Isaac Ohavi

QUASI LINEAR PARABOLIC PDE POSED ON A NETWORK WITH NON LINEAR NEUMANN BOUNDARY CONDITION AT VERTICES

The purpose of this article is to study quasi linear parabolic partial differential equations of second order, posed on a bounded network, satisfying a nonlinear and non dynamical Neumann boundary condition at the vertices. We prove the existence and the uniqueness of a classical solution.

Introduction

The purpose of this article is to study quasi linear parabolic partial differential equations of second order, posed on a bounded network, satisfying a nonlinear and non dynamical Neumann boundary condition at the vertices, also called the junction points. For simplicity of notation, we have focused on a network containing a single junction. More precisely, giving a final time T > 0, if I ∈ N * denotes the number of edges of the single junction, a i > 0 is the length of each edge i (i ∈ {1 . . . I}), and "0" denotes the junction point, the problem is reduced to:

                                       ∂ t u i (t, x) -σ i (x, ∂ x u i (t, x))∂ 2
x u i (t, x)+ H i (x, u i (t, x), ∂ x u i (t, x)) = 0, if (t, x) ∈ (0, T ) × (0, a i ),

F (u(t, 0), ∂ x u(t, 0)) = 0, if t ∈ [0, T ),
with u(t, 0) = (u 1 (t, 0), . . . , u I (t, 0)), ∂ x u(t, 0) = (∂ x u 1 (t, 0), . . . , ∂ x u I (t, 0)), and ∀(i, j) ∈ {1 . . . I} 2 , u i (t, 0) = u j (t, 0), ∀i ∈ {1 . . . I}, u i (t, a i ) = φ i (t), if t ∈ [0, T ],

∀i ∈ {1 . . . I}, u i (0, x) = g i (x), if x ∈ [0, a i ].

(

) 1
To simplify our study, we impose Dirichlet boundary condition (φ i ) at the a i . One can consider classical Neumann boundary conditions, without adding any mathematical technical issues for the problem [START_REF] Camilli | The vanishing viscosity limit for Hamilton-Jacobi equations on networks[END_REF]. The multi-junction setting involving the equations of type (1) can be treated with similar tools, with a Neumann boundary condition F (or a classical Dirichlet boundary condition) at the junction points. The well-known Kirchhoff law corresponds to the case where F is linear in ∂ x u and independent of u, (for instance:

F (∂ x u(t, 0)) = I i=1 ∂ x u i (t, 0) = 0). Originally introduced by Nikol'skii [START_REF] Nikol'skii | The properties of certain classes of functions of many variables on differentiable manifolds[END_REF] and Lumer [START_REF] Lumer | Connecting of local operators and evolution equations on networks[END_REF][START_REF] Lumer | Equations de diffusion sur des réseaux infinis[END_REF], the concept of ramified spaces and the analysis of partial differential equation on these spaces have attracted a lot of attention in the last 30 years. As explained in [START_REF] Nikol'skii | The properties of certain classes of functions of many variables on differentiable manifolds[END_REF], the main motivations are applications in physics, chemistry, and biology (for instance small transverse vibrations in a grid of strings, vibration of a grid of beams, drainage system, electrical equation with Kirchhoff law, wave equation, heat equation,...). Concerning applications in biology, we can cite for instance the recent works ( [START_REF] Du | The Fisher-KPP equation over simple graphs: Varied persistence states in river networks[END_REF], [START_REF] Jin | Population dynamics in river networks[END_REF] and [START_REF] Vasilyeva | Population dynamics in river networks: analysis of steady states[END_REF]), where equations of type [START_REF] Camilli | The vanishing viscosity limit for Hamilton-Jacobi equations on networks[END_REF] are used in the multi-junction case to modelize the dynamic of species in river networks.

Linear diffusions of the form [START_REF] Camilli | The vanishing viscosity limit for Hamilton-Jacobi equations on networks[END_REF], with a Kirchhoff law, are also naturally associated with stochastic processes living on graphs. These processes were introduced in the seminal papers [START_REF] Freidlin | Diffusion processes on graphs: stochastic differential equations, large deviation principle[END_REF] and [START_REF] Freidlin | Diffusion processes on an open book and the averaging principle[END_REF]. Another motivation for studying [START_REF] Camilli | The vanishing viscosity limit for Hamilton-Jacobi equations on networks[END_REF] is the analysis of associated stochastic optimal control problems with a control at the junction.

There have been several works on linear and quasilinear parabolic equations of the form [START_REF] Camilli | The vanishing viscosity limit for Hamilton-Jacobi equations on networks[END_REF]. For linear equations, von Below [START_REF] Below | Classical solvability of linear parabolic equations on networks[END_REF] shows that, under natural smoothness and compatibility conditions, linear boundary value problems posed on a junction with a linear Kirchhoff condition at the junction point is well-posed. The proof consists mainly in showing that the initial boundary value problem posed on a junction is equivalent to a well-posed initial boundary value problem for a parabolic system, where the boundary conditions are such that the classical results on linear parabolic equations [START_REF] Ladyzenskaja | Linear and Quasi-Linear equations of Parabolic type[END_REF] can be applied. The same author investigates in [START_REF] Below | A maximum principle for semi linear parabolic network equations[END_REF] the strong maximum principle for semi linear parabolic operators with Kirchhoff condition, while in [START_REF] Below | An existence result for semi linear parabolic network equations with dynamical node conditions[END_REF] he studies the classical global solvability for a class of semilinear parabolic equations on ramified networks, where a dynamical node condition is prescribed: Namely the Neumann condition at the junction point x = 0 in [START_REF] Camilli | The vanishing viscosity limit for Hamilton-Jacobi equations on networks[END_REF], is replaced by the dynamic one: ∂ t u(t, 0) + F (t, u(t, 0), ∂ x u(t, 0)) = 0.

In this way the application of classical estimates for domains established in [START_REF] Ladyzenskaja | Linear and Quasi-Linear equations of Parabolic type[END_REF] becomes possible. The author then establishes the classical solvability in the class C 1+α,2+α , with the aid of the Leray-Schauder-principle and the maximum principle of [START_REF] Below | A maximum principle for semi linear parabolic network equations[END_REF]. Let us note that this kind of proof fails for equation [START_REF] Camilli | The vanishing viscosity limit for Hamilton-Jacobi equations on networks[END_REF] because in this case one cannot expect an uniform bound for the term |∂ t u(t, 0)| (the proof of Lemma 3.1 of [START_REF] Ladyzenskaja | Linear and Quasi-Linear equations of Parabolic type[END_REF] VI.3 fails). Still in the linear setting, another approach, yielding similar existence results, was developed by Fijavz, Mugnolo and Sikolya in [START_REF] Fijavz | Variational and semigroup methods for waves and diffusion in networks[END_REF]: the idea is to use semi-group theory as well as variational methods to understand how the spectrum of the operator is related to the structure of the network.

Equations of the form (1) can also be analyzed in terms of viscosity solutions. The first results on viscosity solutions for Hamilton-Jacobi equations on networks have been obtained by Schieborn in [START_REF] Schieborn | Viscosity solutions of Hamilton-Jacobi equations of Eikonal type on ramified spaces[END_REF] for the Eikonal equations and later discussed in many contributions on first order problems [START_REF] Camilli | The vanishing viscosity limit for Hamilton-Jacobi equations on networks[END_REF][START_REF] Imbert | Generalized junction conditions for degenerate parabolic equations[END_REF][START_REF] Lions | Lectures at Collège de France[END_REF], elliptic equations [START_REF] Lions | Viscosity solutions for junctions: Well posedness and stability[END_REF] and second order problems with vanishing diffusion at the vertex [START_REF] Lions | Well posedness for multi-dimensional junction problems with Kirchhoff-type conditions[END_REF]. In contrast second order Hamilton-Jacobi equations with a non vanishing viscosity at the boundary have seldom been studied in the literature and our aim is to show the well-posedness of classical solutions for (1) in suitable Höder spaces: see Theorem 2.2 for the existence and Theorem 2.4 for the comparison, and thus the uniqueness. Our main assumptions are that the equation is uniformly parabolic with smooth coefficients and that the term F = F (u, p) at the junction is either decreasing with respect to u or increasing with respect to p. The main idea of the proof is to use a time discretization, exploiting at each step the solvability in C 2+α of the elliptic problem:

                         -σ i (x, ∂ x u i (x))∂ 2 x u i (x) + H i (x, u i (x), ∂ x u i (x)) = 0, if x ∈ (0, a i ) F (u(0), ∂ x u(0)) = 0, with u(0) = (u 1 (0), . . . , u I (0)), ∂ x u(0) = (∂ x u 1 (0), . . . , ∂ x u I (0)),
and

∀(i, j) ∈ {1 . . . I} 2 , u i (0) = u j (0), ∀i ∈ {1 . . . I}, u i (a i ) = φ i .
(

) 2
The paper is organized as follows. In section 2, we introduce the notations and state our main results. In Section 3, we review the mains results of existence and uniqueness of the elliptic problem [START_REF] Du | The Fisher-KPP equation over simple graphs: Varied persistence states in river networks[END_REF]. Finally Section 4, is dedicated to the proof of our main results.

main results

In this section we state our main result Theorem 2.2, on the solvability of the parabolic problem with Neumann boundary condition at the vertex, posed on a bounded junction [START_REF] Camilli | The vanishing viscosity limit for Hamilton-Jacobi equations on networks[END_REF], stated in Introduction:

                                       ∂ t u i (t, x) -σ i (x, ∂ x u i (t, x))∂ 2 x u i (t, x)+ H i (x, u i (t, x), ∂ x u i (t, x)) = 0, if (t, x) ∈ (0, T ) × (0, a i ), F (u(t, 0), ∂ x u(t, 0)) = 0, if t ∈ [0, T ), with u(t, 0) = (u 1 (t, 0), . . . , u I (t, 0)), ∂ x u(t, 0) = (∂ x u 1 (t, 0), . . . , ∂ x u I (t, 0)),
and

∀(i, j) ∈ {1 . . . I} 2 , u i (t, 0) = u j (t, 0), ∀i ∈ {1 . . . I}, u i (t, a i ) = φ i (t), if t ∈ [0, T ], ∀i ∈ {1 . . . I}, u i (0, x) = g i (x), if x ∈ [0, a i ]. (1)
There will be two typical assumptions for F = F (u, p): either F is decreasing with respect to u or F is increasing with respect to p (Kirchhoff conditions).

2.1. Notations and preliminary results. Let us start by introducing the main notation used in this paper as well as an interpolation result. Let I ∈ N * be the number of edges, and a = (a 1 , . . . a I ) ∈ (0, ∞) I be the length of each edge. The bounded junction J a is defined by

J a = I i=1 J a i i , with: ∀i ∈ {1 . . . I} J a i i := [0, a i ], and
∀(i, j) ∈ {1 . . . I} 2 , i = j, J a i i ∩ J a j j = {0}.
The intersection of the (J a i i ) 1≤i≤I is called the junction point and is denoted by 0. We identify all the points of J a by the couples (x, i) (with i ∈ {1 . . . I}, x ∈ |0, max i∈{1...I} a i ]), such that we have: (x, i) ∈ J a if and only if x ∈ J a i i . For T > 0, the time-space domain J a T is defined by

J a T = [0, T ] × J a .
The interior of J a T set minus the junction point 0 is denoted by

• J a
T , and is defined by

• J a T = (0, T ) × I i=1 • J a i i .
For the functionnal spaces that will be used in the sequel, we use here the notations of Chapter 1.1 of [START_REF] Ladyzenskaja | Linear and Quasi-Linear equations of Parabolic type[END_REF]. For the convenience of the reader, we recall these notations in Appendix A. In addition we introduce the parabolic Hölder space on the junction

C l 2 ,l (J a T ), . C l 2 ,l (J a T )
and the space C l 2 ,l b (

• J a T ), defined by (where l > 0, see Annexe A for more details)

C l 2 ,l (J a T ) := f : J a T → R, (t, (x, i)) → f i (t, x), ∀(i, j) ∈ {1 . . . I} 2 , ∀t ∈ (0, T ), f i (t, 0) = f j (t, 0), ∀i ∈ {1 . . . I}, (t, x) → f i (t, x) ∈ C l 2 ,l ([0, T ] × [0, a i ]) , C l 2 ,l b ( • J a T ) := f : J a T → R, (t, (x, i)) → f i (t, x), ∀i ∈ {1 . . . I}, (t, x) → f i (t, x) ∈ C l 2 ,l b ((0, T ) × (0, a i )) , with: u C l 2 ,l (J a T ) = 1≤i≤I u i C l 2 ,l ([0,T ]×[0,a i ])
.

We will use the same notations, when the domain does not depend on time, namely T = 0, Ω T = Ω, removing the dependence on the time variable.

We continue with the definition of a nondecreasing map F : R I → R. Let (x = (x 1 , . . . x I ), y = (y 1 . . . y I )) ∈ R 2I , we say that

x ≤ y, if ∀i ∈ {1 . . . I}, x i ≤ y i , and

x < y, if x ≤ y, and there exists j ∈ {1 . . . I}, x j < y j .

We say that

F ∈ C(R I , R) is nondecreasing if ∀(x, y) ∈ R I , if x ≤ y, then F (x) ≤ F (y), increasing if ∀(x, y) ∈ R I , if x < y, then F (x) < F (y).
Next we recall an interpolation inequality, which will be useful in the sequel.

∀(t, s) ∈ [0, T ] 2 , |t -s| ≤ 1, ∀x ∈ [0, R], |∂ x u(t, x) -∂ x u(s, x)| ≤ 2ν 2 ν 1 γν 2 γ 1+γ + 2ν 1 γν 2 ν 1 -1 1+γ |t -s| αγ 1+γ .
This is a special case of Lemma II.3.1, in [START_REF] Ladyzenskaja | Linear and Quasi-Linear equations of Parabolic type[END_REF], (see also [START_REF] Nikol'skii | The properties of certain classes of functions of many variables on differentiable manifolds[END_REF]). The main difference is that we are able to get global Hölder regularity in [0, T ] × [0, R] for ∂ x u in its first variable.

Let us recall that this kind of result fails in higher dimensions.

Proof. Let (t, s) ∈ [0, T ] 2 , with |t -s| ≤ 1, and x ∈ [0, R]. Suppose first that x ∈ [0, R 2 ]. Let y ∈ [0, R],
with y = x, we write:

∂ x u(t, x) -∂ x u(s, x) = 1 y -x y x (∂ x u(t, x) -∂ x u(t, z)) + (∂ x u(t, z) -∂ x u(s, z)) + (∂ x u(s, z) -∂ x u(s, x)) dz.
Using the Hölder condition in time satisfied by u, we have:

1 y -x y x (∂ x u(t, z) -∂ x u(s, z))dz ≤ 2ν 1 |t -s| α |y -x| .
On the other hand, using the Hölder regularity of ∂ x u in space satisfied, we have:

1 y -x y x (∂ x u(t, x) -∂ x u(t, z)) + (∂ x u(s, z) -∂ x u(s, x))dz ≤ 2ν 2 |y -x| γ .
It follows:

|∂ x u(t, x) -∂ x u(s, x)| ≤ 2ν 2 |y -x| γ + 2ν 1 |t -s| α |y -x| . Assuming that |t -s| ≤ ( 3R 2 ) 1+γ γν 2 ν 1 1 α ∧ 1, minimizing in y ∈ [0, R],
for y > x, the right side of the last equation, we get that the infimum is reached for

y * = x + ν 1 |t -s| α γν 2 1 1+γ
, and then:

|∂ x u(t, x) -∂ x u(s, x)| ≤ C(ν 1 , ν 2 , γ)|t -s| αγ 1+γ ,
where the constant C(ν 1 , ν 2 , γ), depends only on the data (ν 1 , ν 2 , γ), and is given by:

C(ν 1 , ν 2 , γ) = 2ν 2 ν 1 γν 2 γ 1+γ + 2ν 1 γν 2 ν 1 -1 1+γ .
For the cases y < x, and x ∈ [ R 2 , R], we argue similarly, which completes the proof.

2.2. Assumptions and main results. We state in this subsection the central Theorem of this note, namely the solvability and uniqueness of (1) in the class

C α 2 ,1+α (J a T ) ∩ C 1+ α 2 ,2+α b ( • J a T ).
In the rest of these notes, we fix α ∈ (0, 1). We introduce the following data

     F ∈ C 0 (R I × R I , R) g ∈ C 1 (J a ) ∩ C 2 b ( • J a )
,

and for each i ∈ {1 . . . I}            σ i ∈ C 1 ([0, a i ] × R, R) H i ∈ C 1 ([0, a i ] × R 2 , R) φ i ∈ C 1 ([0, T ], R)
.

We suppose furthermore that the data satisfy the following assumption

Assumption (P) (i) Assumption on F            a)
F is decreasing with respect to its first variable, b) F is nondecreasing with respect to its second variable,

c) ∃(b, B) ∈ R I × R I , F (b, B) = 0,
or satisfies the Kirchhoff condition

           a)
F is nonincreasing with respect to its first variable, b) F is increasing with respect to its second variable,

c) ∃(b, B) ∈ R I × R I , F (b, B) = 0.
We suppose moreover that there exists a parameter m ∈ R, m ≥ 2 such that we have (ii) The (uniform) ellipticity condition on the (σ i ) i∈{1...I} : there exist ν, ν, strictly positive constants such that:

∀i ∈ {1 . . . I}, ∀(x, p) ∈ [0, a i ] × R, ν(1 + |p|) m-2 ≤ σ i (x, p) ≤ ν(1 + |p|) m-2 .
(iii) The growth of the (H i ) i∈{1...I} with respect to p exceed the growth of the σ i with respect to p by no more than two, namely there exists µ an increasing real continuous function such that:

∀i ∈ {1 . . . I}, ∀(x, u, p) ∈ [0, a i ] × R 2 , |H i (x, u, p)| ≤ µ(|u|)(1 + |p|) m .
(iv) We impose the following restrictions on the growth with respect to p of the derivatives for the coefficients (σ i , H i ) i∈{1...I} , which are for all i ∈ {1 . . . I}:

a) |∂ p σ i | [0,a i ]×R 2 (1 + |p|) 2 + |∂ p H i | [0,a i ]×R 2 ≤ γ(|u|)(1 + |p|) m-1 , b) |∂ x σ i | [0,a i ]×R 2 (1 + |p|) 2 + |∂ x H i | [0,a i ]×R 2 ≤ ε(|u|) + P (|u|, |p|) (1 + |p|) m+1 , c) ∀(x, u, p) ∈ [0, a i ] × R 3 , -C H ≤ ∂ u H i (x, u, p) ≤ ε(|u|) + P (|u|, |p|) (1 + |p|) m ,
where γ and ε are continuous non negative increasing functions. P is a continuous function, increasing with respect to its first variable, and tends to 0 for p → +∞, uniformly with respect to its first variable, from [0, u 1 ] with u 1 ∈ R, and C H > 0 is real strictly positive number. We assume that (γ, ε, P, C H ) are independent of i ∈ {1 . . . I}.

(v) A compatibility conditions for g and (φ i ) {1...I} :

F (g(0), ∂ x g(0)) = 0 ; ∀i ∈ {1 . . . I}, g i (a i ) = φ i (0). Theorem 2.2. Assume (P). Then system (1) is uniquely solvable in the class C α 2 ,1+α (J a T )∩ C 1+ α 2 ,2+α b ( • J a T ).
There exist constants (M 1 , M 2 , M 3 ), depending only the data introduced in assumption (P),

M 1 = M 1 max i∈{1...I} sup x∈(0,a i ) | -σ i (x, ∂ x g i (x))∂ 2 x g i (x) + H i (x, g i (x), ∂ x g i (x))| + |∂ t φ i | (0,T ) , max i∈{1...I} |g i | (0,a i ) , C H , M 2 = M 2 ν, ν, µ(M 1 ), γ(M 1 ), ε(M 1 ), sup |p|≥0 P (M 1 , |p|), max i∈{1...I} |∂ x g i | (0,a i ) , M 1 , M 3 = M 3 M 1 , ν(1 + |p|) m-2 , µ(|u|)(1 + |p|) m , |u| ≤ M 1 , |p| ≤ M 2 , such that ||u|| C(J a T ) ≤ M 1 , ||∂ x u|| C(J a T ) ≤ M 2 , ||∂ t u|| C(J a T ) ≤ M 1 , ||∂ 2 x u|| C(J a T ) ≤ M 3 .
Moreover, there exists a constant M(α)

depending on α, M 1 , M 2 , M 3 such that ||u|| C α 2 ,1+α (J a T ) ≤ M(α).
We continue this Section by giving the definitions of super and sub solution, and stating a comparison Theorem for our problem.

Definition 2.3. We say that u ∈ C 0,1 (J a T ) ∩ C 1,2 ( • J a T ), is a super solution (resp. sub solution) of            ∂ t u i (t, x) -σ i (x, ∂ x u i (t, x))∂ 2 x u i (t, x)+ H i (x, u i (t, x), ∂ x u i (t, x)) = 0, if (t, x) ∈ (0, T ) × (0, a i ), F (u(t, 0), ∂ x u(t, 0)) = 0, if t ∈ (0, T ), ( 3
) if            ∂ t u i (t, x) -σ i (x, ∂ x u i (t, x))∂ 2 x u i (t, x)+ H i (x, u i (t, x), ∂ x u i (t, x)) ≥ 0, (resp. ≤ 0), ∀(t, x) ∈ (0, T ) × (0, a i ), F (u(t, 0), ∂ x u(t, 0)) ≤ 0, (resp. ≥ 0), ∀t ∈ (0, T ) Theorem 2.4. Parabolic comparison. Assume (P). Let u ∈ C 0,1 (J a T ) ∩ C 1,2 b ( • J a T ) (resp. v ∈ C 0,1 (J a T ) ∩ C 1,2 b ( • J a T )) a super solution (resp. a sub solution) of (3), satisfying for all i ∈ {1 . . . I}, u i (t, a i ) ≥ v i (t, a i ), for all t ∈ [0, T ], and u i (0, x) ≥ v i (0, x), for all x ∈ [0, a i ].
Then for each (t, (x, i)) ∈ J a T :

u i (t, x) ≥ v i (t, x).
Proof. We start by showing that for each 0 ≤ s < T , for all (t, (x, i))

∈ J a s , u i (t, x) ≥ v i (t, x). Let λ > 0. Suppose that λ > C 1 + C 2 ,
where the expression of the constants (C 1 , C 2 ) are given in the sequel (see [START_REF] Freidlin | Diffusion processes on graphs: stochastic differential equations, large deviation principle[END_REF], and ( 5)). We argue by contradiction assuming that sup

(t,(x,i))∈J a s exp(-λt + x) v i (t, x) -u i (t, x) > 0.
Using the boundary conditions satisfied by u and v, the supremum above is reached at a point (t 0 , (x 0 , j 0 )) ∈ (0, s] × J , with 0 ≤ x 0 < a j 0 .

Suppose first that x 0 > 0, the optimality conditions imply that exp(-

λt 0 + x 0 ) -λ(v j 0 (t 0 , x 0 ) -u j 0 (t 0 , x 0 )) + ∂ t v j 0 (t 0 , x 0 ) -∂ t u j 0 (t 0 , x 0 ) ≥ 0, exp(-λt 0 + x 0 )) v j 0 (t 0 , x 0 ) -u j 0 (t 0 , x 0 ) + ∂ x v j 0 (t 0 , x 0 ) -∂ x u j 0 (t 0 , x 0 ) = 0, exp(-λt 0 + x 0 ) v j 0 (t 0 , x 0 ) -u j 0 (t 0 , x 0 ) + 2 ∂ x v j 0 (t 0 , x 0 ) -∂ x u j 0 (t 0 , x 0 ) + ∂ 2 x v j 0 (t 0 , x 0 ) -∂ 2 x u j 0 (t 0 , x 0 ) = exp(-λt 0 + x 0 ) -v j 0 (t 0 , x 0 ) -u j 0 (t 0 , x 0 ) + ∂ 2 x v j 0 (t 0 , x 0 ) -∂ 2 x u j 0 (t 0 , x 0 ) ≤ 0.
Using assumptions (P) (iv) a), (iv) c) and the optimality conditions above we have

H j 0 (x 0 , u i (t 0 , x 0 ), ∂ x u j 0 (t 0 , x 0 )) -H j 0 (x 0 , v j 0 (t 0 , x 0 ), ∂ x v j 0 (t 0 , x 0 )) ≤ v j 0 (t 0 , x 0 ) -u j 0 (t 0 , x 0 ) C H + γ(|∂ x v j 0 (t 0 , x 0 )|) (1 + |∂ x u j 0 (t 0 , x 0 ))| ∨ |∂ x v j 0 (t 0 , x 0 ))|) m-1 ≤ C 1 v j 0 (t 0 , x 0 ) -u j 0 (t 0 , x 0 ) ,
where

C 1 := max i∈{1...I} sup (t,x)∈[0,T ]×[0,a i ] C H + γ(|∂ x v i (t, x)| 1 + |∂ x u i (t, x))| ∨|∂ x v i (t, x))| m-1 . (4)
On the other hand we have using assumption (P) (ii), (iv) a), (iv) c), and the optimality conditions

σ j 0 (x 0 , ∂ x v j 0 (t 0 , x 0 ))∂ 2 x v j 0 (t 0 , x 0 ) -σ j 0 (x 0 , ∂ x u j 0 (t 0 , x 0 ))∂ 2 x u j 0 (t 0 , x 0 ) ≤ v j 0 (t 0 , x 0 ) -u j 0 (t 0 , x 0 ) ν(1 + |∂ x v j 0 (t 0 , x 0 )|) m-2 + ∂ 2 x u j 0 (t 0 , x 0 ) + γ(|∂ x u j 0 (t 0 , x 0 )|)(1 + |∂ x u j 0 (t 0 , x 0 ))| ∨ |∂ x v j 0 (t 0 , x 0 ))|) m-1 ≤ C 2 v j 0 (t 0 , x 0 ) -u j 0 (t 0 , x 0 ) ,
where

C 2 := max i∈{1...I} sup (t,x)∈[0,T ]×[0,a i ] ν(1 + |∂ x v i (t, x)|) m-2 + ∂ 2 x u i (t, x) + γ(|∂ x u i (t, x)|)(1 + |∂ x u i (t, x))| + |∂ x v i (t, x))|) m-1 . ( 5
)
Using now the fact that v is a sub-solution while u is a super-solution, we get

0 ≤ ∂ t u j 0 (t 0 , x 0 ) -σ j 0 (x 0 , ∂ x u j 0 (t 0 , x 0 ))∂ 2 x u j 0 (t 0 , x 0 ) + H j 0 (x 0 , u i (t 0 , x 0 ), ∂ x u j 0 (t 0 , x 0 )) -∂ t v j 0 (t 0 , x 0 ) + σ j 0 (x 0 , ∂ x v j 0 (t 0 , x 0 ))∂ 2 x v j 0 (t 0 , x 0 ) -H j 0 (x 0 , v j 0 (t 0 , x 0 ), ∂ x v j 0 (t 0 , x 0 )) ≤ -(λ -(C 1 + C 2 ))(v j 0 (t 0 , x 0 ) -u j 0 (t 0 , x 0 )) < 0,
which is a contradiction. Therefore the supremum is reached at (t 0 , 0), with t 0 ∈ (0, s].

We apply a first order Taylor expansion in space, in the neighborhood of the junction point 0. Since for all (i, j) ∈ {1 . . . I}, u i (t 0 , 0) = u j (t 0 , 0), and v i (t 0 , 0) = v j (t 0 , 0), we get from

∀(i, j) ∈ {1, . . . I} 2 , ∀h ∈ (0, min i∈{1...I} a i ] v j (t 0 , 0) -u j (t 0 , 0) ≥ exp(h) v i (t 0 , h) -u i (t 0 , h) , that ∀(i, j) ∈ {1, . . . I} 2 , ∀h ∈ (0, min i∈{1...I} a i ] v j (t 0 , 0) -u j (t 0 , 0) ≥ v i (t 0 , 0) -u i (t 0 , 0) + h v i (t 0 , 0) -u i (t 0 , 0) + ∂ x v i (t 0 , 0) -∂ x u i (t 0 , 0) + hε i (h),
where ∀i ∈ {1, . . . I}, lim h→0 ε i (h) = 0.

We get then

∀i ∈ {1, . . . I}, ∂ x v i (t 0 , 0) ≤ ∂ x u i (t 0 , 0) -v i (t 0 , 0) -u i (t 0 , 0) < ∂ x u i (t 0 , 0).
Using the growth assumptions on F (assumption (P)(i)), and the fact that v is a subsolution while u is a super-solution, we get

0 ≤ F (v(t 0 , 0), ∂ x v(t 0 , 0)) < F (u(t 0 , 0), ∂ x u(t 0 , 0)) ≤ 0,
and then a contradiction.

We deduce then for all 0 ≤ s < T , for all (t, (x, i))

∈ [0, s] × J a , exp(-λt + x) v i (t, x) -u i (t, x) ≤ 0.
Using the continuity of u and v, we deduce finally that for all (t, (x, i))

∈ [0, T ] × J a , v i (t, x) ≤ u i (t, x).

The elliptic problem

As explained in the introduction, the construction of a solution for our parabolic problem (1) relies on a time discretization and on the solvability of the associated elliptic problem. We review in this section the well-posedness of the elliptic problem (2):

                   -σ i (x, ∂ x u i (x))∂ 2 x u i (x) + H i (x, u i (x), ∂ x u i (x)) = 0, if x ∈ (0, a i ) F (u(0), ∂ x u(0)) = 0, with ∀(i, j) ∈ {1 . . . I} 2 , u i (0) = u j (0), and
u(0) = (u 1 (0), . . . , u I (0)), ∂ x u(0) = (∂ x u 1 (0), . . . , ∂ x u I (0)), ∀i ∈ {1 . . . I}, u i (a i ) = φ i . (2)
We introduce the following data for i ∈ {1 . . . I}

                   F ∈ C 0 (R I × R I , R) σ i ∈ C 1 ([0, a i ] × R, R) H i ∈ C 1 ([0, a i ] × R 2 , R) φ i ∈ R
, satisfying the following assumption

Assumption (E) (i) Assumption on F            a) F is decreasing with respect to its first variable, b) F is nondecreasing with respect to its second variable, c) ∃(b, B) ∈ R I × R I , such that : F (b, B) = 0, or F satisfy the Kirchhoff condition            a)
F is nonincreasing with respect to its first variable, b) F is increasing with respect to its second variable,

c) ∃(b, B) ∈ R I × R I , such that : F (b, B) = 0.
(ii) The ellipticity condition on the

σ i ∃c > 0, ∀i ∈ {1 . . . I}, ∀(x, p) ∈ [0, a i ] × R, σ i (x, p) ≥ c.
(iii) For the Hamiltonians H i , we suppose

∃C H > 0, ∀i ∈ {1 . . . I}, ∀(x, u, v, p) ∈ (0, a i ) × R 3 , if u ≤ v, C H (u -v) ≤ H i (x, u, p) -H i (x, v, p).
For each i ∈ {1 . . . I}, we define the following differential operators

(δ i , δ i ) i∈{1...I} acting on C 1 ([0, a i ] × R 2 , R), for f = f (x, u, p) by δ i := ∂ u + 1 p ∂ x ; δ i := p∂ p .
(iv) We impose the following restrictions on the growth with respect to p for the coefficients

(σ i , H i ) i∈{1...I} = (σ i (x, p), H i (x, u, p)) i∈{1...I} , which are for all i ∈ {1 . . . I} δ i σ i = o(σ i ), δ i σ i = O(σ i ), H i = O(σ i p 2 ), δ i H i ≤ o(σ i p 2 ), δ i H i ≤ O(σ i p 2 ),
where the limits behind are understood as p → +∞, uniformly in x, for bounded u.

The main result of this section is the following Theorem, for the solvability and uniqueness of the elliptic problem posed on the junction, with non linear Neumann condition at the junction point.

Theorem 3.1. Assume (E). Then system (2), is uniquely solvable in the class C 2+α (J a ).

Theorem 3.1 is stated without proof in [START_REF] Lions | Viscosity solutions for junctions: Well posedness and stability[END_REF]. For the convenience of the reader, we sketch its proof in the Appendix. The uniqueness of the solution of (2), is a consequence of the elliptic comparison Theorem for smooth solutions, for the Neumann problem, stated in this Section, and whose proof uses the same arguments of the proof of the parabolic comparison Theorem 2.4.

We complete this section by recalling the definition of super and sub solution for the elliptic problem (2), and the corresponding elliptic comparison Theorem.

Definition 3.2. Let u ∈ C 2 (J a ). We say that u is a super solution (resp. sub solution) of      -σ i (x, ∂ x f i (x))∂ 2 x f i (x) + H i (x, f i (x), ∂ x f i (x)) = 0, if x ∈ (0, a i ), F (f (0), ∂ x f (0)) = 0, ( 6
) if      -σ i (x, ∂ x u i (x))∂ 2 x u i (x) + H i (x, u i (x), ∂ x u i (x)) ≥ 0, (resp. ≤ 0), if x ∈ (0, a i ), F (u(0), ∂ x u(0)) ≤ 0, (resp. ≥ 0).
Theorem 3.3. Elliptic comparison Theorem, see for instance Theorem 2.1 of [START_REF] Lions | Viscosity solutions for junctions: Well posedness and stability[END_REF].

Assume (E). Let u ∈ C 2 (J a ) (resp. v ∈ C 2 (J a
)) a super solution (resp. a sub solution) of [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF], satisfying for all i ∈ {1 . . . I}, u i (a i ) ≥ v i (a i ). Then for each (x, i) ∈ J a :

u i (x) ≥ v i (x).

The parabolic problem

In this Section, we prove Theorem 2.2. The construction of the solution is based on the results obtained in Section 3 for the elliptic problem, and is done by considering a sequence u n ∈ C 2 (J a ), solving on a time grid an elliptic scheme defined by induction. We will prove that the solution u n converges to the required solution.

4.1. Estimates on the discretized scheme. Let n ∈ N * , we consider the following time grid, (t n k = kT n ) 0≤k≤n of [0, T ], and the following sequence (u k ) 0≤k≤n of C 2+α (J a ), defined recursively by for k = 0, u 0 = g, and for 1 ≤ k ≤ n, u k is the unique solution of the following elliptic problem

                   n(u i,k (x) -u i,k-1 (x)) -σ i (x, ∂ x u i,k (x))∂ 2 x u i,k (x))+ H i (x, u i,k (x), ∂ x u i,k (x)) = 0, if x ∈ (0, a i ), F (u k (0), ∂ x u k (0)) = 0, ∀i ∈ {1 . . . I}, u i,k (a i ) = φ i (t n k ). ( 7
)
The solvability of the elliptic scheme (7) can be proved by induction, using the same arguments as for Theorem 3.1. The next step consists in obtaining uniform estimates of (u k ) 0≤k≤n . We start first by getting uniform bounds for n|u i,k -u i,k-1 | (0,a i ) using the comparison Theorem 3.3.

Lemma 4.1. Assume (P). There exists a constant C > 0, independent of n, depending

only the data C = C max i∈{1...I} sup x∈(0,a i ) |-σ i (x, ∂ x g i (x))∂ 2 x g i (x)+H i (x, g i (x), ∂ x g i (x))|+ |∂ t φ i | (0,T ) , C H , such that: sup n≥0 max k∈{1...n} max i∈{1...I} n|u i,k -u i,k-1 | (0,a i ) ≤ C,
and then

sup n≥0 max k∈{0...n} max i∈{1...I} |u i,k | (0,a i ) ≤ C + max i∈{1...I} |g i | (0,a i ) .
Proof. Let n > ⌊C H ⌋, where C H is defined in assumption (P) (iv) c). Let k ∈ {1 . . . n}, we define the following sequence:

     M 0 = max i∈{1...I} sup x∈(0,a i ) | -σ i (x, ∂ x g i (x))∂ 2 x g i (x) + H i (x, g i (x), ∂ x g i (x))| + |∂ t φ i | (0,T ) , M k,n = n n -C H M k-1,n , k ∈ {1 . . . n}.
We claim that for each k ∈ {1 . . . n}:

max i∈{1...I} n|u i,k -u i,k-1 | (0,a i ) ≤ M k,n .
We give a proof by induction. For this, if k = 1, let us show that the map h defined on the junction by:

h :=      J a → R (x, i) → M 1,n n + g i (x),
is a super solution of [START_REF] Imbert | Generalized junction conditions for degenerate parabolic equations[END_REF], for k = 1. For this we will use the Elliptic Comparison Theorem

3.3.

Using the compatibility conditions satisfied by g, namely assumption (P) (v), and the assumptions of growth on F , assumption (P) (i), we get for the boundary conditions:

F (h(0), ∂ x h(0)) ≤ F (g(0), ∂ x g(0)) = 0, ∀i ∈ {1 . . . I}, h i (a i ) = M 1,n n + g i (a i ) ≥ M 0,n n + g i (a i ) ≥ φ i (t n 1 ).
For all i ∈ {1 . . . I}, and x ∈ (0, a i ), we get using assumption (P) (iii):

n(h i (x) -g i (x)) -σ i (x, ∂ x h i (x))∂ 2 x h i (x) + H i (x, h i (x), ∂ x h i (x)) = M 1,n -σ i (x, ∂ x g i (x))∂ 2 x g i (x) + H i (x, M 1,n n + g i (x), ∂ x g i (x)) ≥ M 1,n -σ i (x, ∂ x g i (x))∂ 2 x g i (x) + H i (x, g i (x), ∂ x g i (x)) - M 1,n C H n ≥ 0.
It follows from the comparison Theorem 3.3, that for all i ∈ {1 . . . I}, and x ∈ [0, a i ]:

u 1,i (x) ≤ M 1,n n + g i (x).
Using the same arguments, we show that:

h :=      J a → R (x, i) → -M 1,n n + g i (x),
is a sub solution of ( 7) for k = 1, and we then get:

max i∈{1...I} sup x∈(0,a i ) n|u 1,i (x) -g i (x)| ≤ M 1,n .
Let 2 ≤ k ≤ n, suppose that the assumption of induction holds true.Let us show that the following map:

h :=      J a → R (x, i) → M k,n n + u i,k-1 (x),
is a super solution of [START_REF] Imbert | Generalized junction conditions for degenerate parabolic equations[END_REF]. For the boundary conditions, using assumption (P) (i), we get:

F (h(0), ∂ x h(0)) ≤ F (u k-1 (0), ∂ x u k-1 (0)) ≤ 0, ∀i ∈ {1 . . . I}, h i (a i ) = M k,n n + u i,k-1 (a i ) ≥ M 0,n n + u i,k-1 (a i ) ≥ φ i (t n k ).
For all i ∈ {1 . . . I}, and x ∈ (0, a i ):

n(h i (x) -u i,k-1 (x)) -σ i (x, ∂ x h(x))∂ 2 x h(x) + H i (x, h(x), ∂ x h(x)) = M k,n -σ i (x, ∂ x u i,k-1 (x))∂ 2 x u i,k-1 (x) + H i (x, M k,n n + u i,k-1 (x), ∂ x u i,k-1 (x)) ≥ M k,n -σ i (x, ∂ x u i,k-1 (x))∂ 2 x u i,k-1 (x) + H i (x, u i,k-1 (x), ∂ x u i,k-1 (x)) - C H M k,n n .
Since we have for all x ∈ (0, a i ):

-σ i (x, ∂ x u i,k-1 (x))∂ 2 x u i,k-1 (x) + H i (x, u i,k-1 (x), ∂ x u i,k-1 (x)) = -n(u i,k-1 (x) -u i,k-2 (x)),
using the induction assumption we get:

n(h i (x) -u i,k-1 (x)) -σ i (x, ∂ x h(x))∂ 2 x h(x) + H i (x, ∂ x h(x), ∂ x h(x)) ≥ M k,n -n(u i,k-1 (x) -u i,k-2 (x)) - C H M k,n n ≥ M k,n n -C H n -M k-1,n ≥ 0.
It follows from the comparison Theorem 3.3, that for all (x, i) ∈ J a :

u i,k (x) ≤ M k,n n + u i,k-1 (x).
Using the same arguments, we show that:

h :=      J a → R (x, i) → - M k,n n + u i,k-1 (x),
is a sub solution of ( 7), and we get:

max i∈{1...I} n|u i,k (x) -u i,k-1 (x)| (0,a i ) ≤ M k,n .
We obtain finally using that for all k ∈ {1 . . . n}:

     M k,n ≤ M n,n , M k,n = n n -C H k M 0 , and M n,n n→+∞ ----→ C := exp(C H ) max i∈{1...I} sup x∈(0,a i ) | -σ i (x, ∂ x g i (x))∂ 2 x g i (x) + H i (x, g i (x), ∂ x g i (x))| + |∂ t φ i | (0,T ) , that sup n≥0 max k∈{1...n} max i∈{1...I} n|u i,k -u i,k-1 | (0,a i ) ≤ C, sup n≥0 max k∈{0...n} max i∈{1...I} |u i,k | (0,a i ) ≤ C + max i∈{1...I} |g i | (0,a i ) .
That completes the proof.

The next step consists in obtaining uniform estimates for |∂ x u i,k | (0,a i ) , in terms of n|u i,ku i,k-1 | (0,a i ) and the quantities (ν, ν, µ, γ, ε, P ) introduced in assumption (P) (ii), (iii) and (iv). More precisely, we use similar arguments as for the proof of Theorem 14.1 of [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF],

using a classical argument of upper and lower barrier functions at the boundary. The assumption of growth (P) (ii) and (iii) are used in a key way to get an uniform bound on the gradient at the boundary. Finally to conclude, we appeal to a gradient maximum principle, using the growth assumption (P) (iv), adapting Theorem 15.2 of [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF]

|u i,k | (0,a i ) + n|u i,k -u i,k-1 | (0,a i ) ≤ M.
We fix i ∈ {1 . . . I}. We apply a barrier method consisting in building two functions

w + i,k , w - i,k
satisfying in a neighborhood of 0, for example [0, κ], with κ ≤ a i :

Q i (x, w + i,k (x), ∂ x w + i,k (x), ∂ 2 x w + i,k (x)) ≥ 0, ∀x ∈ [0, κ], w + i,k (0) = u i,k (0), w + i,k (κ) ≥ M, Q i (x, w - i,k (x), ∂ x w - i,k (x), ∂ 2 x w - i,k (x)) ≤ 0, ∀x ∈ [0, κ], w - i,k (0) = u i,k (0), w - i,k (κ) ≤ -M,
where we recall that for each (x, u, p, S) ∈ [0, a i ] × R 3 :

Q i (x, u, p, S) = n(u -u i,k-1 (x)) -σ i (x, p)S + H i (x, u, p).
For n > ⌊C H ⌋, where C H is defined in assumption P (iv) c), it follows then from the comparison principle that:

w - i,k (x) ≤ u i,k (x) ≤ w + i,k (x), ∀x ∈ [0, κ],
and then:

∂ x w - i,k (0) ≤ ∂ x u i,k (0) ≤ ∂ x w + i,k (0).
We look for w + i,k defined on [0, κ] of the form:

w + i,0 = g i (x) w + i,k : x → u i,k (0) + 1 β ln(1 + θx),
where the constants (β, θ, κ) will be chosen in the sequel independent of k. Remark first that for all x ∈ [0, κ], ∂ 2 x w + i,k (x) = -β∂ x w + i,k (x) 2 , and w + i,k (0) = u i,k (0). Let us choose (θ, κ), such that:

∀k ∈ {1 . . . n}, 0 < κ ≤ min i∈{1...I} a i , w + i,k (κ) ≥ M, ∂ x w + i,k (κ) ≥ β. (8)
We choose for instance:

θ = β 2 exp(2βM) + 1 min i∈{1...I} a i exp(2βM) κ = 1 θ exp(2βM) -1 . (9)
The constant β will be chosen in order to get:

β ≥ sup k∈{1...n} sup x∈[0,κ] µ(w + i,k (x))(1 + ∂ x w + i,k (x)) m + M ν(1 + ∂ x w + i,k (x)) m-2 ∂ x w + i,k (x) 2 , (10)
where (µ(.), ν, m) are defined in assumption (P) (ii) and (iii). Since we have:

∀x ∈ [0, κ], w + i,k (x) ≤ w + i,k (κ) = 2M, β ≤ ∂ x w + i,k (κ) ≤ ∂ x w + i,k (x) ≤ ∂ x w + i,k (0),
we can then choose β large enough to get [START_REF] Lions | Lectures at Collège de France[END_REF], for instance:

β ≥ µ(2M) ν 1 + 1 β 2 + M νβ 2 .
It is easy to show by induction that w + i,k is lower barrier of u i,k in the neighborhood [0, κ]. More precisely, since w + i,0 = u i,0 , and for all k ∈ {1 . . . n}:

w + i,k (0) = u i,k (0), w + i,k (κ) ≥ u i,k (κ), w + i,k (x) = w + i,k-1 (x) + u i,k (0) -u i,k-1 (0) ≥ w + i,k-1 (x) - M n ,
we get using the assumption of induction, assumption (P) (ii) and (iii), and ( 10) that for all x ∈ (0, κ):

n(w + i,k (x) -u i,k-1 (x)) -σ i (x, ∂ x w + i,k (x))∂ 2 x w + i,k (x) + H i (x, w + i,k (x), ∂ x w + i,k (x)) ≥ -M + βσ i (x, ∂ x w + i,k (x))∂ x w + i,k (x) 2 + H i (x, w + i,k (x), ∂ x w + i,k (x)) ≥ -M + βν(1 + ∂ x w + i,k (x)) m-2 ∂ x w + i,k (x) 2 + µ(w + i,k (x))(1 + ∂ x w + i,k (x)) m ≥ 0.
We obtain therefore:

sup n≥0 max k∈{0...n} max i∈{1...I} ∂ x u i,k (0) ≤ θ β ∨ max i∈{1...I} ∂ x g i (0).
With the same arguments we can show that:

w - i,0 = g i (x) w - i,k : x → u i,k (0) - 1 β ln(1 + θx),
is a lower barrier in the neighborhood of 0. Using the same method, we can show that

∂ x u i,k (a i
) is uniformly bounded by the same upper bounds, which completes the proof of

Step 1.

Step 2 : For the convenience of the reader, we do not detail all the computations of this Step, since they can be found in the proof of Theorem 15.2 of [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF]. It follows from Lemma 4.1 that there exists M > 0 such that:

sup n≥0 max k∈{0...n} max i∈{1...I} |u i,k | (0,a i ) ≤ M.
We set furthermore:

∀(x, u, p) ∈ [0, a i ] × R 2 , H n i,k (x, u, p) = n(u -u i,k-1 (x)) + H i (x, u, p).
Let u be a solution of the elliptic equation, for x ∈ (0, a i ):

σ i (x, ∂ x u(x))∂ x u(x) -H n i,k (x, u(x), ∂ x u(x)) = 0,
and assume that |u| (0,a i ) ≤ M. The main key of the proof will be in the use of the following equalities:

δ i H n i,k (x, u, p) = δ i H i (x, u, p) + n(p -∂ x u i,k-1 (x)) p , δ i H n i,k (x, u, p) = δ i H i (x, u, p), ( 11
)
where we recall that the operators δ i and δi are defined in assumption (E) (iii). We follow the proof of Theorem 15.2 in [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF]. We set u = ψ(u), where ψ ∈ C 3 [m, M], is increasing and

m = φ(-M), M = φ(M).
In the sequel, we will set v = ∂ x u 2 and v = ∂ x u 2 . To simplify the notations, we will omit the variables (x, u(x), ∂ x u(x)) in the functions σ i and H n i,k , and the variable u for ψ. We assume first that the solution u ∈ C 3 ([-M, M]), and we follow exactly all the computations that lead to equation of (15.25) of [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF] to get the following inequality:

σ i ∂ 2 x v + B i ∂ x v + G n i,k ≥ 0, (12)
where B i and G n i,k have the same expression in (15.26) of [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF] with (σ i = σ * i , c i = 0). We choose (r = 0, s = 0), since we will see in the sequel [START_REF] Lumer | Connecting of local operators and evolution equations on networks[END_REF], that condition (15.32) of [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF] holds under assumption assumption (P). We have more precisely:

B i = ψ ′ ∂ p σ i ∂ x u -∂ p H i + ω∂ p (σ i p 2 ), G n i,k = ω ′ ψ ′ + κ i ω 2 + β i ω + θ n i,k , ω = ψ ′′ ψ ′2 ∈ C 1 ([m, M]), κ i = 1 σ i p 2 δ i (σ i p 2 ) + p 2 4σ i |(δ i + 1)σ i | 2 , β i = 1 σ i p 2 δ i (σ i p 2 ) -δ i H i + p 2 2σ i ((δ i + 1)σ i )(δ i σ i ) , θ n i,k = 1 σ i p 2 p 2 4σ i |δ i σ i | 2 -δ i H n i,k = θ i - 1 σ i p 2 n(p -∂ x u i,k-1 (x)) p , θ i = 1 σ i p 2 p 2 4σ i |δ i σ i | 2 -δ i H i .
We set in the sequel:

G i = ∂ x ω ∂ x ψ + κ i ω 2 + β i ω + θ i , in order to get G n i,k = G i - 1 σ i p 2 n(p -∂ x u i,k-1 (x)) p .
More precisely, we see from ( 11) that all the coefficients (B i , κ i , β i , θ i ) can be chosen independent of n and u i,k-1 . The main argument then to get a bound of ∂ x u is to apply a maximum principle for v in [START_REF] Lions | Well posedness for multi-dimensional junction problems with Kirchhoff-type conditions[END_REF], and this will be done as soon as we ensure:

G n i,k ≤ 0, for |∂ x u| ≥ L n k .
On the other hand, using assumption (P) (ii) (iii) and (iv), it is easy to check that there exist constants (a, b, c), depending only on the data:

ν, ν, µ(M), γ(M), ε(M), sup |p|≥0 P (M, |p|) ,
such that:

sup x∈[0,a i ],|u|≤M lim sup |p|→+∞ κ i (x, u, p) ≤ a, sup x∈[0,a i ],|u|≤M lim sup |p|→+∞ β i (x, u, p) ≤ b, sup x∈[0,a i ],|u|≤M lim sup |p|→+∞ θ i (x, u, p) ≤ c, (13)
where,

a = 1 ν (γ(M) + 2ν) + 1 2 + γ(M) 2 ν 2 , b = ε(M) + sup |p|≥0 P (M, |p|) + γ(M) ν + (ε(M) + sup |p|≥0 P (M, |p|))(ν + γ(M)) ν 2 , c = (ε(M) + sup |p|≥0 P (M, |p|)) 2 4ν 2 + 2(ε(M) + sup |p|≥0 P (M, |p|)) ν .
As it has been on the proof of Theorem 15.2 of [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF], we choose then L = L(a, b, c), and

ψ(•) = ψ(a, b, c)(•
) such that we have:

G i ≤ 0, if |∂ x u(x)| ≥ L(a, b, c).
We see then from the expression of θ n i,k that we get

G n i,k ≤ 0, if |∂ x u(x)| ≥ L(a, b, c) ∨ |∂ x u i,k-1 (x)|.
Therefore applying the maximum principle to v in [START_REF] Lions | Well posedness for multi-dimensional junction problems with Kirchhoff-type conditions[END_REF], and from the relation u = ψ(u), v = ∂ x u 2 we get finally:

|∂ x u| (0,a i ) ≤ max max ψ ′ (a, b, c)(•) min ψ ′ (a, b, c)(•) , |∂ x u| ∂(0,a i ) , L(a, b, c), |∂ x u i,k-1 | (0,a i ) .
This upper bound still holds if u ∈ C 2 ([0, a i ]), (cf. (15.30) and (15.31) of the proof of Theorem 15.2 in [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF]). Finally applying the upper bound above to the solution u k , we get by induction that:

sup n≥0 max k∈{0...n} max i∈{1...I} |∂ x u i,k | (0,a i ) ≤ max i∈{1...I} max max ψ ′ (a, b, c)(•) min ψ ′ (a, b, c)(•) , |∂ x u i,k | ∂(0,a i ) , L(a, b, c), |∂ x g i | (0,a i ) .
This completes the proof.

The following Proposition follows from Lemmas 4.1 and 4.2, assumption (P) (ii) (iii), and from the relation:

∀x ∈ [0, a i ], |∂ 2 x u i,k (x))| ≤ |n(u i,k (x) -u i,k-1 (x))| + |H i (x, u i,k (x), ∂ x u i,k (x))| σ i (x, ∂ x u i,k (x)) ≤ |n(u i,k (x) -u i,k-1 (x))| + µ(|u i,k (x)|)(1 + |∂ x u i,k (x)| m ) ν(1 + |∂ x u i,k (x)| m-2
) .

Proposition 4.3. Assume (P)

. There exist constants (M 1 , M 2 , M 3 ), depending only the data introduced in assumption (P)

M 1 = M 1 max i∈{1...I} sup x∈(0,a i ) | -σ i (x, ∂ x g i (x))∂ 2 x g i (x) + H i (x, g i (x), ∂ x g i (x))| + |∂ t φ i | (0,T ) , max i∈{1...I} |g i | (0,a i ) , C H , M 2 = M 2 ν, ν, µ(M 1 ), γ(M 1 ), ε(M 1 ), sup |p|≥0 P (M 1 , |p|), max i∈{1...I} |∂ x g i | (0,a i ) , M 1 , M 3 = M 3 M 1 , ν(1 + |p|) m-2 , µ(|u|)(1 + |p|) m , |u| ≤ M 1 , |p| ≤ M 2 ,
such that:

sup n≥0 max k∈{0...n} max i∈{1...I} |u i,k | (0,a i ) ≤ M 1 , sup n≥0 max k∈{0...n} max i∈{1...I} |∂ x u i,k | (0,a i ) ≤ M 2 , sup n≥0 max k∈{1...n} max i∈{1...I} |n(u i,k -u i,k-1 )| (0,a i ) ≤ M 1 , sup n≥0 max k∈{0...n} max i∈{1...I} |∂ 2 x u i,k | (0,a i ) ≤ M 3 .
Unfortunately, we are unable to give an upper bound of the modulus of continuity of

∂ 2 x u i,k in C α ([0, a]
) independent of n. However, we are able to formulate in the weak sense a limit solution. From the regularity of the coefficients, using some tools introduced in Section 1, Lemma 2.1, we get interior regularity, and a smooth limit solution.

Proof of Theorem 2.2.

Proof. The uniqueness is a result of the comparison Theorem 2.4. To simplify the notations, we set for each i ∈ {1 . . . I}, and for each (x, q, u, p, S) ∈ [0, a i ] × R 4 Q i (x, u, q, p, S) = q -σ i (x, p)S + H i (x, u, p).

Let n ≥ 0. Consider the subdivision (t n k = kT n ) 0≤k≤n of [0, T ], and (u k ) 0≤k≤n the solution of [START_REF] Imbert | Generalized junction conditions for degenerate parabolic equations[END_REF].

From estimates of Proposition 4.3, there exists a constant M > 0 independent of n, such that:

sup n≥0 max k∈{1...n} max i∈{1...I} |u i,k | (0,a i ) + |n(u i,k -u i,k-1 )| (0,a i ) + |∂ x u i,k | (0,a i ) + |∂ 2 x u i,k | (0,a i ) ≤ M. (14)
We define the following sequence (v n ) n≥0 in C 0,2 (J a T ), piecewise differentiable with respect to its first variable by:

∀i ∈ {1 . . . I}, v i,0 (0, x) = g i (x) if x ∈ [0, a i ], v i,n (t, x) = u i,k (x) + n(t -t n k )(u i,k+1 (x) -u i,k (x)) if (t, x) ∈ [t n k , t n k+1 ) × [0, a i ].
We deduce then from [START_REF] Lumer | Equations de diffusion sur des réseaux infinis[END_REF], that there exists a constant M 1 independent of n, depending only on the data of the system, such that for all i ∈ {1 . . . I}

|v i,n | α [0,T ]×[0,a i ] + |∂ x v i,n | α x,[0,T ]×[0,a i ] ≤ M 1 .
Using Lemma 2.1, we deduce that there exists a constant M 2 (α) > 0, independent of n, such that for all i ∈ {1 . . . I}, we have the following global Hölder condition:

|∂ x v i,n | α 2 t,[0,T ]×[0,a i ] + |∂ x v i,n | α x,[0,T ]×[0,a i ] ≤ M 2 (α).
We deduce then from Ascoli's Theorem, that up to a sub sequence n,

(v i,n ) n≥0 converge in C 0,1 ([0, T ] × [0, a i ]) to v i , and then v i ∈ C α 2 ,1+α ([0, T ] × [0, a i ]
). Since v n satisfies the following continuity condition at the junction point

∀(i, j) ∈ {1 . . . I} 2 , ∀n ≥ 0, ∀t ∈ [0, T ], v i,n (t, 0) = v j,n (t, 0), we deduce then v ∈ C α 2 ,1+α (J a T
). We now focus on the regularity of v in

• J a
T , and we will prove that v ∈ C 1+ α 2 ,2+α (

• J a T ), and satisfies on each edge:

Q i (x, v i (t, x), ∂ t v i (t, x), ∂ x v i (t, x), ∂ 2 x v i (t, x)) = 0, if (t, x) ∈ (0, T ) × (0, a i ).
Using once again [START_REF] Lumer | Equations de diffusion sur des réseaux infinis[END_REF], there exists a constant M 3 independent of n, such that for each i ∈ {1 . . . I}:

∂ t v i,n L 2 ((0,T )×(0,a i )) ≤ M 3 , ∂ 2 x v i,n L 2 ((0,T )×(0,a i )) ≤ M 3 .
Hence we get up to a sub sequence, that:

∂ t v i,n ⇀ ∂ t v i , ∂ 2 x v i,n ⇀ ∂ 2 x v i , weakly in L 2 ((0, T ) × (0, a i )).
The continuity of the coefficients (σ i , H i ) i∈{1...I} , Lebesgue Theorem, the linearity of Q i in the variable ∂ t and ∂ 2 x , allows us to get for each i ∈ {1 . . . I}, up to a subsequence n p :

T 0 a i 0 Q i (x, v i,np (t, x), ∂ t v i,np (t, x), ∂ x v i,np (t, x), ∂ 2 x v i,np (t, x)) ψ(t, x)dxdt p→+∞ ----→ T 0 a i 0 Q i (x, v i (t, x), ∂ t v i (t, x), ∂ x v i (t, x), ∂ 2 x v i (t, x)) ψ(t, x)dxdt, ∀ψ ∈ C ∞ c ((0, T ) × (0, a i )).
We now prove that for any ψ ∈ C ∞ c ((0, T ) × (0, a i )):

T 0 a i 0 Q i (x, v i,np (t, x), ∂ t v i,np (t, x), ∂ x v i,np (t, x), ∂ 2 x v i,np (t, x)) ψ(t, x)dxdt p→+∞ ----→ 0.
Using that (u k ) 0≤k≤n is the solution of ( 7), we get for any ψ ∈ C ∞ c ((0, T ) × (0, a i )):

T 0 a i 0 Q i (x, v i,n (t, x), ∂ t v i,n (t, x), ∂ x v i,n (t, x), ∂ 2 x v i,n (t, x)) ψ(t, x)dxdt = n-1 k=0 t n k+1 t n k a i 0 σ i (x, ∂ x u i,k+1 (x))∂ 2 x u i,k+1 (x) -σ i (x, ∂ x v i,n (t, x))∂ 2 x v i,n (t, x) +H i (x, v i,n (t, x), ∂ x v i,n (t, x)) -H i (x, u i,k+1 (x), ∂ x u i,k+1 (x)) ψ(t, x)dxdt. (15)
Using assumption (P) more precisely the Lipschitz continuity of the Hamiltonians H i , the Hölder equicontinuity in time of (v i,n , ∂ x v i,n ), there exists a constant M 4 (α) independent of n, such that for each i ∈ {1 . . . I}, for each (t, x) ∈ [t n k , t n k+1 ] × [0, a i ]:

|H i (x, u i,k+1 (x), ∂ x u i,k+1 (x)) -H i (x, v i,n (t, x), ∂ x v i,n (t, x))| ≤ M 4 (α)(t -t n k ) α 2 ,
and therefore for any ψ ∈ C ∞ c ((0, T ) × (0, a i )):

n-1 k=0 t n k+1 t n k a i 0 H i (x, u i,k+1 (x), ∂ x u i,k+1 (x)) -H i (x, v i,n (t, x), ∂ x v i,n (t, x)) ψ(t, x)dxdt ≤ a i M 4 (α)|ψ| (0,T )×(0,a i ) n -α 2 n→+∞ ----→ 0.
For the last term in (15), we write for each i ∈ {1 . . . I}, for each (t, x) ∈ (t n k , t n k+1 )×(0, a i ):

σ i (x, ∂ x u i,k+1 (x))∂ 2 x u i,k+1 (x) -σ i (x, ∂ x v i,n (t, x))∂ 2 x v i,n (t, x) = σ i (x, ∂ x u i,k+1 (x)) -σ i (x, ∂ x v i,n (t, x)) ∂ 2 x u i,k (x) + (16) σ i (x, ∂ x u i,k+1 (x)) -n(t -t n k )σ i (x, ∂ x v i,n (t, x)) ∂ 2 x u i,k+1 (x) -∂ 2 x u i,k (x) . ( 17
)
Using again the Hölder equicontinuity in time of (v i,n , ∂ x v i,n ) as well as the uniform bound ), we can show that for [START_REF] Schieborn | Viscosity solutions of Hamilton-Jacobi equations of Eikonal type on ramified spaces[END_REF], for any ψ ∈ C ∞ c ((0, T ) × (0, a i )):

on |∂ 2 x u i,k | [0,a i ] ( 14
n-1 k=0 t n k+1 t n k a i 0 σ i (x, ∂ x u i,k+1 (x)) -σ i (x, ∂ x v i,n (t, x)) ∂ 2 x u i,k (x)ψ(t, x)dxdt n→+∞ ----→ 0.
Finally, from assumptions (P), for all i ∈ {1 . . . I}, σ i is differentiable with respect to all its variable, integrating by part we get for [START_REF] Vasilyeva | Population dynamics in river networks: analysis of steady states[END_REF]:

n-1 k=0 t n k+1 t n k a i 0 σ i (x, ∂ x u i,k+1 (x)) -n(t -t n k )σ i (x, ∂ x v i,n (t, x)) ∂ 2 x u i,k+1 (x) -∂ 2 x u i,k (x) ψ(t, x)dxdt = n-1 k=0 t n k+1 t n k a i 0 ∂ x σ i (x, ∂ x u i,k+1 (t, x))ψ(t, x) -n(t -t n k )∂ x σ i (x, ∂ x v i,n (t, x))ψ(t, x) ∂ x u i,k+1 (x) -∂ x u i,k (x) dxdt n→+∞ ----→ 0.
We conclude that for any ψ ∈ C ∞ c ((0, T ) × (0, a i )):

T 0 a i 0 Q i (x, v i (t, x), ∂ t v i (t, x), ∂ x v i (t, x), ∂ 2 x v i (t, x))) ψ(t, x)dxdt = 0.
It is then possible to consider the last equation as a linear one, with coefficients σi (t, x) =

σ i (x, ∂ x v i (t, x)), Hi (t, x) = H i (x, v i (t, x), ∂ x v i (t, x)) belonging to the class C α 2
,α ((0, T ) × (0, a i )), and using Theorem III.12.2 of [START_REF] Ladyzenskaja | Linear and Quasi-Linear equations of Parabolic type[END_REF], we get finally that for all i ∈ {1 . . . I}, v i ∈ C 1+ α 2 ,2+α ((0, T ) × (0, a i )), which means that v ∈ C 1+ α 2 ,2+α (

• J a T ). We deduce that v i satisfies on each edge:

Q i (x, v i (t, x), ∂ t v i (t, x), ∂ x v i (t, x), ∂ 2 x v i (t, x))) = 0, if (t, x) ∈ (0, T ) × (0, a i ).
From the estimates ( 14), we know that ∂ t v i,n and ∂ 2 x v i,n are uniformly bounded by n. We deduce finally that v ∈ C

1+ α 2 ,2+α b ( • J a T ).
We conclude by proving that v satisfies the non linear Neumann boundary condition at the vertex. For this, let t ∈ (0, T ); we have up to a sub sequence n p :

F (v np (t, 0), ∂ x v np (t, 0)) ----→ p→+∞ F (v(t, 0), ∂ x v(t, 0)).
On the other hand, using that F (u k (0), ∂ 0 u k (x)) = 0, we know from the continuity of F

(assumption (P)), the Hölder equicontinuity in time of t → v n (t, 0), and t → ∂ x v(t, 0), that there exists a constant M 5 (α) independent of n, such that if t ∈ [t n k , t n k+1 ):

|F (v n (t, 0), ∂ x v n (t, 0))| = |F (v n (t, 0), ∂ x v n (t, 0)) -F (u k (0), ∂ x u k (0))| ≤ sup |F (u, x) -F (v, y)|, u -v R I + x -y R I ≤ M 5 (α)n -α 2 n→+∞ ----→ 0.
Therefore, we conclude once more from the continuity of F (assumption (P)), the compatibility condition (assumption (P) (v)), that for each t ∈ [0, T ):

F (v(t, 0), ∂ x v(t, 0)) = 0.
On the other hand, it is easy to get:

∀i ∈ {1 . . . I}, ∀x ∈ [0, a i ], v i (0, x) = g i (x), ∀t ∈ [0, T ], v i (t, a i ) = φ i (t).
Finally, the expression of the upper bounds of the solution given in Theorem 2.2, are a consequence of Proposition 4.3, and Lemma 2.1, which completes the proof.

4.3. On the existence for unbounded junction. We give in this subsection a result on the existence and the uniqueness of the solution for the parabolic problem (1), posed on an unbounded junction J defined for I ∈ N * edges by:

J = I i=1
J i , with: ∀i ∈ {1 . . . I} J i = [0, +∞), and ∀(i, j) ∈ {1 . . . I} 2 , i = j, J i ∩ J i = {0}.

In the sequel, C 0,1 (J T ) ∩ C 1,2 ( • J T ) is the class of function with regularity C 0,1 ([0, T ] × [0, +∞)) ∩ C 1,2 ((0, T ) × (0, +∞)) on each edge, and L ∞ (J T ) is the set of measurable real bounded maps defined on J T .

We introduce the following data

     F ∈ C 0 (R I × R I , R) g ∈ C 1 b (J ) ∩ C 2 b ( • J )
,

and for each i ∈ {1 . . . I}            σ i ∈ C 1 (R + × R, R) H i ∈ C 1 (R + × R 2 , R) φ i ∈ C 1 ([0, T ], R)
.

We suppose furthermore that the data satisfy the following assumption

Assumption (P ∞ ) (i) Assumption on F :            a)
F is decreasing with respect to its first variable, b) F is nondecreasing with respect to its second variable,

c) ∃(b, B) ∈ R I × R I , F (b, B) = 0,
or the Kirchhoff condition:

           a)
F is nonincreasing with respect to its first variable, b) F is increasing with respect to its second variable,

c) ∃(b, B) ∈ R I × R I , F (b, B) = 0.
We suppose moreover that there exists a parameter m ∈ R, m ≥ 2 such that we have (ii) The (uniform) ellipticity condition on the (σ i ) i∈{1...I} : there exist ν, ν, strictly positive constants such that:

∀i ∈ {1 . . . I}, ∀(x, p) ∈ R + × R, ν(1 + |p|) m-2 ≤ σ i (x, p) ≤ ν(1 + |p|) m-2 .
(iii) The growth of the (H i ) i∈{1...I} with respect to p exceed the growth of the σ i with respect to p by no more than two, namely there exists µ an increasing real continuous function such that:

∀i ∈ {1 . . . I}, ∀(x, u, p) ∈ R + × R 2 , |H i (x, u, p)| ≤ µ(|u|)(1 + |p|) m .
(iv) We impose the following restrictions on the growth with respect to p of the derivatives for the coefficients (σ i , H i ) i∈{1...I} , which are for all i ∈ {1 . . . I}:

a) |∂ p σ i | R + ×R 2 (1 + |p|) 2 + |∂ p H i | R + ×R 2 ≤ γ(|u|)(1 + |p|) m-1 , b) |∂ x σ i | R + ×R 2 (1 + |p|) 2 + |∂ x H i | R + ×R 2 ≤ ε(|u|) + P (|u|, |p|) (1 + |p|) m+1 , c) ∀(x, u, p) ∈ R + × R 2 , -C H ≤ ∂ u H i (x, u, p) ≤ ε(|u|) + P (|u|, |p|) (1 + |p|) m ,
where γ and ε are continuous non negative increasing functions. P is a continuous function, increasing with respect to its first variable, and tends to 0 for p → +∞, uniformly with respect to its first variable, from [0, u 1 ] with u 1 ∈ R, and C H > 0 is real strictly positive number. We assume that (γ, ε, P, C H ) are independent of i ∈ {1 . . . I}.

(v) A compatibility conditions for g:

F (g(0), ∂ x g(0)) = 0.
We state here a comparison Theorem for the problem 1, posed on an unbounded junction.

Theorem 4.4.

Assume (P ∞ ). Let u ∈ C 0,1 (J T )∩C 1,2 ( • J T )∩L ∞ (J T ) (resp. v ∈ C 0,1 (J T )∩ C 1,2 ( • J T ) ∩ L ∞ (J T
)) be a super solution (resp. a sub solution) of (3) (where a i = +∞), satisfying for all i ∈ {1 . . . I} for all x ∈ [0, +∞), u i (0, x) ≥ v i (0, x). Then for each

(t, (x, i)) ∈ J T : u i (t, x) ≥ v i (t, x). Proof. Let s ∈ [0, T ), K = (K . . . K) > (1, . . . 1
) in R I , and λ = λ(K) > 0, that will be chosen in the sequel. We argue as in the proof of Theorem 2.4, assuming sup

(t,(x,i))∈J K s exp(-λt - (x -1) 2 2 ) v i (t, x) -u i (t, x) > 0.
Using the boundary conditions satisfied by u and v, the above supremum is reached at a point (t 0 , (x 0 , j 0 )) ∈ (0, s] × J , with 0 ≤ x 0 ≤ K.

If x 0 ∈ [0, K), the optimality conditions are given for x 0 = 0 by:

-λ(v j 0 (t 0 , x 0 ) -u j 0 (t 0 , x 0 )) + ∂ t v j 0 (t 0 , x 0 ) -∂ t u j 0 (t 0 , x 0 ) ≥ 0, -(x 0 -1) v j 0 (t 0 , x 0 ) -u j 0 (t 0 , x 0 ) + ∂ x v j 0 (t 0 , x 0 ) -∂ x u j 0 (t 0 , x 0 ) = 0, v j 0 (t 0 , x 0 ) -u j 0 (t 0 , x 0 ) -2(x 0 -1) 2 v j 0 (t 0 , x 0 ) -u j 0 (t 0 , x 0 ) + ∂ 2 x v j 0 (t 0 , x 0 ) -∂ 2 x u j 0 (t 0 , x 0 ) ≤ 0, and if x 0 = 0: ∀i ∈ {1, . . . I}, ∂ x v i (t 0 , 0) ≤ ∂ x u i (t 0 , 0) -v i (t 0 , 0) -u i (t 0 , 0) < ∂ x u i (t 0 , 0).
If x 0 = 0, we obtain a contradiction exactly as in the proof of Theorem 2.4. On the other hand if x 0 ∈ (0, K), using assumptions (P) (iv) a), (iv) c) and the optimality conditions, we can choose λ(K) of the form λ(K) = C(1 + K 2 ), (see ( 4) and ( 5)), where C > 0 is a constant independent of K, to get again a contradiction. We deduce that, if:

sup (t,(x,i))∈J K s exp(-λ(K)t - (x -1) 2 2 ) v i (t, x) -u i (t, x) > 0, then for all (t, (x, i)) ∈ [0, T ] × J K : exp(-λ(K)t - (x -1) 2 2 ) v i (t, x) -u i (t, x) ≤ exp(-λ(K)t - (K -1) 2 2 ) v i (t, K) -u i (t, K) .
Hence for all (t, (x, i))

∈ [0, T ] × J K : exp(- (x -1) 2 2 ) v i (t, x) -u i (t, x) ≤ exp(- (K -1) 2 2 ) v i (t, K) -u i (t, K) .
On the other hand, if:

sup (t,(x,i))∈J K s exp(-λ(K)t - (x -1) 2 2 ) v i (t, x) -u i (t, x) ≤ 0, then for all (t, (x, i)) ∈ [0, T ] × J K : exp(-λ(K)t - (x -1) 2 2 ) v i (t, x) -u i (t, x) ≤ 0. So exp(- (x -1) 2 2 ) v i (t, x) -u i (t, x) ≤ 0.
Finally we have, for all (t, (x, i))

∈ [0, T ] × J K : max 0, exp(- (x -1) 2 2 ) v i (t, x) -u i (t, x) ≤ exp(- (K -1) 2 2 ) ||u|| L ∞ (J T ) + ||v|| L ∞ (J T ) .
Sending K → ∞ and using the boundedness of u and v, we deduce the inequality v ≤ u in [0, T ] × J .

                                 ∂ t u i (t, x) -σ i (x, ∂ x u i (t, x))∂ 2 x u i (t, x)+ H i (x, u i (t, x), ∂ x u i (t, x)) = 0, if (t, x) ∈ (0, T ) × (0, +∞), F (u(t, 0), ∂ x u(t, 0)) = 0, if t ∈ [0, T ), with u(t, 0) = (u 1 (t, 0), . . . , u I (t, 0)), ∂ x u(t, 0) = (∂ x u 1 (t, 0), . . . , ∂ x u I (t, 0)), and
∀(i, j) ∈ {1 . . . I} 2 , u i (t, 0) = u j (t, 0), ∀i ∈ {1 . . . I}, u i (0, x) = g i (x), if x ∈ [0, +∞), (18)
is uniquely solvable in the class C For R > 0, we denote by L 2 ((0, T ) × (0, R)) the usual space of square integrable maps and by C ∞ c ((0, T ) × (0, R)) the set of infinite continuous differentiable functions on (0, T ) × (0, R), with compact support. For all i ∈ {1 . . . I}, each elliptic problem is uniquely solvable on each edge in C 2+α ([0, a i ]), then (21) is uniquely solvable in the class C 2+α (J a ), and we denote by u θ its solution.

We turn to the Neumann boundary condition at the vertex. Let us recall assumption (E)(i)

          
F is decreasing in its first variable, nondecreasing in its second variable, or F is nonincreasing in its first variable, increasing in its second variable, ∃(b, B) ∈ R I × R I , such that : F (b, B) = 0.

Fix now:

K i = sup (x,u)∈(0,a i )×(-a i B i ,a i B i ) |H i (x, u, B i )|, θ ≥ max i∈{1...I} |b i | + |φ i | + |a i B i | + K i C H ,
and let us show that f : x → θ + B i x, is a super solution on each edge J a i i of (21). We have the boundary conditions

f (0) = θ, f (a i ) = θ + a i B i ≥ |φ i | + |a i B i | + a i B i ≥ φ i ,
and using assumption (E) (iii), we have for all x ∈ (0, a i ) -σ i (x, ∂ x f (x))∂ 2

x f (x) + H i (x, f (x), ∂ x f (x)) = H i (x, θ + B i x, B i ) ≥ H i (x, B i x, B i )

+ C H θ ≥ H i (x, B i x, B i ) + K i ≥ 0.
We then get that for each i ∈ {1 . . . I}, x ∈ [0, a i ], u θ i (x) ≤ θ + B i x. A Taylor expansion in the neighborhood of the junction point gives that for each i ∈ {1 . . . I}, ∂ x u θ i (0) ≤ B i . Since u θ (0) ≥ b, we then get from assumption (E) (i):

F (u θ (0), ∂ x u θ (0)) ≤ F (b, B) ≤ 0.
Similarly, fixing:

θ ≤ min i∈{1...I} -|b i | -|φ i | -|a i B i | - K i C H ,
the map f : x → θ + xB i is a sub solution on each vertex J a i i of (21), then for each i ∈ {1 . . . I}, ∂ x u θ i (0) ≥ B i , which means:

F (u θ (0), ∂ x u θ (0)) ≥ 0.

From Proposition B.1, we know that the real maps θ → u θ (0) and θ → ∂ x u θ (0) are continuous. Using the continuity of F (assumption (E)), we get that θ → F (u θ (0), ∂ x u θ (0)) is continuous, and therefore there exists θ * ∈ R such that:

F (u θ * (0), ∂ x u θ * (0)) = 0.
We remark that θ * is bounded by the data, namely θ * belongs to the following interval: This completes the proof. Finally, since the solution u θ * of ( 2) is unique, we get the uniqueness of θ * .

Theorem 4 . 5 .

 45 Assume (P ∞ ). The following parabolic problem with Neumann boundary condition at the vertex:

α 2 ,M 1 =M 3 =

 213 1+α (J T )∩C 1+ α 2 ,2+α ( • J T ). There exist constants (M 1 , M 2 , M 3 ), depending only the data introduced in assumption(P ∞ ) M 1 max i∈{1...I} sup x∈(0,+∞) | -σ i (x, ∂ x g i (x))∂ 2 x g i (x) + H i (x, g i (x), ∂ x g i (x))| , max i∈{1...I} |g i | (0,+∞) , C H , M 2 = M 2 ν, ν, µ(M 1 ), γ(M 1 ), ε(M 1 ), sup |p|≥0 P (M 1 , |p|), max i∈{1...I} |∂ x g i | (0,+∞) , M 1 , M 3 M 1 , ν(1 + |p|) m-2 , µ(|u|)(1 + |p|) m , |u| ≤ M 1 , |p| ≤ M 2 ,such that:||u|| C(J T ) ≤ M 1 , ||∂ x u|| C(J T ) ≤ M 2 , ||∂ t u|| C(J T ) ≤ M 1 , ||∂ x u|| C(J T ) ≤ M 3 .Moreover, there exists a constant M(α) depending on α, M 1 , M 2 , M 3 such that for any a ∈ (0, +∞) I :||u|| C α 2 ,1+α (J a T ) ≤ M(α).

  2r + s < l, and satisfying an Hölder condition with exponent l-2r-s 2 in their first variable, and with exponent (l -⌊l⌋) in their second variable, over all the connected components of Ω T whose radius is smaller than 1.The norm• C l 2 ,l (Ω T ) is defined for all u ∈ C l 2 ,l (Ω T ) by: ,x =y,|x-y|≤1 |u(t, x) -u(t, y)| |x -y| α , 0 < α < 1, (0,T ),t =s,|t-s|≤1 |u(t, x) -u(s, x)| |t -s| α , 0 < α < 1.-C l 2 ,l (Ω T ) is the set whose elements f belong to C l 2 ,l (O T ) for any open set O T separated from the boundary of Ω T by a strictly positive distance, namely:inf y∈∂Ω T ,x∈O T ||x -y|| R n > 0. T ) is the subset of C l 2 ,l (Ω T )consisting in maps u such that the derivatives of the form ∂ r t ∂ s x u, (with 2r + s < l) are bounded, namely sup (t,x)∈Ω T |∂ r t ∂ s x u(t, x)| < +∞. We use the same notations when the domain does not depend on time, namely T = 0, Ω T = Ω, just removing the dependence on the time variable.

  min i∈{1...I} -|b i | -|φ i | -|a i B i | -sup (x,u)∈(0,a i ) |H i (x,B i x,B i )| C H , max i∈{1...I} |b i | + |φ i | + |a i B i | + sup (x,u)∈(0,a i ) |H i (x,B i x,B i )| C H .

  Lemma 2.1. Suppose that u ∈ C 0,1 ([0, T ] × [0, R]) satisfies an Hölder condition in t in [0, T ] × [0, R], with exponent α ∈ (0, 1], constant ν 1 , and has derivative ∂ x u, which for any t ∈ [0, T ] are Hölder continuous in the variable x, with exponent γ ∈ (0, 1], and constant ν 2 . Then the derivative ∂ x u satisfies in [0, T ] × [0, R], an Hölder condition in t, with

	exponent αγ 1+γ , and constant depending only on ν 1 , ν 2 , γ. More precisely

Proof. Assume (P ∞ ) and let a = (a, . . . , a) ∈ (0, +∞) I . Applying Theorem 2.2, we can define u a ∈ C 0,1 (J a T ) ∩ C 1,2 (

• J a T ) as the unique solution of:

Using assumption (P ∞ ) and Theorem 2.2, we get that there exists a constant C > 0 independent of a such that:

We are going to send a to +∞ in [START_REF] Below | A maximum principle for semi linear parabolic network equations[END_REF]. Following the same argument as for the proof of Theorem 2.2, we get that, up to a sub sequence, u a converges locally uniformly to some map u which solves [START_REF] Below | Classical solvability of linear parabolic equations on networks[END_REF]. On the other hand, uniqueness of u is a direct consequence of the comparison Theorem 4.4, since u ∈ L ∞ (J T ). Finally the expression of the upper bounds of the derivatives of u given in Theorem 4.5, are a consequence of Theorem 2.2

and assumption (P ∞ ).

Appendix A. Functionnal spaces

In this section, we recall several classical notations from [START_REF] Ladyzenskaja | Linear and Quasi-Linear equations of Parabolic type[END_REF]. Let l, T ∈ (0, +∞) and Ω be an open and bounded subset of R n with smooth boundary (n > 0). We set Ω T = (0, T )×Ω, and we introduce the following spaces :

is the Banach space whose elements are continuous functions (t, x) → u(t, x) in Ω T , together with all its derivatives of the form ∂ r t ∂ s x u, with 2r + s < l. The norm

is defined for all u ∈ C l 2 ,l (Ω T ) by:

is the Banach space whose elements are continuous functions (t, x) → u(t, x) in Ω T , together with all its derivatives of the form ∂

Then the following map:

Proof. Let θ n a sequence converging to θ. Using the Schauder estimates Theorem 6.6 of [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF], we get that there exists a constant M > 0 independent of n, depending only the data, such that for all α ∈ (0, 1):

From Ascoli's Theorem, u θn i converges up to a subsequence to v in C 2 ([0, a i ]) solution of [START_REF] Below | An existence result for semi linear parabolic network equations with dynamical node conditions[END_REF]. By uniqueness of the solution of (20), u θn i converges necessary to the solution u θ i of (20) in C 2 ([0, a i ]), which completes the proof.

Proof of Theorem 3.1.

Proof. The uniqueness of (2) results from the elliptic comparison Theorem 3.3.

We turn to the solvalbility, and for this let θ ∈ R. We consider the elliptic Dirichlet problem posed on the the junction:

x u i (x) + H i (x, u i (x), ∂ x u i (x)) = 0, if x ∈ (0, a i ), ∀i ∈ {1 . . . I}, u i (0) = u(0) = θ, u i (a i ) = φ i .

(21)