost twice as heavy and are carried over longer distances than downward ones.

The catchment areas of urban areas are linked to population size and reveal the central place structure. The average distance of freight flows is here interpreted as a proxy of the scope of the catchment areas of urban areas. These catchment areas vary directly with the population size. This relationship has been verified both for shipments and tonnes. Vertical ties are predominant: more than one in two shipments go to a lower-ranked entity and more than one in three to a higher-ranked entity. Comparatively, horizontal flows are less important than one would expect from recent works on urban networks.

In order to be statistically representative, the urban area of Paris has been considered here as a whole, even if it contains several important urban centers. The existence of a hierarchy of urban centers inside the urban area of Paris should not be excluded. This metropolitan issue will be tackled by a recent urban goods movement survey carried out in Paris.

The outcomes of the research confirm that productive systems are not disconnected from territories. Obviously, not all activities are equally organized. Further research is needed in order to understand how goods are carried between places, perhaps by zooming in a more disaggregated way on economic activities. Research improvements may be obtained by more reflection on the main activities of consignees. In fact, manufacturing flows can reflect both intra-sector links (intermediate products) and inter-sector links (finished products going to markets). Taking into account the activities of consignees will lead to a sharper differentiation of the patterns of manufacturing flows between urban areas [START_REF] Guerrero | Freight flows and urban hierarchy[END_REF]. Finally, these results are representative of the French situation and new research needs to be carried in other countries in order to provide more general conclusions.

Figure 1 :

 1 Figure 1: Share of Each Size Class in Jobs, Shipments and Tons. (A) Outbound (B) Inbound (Source: Authors, 2015)

Table 1 :

 1 Background Data on Urban Areas, Aggregated by Size Class (Source: Authors, 2015)

Table 2 :

 2 Scheme Explaining How Flows Have Been Aggregated in Our Analysis (Source: Authors, 2015)

Table 4 : Ratios between Flows and Employments (Source: Authors, 2015)

 4

Table 5 :

 5 Freight Flows between the Levels of the Urban Hierarchy; All Activities(Source: Authors, 2015)

Laurent Proulhac is a study engineer at the Université Paris Est (France) and Ifsttar in the Laboratory City Mobility Transport (LVMT). He also teaches the geography of transport systems and geographic information systems (GIS). He has published several papers on these topics in international peerreviewed academic journals and books. ,