Quasi linear parabolic pde posed on a network with non linear Neumann boundary condition at vertices
aa r X i v : . [ m a t h . A P ] J u l QUASI LINEAR PARABOLIC PDE IN A JUNCTION WITH NONLINEAR NEUMANN VERTEX CONDITION
ISAAC WAHBI
Version: July 12, 2018
Abstract.
The purpose of this article is to study quasi linear parabolic partial differ-ential equations of second order, on a bounded junction, satisfying a nonlinear and nondynamical Neumann boundary condition at the junction point. We prove the existenceand the uniqueness of a classical solution. Introduction
In this paper, we study non degenerate quasi linear parabolic partial differential equa-tions on a junction, satisfying a non linear Neumann boundary condition at the junctionpoint x = 0: ∂ t u i ( t, x ) − σ i ( x, ∂ x u i ( t, x )) ∂ x,x u i ( t, x ) + H i ( x, u i ( t, x ) , ∂ x u i ( t, x )) = 0 , for all x > , and for all i ∈ { . . . I } ,F ( u ( t, , ∂ x u ( t, . (1)The well-known Kirchhoff law corresponds to the case where F is linear in ∂ x u andindependent of u .Originally introduced by Nikol’skii [13] and Lumer [11, 12], the concept of ramifiedspaces and the analysis of partial differential equation on these spaces have attracteda lot of attention in the last 30 years. As explained in [13], the main motivations areapplications in physics, chemistry, and biology (for instance small transverse vibrationsin a grid of strings, vibration of a grid of beams, drainage system, electrical equation withKirchhoff law, wave equation, heat equation,...). Linear diffusions of the form (1), witha Kirchhoff law, are also naturally associated with stochastic processes living on graphs.These processes were introduced in the seminal papers [3] and [4]. Another motivationfor studying (1) is the analysis of associated stochastic optimal control problems with a control at the junction. The result of this paper will allow us in a future work tocharacterize the value function of such problems.There has been several works on linear and quasilinear parabolic equations of the form(1). For linear equations, von Below [15] shows that, under natural smoothness andcompatibility conditions, linear boundary value problems on a network with a linearKirchhoff condition and an additional Dirichlet boundary condition at the vertex point,are well-posed. The proof consists mainly in showing that the initial boundary valueproblem on a junction is equivalent to a well-posed initial boundary value problem fora parabolic system, where the boundary conditions are such that the classical results onlinear parabolic equations [7] can be applied. The same author investigates in [16] thestrong maximum principle for semi linear parabolic operators with Kirchhoff condition,while in [17] he studies the classical global solvability for a class of semilinear parabolicequations on ramified networks, where a dynamical node condition is prescribed: Namelythe Neumann condition at the junction point x = 0 in (1), is replaced by the dynamicone ∂ t u ( t,
0) + F ( t, u ( t, , ∂ x u ( t, . In this way the application of classical estimates for domains established in [7] becomespossible. The author then establish the classical solvability in the class C α, α , withthe aid of the Leray-Schauder-principle and the maximum principle of [16]. Let us notethat this kind of proof fails for equation (1) because in this case one cannot expect anuniform bound for the term | ∂ t u ( t, | (the proof of Lemma 3.1 of [7] VI.3 fails). Stillin the linear setting, another approach, yielding similar existence results, was developedby Fijavz, Mugnolo and Sikolya in [2]: the idea is to use semi-group theory as well asvariational methods to understand how the spectrum of the operator is related to thestructure of the network.Equations of the form (1) can also be analyzed in terms of viscosity solutions. Thefirst results on viscosity solutions for Hamilton-Jacobi equations on networks have beenobtained by Schieborn in [14] for the Eikonal equations and later discussed in manycontributions on first order problems [1, 6, 8], elliptic equations [9] and second orderproblems with vanishing diffusion at the vertex [10]. UASI LINEAR PARABOLIC PDE IN A JUNCTION 3
In contrast second order Hamilton-Jacobi equations with a non vanishing viscosity atthe boundary have seldom been studied in the literature and our aim is to show thewell-posedness of classical solutions for (1) in suitable H¨oder spaces: see Theorem 2.2for the existence and Theorem 2.4 for the comparison, and thus the uniqueness. Ourmain assumptions are that the equation is uniformly parabolic with smooth coefficientsand that the term F = F ( u, p ) at the junction is either decreasing with respect to u orincreasing with respect to p .The main idea of the proof is to use a time discretization, exploiting at each step thesolvability in C α of the elliptic problem − σ i ( x, ∂ x u i ( x )) ∂ x,x u i ( x ) + H i ( x, u i ( x ) , ∂ x u i ( x )) = 0 ,F ( u (0) , ∂ x u (0)) = 0 . (2)The paper is organized as follows. In section 2, we introduce the notations and state ourmain results. In Section 3, we review the mains results of existence and uniqueness of theelliptic problem (2). Finally Section 4, is dedicated to the proof of our main results.2. main results In this section we state our main result Theorem 2.2, on the solvability of the parabolicproblem with Neumann boundary condition at the vertex, on a bounded junction ∂ t u i ( t, x ) − σ i ( x, ∂ x u i ( t, x )) ∂ x,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, , ∂ x u ( t, , if t ∈ [0 , T ) , ∀ i ∈ { . . . I } , u i ( t, a i ) = φ i ( t ) , if t ∈ [0 , T ] , ∀ i ∈ { . . . I } , u i (0 , x ) = g i ( x ) , if x ∈ [0 , a i ] . (3)There will be two typical assumptions for F = F ( u, p ): either F is decreasing with respectto u or F is increasing with respect to p (Kirchhoff conditions).2.1. Notations and preliminary results.
Let us start by introducing the main nota-tion used in this paper as well as an interpolation result.Let I ∈ N ∗ be the number of edges, and a = ( a , . . . a I ) ∈ (0 , ∞ ) I be the length of each ISAAC WAHBI edge.The bounded junction is defined by J a = n X = ( x, i ) , x ∈ [0 , a i ] and i ∈ { , . . . , I } o , where all the points (0 , i ), i = 1 , . . . , I , are identified to the vertex denoted by 0. We canthen write J a = I [ i =1 J a i i , with J a i i := [0 , a i ] × { i } , J a i i ∩ J a j j = { } . For T >
0, the time-space domain J aT is definedby J aT = [0 , T ] × J a . The interior of J aT set minus the junction point 0 is denoted by ◦ J aT , and is defined by ◦ J aT = (0 , T ) × (cid:16) I [ i =1 ◦ J a i i (cid:17) . For the functionnal spaces that will be used in the sequel, we use here the notationsof Chapter 1.1 of [7]. For the convenience of the reader, we recall these notations inAppendix A.In addition we introduce the parabolic H¨older space on the junction (cid:16) C l ,l ( J aT ) , k . k C l ,l ( J aT ) (cid:17) and the space C l ,lb ( ◦ J aT ), defined by (where l >
0, see Annexe A for more details) C l ,l ( J aT ) := n f : J aT → R , ( t, ( x, i )) f i ( t, x ) , ∀ ( i, j ) ∈ { . . . I } , ∀ t ∈ (0 , T ) ,f i ( t,
0) = f j ( t,
0) = f ( t, , ∀ i ∈ { . . . I } , ( t, x ) f i ( t, x ) ∈ C l ,l ([0 , T ] × [0 , a i ]) o , C l ,lb ( ◦ J aT ) := n f : J aT → R , ( t, ( x, i )) f i ( t, x ) , ∀ i ∈ { . . . I } , ( t, x ) f i ( t, x ) ∈ C l ,lb ((0 , T ) × (0 , a i )) o , with k u k C l ,l ( J aT ) = X ≤ i ≤ I k u i k C l ,l ([0 ,T ] × [0 ,a i ]) . UASI LINEAR PARABOLIC PDE IN A JUNCTION 5
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 maps F : R I → R .Let ( x = ( x , . . . x I ) , y = ( y . . . y I )) ∈ R I , we say that x ≤ y, if ∀ i ∈ { . . . I } , x i ≤ y i , and x < y, if x ≤ y, and there exists j ∈ { . . . 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.
Lemma 2.1.
Suppose that u ∈ C , ([0 , T ] × [0 , R ]) satisfies an H¨older condition in t in [0 , T ] × [0 , R ] , with exponent α ∈ (0 , , constant ν , and has derivative ∂ x u , which for any t ∈ [0 , T ] are H¨older continuous in the variable x , with exponent γ ∈ (0 , , and constant ν . Then the derivative ∂ x u satisfies in [0 , T ] × [0 , R ] , an H¨older condition in t , withexponent αγ γ , and constant depending only on ν , ν , γ . More precisely ∀ ( t, s ) ∈ [0 , T ] , | t − s | ≤ , ∀ x ∈ [0 , R ] , | ∂ x u ( t, x ) − ∂ x u ( s, x ) | ≤ (cid:16) ν (cid:16) ν γν (cid:17) γ γ + 2 ν (cid:16) γν ν (cid:17) − γ (cid:17) | t − s | αγ γ . This is a special case of Lemma II.3.1, in [7], (see also [13]). The main difference is thatwe are able to get global H¨older 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. ISAAC WAHBI
Proof.
Let ( t, s ) ∈ [0 , T ] , with | t − s | ≤
1, and x ∈ [0 , R ]. Suppose first that x ∈ [0 , R ].Let y ∈ [0 , R ], with y = x , we write ∂ x u ( t, x ) − ∂ x u ( s, x ) =1 y − x Z yx ( ∂ 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 H¨older condition in time satisfied by u , we have (cid:12)(cid:12)(cid:12) y − x Z yx ( ∂ x u ( t, z ) − ∂ x u ( s, z )) dz (cid:12)(cid:12)(cid:12) ≤ ν | t − s | α | y − x | . On the other hand, using the H¨older regularity of ∂ x u in space satisfied, we have (cid:12)(cid:12)(cid:12) y − x Z yx ( ∂ x u ( t, x ) − ∂ x u ( t, z )) + ( ∂ x u ( s, z ) − ∂ x u ( s, x )) dz (cid:12)(cid:12)(cid:12) ≤ ν | y − x | γ . It follows | ∂ x u ( t, x ) − ∂ x u ( s, x ) | ≤ ν | y − x | γ + 2 ν | t − s | α | y − x | . Assuming that | t − s | ≤ (cid:16) ( R ) γ γν ν (cid:17) α ∧
1, minimizing in y ∈ [0 , R ], for y > x , the rightside of the last equation, we get that the infimum is reached for y ∗ = x + (cid:16) ν | t − s | α γν (cid:17) γ , and then | ∂ x u ( t, x ) − ∂ x u ( s, x ) | ≤ C ( ν , ν , γ ) | t − s | αγ γ , where the constant C ( ν , ν , γ ), depends only on the data ( ν , ν , γ ), and is given by C ( ν , ν , γ ) = 2 ν (cid:16) ν γν (cid:17) γ γ + 2 ν (cid:16) γν ν (cid:17) − γ . For the cases y < x , and x ∈ [ R , R ], we argue similarly, which completes the proof. (cid:3) Assumptions and main results.
We state in this subsection the central Theo-rem of this note, namely the solvability and uniqueness of (1) in the class C α , α ( J aT ) ∩C α , αb ( ◦ J aT ). In the rest of these notes, we fix α ∈ (0 , UASI LINEAR PARABOLIC PDE IN A JUNCTION 7
Let us state the assumptions we will work on.
Assumption ( P )We introduce the following data F ∈ C ( R × R I , R ) g ∈ C ( J a ) ∩ C b ( ◦ J a ) , and for each i ∈ { . . . I } σ i ∈ C ([0 , a i ] × R , R ) H i ∈ C ([0 , a i ] × R , R ) φ i ∈ C ([0 , T ] , R ) . We suppose furthermore that the data satisfy(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 × 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 × R I , F ( b, B ) = 0 . We suppose moreover that there exist a parameter m ∈ R , m ≥ σ i ) i ∈{ ...I } : there exists ν, ν , strictly positiveconstants such that ∀ i ∈ { . . . I } , ∀ ( x, p ) ∈ [0 , a i ] × R ,ν (1 + | p | ) m − ≤ σ i ( x, p ) ≤ ν (1 + | p | ) m − . (iii) The growth of the ( H i ) i ∈{ ...I } with respect to p exceed the growth of the σ i withrespect to p by no more than two, namely there exists µ an increasing real continuous ISAAC WAHBI function such that ∀ i ∈ { . . . I } , ∀ ( x, u, p ) ∈ [0 , a i ] × R , | 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 derivativesfor the coefficients ( σ i , H i ) i ∈{ ...I } , which are for all i ∈ { . . . I } , a ) | ∂ p σ i | [0 ,a i ] × R (1 + | p | ) + | ∂ p H i | [0 ,a i ] × R ≤ γ ( | u | )(1 + | p | ) m − ,b ) | ∂ x σ i | [0 ,a i ] × R (1 + | p | ) + | ∂ x H i | [0 ,a i ] × R ≤ (cid:16) ε ( | u | ) + P ( | u | , | p | ) (cid:17) (1 + | p | ) m +1 ,c ) ∀ ( x, u, p ) ∈ [0 , a i ] × R , − C H ≤ ∂ u H i ( x, u, p ) ≤ (cid:16) ε ( | u | ) + P ( | u | , | p | ) (cid:17) (1 + | p | ) m , where γ and ε are continuous non negative increasing functions. P is a continuous func-tion, increasing with respect to its first variable, and tends to 0 for p → + ∞ , uniformlywith respect to its first variable, from [0 , u ] with u ∈ R , and C H > γ, ε, P, C H ) are independent of i ∈ { . . . I } .(v) A compatibility conditions for g and ( φ i ) { ...I } F ( g (0) , ∂ x g (0)) = 0 ; ∀ i ∈ { . . . I } , g i ( a i ) = φ i (0) . Theorem 2.2.
Assume ( P ) . Then system (3) is uniquely solvable in the class C α , α ( J aT ) ∩C α , αb ( ◦ J aT ) . There exist constants ( M , M , M ) , depending only the data introduced inassumption ( P ) , M = M (cid:16) max i ∈{ ...I } n sup x ∈ (0 ,a i ) | − σ i ( x, ∂ x g i ( x )) ∂ x g i ( x ) + H i ( x, g i ( x ) , ∂ x g i ( x )) | + | ∂ t φ i | (0 ,T ) o , max i ∈{ ...I } | g i | (0 ,a i ) , C H (cid:17) ,M = M (cid:16) ν, ν, µ ( M ) , γ ( M ) , ε ( M ) , sup | p |≥ P ( M , | p | ) , | ∂ x g i | (0 ,a i ) , M (cid:17) ,M = M (cid:16) M , ν (1 + | p | ) m − , µ ( | u | )(1 + | p | ) m , | u | ≤ M , | p | ≤ M (cid:17) , such that || u || C ( J aT ) ≤ M , || ∂ x u || C ( J aT ) ≤ M , || ∂ t u || C ( J aT ) ≤ M , || ∂ x,x u || C ( J aT ) ≤ M . UASI LINEAR PARABOLIC PDE IN A JUNCTION 9
Moreover, there exists a constant M ( α ) depending on (cid:16) α, M , M , M (cid:17) such that || u || C α , α ( J aT ) ≤ M ( α ) . We continue this Section by giving the definitions of super and sub solution, and statinga comparison Theorem for our problem.
Definition 2.3.
We say that u ∈ C , ( J aT ) ∩ C , ( ◦ J aT ) , is a super solution (resp. subsolution) of ∂ t u i ( t, x ) − σ i ( x, ∂ x u i ( t, x )) ∂ x,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, , ∂ x u ( t, , if t ∈ (0 , T ) , (4) if ∂ t u i ( t, x ) − σ i ( x, ∂ x u i ( t, x )) ∂ x,x u i ( t, x )+ H i ( x, u i ( t, x ) , ∂ x u i ( t, x )) ≥ , ( resp. ≤ , ∀ ( t, x ) ∈ (0 , T ) × (0 , a i ) ,F ( u ( t, , ∂ x u ( t, ≤ , ( resp. ≥ , ∀ t ∈ (0 , T ) Theorem 2.4.
Parabolic comparison.Assume ( P ) . Let u ∈ C , ( J aT ) ∩ C , b ( ◦ J aT ) (resp. v ∈ C , ( J aT ) ∩ C , b ( ◦ J aT ) ) a super solution(resp. a sub solution) of (4) , satisfying for all i ∈ { . . . 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 aT : 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 as , u i ( t, x ) ≥ v i ( t, x ).Let λ >
0. Suppose that λ > C + C , where the expression of the constants ( C , C ) aregiven in the sequel (see (5), and (6)). We argue by contradiction assuming thatsup ( t, ( x,i )) ∈J as exp( − λt + x ) (cid:16) v i ( t, x ) − u i ( t, x ) (cid:17) > . Using the boundary conditions satisfied by u and v , the supremum above is reached at apoint ( t , ( x , j )) ∈ (0 , s ] × J , with 0 ≤ x < a j . Suppose first that x >
0, the optimality conditions imply thatexp( − λt + x ) (cid:16) − λ ( v j ( t , x ) − u j ( t , x )) + ∂ t v j ( t , x ) − ∂ t u j ( t , x ) (cid:17) ≥ , exp( − λt + x )) (cid:16) v j ( t , x ) − u j ( t , x ) + ∂ x v j ( t , x ) − ∂ x u j ( t , x ) (cid:17) = 0 , exp( − λt + x ) (cid:16) v j ( t , x ) − u j ( t , x ) + 2 (cid:16) ∂ x v j ( t , x ) − ∂ x u j ( t , x ) (cid:17) + (cid:16) ∂ x,x v j ( t , x ) − ∂ x,x u j ( t , x ) (cid:17)(cid:17) =exp( − λt + x ) (cid:16) − (cid:16) v j ( t , x ) − u j ( t , x ) (cid:17) + ∂ x,x v j ( t , x ) − ∂ x,x u j ( t , x ) (cid:17) ≤ . Using assumptions ( P ) (iv) a), (iv) c) and the optimality conditions above we have H j ( x , u i ( t , x ) , ∂ x u j ( t , x )) − H j ( x , v j ( t , x ) , ∂ x v j ( t , x )) ≤ (cid:16) v j ( t , x ) − u j ( t , x ) (cid:17)(cid:16) C H + γ ( | ∂ x v j ( t , x ) | ) (cid:17)(cid:16) (1 + | ∂ x u j ( t , x )) | ∨ | ∂ x v j ( t , x )) | ) m − (cid:17) ≤ C (cid:16) v j ( t , x ) − u j ( t , x ) (cid:17) , where C := max i ∈{ ...I } n sup ( t,x ) ∈ [0 ,T ] × [0 ,a i ] n (cid:16) C H + γ ( | ∂ x v i ( t, x ) | (cid:17)(cid:16) | ∂ x u i ( t, x )) |∨| ∂ x v i ( t, x )) | (cid:17) m − o o . (5)On the other hand we have using assumption ( P ) (ii), (iv) a), (iv) c), and the optimalityconditions σ j ( x , ∂ x v j ( t , x )) ∂ x,x v j ( t , x ) − σ j ( x , ∂ x u j ( t , x )) ∂ x,x u j ( t , x ) ≤ (cid:16) v j ( t , x ) − u j ( t , x ) (cid:17)(cid:16) ν (1 + | ∂ x v j ( t , x ) | ) m − + (cid:12)(cid:12)(cid:12) ∂ x,x u j ( t , x ) (cid:12)(cid:12)(cid:12) + γ ( | ∂ x u j ( t , x ) | )(1 + | ∂ x u j ( t , x )) | ∨ | ∂ x v j ( t , x )) | ) m − (cid:17) ≤ C (cid:16) v j ( t , x ) − u j ( t , x ) (cid:17) , where C := max i ∈{ ...I } n sup ( t,x ) ∈ [0 ,T ] × [0 ,a i ] n ν (1 + | ∂ x v i ( t, x ) | ) m − + (cid:12)(cid:12)(cid:12) ∂ x,x u i ( t, x ) (cid:12)(cid:12)(cid:12) + γ ( | ∂ x u i ( t, x ) | )(1 + | ∂ x u i ( t, x )) | + | ∂ x v i ( t, x )) | ) m − o o . (6) UASI LINEAR PARABOLIC PDE IN A JUNCTION 11
Using now the fact that v is a sub-solution while u is a super-solution, we get0 ≤ ∂ t u j ( t , x ) − σ j ( x , ∂ x u j ( t , x )) ∂ x,x u j ( t , x ) + H j ( x , u i ( t , x ) , ∂ x u j ( t , x )) − ∂ t v j ( t , x ) + σ j ( x , ∂ x v j ( t , x )) ∂ x,x v j ( t , x ) − H j ( x , v j ( t , x ) , ∂ x v j ( t , x )) ≤ − ( λ − ( C + C ))( v j ( t , x ) − u j ( t , x )) < , which is a contradiction. Therefore the supremum is reached at ( t , t ∈ (0 , s ].We apply a first order Taylor expansion in space, in the neighborhood of the junctionpoint 0. Since for all ( i, j ) ∈ { . . . I } , u i ( t ,
0) = u j ( t ,
0) = u ( t , v i ( t ,
0) = v j ( t ,
0) = v ( t , ∀ ( i, j ) ∈ { , . . . I } , ∀ h ∈ (0 , min i ∈{ ...I } a i ] v j ( t , − u j ( t , ≥ exp( h ) (cid:16) v i ( t , h ) − u i ( t , h ) (cid:17) , that ∀ ( i, j ) ∈ { , . . . I } , ∀ h ∈ (0 , min i ∈{ ...I } a i ] v j ( t , − u j ( t , ≥ v i ( t , − u i ( t ,
0) + h (cid:16) v i ( t , − u i ( t ,
0) + ∂ x v i ( t , − ∂ x u i ( t , (cid:17) + hε i ( h ) , where ∀ i ∈ { , . . . I } , lim h → ε i ( h ) = 0 . We get then ∀ i ∈ { , . . . I } , ∂ x v i ( t , ≤ ∂ x u i ( t , − (cid:16) v i ( t , − u i ( t , (cid:17) < ∂ x u i ( t , . Using the growth assumptions on F (assumption ( P )(i)), and the fact that v is a sub-solution while u is a super-solution, we get0 ≤ F ( t , v ( t , , ∂ x v ( t , < F ( t , u ( t , , ∂ x u ( t , ≤ , and then a contradiction.We deduce then for all 0 ≤ s < T , for all ( t, ( x, i )) ∈ [0 , s ] × J a ,exp( − λt + x ) (cid:16) v i ( t, x ) − u i ( t, x ) (cid:17) ≤ . 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 ) . (cid:3) The elliptic problem
As explained in the introduction, the construction of a solution for our parabolic prob-lem (3) relies on a time discretization and on the solvability of the associated ellipticproblem. We review in this section the well-posedness of the elliptic problem (2), whichis formulated for regular maps ( x, i ) u i ( x ), continuous at the junction point, namelyeach i = j ∈ { . . . I } , u i (0) = u j (0) = u (0), that follows at each edge − σ i ( x, ∂ x u i ( x )) ∂ x,x u i ( x ) + H i ( x, u i ( x ) , ∂ x u i ( x )) = 0 , and u i satisfy the following non linear Neumann boundary condition at the vertex F ( u (0) , ∂ x u (0)) = 0 , where ∂ x u (0) = ( ∂ x u (0) , . . . , ∂ x u I (0)) . We introduce the following data for i ∈ { . . . I } F ∈ C ( R × R I , R ) ,σ i ∈ C ([0 , a i ] × R , R ) H i ∈ C ([0 , a i ] × R , R ) φ i ∈ R , satisfying the following assumptions Assumption ( E ) UASI LINEAR PARABOLIC PDE IN A JUNCTION 13 (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 × 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 × R I , such that : F ( b, B ) = 0 . (ii) The ellipticity condition on the σ i ∃ c > , ∀ i ∈ { . . . I } , ∀ ( x, p ) ∈ [0 , a i ] × R , σ i ( x, p ) ≥ c. (iii) For the Hamiltonians H i , we suppose ∃ C H > , ∀ i ∈ { . . . I } , ∀ ( x, u, v, p ) ∈ (0 , a i ) × R , if u ≤ v, C H ( u − v ) ≤ H i ( x, u, p ) − H i ( x, v, p ) . For each i ∈ { . . . I } , we define the following differential operators ( δ i , δ i ) i ∈{ ...I } actingon C ([0 , a i ] × R , 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 ∈{ ...I } = ( σ i ( x, p ) , H i ( x, u, p )) i ∈{ ...I } , which are for all i ∈ { . . . I } δ i σ i = o ( σ i ) ,δ i σ i = O ( σ i ) ,H i = O ( σ i p ) ,δ i H i ≤ o ( σ i p ) ,δ i H i ≤ O ( σ i p ) , 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 uniquenessof the elliptic problem at the junction, with non linear Neumann condition at the junctionpoint. Theorem 3.1.
Assume ( E ) . The following elliptic problem at the junction, with Neumannboundary condition at the vertex − σ i ( x, ∂ x u i ( x )) ∂ x,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 , ∀ i ∈ { . . . I } , u i ( a i ) = φ i , (7) is uniquely solvable in the class C α ( J a ) . Theorem 3.1 is stated without proof in [9]. For the convenience of the reader, we sketchits proof in the Appendix.The uniqueness of the solution of (7), is a consequence of the elliptic comparison Theoremfor smooth solutions, for the Neumann problem, stated in this Section, and whose proofuses the same arguments of the proof of the parabolic comparison Theorem 2.4. Wecomplete this section by recalling the definition of super and sub solution for the ellipticproblem (7), and the corresponding elliptic comparison Theorem.
Definition 3.2.
Let u ∈ C ( J a ) . We say that u is a super solution (resp. sub solution)of − σ i ( x, ∂ x f i ( x )) ∂ x,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 , (8) if − σ i ( x, ∂ x u i ( x )) ∂ x,x u i ( x ) + H i ( x, u i ( x ) , ∂ x u i ( x )) ≥ , ( resp. ≤ , if x ∈ (0 , a i ) ,F ( u (0) , ∂ x u (0)) ≤ , ( resp. ≥ . Theorem 3.3.
Elliptic comparison Theorem, see for instance Theorem 2.1 of [9] .Assume ( E ). Let u ∈ C ( J a ) (resp. v ∈ C ( J a ) ) a super solution (resp. a sub solution) UASI LINEAR PARABOLIC PDE IN A JUNCTION 15 of (8) , satisfying for all i ∈ { . . . 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 onthe results obtained in Section 3 for the elliptic problem, and is done by considering asequence u n ∈ C ( J a ), solving on a time grid an elliptic scheme defined by induction. Wewill prove that the solution u n converges to the required solution.4.1. Estimates on the discretized scheme.
Let n ∈ N ∗ , we consider the followingtime grid, ( t nk = kTn ) ≤ k ≤ n of [0 , T ], and the following sequence ( u k ) ≤ k ≤ n of C α ( J a ),defined recursively by for k = 0 , u = 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 − ( x )) − σ i ( x, ∂ x u i,k ( x )) ∂ x,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 ∈ { . . . I } , u i,k ( a i ) = φ i ( t nk ) . (9)The solvability of the elliptic scheme (9) can be proved by induction, using the samearguments as for Theorem 3.1. The next step consists in obtaining uniform estimatesof ( u k ) ≤ k ≤ n . We start first by getting uniform bounds for n | u k − u k − | (0 ,a i ) using thecomparison Theorem 3.3. Lemma 4.1.
Assume ( P ) . There exists a constant C > , independent of n , dependingonly the data C = C (cid:16) max i ∈{ ...I } n sup x ∈ (0 ,a i ) |− σ i ( x, ∂ x g i ( x )) ∂ x g i ( x )+ H i ( x, g i ( x ) , ∂ x g i ( x )) | + | ∂ t φ i | (0 ,T ) o , C H (cid:17) , such that sup n ≥ max k ∈{ ...n } max i ∈{ ...I } n n | u i,k − u i,k − | (0 ,a i ) o ≤ C, and then sup n ≥ max k ∈{ ...n } max i ∈{ ...I } n | u i,k | (0 ,a i ) o ≤ C + max i ∈{ ...I } n | g i | (0 ,a i ) o . Proof.
Let n > ⌊ C H ⌋ , where C H is defined in assumption ( P ) (iv) c). Let k ∈ { . . . n } ,we define the following sequence M = max i ∈{ ...I } n sup x ∈ (0 ,a i ) | − σ i ( x, ∂ x g i ( x )) ∂ x g i ( x ) + H i ( x, g i ( x ) , ∂ x g i ( x )) | + | ∂ t φ i | (0 ,T ) o ,M k,n = nn − C H M k − ,n , k ∈ { . . . n } . We claim that for each k ∈ { . . . n } max i ∈{ ...I } n n | u i,k − u i,k − | (0 ,a i ) o ≤ M k,n . We give a proof by induction. For this, if k = 1, let us show that the map h defined onthe junction by h := J a → R ( x, i ) M ,n n + g i ( x ) , is a super solution of (9), for k = 1. For this we will use the Elliptic Comparison Theorem3.3.Using the compatibility conditions satisfied by g , namely assumption ( P ) (v), and theassumptions of growth on F , assumption ( P ) (i), we get for the boundary conditions F ( h (0) , ∂ x h (0)) = F ( M ,n n + g (0) , ∂ x g (0)) ≤ F ( g (0) , ∂ x g (0)) = 0 ,h ( a i ) = M ,n n + g i ( a i ) ≥ M ,n n + g i ( a i ) ≥ φ i ( t n ) . For all i ∈ { . . . 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 )) ∂ x h i ( x ) + H i ( x, h i ( x ) , ∂ x h i ( x )) = M ,n − σ i ( x, ∂ x g i ( x )) ∂ x g i ( x ) + H i ( x, M ,n n + g i ( x ) , ∂ x g i ( x )) ≥ M ,n − σ i ( x, ∂ x g i ( x )) ∂ x g i ( x ) + H i ( x, g i ( x ) , ∂ x g i ( x )) − M ,n C H n ≥ . UASI LINEAR PARABOLIC PDE IN A JUNCTION 17
It follows from the comparison Theorem 3.3, that for all i ∈ { . . . I } , and x ∈ [0 , a i ] u ,i ( x ) ≤ M ,n n + g i ( x ) . Using the same arguments, we show that h := J a → R ( x, i )
7→ − M ,n n + g i ( x ) , is a sub solution of (9) for k = 1, and we then getmax i ∈{ ...I } n sup x ∈ (0 ,a ) n | u ,i ( x ) − g i ( x ) | o ≤ M ,n . Let 2 ≤ k ≤ n , suppose that the assumption of induction holds true. Let us show thatthe following map h := J a → R ( x, i ) M k,n n + u i,k − ( x ) , is a super solution of (9). For the boundary conditions, using assumption ( P ) (i), we get F ( h (0) , ∂ x h (0)) = F ( M k,n n + u k − (0) , ∂ x u k − (0)) ≤ F ( u k − (0) , ∂ x u k − (0)) ≤ ,h ( a i ) = M k,n n + u i,k − ( a i ) ≥ M ,n n + u i,k − ( a i ) ≥ φ i ( t nk ) . For all i ∈ { . . . I } , and x ∈ (0 , a i ) n ( h i ( x ) − u i,k − ( x )) − σ i ( x, ∂ x h ( x )) ∂ x h ( x ) + H i ( x, h ( x ) , ∂ x h ( x )) = M k,n − σ i ( x, ∂ x u i,k − ( x )) ∂ x u i,k − ( x ) + H i ( x, M k,n n + u i,k − ( x ) , ∂ x u k − ( x )) ≥ M k,n − σ i ( x, ∂ x u i,k − ( x )) ∂ x u i,k − ( x ) + H i ( x, u i,k − ( x ) , ∂ x u k − ( x )) − C H M k,n n . Since we have for all x ∈ (0 , a i ) − σ i ( x, ∂ x u i,k − ( x )) ∂ x u i,k − ( x ) + H i ( x, u i,k − ( x ) , ∂ x u i,k − ( x )) = − n ( u i,k − ( x ) − u i,k − ( x )) , using the induction assumption we get n ( h i ( x ) − u i,k − ( x )) − σ i ( x, ∂ x h ( x )) ∂ x h ( x ) + H i ( x, ∂ x h ( x ) , ∂ x h ( x )) ≥ M k,n − n ( u i,k − ( x ) − u i,k − ( x )) − C H M k,n n ≥ M k,n n − C H n − M k − ,n ≥ . 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 − ( x ) . Using the same arguments, we show that h := J a → R ( x, i )
7→ − M k,n n + u i,k − ( x ) , is a sub solution of (9), and we getmax i ∈{ ...I } n n | u i,k ( x ) − u i,k − ( x ) | (0 ,a i ) o ≤ M k,n . We obtain finally using that for all k ∈ { . . . n } M k,n ≤ M n,n ,M k,n = (cid:16) nn − C H (cid:17) k M , and M n,n n → + ∞ −−−−→ M := exp( C H ) max i ∈{ ...I } n sup x ∈ (0 ,a i ) | − σ i ( x, ∂ x g i ( x )) ∂ x g i ( x ) + H i ( x, g i ( x ) , ∂ x g i ( x )) | + | ∂ t φ i | (0 ,T ) o , that sup n ≥ max k ∈{ ...n } max i ∈{ ...I } n n | u i,k − u i,k − | (0 ,a i ) o ≤ C, sup n ≥ max k ∈{ ...n } max i ∈{ ...I } n | u i,k | (0 ,a i ) o ≤ C + max i ∈{ ...I } n | g i | (0 ,a i ) o . That completes the proof. (cid:3)
The next step consists in obtaining uniform estimates for | ∂ x u k | (0 ,a i ) , in terms of n | u k − u k − | (0 ,a i ) and the quantities ( ν, ν, µ, γ, ε, P ) introduced in assumption ( P ) (ii), (iii) and UASI LINEAR PARABOLIC PDE IN A JUNCTION 19 (iv). More precisely, we use similar arguments as for the proof of Theorem 14.1 of [5],using a classical argument of upper and lower barrier functions at the boundary. Theassumption of growth ( P ) (ii) and (iii) are used in a key way to get an uniform boundon the gradient at the boundary. Finally to conclude, we appeal to a gradient maximumprinciple, using the growth assumption ( P ) (iv), adapting Theorem 15.2 of [5] to ourelliptic scheme. Lemma 4.2.
Assume ( P ). There exists a constant C > , independent of n , dependingonly the data (cid:16) ν, ν, µ ( | u | ) , γ ( | u | ) , ε ( | u | ) , sup | p |≥ P ( | u | , | p | ) , | ∂ x g i | (0 ,a i ) , | u | ≤ sup n ≥ max k ∈{ ...n } max i ∈{ ...I } n | u i,k | (0 ,a i ) o , sup n ≥ max k ∈{ ...n } max i ∈{ ...I } n n | u i,k − u i,k − | (0 ,a i ) o(cid:17) , such that sup n ≥ max k ∈{ ...n } max i ∈{ ...I } n | ∂ x u i,k | (0 ,a i ) o ≤ C. Proof.
Step 1 :
We claim that, for each k ∈ { . . . n } , max i ∈{ ...I } n | ∂ x u i,k | ∂ (0 ,a i ) o isbounded by the data, uniformly in n .It follows from Lemma 4.1, that there exists M > n ≥ max k ∈{ ...n } max i ∈{ ...I } n | u i,k | (0 ,a i ) + n | u i,k − u i,k − | (0 ,a i ) o ≤ M. We fix i ∈ { . . . 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 ) , ∂ x w + i,k ( x )) ≥ , ∀ 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 ) , ∂ x w − i,k ( x )) ≤ , ∀ 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 Q i ( x, u, p, S ) = n ( u − u i,k − ( 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 thecomparison 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, = 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 ], ∂ x w + i,k ( x ) = − β∂ x w + i,k ( x ) , and w + i,k (0) = u i,k (0). Letwe choose ( θ, κ ), such that ∀ k ∈ { . . . n } , < κ ≤ min i ∈{ ...I } a i , w + i,k ( κ ) ≥ M, ∂ x w + i,k ( κ ) ≥ β. (10)We choose for instance θ = β exp(2 βM ) + 1min i ∈{ ...I } a i exp(2 βM ) κ = 1 θ (cid:16) exp(2 βM ) − (cid:17) . (11)The constant β will be chosen in order to get β ≥ sup k ∈{ ...n } sup x ∈ [0 ,κ ] µ ( w + i,k ( x ))(1 + ∂ x w + i,k ( x )) m + Mν (1 + ∂ x w + i,k ( x )) m − ∂ x w + i,k ( x ) , (12)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 ) = 2 M,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 (12), for instance β ≥ µ (2 M ) ν (cid:16) β (cid:17) + Mνβ . UASI LINEAR PARABOLIC PDE IN A JUNCTION 21
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, = u i, , and for all k ∈ { . . . n } w + i,k (0) = u i,k (0) , w + i,k ( κ ) ≥ u i,k ( κ ) ,w + i,k ( x ) = w + i,k − ( x ) + u i,k (0) − u i,k − (0) ≥ w + i,k − ( x ) − Mn , we get using the assumption of induction, assumption ( P ) (ii) and (iii), and (12) that forall x ∈ (0 , κ ) n ( w + i,k ( x ) − u i,k − ( x )) − σ i ( x, ∂ x w + i,k ( x )) ∂ x,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 ) + H i ( x, w + i,k ( x ) , ∂ x w + i,k ( x )) ≥− M + βν (1 + ∂ x w + i,k ( x )) m − ∂ x w + i,k ( x ) + µ ( w + i,k ( x ))(1 + ∂ x w + i,k ( x )) m ≥ . We obtain thereforesup n ≥ max k ∈{ ...n } max i ∈{ ...I } ∂ x u i,k (0) ≤ θβ ∨ ∂ x g i (0) . With the same arguments we can show that w − i, = g i ( x ) w − i,k : x u i,k (0) − 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 thisStep, since they can be found in the proof of Theorem 15.2 of [5]. It follows from Lemma4.1 that there exists
M > n ≥ max k ∈{ ...n } max i ∈{ ...I } n | u i,k | (0 ,a i ) o ≤ M. We set furthermore ∀ ( x, u, p ) ∈ [0 , a i ] × R , H ni,k ( x, u, p ) = n ( u − u i,k − ( 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,x u ( x ) − H ni,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 followingequalities δ i H ni,k ( x, u, p ) = δ i H i ( x, u, p ) + n ( p − ∂ x u i,k − ( x )) p , δ i H ni,k ( x, u, p ) = δ i H i ( x, u, p ) , (13)where we recall that the operators δ i and ¯ δ i are defined in assumption ( E ) (iii). We followthe proof of Theorem 15.2 in [5]. We set u = ψ ( u ), where ψ ∈ C [ m, M ], is increasingand m = φ ( − M ), M = φ ( M ). In the sequel, we will set v = ∂ x u and v = ∂ x u . Tosimplify the notations, we will omit the variables ( x, u ( x ) , ∂ x u ( x )) in the functions σ i and H ni,k , and the variable u for ψ . We assume first that the solution u ∈ C ([ − M, M ]), andwe follow exactly all the computations that lead to equation of (15.25) of [5] to get thefollowing inequality σ i ∂ x,x v + B i ∂ x v + G ni,k ≥ , (14)where B i and G ni,k have the same expression in (15.26) of [5] with ( σ i = σ ∗ i , c i = 0). Wechoose ( r = 0 , s = 0), since we will see in the sequel (15), that condition (15.32) of [5]holds under assumption assumption ( P ). We have more precisely B i = ψ ′ ∂ p σ i ∂ x,x u − ∂ p H i + ω∂ p ( σ i p ) ,G ni,k = ω ′ ψ ′ + κ i ω + β i ω + θ ni,k ,ω = ψ ′′ ψ ′ ∈ C ([ m, M ]) ,κ i = 1 σ i p (cid:16) δ i ( σ i p ) + p σ i | ( δ i + 1) σ i | (cid:17) ,β i = 1 σ i p (cid:16) δ i ( σ i p ) − δ i H i + p σ i (( δ i + 1) σ i )( δ i σ i ) (cid:17) ,θ ni,k = 1 σ i p (cid:16) p σ i | δ i σ i | − δ i H ni,k (cid:17) = θ i − σ i p (cid:16) n ( p − ∂ x u i,k − ( x )) p (cid:17) ,θ i = 1 σ i p (cid:16) p σ i | δ i σ i | − δ i H i (cid:17) . UASI LINEAR PARABOLIC PDE IN A JUNCTION 23
We set in the sequel G i = ∂ x ω∂ x ψ + κ i ω + β i ω + θ i , in order to get G ni,k = G i − σ i p (cid:16) n ( p − ∂ x u i,k − ( x )) p (cid:17) . More precisely, we see from (13) that all the coefficients ( B i , κ i , β i , θ i ) can be chosenindependent of n and u i,k − . The main argument then to get a bound of ∂ x u is to applya maximum principle for v in (14), and this will be done as soon as we ensure G ni,k ≤ , for | ∂ x u | ≥ L nk . On the other hand, using assumption ( P ) (ii) (iii) and (iv), it is easy to check that thereexists a constants ( a, b, c ), depending only on the data (cid:16) ν, ν, µ ( M ) , γ ( M ) , ε ( M ) , sup | p |≥ P ( M, | p | ) (cid:17) , 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, where a = 1 ν ( γ ( M ) + ν ) + 12 + γ ( M ) ν ,b = ε ( M ) + sup | p |≥ P ( M, | p | ) + γ ( M ) ν + ( ε ( M ) + sup | p |≥ P ( M, | p | ))( ν + γ ( M )) ν ,c = ( ε ( M ) + sup | p |≥ P ( M, | p | )) ν + 2( ε ( M ) + sup | p |≥ P ( M, | p | )) ν . As it has been on the proof of Theorem 15.2 of [5], we choose then L = L ( a, b, c ), and ψ ( · ) = ψ ( a, b, c )( · ) such that we have G i ≤ , if | ∂ x u ( x ) | ≥ L ( a, b, c ) . We see then from the expression of θ ni,k that we get G ni,k ≤ , if | ∂ x u ( x ) | ≥ L ( a, b, c ) ∨ | ∂ x u i,k − ( x ) | . Therefore applying the maximum principle to v in (14), and from the relation u = ψ ( u ), v = ∂ x u we get finally | ∂ x u | (0 ,a i ) ≤ max (cid:16) max ψ ′ ( a, b, c )( · )min ψ ′ ( a, b, c )( · ) , | ∂ x u | ∂ (0 ,a i ) , L ( a, b, c ) , | ∂ x u i,k − | (0 ,a i ) (cid:17) . This upper bound still holds if u ∈ C ([0 , a i ]), (cf. (15.30) and (15.31) of the proof ofTheorem 15.2 in [5]). Finally applying the upper bound above to the solution u k , we getby induction that sup n ≥ max k ∈{ ...n } max i ∈{ ...I } n | ∂ x u i,k | (0 ,a i ) o ≤ max (cid:16) 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 ) (cid:17) . This completes the proof. (cid:3)
The following Proposition follows from Lemmas 4.1 and 4.2, assumption ( P ) (ii) (iii),and from the relation ∀ x ∈ [0 , a i ] , | ∂ x,x u i,k ( x )) | ≤ | n ( u i,k ( x ) − u i,k − ( 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 − ( x )) | + µ ( | u i,k ( x ) | )(1 + | ∂ x u i,k ( x ) | m ) ν (1 + | ∂ x u i,k ( x ) | m − ) . Proposition 4.3.
Assume ( P ). There exist constants ( M , M , M ) , depending only thedata introduced in assumption ( P ) M = M (cid:16) max i ∈{ ...I } n sup x ∈ (0 ,a i ) | − σ i ( x, ∂ x g i ( x )) ∂ x g i ( x ) + H i ( x, g i ( x ) , ∂ x g i ( x )) | + | ∂ t φ i | (0 ,T ) o , max i ∈{ ...I } | g i | (0 ,a i ) , C H (cid:17) ,M = M (cid:16) ν, ν, µ ( M ) , γ ( M ) , ε ( M ) , sup | p |≥ P ( M , | p | ) , | ∂ x g i | (0 ,a i ) , M (cid:17) ,M = M (cid:16) M , ν (1 + | p | ) m − , µ ( | u | )(1 + | p | ) m , | u | ≤ M , | p | ≤ M (cid:17) , UASI LINEAR PARABOLIC PDE IN A JUNCTION 25 such that sup n ≥ max k ∈{ ...n } max i ∈{ ...I } n | u i,k | (0 ,a i ) o ≤ M , sup n ≥ max k ∈{ ...n } max i ∈{ ...I } n | ∂ x u i,k | (0 ,a i ) o ≤ M , sup n ≥ max k ∈{ ...n } max i ∈{ ...I } n | n ( u i,k − u i,k − ) | (0 ,a i ) o ≤ M , sup n ≥ max k ∈{ ...n } max i ∈{ ...I } n | ∂ x,x u i,k | (0 ,a i ) o ≤ M . Unfortunately, we are unable to give an upper bound of the modulus of continuity of ∂ x,x u i,k in C α ([0 , a ]) independent of n . However, we are able to formulate in the weaksense a limit solution. From the regularity of the coefficients, using some tools introducedin Section 1, Lemma 2.1, we get interior regularity, and a smooth limit solution.4.2. Proof of Theorem 2.2.
Proof.
The uniqueness is a result of the comparison Theorem 2.4. To simplify the nota-tions, we set for each i ∈ { . . . I } , and for each ( x, q, u, p, S ) ∈ [0 , a i ] × R 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 nk = kTn ) ≤ k ≤ n of [0 , T ], and ( u k ) ≤ k ≤ n the solutionof (9).From estimates of Proposition 4.3, there exists a constant M > n , suchthat sup n ≥ max k ∈{ ...n } max i ∈{ ...I } n | u i,k | (0 ,a i ) + | n ( u i,k − u i,k − ) | (0 ,a i ) + | ∂ x u i,k | (0 ,a i ) + | ∂ x,x u i,k | (0 ,a i ) o ≤ M. (15)We define the following sequence ( v n ) n ≥ in C , ( J aT ), piecewise differentiable with respectto its first variable by ∀ i ∈ { . . . I } , v i, (0 , x ) = g i ( x ) if x ∈ [0 , a i ] ,v i,n ( t, x ) = u i,k ( x ) + n ( t − t nk )( u i,k +1 ( x ) − u i,k ( x )) if ( t, x ) ∈ [ t nk , t nk +1 ) × [0 , a i ] . We deduce then from (15), that there exists a constant M independent of n , dependingonly on the data of the system, such that for all i ∈ { . . . I }| v i,n | α [0 ,T ] × [0 ,a i ] + | ∂ x v i,n | αx, [0 ,T ] × [0 ,a i ] ≤ M . Using Lemma 2.1, we deduce that there exists a constant M ( α ) >
0, independent of n ,such that for all i ∈ { . . . I } , we have the following global H¨older condition | ∂ x v i,n | α t, [0 ,T ] × [0 ,a i ] + | ∂ x v i,n | αx, [0 ,T ] × [0 ,a i ] ≤ M ( α ) . We deduce then from Ascoli’s Theorem, that up to a sub sequence n , ( v i,n ) n ≥ convergein C , ([0 , T ] × [0 , a i ]) to v i , and then v i ∈ C α , α ([0 , T ] × [0 , a i ]).Since v n satisfies the following continuity condition at the junction point ∀ ( i, j ) ∈ { . . . I } , ∀ n ≥ , ∀ t ∈ [0 , T ] , v i,n ( t,
0) = v j,n ( t,
0) = v n ( t, , we deduce then v ∈ C α , α ( J aT ).We now focus on the regularity of v in ◦ J aT , and we will prove that v ∈ C α , α ( ◦ J aT ), andsatisfies on each edge Q i ( x, v i ( t, x ) , ∂ t v i ( t, x ) , ∂ x v i ( t, x ) , ∂ x,x v i ( t, x )) = 0 , if ( t, x ) ∈ (0 , T ) × (0 , a i ) . Using once again (15), there exists a constant M independent of n , such that for each i ∈ { . . . I }k ∂ t v i,n k L ((0 ,T ) × (0 ,a i )) ≤ M , k ∂ x,x v i,n k L ((0 ,T ) × (0 ,a i )) ≤ M . Hence we get up to a sub sequence, that ∂ t v i,n ⇀ ∂ t v i , ∂ x,x v i,n ⇀ ∂ x,x v i , weakly in L ((0 , T ) × (0 , a i )).The continuity of the coefficients ( σ i , H i ) i ∈{ ...I } , Lebesgue Theorem, the linearity of Q i in UASI LINEAR PARABOLIC PDE IN A JUNCTION 27 the variable ∂ t and ∂ x,x , allows us to get for each i ∈ { . . . I } , up to a subsequence n p Z T Z a i (cid:16) Q i ( x, v i,n p ( t, x ) , ∂ t v i,n p ( t, x ) , ∂ x v i,n p ( t, x ) , ∂ x,x v i,n p ( t, x )) (cid:17) ψ ( t, x ) dxdt p → + ∞ −−−−→ Z T Z a i (cid:16) Q i ( x, v i ( t, x ) , ∂ t v i ( t, x ) , ∂ x v i ( t, x ) , ∂ x,x v i ( t, x )) (cid:17) 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 )) Z T Z a i (cid:16) Q i ( x, v i,n p ( t, x ) , ∂ t v i,n p ( t, x ) , ∂ x v i,n p ( t, x ) , ∂ x,x v i,n p ( t, x ))) (cid:17) ψ ( t, x ) dxdt p → + ∞ −−−−→ . Using that ( u k ) ≤ k ≤ n is the solution of (9), we get for any ψ ∈ C ∞ c ((0 , T ) × (0 , a i )) Z T Z a i (cid:16) Q i ( x, v i,n ( t, x ) , ∂ t v i,n ( t, x ) , ∂ x v i,n ( t, x ) , ∂ x,x v i,n ( t, x )) (cid:17) ψ ( t, x ) dxdt = n − X k =0 Z t nk +1 t nk Z a i (cid:16) σ i ( x, ∂ x u i,k +1 ( x )) ∂ x,x u i,k +1 ( x ) − σ i ( x, ∂ x v i,n ( t, x )) ∂ x,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 )) (cid:17) ψ ( t, x ) dxdt. (16)Using assumption ( P ) more precisely the Lipschitz continuity of the Hamiltonians H i , theH¨older equicontinuity in time of ( v i,n , ∂ x v i,n ), there exists a constant M ( α ) independentof n , such that for each i ∈ { . . . I } , for each ( t, x ) ∈ [ t nk , t nk +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 ( α )( t − t nk ) α , and therefore for any ψ ∈ C ∞ c ((0 , T ) × (0 , a i )) (cid:12)(cid:12)(cid:12) n − X k =0 Z t nk +1 t nk Z a i (cid:16) 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 )) (cid:17) ψ ( t, x ) dxdt (cid:12)(cid:12)(cid:12) ≤ a i M ( α ) | ψ | (0 ,T ) × (0 ,a i ) n − α n → + ∞ −−−−→ . For the last term in (16), we write for each i ∈ { . . . I } , for each ( t, x ) ∈ ( t nk , t nk +1 ) × (0 , a i ) σ i ( x, ∂ x u i,k +1 ( x )) ∂ x,x u i,k +1 ( x ) − σ i ( x, ∂ x v i,n ( t, x )) ∂ x,x v i,n ( t, x ) = (cid:16) σ i ( x, ∂ x u i,k +1 ( x )) − σ i ( x, ∂ x v i,n ( t, x )) (cid:17) ∂ x,x u i,k ( x ) + (17) (cid:16) σ i ( x, ∂ x u i,k +1 ( x )) − n ( t − t nk ) σ i ( x, ∂ x v i,n ( t, x )) (cid:17)(cid:16) ∂ x,x u i,k +1 ( x ) − ∂ x,x u i,k ( x ) (cid:17) . (18) Using again the H¨older equicontinuity in time of ( v i,n , ∂ x v i,n ) as well as the uniform boundon | ∂ x,x u i,k | [0 ,a i ] (15), we can show that for (17), for any ψ ∈ C ∞ c ((0 , T ) × (0 , a i )), (cid:12)(cid:12)(cid:12) n − X k =0 Z t nk +1 t nk Z a i (cid:16) σ i ( x, ∂ x u i,k +1 ( x )) − σ i ( x, ∂ x v i,n ( t, x )) (cid:17) ∂ x,x u i,k ( x ) ψ ( t, x ) dxdt (cid:12)(cid:12)(cid:12) n → + ∞ −−−−→ . Finally, from assumptions ( P ), for all i ∈ { . . . I } , σ i is differentiable with respect to allits variable, integrating by part we get for (18) (cid:12)(cid:12)(cid:12) n − X k =0 Z t nk +1 t nk Z a i (cid:16) σ i ( x, ∂ x u i,k +1 ( x )) − n ( t − t nk ) σ i ( x, ∂ x v i,n ( t, x )) (cid:17)(cid:16) ∂ x,x u i,k +1 ( x ) − ∂ x,x u i,k ( x ) (cid:17) ψ ( t, x ) dxdt (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) n − X k =0 Z t nk +1 t nk Z a i (cid:16) ∂ x (cid:16) σ i ( x, ∂ x u i,k +1 ( t, x )) ψ ( t, x ) (cid:17) − n ( t − t nk ) ∂ x (cid:16) σ i ( x, ∂ x v i,n ( t, x )) ψ ( t, x ) (cid:17)(cid:17)(cid:16) ∂ x u i,k +1 ( x ) − ∂ x u i,k ( x ) (cid:17) dxdt (cid:12)(cid:12)(cid:12) n → + ∞ −−−−→ . We conclude that for any ψ ∈ C ∞ c ((0 , T ) × (0 , a i )) Z T Z a i (cid:16) Q i ( x, v i ( t, x ) , ∂ t v i ( t, x ) , ∂ x v i ( t, x ) , ∂ x,x v i ( t, x ))) (cid:17) ψ ( t, x ) dxdt = 0 . It is then possible to consider the last equation as a linear one, with coefficients ˜ σ i ( t, x ) = σ i ( x, ∂ x v i ( t, x )), ˜ H i ( t, x ) = H i ( x, v i ( t, x ) , ∂ x v i ( t, x )) belonging to the class C α ,α ((0 , T ) × (0 , a i )), and using Theorem III.12.2 of [7], we get finally that for all i ∈ { . . . I } , v i ∈C α , α ((0 , T ) × (0 , a i )), which means that v ∈ C α , α ( ◦ J aT ).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 ) , ∂ x,x v i ( t, x ))) = 0 , if ( t, x ) ∈ (0 , T ) × (0 , a i ) . From the estimates (15), we know that ∂ t v i,n and ∂ x,x v i,n are uniformly bounded by n .We deduce finally that v ∈ C α , αb ( ◦ J aT ).We conclude by proving that v satisfies the non linear Neumann boundary condition atthe vertex. For this, let t ∈ (0 , T ); we have up to a sub sequence n p F ( v n p ( t, , ∂ x v n p ( t, −−−−→ p → + ∞ F ( v ( t, , ∂ x v ( t, . UASI LINEAR PARABOLIC PDE IN A JUNCTION 29
On the other hand, using that F ( u k (0) , ∂ u k ( x )) = 0, we know from the continuity of F (assumption ( P )), the H¨older equicontinuity in time of t v n ( t, t ∂ x v ( t, M ( α ) independent of n , such that if t ∈ [ t nk , t nk +1 ) | F ( v n ( t, , ∂ x v n ( t, | = | F ( v n ( t, , ∂ x v n ( t, − F ( u k (0) , ∂ x u k (0)) | ≤ sup n | F ( u, x ) − F ( v, y ) | , | u − v | + k x − y k R I ≤ M ( α ) n − α o n → + ∞ −−−−→ . Therefore, we conclude once more from the continuity of F (assumption ( P )), the com-patibility condition (assumption ( P ) (v)), that for each t ∈ [0 , T ) F ( v ( t, , ∂ x v ( t, . On the other hand, it is easy to get ∀ i ∈ { . . . 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 aconsequence of Proposition 4.3, and Lemma 2.1, which completes the proof. (cid:3)
On the existence for unbounded junction.
We give in this subsection a resulton the existence and the uniqueness of the solution for the parabolic problem (1), in aunbounded junction J defined for I ∈ N ∗ edges by J = n X = ( x, i ) , x ∈ R + and i ∈ { , . . . , I } o . In the sequel, C , ( J T ) ∩ C , ( ◦ J T ) is the class of function with regularity C , ([0 , T ] × [0 , + ∞ )) ∩ C , ((0 , T ) × (0 , + ∞ )) on each edge, and L ∞ ( J T ) is the set of measurable realbounded maps defined on J T .We introduce the following data F ∈ C ( R × R I , R ) g ∈ C b ( J ) ∩ C b ( ◦ J ) , and for each i ∈ { . . . I } σ i ∈ C ( R + × R , R ) H i ∈ C ( R + × R , R ) φ i ∈ C ([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 × 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 × R I , F ( b, B ) = 0 . We suppose moreover that there exist a parameter m ∈ R , m ≥ σ i ) i ∈{ ...I } : there exists ν, ν , strictly positiveconstants such that ∀ i ∈ { . . . I } , ∀ ( x, p ) ∈ R + × R ,ν (1 + | p | ) m − ≤ σ i ( x, p ) ≤ ν (1 + | p | ) m − . (iii) The growth of the ( H i ) i ∈{ ...I } with respect to p exceed the growth of the σ i withrespect to p by no more than two, namely there exists µ an increasing real continuousfunction such that ∀ i ∈ { . . . I } , ∀ ( x, u, p ) ∈ R + × R , | H i ( x, u, p ) | ≤ µ ( | u | )(1 + | p | ) m . UASI LINEAR PARABOLIC PDE IN A JUNCTION 31 (iv) We impose the following restrictions on the growth with respect to p of the derivativesfor the coefficients ( σ i , H i ) i ∈{ ...I } , which are for all i ∈ { . . . I } , a ) | ∂ p σ i | R + × R (1 + | p | ) + | ∂ p H i | R + × R ≤ γ ( | u | )(1 + | p | ) m − ,b ) | ∂ x σ i | R + × R (1 + | p | ) + | ∂ x H i | R + × R ≤ (cid:16) ε ( | u | ) + P ( | u | , | p | ) (cid:17) (1 + | p | ) m +1 ,c ) ∀ ( x, u, p ) ∈ R + × R , − C H ≤ ∂ u H i ( x, u, p ) ≤ (cid:16) ε ( | u | ) + P ( | u | , | p | ) (cid:17) (1 + | p | ) m , where γ and ε are continuous non negative increasing functions. P is a continuous func-tion, increasing with respect to its first variable, and tends to 0 for p → + ∞ , uniformlywith respect to its first variable, from [0 , u ] with u ∈ R , and C H > γ, ε, P, C H ) are independent of i ∈ { . . . I } .(v) A compatibility conditions for gF ( g (0) , ∂ x g (0)) = 0 . We state here a comparison Theorem for the problem 1, in a unbounded junction.
Theorem 4.4.
Assume ( P ∞ ) . Let u ∈ C , ( J T ) ∩C , ( ◦ J T ) ∩ L ∞ ( J T ) (resp. v ∈ C , ( J T ) ∩C , ( ◦ J T ) ∩ L ∞ ( J T ) ) be a super solution (resp. a sub solution) of (4) (where a i = + ∞ ),satisfying for all i ∈ { . . . 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 bechosen in the sequel. We argue as in the proof of Theorem 2.4, assumingsup ( t, ( x,i )) ∈J Ks exp( − λt − ( x − (cid:16) v i ( t, x ) − u i ( t, x ) (cid:17) > . Using the boundary conditions satisfied by u and v , the above supremum is reached at apoint ( t , ( x , j )) ∈ (0 , s ] × J , with 0 ≤ x ≤ K . If x ∈ [0 , K ), the optimality conditions are given for x = 0 by − λ ( v j ( t , x ) − u j ( t , x )) + ∂ t v j ( t , x ) − ∂ t u j ( t , x ) ≥ , − ( x − (cid:16) v j ( t , x ) − u j ( t , x ) (cid:17) + ∂ x v j ( t , x ) − ∂ x u j ( t , x ) = 0 , (cid:16) v j ( t , x ) − u j ( t , x ) (cid:17) − x − (cid:16) v j ( t , x ) − u j ( t , x ) (cid:17) + (cid:16) ∂ x,x v j ( t , x ) − ∂ x,x u j ( t , x ) (cid:17) ≤ , and if x = 0, ∀ i ∈ { , . . . I } , ∂ x v i ( t , ≤ ∂ x u i ( t , − (cid:16) v i ( t , − u i ( t , (cid:17) < ∂ x u i ( t , . If x = 0, we obtain a contradiction exactly as in the proof of Theorem 2.4. On the otherhand if x ∈ (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 ), (see (5) and (6)), where C > K , to get again a contradiction. We deduce that, ifsup ( t, ( x,i )) ∈J Ks exp( − λ ( K ) t − ( x − (cid:16) v i ( t, x ) − u i ( t, x ) (cid:17) > , then for all ( t, ( x, i )) ∈ [0 , T ] × J K exp( − λ ( K ) t − ( x − (cid:16) v i ( t, x ) − u i ( t, x ) (cid:17) ≤ exp( − λ ( K ) t − ( K − (cid:16) v i ( t, K ) − u i ( t, K ) (cid:17) . Hence for all ( t, ( x, i )) ∈ [0 , T ] × J K exp( − ( x − (cid:16) v i ( t, x ) − u i ( t, x ) (cid:17) ≤ exp( − ( K − (cid:16) v i ( t, K ) − u i ( t, K ) (cid:17) . On the other hand, ifsup ( t, ( x,i )) ∈J Ks exp( − λ ( K ) t − ( x − (cid:16) v i ( t, x ) − u i ( t, x ) (cid:17) ≤ , then for all ( t, ( x, i )) ∈ [0 , T ] × J K exp( − λ ( K ) t − ( x − (cid:16) v i ( t, x ) − u i ( t, x ) (cid:17) ≤ . So exp( − ( x − (cid:16) v i ( t, x ) − u i ( t, x ) (cid:17) ≤ . UASI LINEAR PARABOLIC PDE IN A JUNCTION 33
Finally we have, for all ( t, ( x, i )) ∈ [0 , T ] × J K max (cid:16) , exp( − ( x − (cid:16) v i ( t, x ) − u i ( t, x ) (cid:17)(cid:17) ≤ exp( − ( K − (cid:16) || u || L ∞ ( J T ) + || v || L ∞ ( J T ) (cid:17) . Sending K → ∞ and using the boundedness of u and v , we deduce the inequality v ≤ u in [0 , T ] × J . (cid:3) Theorem 4.5.
Assume ( P ∞ ) . The following parabolic problem with Neumann boundarycondition at the vertex ∂ t u i ( t, x ) − σ i ( x, ∂ x u i ( t, x )) ∂ x,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, , ∂ x u ( t, , if t ∈ [0 , T ) , ∀ i ∈ { . . . I } , u i (0 , x ) = g i ( x ) , if x ∈ [0 , + ∞ ) , (19) is uniquely solvable in the class C α , α ( J T ) ∩C α , α ( ◦ J T ) . There exist constants ( M , M , M ) ,depending only the data introduced in assumption ( P ∞ ) M = M (cid:16) max i ∈{ ...I } n sup x ∈ (0 , + ∞ ) | − σ i ( x, ∂ x g i ( x )) ∂ x g i ( x ) + H i ( x, g i ( x ) , ∂ x g i ( x )) | o , max i ∈{ ...I } | g i | (0 , + ∞ ) , C H (cid:17) ,M = M (cid:16) ν, ν, µ ( M ) , γ ( M ) , ε ( M ) , sup | p |≥ P ( M , | p | ) , | ∂ x g i | (0 , + ∞ ) , M (cid:17) ,M = M (cid:16) M , ν (1 + | p | ) m − , µ ( | u | )(1 + | p | ) m , | u | ≤ M , | p | ≤ M (cid:17) , such that || u || C ( J T ) ≤ M , || ∂ x u || C ( J T ) ≤ M , || ∂ t u || C ( J T ) ≤ M , || ∂ x,x u || C ( J T ) ≤ M . Moreover, there exists a constant M ( α ) depending on (cid:16) α, M , M , M (cid:17) such that for any a ∈ (0 , + ∞ ) I || u || C α , α ( J aT ) ≤ M ( α ) . Proof.
Assume ( P ∞ ) and let a = ( a, . . . , a ) ∈ (0 , + ∞ ) I . Applying Theorem 2.2, we candefine u a ∈ C , ( J aT ) ∩ C , ( ◦ J aT ) as the unique solution of ∂ t u i ( t, x ) − σ i ( x, ∂ x u i ( t, x )) ∂ x,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 ) ,F ( u ( t, , ∂ x u ( t, , if t ∈ [0 , T ) , ∀ i ∈ { . . . I } , u i ( t, a ) = g i ( a ) , if t ∈ [0 , T ] , ∀ i ∈ { . . . I } , u i (0 , x ) = g i ( x ) , if x ∈ [0 , a ] . (20)Using assumption ( P ∞ ) and Theorem 2.2, we get that there exists a constant C > a such that sup a ≥ || u a || C , ( J aT ) ≤ C. We are going to send a to + ∞ in (20).Following the same argument as for the proof of Theorem 2.2, we get that, up to asub sequence, u a converges locally uniformly to some map u which solves (19). On theother 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 inTheorem 4.5, are a consequence of Theorem 2.2 and assumption ( P ∞ ). (cid:3) Appendix A. Functionnal spaces
In this section, we recall several classical notations from [7]. Let l, T ∈ (0 , + ∞ ) and Ω bean open and bounded subset of R n with smooth boundary ( n > T = (0 , T ) × Ω,and we introduce the following spaces :-if l ∈ N ∗ , (cid:16) C l ,l (Ω T ) , k · k C l ,l (Ω T ) (cid:17) is the Banach space whose elements are continuousfunctions ( t, x ) u ( t, x ) in Ω T , together with all its derivatives of the form ∂ rt ∂ sx u , with2 r + s < l . The norm k · k C l ,l (Ω T ) is defined for all u ∈ C l ,l (Ω T ) by k u k C l ,l (Ω T ) = X r + s = j sup ( t,x ) ∈ Ω T | ∂ rt ∂ sx u ( t, x ) | . -if l / ∈ N ∗ , (cid:16) C l ,l (Ω T ) , k . k C l ,l (Ω T ) (cid:17) is the Banach space whose elements are continuous UASI LINEAR PARABOLIC PDE IN A JUNCTION 35 functions ( t, x ) u ( t, x ) in Ω T , together with all its derivatives of the form ∂ rt ∂ sx u , with2 r + s < l , and satisfying an H¨older condition with exponent l − r − s in their first variable,and with exponent ( l − ⌊ l ⌋ ) in their second variable, over all the connected componentsof Ω T whose radius is smaller than 1.The norm k · k C l ,l (Ω T ) is defined for all u ∈ C l ,l (Ω T ) by k u k C l ,l (Ω T ) = | u | l Ω T + ⌊ l ⌋ X j =0 | u | j Ω T , with ∀ j ∈ { , . . . , l } , | u | j Ω T = X r + s = j sup ( t,x ) ∈ Ω T | ∂ rt ∂ sx u ( t, x ) | , | u | l Ω T = | u | lx, Ω T + | u | l t, Ω T , | u | lx, Ω T = X r + s = ⌊ l ⌋ | ∂ rt ∂ sx u ( t, x ) | l −⌊ l ⌋ x, Ω T , | u | lt, Ω T = X
0, we denote by L ((0 , T ) × (0 , R )) the usual space of square integrable maps andby C ∞ c ((0 , T ) × (0 , R )) the set of infinite continuous differentiable functions on (0 , T ) × (0 , R ), with compact support. Appendix B. The Elliptic problem
Proposition B.1.
Let θ ∈ R , i ∈ { , . . . , I } and assume ( E ) holds. Let u θi ∈ C ([0 , a i ]) be the solution to − σ i ( x, ∂ x u θi ( x )) ∂ x,x u θi ( x ) + H i ( x, u θi ( x ) , ∂ x u θi ( x )) = 0 , if x ∈ (0 , a i ) u θi (0) = u θ (0) = θ,u θi ( a i ) = φ i . (21) Then the following map
Ψ := R → C ([0 , a i ]) θ u θi is continuous.Proof. Let θ n a sequence converging to θ . Using the Schauder estimates Theorem 6.6 of[5], we get that there exists a constant M > n , depending only the data,such that for all α ∈ (0 , k u θ n i k C α ([0 ,a i ]) ≤ M. From Ascoli’s Theorem, u θ n i converges up to a subsequence to v in C ([0 , a i ]) solution of(21). By uniqueness of the solution of (21), u θ n i converges necessary to the solution u θi of(21) in C ([0 , a i ]), which completes the proof. (cid:3) Proof of Theorem 3.1.
Proof.
The uniqueness of (7) results from the elliptic comparison Theorem 3.3.We turn to the solvalbility, and for this let θ ∈ R . We consider the elliptic Dirichletproblem at the junction − σ i ( x, ∂ x u i ( x )) ∂ x,x u i ( x ) + H i ( x, u i ( x ) , ∂ x u i ( x )) = 0 , if x ∈ (0 , a i ) , ∀ i ∈ { . . . I } , u i (0) = u (0) = θ,u i ( a i ) = φ i . (22) UASI LINEAR PARABOLIC PDE IN A JUNCTION 37
For all i ∈ { . . . I } , each elliptic problem is uniquely solvable on each edge in C α ([0 , a i ]),then (22) is uniquely solvable in the class C 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 × 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 ≥ | b | + max i ∈{ ...I } n | φ i | + | a i B i | + K i C H o , and let us show that f : x θ + B i x , is a super solution on each edge J a i i of (22).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 )) ∂ x,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 ≥ . We then get that for each i ∈ { . . . I } , x ∈ [0 , a i ], u θi ( x ) ≤ θ + B i x , and a Taylor expansionin the neighborhood of the junction point gives that for each i ∈ { . . . I } , ∂ x u θi (0) ≤ B i .Since u θ (0) = θ ≥ b , we then get from assumption ( E ) (i) F ( u θ (0) , ∂ x u θ (0)) ≤ F ( b, B ) ≤ . Similarly, fixing θ ≤ − | b | − min i ∈{ ...I } n − | φ i | − | a i B i | − K i C H o , the map f : x θ + xB i is a sub solution on each vertex J a i i of (22), then for each i ∈ { . . . I } , ∂ x u θi (0) ≥ B i , which means F ( u θ (0) , ∂ x u θ (0)) ≥ . From Proposition B.1, we know that the real maps θ u θ (0) and θ ∂ x u θ (0) arecontinuous. 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 h − | b | − max i ∈{ ...I } n | φ i | + | a i B i | + sup ( x,u ) ∈ (0 ,ai ) | H i ( x,B i x,B i ) | C H o , | b | + max i ∈{ ...I } n | φ i | + | a i B i | + sup ( x,u ) ∈ (0 ,ai ) | H i ( x,B i x,B i ) | C H o i . This completes the proof. Finally, since the solution u θ ∗ of (7) is unique, we get theuniqueness of θ ∗ . (cid:3) References [1] F.Camilli, C.Marchi, and D.Schieborn. The vanishing viscosity limit for Hamilton-Jacobiequations on networks. J. Differential Equations, 254(10), 4122-4143, 2013.[2] M.K.Fijavz, D.Mugnolo, and E.Sikolya. Variational and semigroup methods for waves anddiffusion in networks. Appl. Math. Optim., 55(2), 219-240, 2007.[3] M.Freidlin and S-J.Sheu. Diffusion processes on graphs: stochastic differential equations, largedeviation principle. Probability Theory and Related Fields 116(2), 181-220, 2000.[4] M.Freidlin and A. D.Wentzell. Diffusion processes on an open book and the averaging prin-ciple. Stochastic Process. Appl., 113(1), 101-126, 2004.[5] D.Gilbarg and N.S.Trudinger. Elliptic partial differential equations of second order, 2001.[6] C.Imbert, V.Nguyen, Generalized junction conditions for degenerate parabolic equations.arXiv:1601.01862, 2016.[7] O.A.Ladyzenskaja, V.A.Solonnikov, and N.N.Ural’ceva. Linear and Quasi-Linear equationsof Parabolic type, 1968.[8] P.L.Lions. Lectures at Coll`ege de France, 2015-2017.
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