Gilles Carbou

Bernard Hanouzet

RELAXATION APPROXIMATION OF THE KERR MODEL FOR THE IMPEDANCE INITIAL-BOUNDARY VALUE PROBLEM

Keywords: 2000 Mathematics Subject Classification. 35L50, 35Q60 Nonlinear Maxwell Equation, Relaxation. 1

The Kerr-Debye model is a relaxation of the nonlinear Kerr model in which the relaxation coefficient is a finite response time of the nonlinear material. We establish the convergence of the Kerr-Debye model to the Kerr model when this relaxation coefficient tends to zero.

1. Physical context. In this paper we study different models for the electromagnetic waves propagation in an isotropic nonlinear material (a cristal for example). The wave propagation is described by Maxwell's equations

∂ t D -curl H = 0, ∂ t B + curl E = 0, div D = div B = 0.
where E is the electric field, H is the magnetic field, D is the electric displacement and B is the magnetic induction. Once nondimensionalized, the constitutive relations for a nonlinear Kerr medium are given by B = H and D = E + P, where P is the nonlinear polarization.

For the Kerr model, the medium exhibits an instantaneous response :

P = |E| 2 E.
For the Kerr-Debye model, the medium exhibits a finite response time ε :

P = χE,
where

∂ t χ + 1 ε χ = 1 ε |E| 2
(see [START_REF] Shen | The Principles of Nonlinear Optics[END_REF][START_REF] Ziolkowski | The incorporation of microscopic material models into FDTD approach for ultrafast optical pulses simulations[END_REF]). Formally, when ε tends to zero, χ tends to |E| 2 , that is the Kerr-Debye model is a relaxation approximation of the Kerr model.

Concerning the Cauchy problem with initial data (D 0 , H 0 , χ 0 ) satisfying div D 0 = div H 0 = 0 and χ 0 ≥ 0, the convergence for the smooth solutions is proved by [START_REF] Hanouzet | Approximation par relaxation d'un système de Maxwell non linéaire[END_REF] using the results of [START_REF] Yong | Singular perturbations of first-order hyperbolic systems with stiff source terms[END_REF]. Generally a boundary layer in time appears because of the non compatibility of the initial data with the equilibrium condition χ = |E| 2 .

In order to describe realistic physical situations it is more convenient to study the initial boundary value problem. We denote by Ω = IR + × IR 2 the domain in which the nonlinear material is confined, and by Γ = {0} × IR 2 its boundary. We consider the Kerr and the Kerr-Debye models in the domain IR + t × Ω with the impedance boundary condition on IR + t × Γ and with null initial data. In this case the Kerr model, denoted by (K), becomes, for (t,

x) ∈ R + × Ω ∂ t D -curl H = 0, ∂ t H + curl E = 0, (1)
with the constitutive relation :

D = (1 + |E| 2 )E. ( 2
)
We suppose that the initial data vanishes :

D(0, x) = H(0, x) = 0 for x ∈ Ω (3)
so that we obtain the conservative relations

div D = div H = 0. ( 4
)
We denote by n = t (-1, 0, 0) the outer unit normal on Γ. We consider the impedance boundary condition

H ∧ n + a((E ∧ n) ∧ n) = ϕ for (t, x) ∈ IR + × Γ, ( 5
)
where a is a positive endomorphism acting on Γ. The initial boundary value problem for the Kerr-Debye model (KD) writes, for

(t, x) ∈ R + × Ω    ∂ t D ε -curl H ε = 0, ∂ t H ε + curl E ε = 0, ∂ t χ ε = 1 ε (|E ε | 2 -χ ε ), (6)
with the constitutive relation :

D ε = (1 + χ ε )E ε . (7)
We suppose that the initial data vanishes :

D ε (0, x) = H ε (0, x) = 0, χ ε (0, x) = 0 for x ∈ Ω, (8)
and we have also div D ε = div H ε = 0. (9) In addition we suppose that we have the same impedance boundary condition

H ε ∧ n + a((E ε ∧ n) ∧ n) = ϕ for (t, x) ∈ IR + × Γ. ( 10
)

Two dimensional models

Following [START_REF] Ziolkowski | Full wave vector Maxwell equation modeling of the selffocusing of ultrashort optical pulses in a nonlinear Kerr medium exhibiting a finite response time[END_REF] we can also introduce the two-dimensional transverse magnetic (TM) and transverse electric (TE) models.

For the transverse magnetic case we assume that

H(x 1 , x 2 , x 3 ) = t (0, H 2 (x 1 , x 3 ), 0), E(x 1 , x 2 , x 3 ) = t (E 1 (x 1 , x 3 ), 0, E 3 (x 1 , x 3 )), ( 11
)
in the domain (x 1 , x 3 ) ∈ {x 1 > 0} × IR. The Maxwell system becomes    ∂ t D 1 + ∂ 3 H 2 = 0, ∂ t D 3 -∂ 1 H 2 = 0, ∂ t H 2 + ∂ 3 E 1 -∂ 1 E 3 = 0, (12)
with the divergence conservation condition

∂ 1 D 1 + ∂ 3 D 3 = 0 (13)
(in this case the divergence condition for H is irrelevant). The impedance boundary condition writes H 2 -aE 3 = ϕ with a ≥ 0. (14) This system is coupled with (2) for the Kerr model and with [START_REF] Huynh | Etudes théorique et numérique de modèles de Kerr[END_REF] and the third equation in [START_REF] Hanouzet | Approximation par relaxation d'un système de Maxwell non linéaire[END_REF] for the Kerr-Debye model.

In the transverse electric case, we assume that

E(x 1 , x 2 , x 3 ) = t (0, E 2 (x 1 , x 3 ), 0), H(x 1 , x 2 , x 3 ) = t (H 1 (x 1 , x 3 ), 0, H 3 (x 1 , x 3 )). ( 15
)
We obtain

   ∂ t D 2 -∂ 3 H 1 + ∂ 1 H 3 = 0, ∂ t H 1 -∂ 3 E 2 = 0, ∂ t H 3 + ∂ 1 E 2 = 0, (16)
with the divergence conservation condition

∂ 1 H 1 + ∂ 3 H 3 = 0, (17)
and the impedance boundary condition becomes

H 3 + aE 2 = ϕ with a ≥ 0. ( 18
)
In the case of a fixed finite response time, numerical simulations are obtained for these two-dimensional models by finite-difference methods in [START_REF] Ziolkowski | Full wave vector Maxwell equation modeling of the selffocusing of ultrashort optical pulses in a nonlinear Kerr medium exhibiting a finite response time[END_REF] and by finiteelement methods in [START_REF] Huynh | Etudes théorique et numérique de modèles de Kerr[END_REF].

One dimensional model

In [START_REF] Carbou | Relaxation approximation of some nonlinear Maxwell initial-boundary value problem[END_REF] the one dimensionnal model is introduced :

E(x 1 , x 2 , x 3 ) = t (0, e(x 1 ), 0), H(x 1 , x 2 , x 3 ) = t (0, 0, h(x 1 )). ( 19
)
In this case the Maxwell system becomes :

∂ t d + ∂ 1 h = 0, ∂ t h + ∂ 1 e = 0, (20)
with the impedance boundary condition

h(t, 0) + ae(t, 0) = ϕ(t), a ≥ 0. ( 21
)
We can also remark that the divergence conditions on h and d are irrelevant.

2. Mathematical Properties.

Properties of the Kerr model:

We recall the initial-boundary value problem for the general Kerr model.

   ∂ t D -curl H = 0, ∂ t H + curl E = 0, D = (1 + |E| 2 )E, (22)
for (t, x) ∈ IR + × Ω,

D(t = 0) = E(t = 0) = 0 for x ∈ Ω, (23)
H ∧ n + a((E ∧ n) ∧ n) = ϕ for (t, x) ∈ IR + × Γ. ( 24
)
The energy density given by

E K (E, H) = 1 2 (|E| 2 + |H| 2 + 3 2 |E| 4 ) (25)
is a strictly convex entropy, so ( 22) is a quasilinear hyperbolic symmetrizable system. In the three dimensional case, the eigenvalues are, for ξ = 0,

λ 1 (E, ξ) ≤ λ 2 (E, ξ) < λ 3 = λ 4 = 0 < λ 5 = -λ 2 ≤ λ 6 = -λ 1 ,
so the boundary IR + × Γ is characteristic of constant multiplicity equal to two.

In the two dimensional cases, TM and TE, the eigenvalues are of the form:

λ 1 (E, ξ) < λ 2 = 0 < λ 3 = -λ 1 ,
so the boundary IR + × Γ is characteristic of constant multiplicity equal to one.

In the one dimensional case, the system is strictly hyperbolic and the boundary is non characteristic. We have

λ 1 (E) < 0 < λ 2 = -λ 1 .
The impedance boundary condition ( 24) is maximal dissipative thus we can apply the existence results in [START_REF] Picard | Local existence of solutions of impedance initialboundary value problem for non-linear Maxwell equations[END_REF] and also the more general results in [START_REF] Guès | Problème mixte hyperbolique quasi-linéaire caractéristique[END_REF]. We precise these results in the 3-d case. We assume that the source term ϕ is compactly supported in IR + t × Γ. We denote by H s the classical Sobolev spaces and we suppose that ϕ belongs to H s (IR t × Γ) for s great enough. So the boundary condition (24) and the initial data (23) match one each other and we obtain smooth local solutions. Proposition 1. Under the previous assumptions there exists a maximal smooth solution (E, H) for the IBVP ( 22)-( 23)-( 24) which lifespan is denoted by T * and such that

∂ i t (E, H) ∈ C 0 ([0, T * [; H 3-i (Ω)) for i = 0, 1, 2, 3. ( 26
)
We have analogous results in the 2-d and the 1-d cases.

Properties of the Kerr-Debye models.

These models write, for (t, x)

∈ IR + × Ω        ∂ t D ε -curl H ε = 0, ∂ t H ε + curl E ε = 0, ∂ t χ ε = 1 ε (|E ε | 2 -χ ε ), D ε = (1 + χ ε )E ε . (27) D ε (0, x) = H ε (0, x) = 0, χ ε (0, x) = 0 for x ∈ Ω, (28)
H ε ∧ n + a((E ε ∧ n) ∧ n) = ϕ for (t, x) ∈ IR + × .Γ, (29)
By the third equation in ( 27) we observe that we have

χ ε ≥ 0. ( 30
)
The energy density given by

E KD (E, H, χ) = 1 2 (1 + χ) -1 |D| 2 + 1 2 |H| 2 + 1 4 χ 2 (31)
is a strictly convex entropy in the domain {χ ≥ 0}. So ( 27) is a quasilinear symmetrizable hyperbolic system.

In the three dimensional case the eigenvalues are, for ξ = 0,

µ 1 (E, χ, ξ) = µ 2 < µ 3 = µ 4 = µ 5 = 0 < µ 6 = µ 7 = -µ 1 ,
so the boundary is characteristic of constant multiplicity equal to three.

In the two dimensional cases we obtain

µ 1 (E, χ, ξ) < µ 2 = µ 3 = 0 < µ 4 = -µ 1 ,
so the boundary is characteristic of constant multiplicity equal to two.

In the one dimensional case, the system is strictly hyperbolic and the boundary is characteristic of constant multiplicity one. The eigenvalues are

µ 1 (E, χ) < µ 2 = 0 < µ 3 = -µ 1 .
The impedance boundary condition ( 29) is maximal dissipative and we apply the existence results of [START_REF] Guès | Problème mixte hyperbolique quasi-linéaire caractéristique[END_REF].

Proposition 2. Under the previous assumptions there exists a maximal smooth solution (D ε , H ε , χ ε ) for the IBVP ( 27)-( 28)-(29) which lifespan is denoted by T * ε and such that

∂ i t (D ε , H ε , χ ε ) ∈ C 0 ([0, T * ε [; H 3-i (Ω)) for i = 0, 1, 2, 3. (32)
If for a fixed ε we have T * ε < +∞ the behaviour of the solution is described in [START_REF] Guès | Problème mixte hyperbolique quasi-linéaire caractéristique[END_REF] extending the results of Majda [START_REF] Majda | Compressible fluid flow and systems of conservation laws in several space variables[END_REF] on the general quasilinear hyperbolic systems. In the one dimensional case, the Kerr-Debye model behaves like a semilinear system. We prove in [START_REF] Carbou | Comportement semi-linéaire d'un système hyperbolique quasi-linéaire : le modèle de Kerr Debye[END_REF] that it does not exhibit schock waves: if the gradient of the solution blows up, the solution itself blows up:

sup [0,T * ε [ d ε L ∞ (I R + ) + h ε L ∞ (I R + ) + χ ε L ∞ (I R + ) = +∞. (33)
3. Convergence Results. In order to prove the convergence results it is more convenient to use the entropic variables as it is proposed in [START_REF] Hanouzet | Global existence of smooth solutions for partially dissipative hyperbolic systems with a convex entropy[END_REF]. These variables are obtained taking the gradient of the convex entropy (31).

             ∂ D E KD = (1 + χ) -1 D = E, ∂ H E KD = H, ∂ χ E KD = 1 2 (χ -|E| 2 ) := v,
The IBVP (27) (28) (29) becomes

A 0 (W ε )∂ t W ε + 3 j=1 A j ∂ j W ε = 1 ε Q(W ε ) (34)
for (t, x) ∈ IR + × Ω, where

• W ε =   E ε H ε v ε   • A 0 (W ε ) =   (|E ε | 2 + 2v ε + 1)I 3 + 2E ε t E ε 0 2E ε 0 I 3 0 2 t E ε 0 2   • 3 j=1 A j ∂ j =   0 -curl 0 curl 0 0 0 0 0   Q(W ε ) =   0 0 -2v ε   with the initial data E ε (0, x) = H ε (0, x) = 0, v ε (0, x) = 0 for x ∈ Ω, ( 35
)
and with the boundary condition

H ε ∧ n + a((E ε ∧ n) ∧ n) = ϕ for (t, x) ∈ IR + × Γ. ( 36
)
We observe that the boundary condition is linear for the variables (E, H). In addition the equilibrium manifold (D, H, χ), χ = |E| 2 is linearized as (E, H, v), v = 0 and the relaxation term is linear.

For the three dimensional case the main convergence results are the two following theorems (see [START_REF] Carbou | Relaxation Approximation of the Kerr Model for the three Dimensional Initial-Boundary Value Problem[END_REF]):

Theorem 1. There exist T > 0 and a constant K > 0 such that for all ε > 0, T * ε ≥ T and the solution W ε of the (KD) boundary value problem ( 34)-( 35)-(36) satisfies

∂ i t W ε C 0 ([0, T ];H 3-i (I R + )) ≤ K for i = 0, 1, 2, 3, (37)
∂ i t v ε C 0 ([0, T ];H 2-i (I R + )) ≤ Kε for i = 0, 1, 2. ( 38
)
Theorem 1 shows the strong convergence of 1)-( 2)-( 3)-( 5) is described in the following statement:

v ε = 1 2 (χ ε -|E ε | 2 ) to zero. The con- vergence of (E ε , H ε ) to the solution (E, H) of the IBVP (
Theorem 2. For T ≤ T and T < T * , there exists a constant K > 0 such that for all ε > 0,

(E ε , H ε ) -(E, H) C 0 ([0,T ];H 1 (I R + )) ≤ Kε. (39)
In our study we remark that no boundary layer appears in the time variable because the null initial data belongs to the equilibrium manifold defined by

V = (D, H, χ) such that χ -(1 + χ) -2 |D| 2 = 2v = 0 .
For the space variable, we have the same boundary condition for the system (K) and for the system (KD), so no space boundary layer appears again.

For the one dimensionnal case the analogous results are proved in [START_REF] Carbou | Relaxation approximation of some nonlinear Maxwell initial-boundary value problem[END_REF]. This case is simpler because we can use the semi-linear behaviour of the K-D model (see (33)) and because the boundary is characteristic of multiplicity one. In the 3-d case, the boundary is characteristic of multiplicity three, so we have to use the conservation conditions [START_REF] Picard | Local existence of solutions of impedance initialboundary value problem for non-linear Maxwell equations[END_REF]. The second one div H ε = 0 is linear, but the first one div D ε = 0 is nonlinear in the entropic variables, which entails additional technical difficulties. We guess that in the 2-d case, the TE case is similar to the 1-d case, and the TM case is analogous to the 3-d case.

4. Sketch of the proof for the 3-d case. Replacing the magnetic field H ε by H ε + ϕ(t, x 2 , x 3 )η(x 1 ), where η is a cut off function we are led to study the following homogeneous IBVP:

A 0 (W ε )∂ t W ε + 3 j=1 A j ∂ j W ε = 1 ε Q(W ε ) + G for (t, x) ∈ IR + × Ω, (40)
W ε (0, x) = 0 for x ∈ Ω, (41)

H ε ∧ n + a((E ε ∧ n) ∧ n) = 0 for (t, x) ∈ IR + × Γ, (42)
So the conservation properties are preserved:

div ((|E ε | 2 + 2v ε + 1)E ε ) = div H ε = 0. ( 43
)
The tangential derivatives ∂ t = ∂ 0 , ∂ 2 , ∂ 3 and the normal derivative ∂ 1 are estimated by different ways. The tangential derivatives are measured by ψ ε :

ψ ε (t) =   W ε (t) 2 L 2 (Ω) + i =1 ∂ i W ε 2 L 2 (Ω) + ... + i,j,k =1 ∂ ijk W ε (t) 2 L 2 (Ω)   1 2
.

(44) The normal derivatives are measured by λ ε :

λ ε (t) =   ∂ 1 W ε 2 L 2 (Ω) + i ∂ 1i W ε (t) 2 L 2 (Ω) + i,j ∂ 1ij W ε (t) 2 L 2 (Ω)   1 2 . ( 45
)
The source term G is measured by γ:

γ(t) =   G 2 L 2 (Ω) + . . . + i,j ∂ ij G 2 L 2 (Ω)   1 2 . ( 46
)
We also introduce :

Ψ ε (t) = sup [0,t] ψ ε , Λ ε (t) = sup [0,t] λ ε , Γ(t) = sup [0,t] γ. (47)
Let us introduce T 1 ε defined by

T 1 ε = max {t ≤ T * ε , Γ(t) ≤ 1, Ψ ε (t) ≤ 1, Λ ε (t) ≤ 1} . ( 48
)
Variational estimates for the tangential derivatives

Taking the inner product of (40) with W ε we first obtain the L 2 estimate on [0, T

1 ε ]: 1 2 d dt Ω A 0 (W ε )W ε • W ε + 1 ε Ω |v ε | 2 ≤ K. (49)
Let us remark that we have used the dissipative property of the boundary condition (42):

Ω 3 j=1 A j ∂ j W ε • W ε = Γ aE T ε • E T ε ≥ 0, where E T ε = t (E 2 ε , E 3 ε ).
We can derivate the IBVP with respect to the tangential derivatives ∂ i , i = 1, We obtain the same initial data and same boundary condition for ∂ i W and the same form for the relaxation term . Using also the uniform elliptic relation

C 1 |ξ| 2 ≤ A 0 (W )ξ • ξ ≤ C 2 |ξ| 2 ,
we obtain the estimation on ψ ε : there exists K 1 such that for all ε and for t ∈ [0, T

1 ε ] we have d dt ψ 2 ε + 2 ε Ω (|v ε | 2 + . . . + |∂ ijk v ε | 2 ) ≤ K 1 . (50)
Estimates for the normal derivatives

In order to estimate v ε we solve the last equation in (40) using Duhamel formula:

v ε (t, x) = - 1 2 |E ε | 2 (t, x) + 1 2ε t 0 exp s -t ε |E ε | 2 (s, x)ds,
and from (50) we obtain that

3 i=0 ∂ i t v ε H 3-i (Ω) ≤ K(Ψ ε + Λ ε ) 2 . ( 51
)
We estimate the normal derivatives using either the equation ( 40) or the divergence equations (43). For example, from curl E ε = G 2 -∂ t H ε we deduce that

∂ 1 E 2 ε L 2 (Ω) ≤ Kψ + γ. From div E ε = -div (|E ε | 2 + 2v ε )E ε we obtain ∂ 1 E 1 ε L 2 (Ω) ≤ Kψ ε + K(ψ ε + λ ε ) 2 .
To sum up we obtain the following estimates for the normal derivatives : there exists a constant K 2 such that on [0,

T 1 ε ], Λ ε (t) ≤ K 2 (Ψ ε (t) + Γ(t)) + K 2 Λ 2 ε (t). ( 52
)
End of the proof.

Integrating (49) we obtain that on [0,

T 1 ε ] Ψ 2 ε (t) ≤ K 1 t. ( 53
)
Let us define the polynomial map P δ (ξ) = K 2 ξ 2ξ + K 2 δ. If δ ≤ 1 2K 2 , the smallest rooth of P δ is less than 1. So while Γ(t) ≤ 1 4K2 and Ψ ε (t) ≤ 1 4K2 , then Λ ε (t) ≤ 1. We fix T 1 > 0 such that

∀ t ≤ T 1 , Γ(t) ≤ 1 4K 2 ,
and we introduce

T 2 = 1 16K 1 K 2 2
. By (53) we have that

∀ t ≤ T 2 , Ψ ε (t) ≤ 1 4K 2 .
Then we obtain that T 1 ε ≥ T = min{ T 1 , T 2 }, and we deduce (37). The estimate (38) is obtained by

v ε (t) = t 0 exp s -t ε E ε ∂ t E ε (s, x)ds.
Theorem 2 is a straightforward corollary of Theorem 1.


Thomas Boivin

Jean-Noël Candau

Alain Chalon

How to invade a Mediterranean forest ecosystem ? A lesson from

published or not. The documents may come   L'archive ouverte pluridisciplinaire

How to invade a Mediterranean forest ecosystem ? A lesson from seed insects in French Cedrus atlantica (Pinaceae) forests. Thomas Boivin (INRA, France, boivin@avignon.inra.fr), Alain Chalon (INRA, France, chalon@avignon.inra.fr), Marion Sondo (INRA, France, sondo@avignon.inra.fr) and Jean-Noël Candau (Jean-Noel.Candau@avignon.inra.fr) Worldwide exchange of tree materials is a major factor favouring insects' invasions in forest ecosystems through considerable extensions of their distribution ranges. Here we describe the biological and ecological mechanisms involved in the recent invasion of southeastern French cedar (Cedrus atlantica) stands by the invasive seed predator Megastigmus schimitscheki (Hymenoptera: Torymidae). Despite the presence of resident direct competitor (M. pinsapinis), French M. schimitscheki populations display a continuous increase in abundance due to multiple adequations of its life cycle with its new environment. Specifically, adults emergence is well synchronized with the timing of cedar's fructification and interannual variations in resource abundance may be counterbalanced by prolonged diapause, temporally dispersing individuals of a cohort. Adult emergence and realized fecundity in M. schimitscheki were found significantly earlier and higher (respectively) than in M. pinsapinis, suggesting enhanced abilities of M. schimitscheki to exploit local resources compared to M. pinsapinis. Consequently, we observe recurrent local exclusion by the invasive species. Population genetics of M. schimitscheki also suggest high long distance dispersal abilities. Our analysis of key components of both local and regional dynamics of this species shed a critical light on the processes involved in a successful insect invasion in a French Mediterranean ecosystem.


G Carbou
email: g.carbou@math.u-bordeaux.fr

B Hanouzet
email: hanouzet@math.u-bordeaux.fr

R Natalini
email: r.natalini@iac.cnr.it.

SEMILINEAR BEHAVIOR FOR TOTALLY LINEARLY DEGENERATE HYPERBOLIC SYSTEMS WITH RELAXATION

Keywords: 1991 Mathematics Subject Classification. Primary: 35L65. Key words and phrases. Dissipative hyperbolic systems, linear degeneration, relaxation systems, semilinear behavior, Suliciu model, Kerr-Debye model

We investigate totally linearly degenerate hyperbolic systems with relaxation. We aim to study their semilinear behavior, which means that the local smooth solutions cannot develop shocks, and the global existence is controlled by the supremum bound of the solution.

In this paper we study two specific examples: the Suliciu-type and the Kerr-Debye-type models. For the Suliciu model, which arises from the numerical approximation of isentropic flows, the semilinear behavior is obtained using pointwise estimates of the gradient. For the Kerr-Debye systems, which arise in nonlinear optics, we show the semilinear behavior via energy methods. For the original Kerr-Debye model, thanks to the special form of the interaction terms, we can show the global existence of smooth solutions.

Introduction

We study the behavior of smooth solutions to the Cauchy problem for some hyperbolic operators in one space dimension. We consider N × N systems which are in the form (1.1)

∂ t u + A(u)∂ x u = F (u), t > 0, x ∈ R,
with the initial condition (1.2) u(x, 0) = u 0 (x), x ∈ R.

Here u = (u 1 , . . . , u N ) ∈ R N , A(u) is a N × N -matrix with smooth coefficients, F (u) is a smooth vector function of the unknown u; u 0 is a sufficiently smooth function. Furthermore, we assume that the operator ∂ t + A(u)∂ x is strictly hyperbolic, i.e. the N eigenvalues of the matrix A(u) are real and distinct,

(1.3) λ 1 (u) < λ 2 (u) < • • • < λ N (u).
In the following we are going to assume always that the C 1 -norm of u 0 is bounded:

(1.4)

u 0 C 1 (R) := u 0 L ∞ (R) + u ′ 0 L ∞ (R) < +∞.
Under these assumptions, it is well known that the Cauchy problem (1.1), (1.2) has a unique local (in time) smooth solution, see for instance [START_REF] Majda | Compressible fluid flow and systems of conservation laws in several space variables[END_REF][START_REF] Li | Global classical solutions for quasilinear hyperbolic systems[END_REF]. This solution can be globally defined, as for instance in the linear case. However, if the matrix A depends on u or F depends in a nonlinear way from u, singularities can appear in the solution, even for smooth initial data, in a finite time T * (u 0 ), the so-called blow-up time. The following results are now classical, see [START_REF] Alinhac | Blowup for nonlinear hyperbolic equations[END_REF][START_REF] Majda | Compressible fluid flow and systems of conservation laws in several space variables[END_REF]: i) If T * (u 0 ) < +∞, then:

(1.5)      for all t < T * (u 0 ), u(t, •) C 1 (R) < +∞ and sup 0≤t<T * (u 0 ) u(t, •) C 1 (R) = +∞.
ii) If the system (1.1) is semilinear, i.e.: A does not depend on u, and T * (u 0 ) < +∞, then:

(

     for all t < T * (u 0 ), u(t, •) C 1 (R) < +∞ and sup 0≤t<T * (u 0 ) u(t, •) L ∞ (R) = +∞. 1.6)
iii) If the system (1.1) is truly quasilinear, i.e.: A depends effectively on u, and T * (u 0 ) < +∞, then singularities have a different nature and shock waves can appear. Namely, the following situation is allowed:

(1.7)

               for all t < T * (u 0 ), u(t, •) C 1 (R) < +∞, sup 0≤t<T * (u 0 ) u(t, •) L ∞ (R) < +∞ and sup 0≤t<T * (u 0 ) ∂ x u(t, •) L ∞ (R) = +∞.
However, in the quasilinear case, (1.7) is not true for every system.

Definition 1.1. We say the system (1.1) has a semilinear behavior if, for every smooth initial datum which satisfies (1.4) and such that T * (u 0 ) < +∞, we have that (1.5) implies (1.6).

Therefore, for a system with a semilinear behavior, shock waves cannot appear. Actually, for such a system, if for a local smooth solution, defined on an interval [0, T [, we have

sup 0≤t<T u(t, •) L ∞ (R) < +∞, then sup 0≤t<T ∂ x u(t, •) L ∞ (R) < +∞.
Let us now introduce the right and left eigenvectors of A(u):

(1.8) A(u)r i (u) = λ i (u)r i (u),

t A(u)l i (u) = λ i (u)l i (u), i = 1, . . . , N.

They depend smoothly on u and they are normalized such that t l i (u)r j (u) = δ ij , i, j = 1, , . . . , N, where δ ij is the standard Kronecker's symbol.

Following the classical definitions, first introduced by P.D. Lax [START_REF] Lax | Developement of singularities of solutions of nonlinear hyperbolic partial differential equations[END_REF], the i-characteristic field is genuinely nonlinear at u ∈ R N if (1.9) λ ′ i (u)r i (u) = 0. A characteristic field which is not genuinely nonlinear for all u ∈ R N is called linearly degenerate. If this is the case for the i-field, then (1.10) λ ′ i (u)r i (u) ≡ 0. Definition 1.2. The system (1.1) is called totally linearly degenerate (TLD) if all of the characteristic fields of the matrix A(u) are linearly degenerate.

Consider the following problem, already proposed by Majda [START_REF] Majda | Compressible fluid flow and systems of conservation laws in several space variables[END_REF] and more recently by Brenier [START_REF] Brenier | Hydrodynamic structure of the augmented Born-Infeld equations[END_REF]: have the TLD systems the semilinear behavior?

In the following, we are going to investigate this problem for some relaxation approximation models to quasilinear hyperbolic systems (for an introduction to this topic see for instance [START_REF] Natalini | Recent results on hyperbolic relaxation problems[END_REF][START_REF] Bouchut | Nonlinear stability of finite volume methods for hyperbolic conservation laws and well-balanced schemes for sources[END_REF] and references therein). These models have the form (1.1). Even if most of the examples of relaxation approximations are written as semilinear systems, which trivially verify the conjecture, this is not the most general case. Recently some quasilinear relaxation approximations, which verify the TLD property, have been proposed as quite effective approximations for various hyperbolic systems, see [START_REF] Bouchut | Entropy satisfying flux vector splittings and kinetic BGK models[END_REF][START_REF] Bouchut | Nonlinear stability of finite volume methods for hyperbolic conservation laws and well-balanced schemes for sources[END_REF][START_REF] Chalons | Navier-Stokes equations with several independent pressure laws and explicit predictor-corrector schemes[END_REF]]. These models yield numerical schemes such that the solution of the corresponding Riemann problem is quite simple, since only contact discontinuities are allowed. In this class, the most interesting example is given by the Suliciu-type relaxation model, which will be investigated in Section 3. Another interesting and more physically motivated model, is the Kerr-Debye relaxation system, see [START_REF] Carbou | Relaxation approximation of some nonlinear Maxwell initial-boundary value problem[END_REF][START_REF] Carbou | Comportement semi-linéaire d'un système hyperbolique quasi-linéaire : le modèle de Kerr Debye[END_REF][START_REF] Carbou | Relaxation approximation of the Kerr Model for the three dimensional initial-boundary value problem[END_REF] and references therein, which arises in nonlinear optics and will be investigated in Section 4.

Let us now present a short review of the state of the art for the general case of TLD systems of the form (1.1). For N = 2 the situation is mostly clear, since in that case the systems are diagonalizable by Riemann invariants. In [START_REF] Rozhdestvenskii | Impossibility of the "gradient catastrophe" for slightly non-linear systems[END_REF], it is proved that a 2 × 2 strictly hyperbolic TLD system has the semilinear behavior. Otherwise, if system (1.1) is homogeneous, namely F ≡ 0, and one of the two eigenvalues is genuinely nonlinear in one point, there exist C ∞ initial data with compact support, such that the corresponding solutions have shocks in finite time, see [START_REF] Majda | Compressible fluid flow and systems of conservation laws in several space variables[END_REF][START_REF] Alinhac | Blowup for nonlinear hyperbolic equations[END_REF]. Let us also point out that, according to a counterexample in [START_REF] Neves | Ill-posedness of the Cauchy problem for totally degenerate system of conservation laws[END_REF], shocks can appear even for TLD 2 × 2 systems, if the strictly hyperbolicity assumption fails.

The situation for N ≥ 3 is not yet completely understood. In the case of homogeneous diagonal TLD systems, the results obtained in [START_REF] Rozhdestvenskii | Impossibility of the "gradient catastrophe" for slightly non-linear systems[END_REF] imply the global existence of smooth solutions for all initial data, so T * (u 0 ) = +∞ and there is nothing to prove. For the general (non diagonal) homogeneous TLD case, many results are known about the global existence of solution for small initial data, see [START_REF] Bressan | Contractive metrics for nonlinear hyperbolic systems[END_REF][START_REF] Kong | Cauchy problem for quasilinear hyperbolic systems[END_REF][START_REF] Dai | Global existence and asymptotic behavior of classical solutions of quasilinear hyperbolic systems with linearly degenerate characteristic fields[END_REF] and references therein. However, from an example in [START_REF] Jeffrey | Breakdown of the solution to a completely exceptional system of hyperbolic equations[END_REF], it is known that finite time blow-up of solutions can occur for some (suitably large) initial data. So, at least for these initial data, it is still possible to address the problem of the semilinear behavior.

For systems with a non vanishing source, both in the TLD and in the general case, all kinds of behavior are possible, since the source term can be sufficiently dissipative to avoid the formation of singularities and to yield global existence of smooth solutions for small initial data, see for instance [START_REF] Hanouzet | Global existence of smooth solutions for partially dissipative hyperbolic systems with a convex entropy[END_REF]. On the other hand, for some choices of the source term, smooth solutions can blow up for all initial data.

In this paper we aim to investigate the semilinear behavior for some specific models with relaxation we mentioned before: the Suliciu-type and the Kerr-Debye-type models. These models are both written as strictly hyperbolic TLD models on a open domain in R 3 , with a partially dissipative source term.

The plan of our paper is as follows.

In Section 2, first we investigate the properties due to the linear degeneracy using the John's decomposition [START_REF] John | Nonlinear wave equations, formation of singularities[END_REF], which yield a fast conclusion on some quite academic examples. We also present the Suliciu and Kerr-Debye models.

The following section is devoted to the study of the Suliciu model. The semilinear behavior is obtained since the system is rich according to the definition in [START_REF] Serre | Systems of conservation laws. 2. Geometric structures, oscillations, and initial-boundary value problems[END_REF][START_REF] Sévennec | Géométrie des systèmes de lois de conservation[END_REF]: there exists a regular change of variable which makes the system diagonal and the differential part has a conservative form. Therefore we can apply a general result: rich strictly hyperbolic TLD systems have the semilinear behavior. Let us remark that recently this paticular result has been independently obtained in [START_REF] Li | Mechanism of the formation of singularities for quasilinear hyperbolic systems with linearly degenerate characteristic fields[END_REF].

The Kerr-Debye system is not rich, and so it does not fit in the previous framework. In the last section, we extend the previous results of [START_REF] Carbou | Comportement semi-linéaire d'un système hyperbolique quasi-linéaire : le modèle de Kerr Debye[END_REF], to deal with a more general class of TLD systems and for general source terms, by showing the semilinear behavior via energy methods. For the original Kerr-Debye model, thanks to the special form of the interaction terms, we can show the global existence of smooth solutions.

The John's decomposition and some examples

2.1. The John's formula. The John's formula, see [START_REF] John | Nonlinear wave equations, formation of singularities[END_REF] and also [START_REF] Hörmander | The lifespan of classical solutions of nonlinear hyperbolic equations[END_REF][START_REF] Kong | Cauchy problem for quasilinear hyperbolic systems[END_REF], is a key ingredient for the study of singularities of systems of type (1.1). Using this formula it is possible to highlight the rôle of linear degeneration phenomena. To obtain the formula, we decompose the spatial gradient of u on the right eigenvectors of A(u)

(2.1) ∂ x u = N j=1
p j r j (u), with p j = t l j ∂ x u.

Therefore, (1.1) reads (2.2)

∂ t u + N j=1 p j λ j (u)r j (u) = F (u).
Differentiating (2.1) with respect to t and using (2.2) to evaluate ∂ t u, we obtain (2.3)

∂ 2 xt u = N j=1 ∂ t p j r j (u) - N j,k=1 p j p k λ k (u)r ′ j (u)r k (u) + N j=1 p j r ′ j (u)F (u).
On the other hand, we differentiate (2.2) with respect to x, to find (2.4)

∂ 2 tx u = - N j=1 λ j (u)∂ x p j r j (u) - N j=1 λ ′ j (u)( N k=1 p k r k (u))p j r j (u) - N j=1 λ j (u)p j r ′ j (u)( N k=1 p k r k (u)) + F ′ (u)( N k=1 p k r k (u)).
Finally, taking the scalar product (2.2) and (2.3) by the left eigenvalue l i (u), we find the John's formula:

(2.5)

∂ t p i + λ i (u)∂ x p i = - N k=1 p i p k λ ′ i (u)r k (u) + N j,k=1 (λ k (u) -λ j (u))p j p k t l i (u)r ′ j (u)r k (u) + N k=1 p k t l i (u) F ′ (u)r k (u) -r ′ k (u)F (u) , i = 1, . . . , N.
For homogeneous systems the last term vanishes. The first two terms are quadratic in p = t (p 1 , . . . , p N ), with variable coefficients depending on u.

The F (u)'s contribution is concentrated in the third term, which is linear in p, with coefficients depending on u. If the system (1.1) is diagonal, the decomposition reduces to (2.6)

∂ t p i + λ i (u)∂ x p i = - N k=1 p i p k ∂ u k λ i (u) + N k=1 p k ∂ u k F i (u).
When the system (1.1) is TLD, there is no squared term in (2.5) or (2.6), i.e.: no term of the form p 2 i . It is well-know that this property plays a fundamental rôle in the analysis of semilinear hyperbolic problems with quadratic interactions, see for instance [START_REF] Tartar | Some existence theorems for semilinear hyperbolic systems in one space variable[END_REF][START_REF] Aregba-Driollet | Cauchy problem for one-dimensional semilinear hyperbolic systems: global existence, blow up[END_REF]. We are going to see that in some simple examples, this is enough to conclude for the semilinear behavior.

2.2. Some examples. First, let us consider two homogeneous 2 × 2 TLD systems introduced by T.T. Li and F.G. Liu, which show a C 1 blow-up of solutions for some smooth initial data, see [START_REF] Li | Singularity caused by eigenvectors for quasilinear hyperbolic systems[END_REF] and also [START_REF] Kong | Cauchy problem for quasilinear hyperbolic systems[END_REF]. Let us consider a system of the form (2.7)

∂ t u + A(u)∂ x u = 0
We take, for the first example,

(2.8)

A(u) =       -1 0 0 -e u 2 0 0 -2e -u 2 0 1       .
This system is strictly hyperbolic and TLD, since the eigenvalues are given by (2.9)

λ 1 = -1 < λ 2 = 0 < λ 3 = +1.
The right and left eigenvalues are given, respectively, by (2.10) r 1 (u) = t (1, e u 2 , e -u 2 ), r 2 (u) = t (0, 1, 0), r 3 (u) = t (0, 0, 1), l 1 (u) = t (1, 0, 0), l 2 (u) = t (-e u 2 , 1, 0), l 3 (u) = t (-e -u 2 , 0, 1).

The corresponding John's decomposition is

(2.11)            ∂ t p 1 -∂ x p 1 = 0, ∂ t p 2 = e u 2 p 1 p 2 , ∂ t p 3 + ∂ x p 3 = -e -u 2 p 1 p 2 .
The second example uses the matrix (2.12)

A(u) =       -1 0 0 -(1 + u 2 2 ) 0 0 u 2 0 1       .
The eigenvalues are still given by (2.13)

λ 1 = -1 < λ 2 = 0 < λ 3 = +1,
and the right and left eigenvalues are given, respectively, by (2.14)

r 1 (u) = t (1, 1 + u 2 2 , -1 2 u 2 ), r 2 (u) = t (0, 1, 0), r 3 (u) = t (0, 0, 1), l 1 (u) = t (1, 0, 0), l 2 (u) = t (-(1 + u 2 2 ), 1, 0), l 3 (u) = t ( 1 2 u 2 , 0, 1).
The corresponding John's decomposition is now given by (2.15)

           ∂ t p 1 -∂ x p 1 = 0, ∂ t p 2 = 2u 2 p 1 p 2 , ∂ t p 3 + ∂ x p 3 = -1 2 u 2 p 1 p 2 .
For these two examples it is easy to establish the semilinear behavior. More precisely, let T > 0 be such that

u 2 ∈ L ∞ ([0, T [×R), so that also r 1 (u) ∈ L ∞ ([0, T [×R)
. Therefore, by some straightforward computations in (2.11) or in (2.15), it is easy to see that

p 1 , p 2 , p 3 ∈ L ∞ ([0, T [×R) and then ∂ x u ∈ L ∞ ([0, T [×R).
On the other hand, it is possible to show, see [START_REF] Li | Singularity caused by eigenvectors for quasilinear hyperbolic systems[END_REF], that the L ∞ -norm of the considered smooth solution blows up in T * .

Finally we consider a system introduced by A. Jeffrey [START_REF] Jeffrey | Breakdown of the solution to a completely exceptional system of hyperbolic equations[END_REF]. The matrix

A(u) in system (2.7) is (2.16) A(u) =       -cosh(2u 2 ) 0 -sinh(2u 2 ) cosh(u 2 ) 0 sinh(u 2 ) sinh(2u 2 ) 0 cosh(2u 2 )      
.

The eigenvalues are still given by (2.17)

λ 1 = -1 < λ 2 = 0 < λ 3 = +1,
and the right and left eigenvalues are given, respectively, by (2.18)

r 1 (u) = t (-cosh(u 2 ), 1, sinh(u 2 )), r 2 (u) = t (0, 1, 0), r 3 (u) = t (-sinh(u 2 ), 0, cosh(u 2 )), l 1 (u) = t (-cosh(u 2 ), 0, sinh(u 2 )), l 2 (u) = t (cosh(u 2 ), 1, sinh(u 2 )), l 3 (u) = t (sinh(u 2 ), 0, cosh(u 2 )).
The corresponding John's decomposition is

(2.19)            ∂ t p 1 -∂ x p 1 = -2p 1 p 3 -p 2 p 3 , ∂ t p 2 = 2p 1 p 3 + p 2 p 3 , ∂ t p 3 + ∂ x p 3 = p 1 p 2 .
This system is semilinear and the right-hand side is a quadratic constant coefficients form. Therefore, we can use the Tartar's result in [START_REF] Tartar | Some existence theorems for semilinear hyperbolic systems in one space variable[END_REF], which show the existence of global solutions for (2.19) for small initial data in L 1 (R) (for p). For this system, however, Jeffrey has shown in [START_REF] Jeffrey | Breakdown of the solution to a completely exceptional system of hyperbolic equations[END_REF] the blowup of smooth solution in finite time, at least for some special (large) initial data.

Here we want to show, using the methods introduced in [START_REF] Aregba-Driollet | Cauchy problem for one-dimensional semilinear hyperbolic systems: global existence, blow up[END_REF], the existence of blow-up solutions to system (2. [START_REF] John | Nonlinear wave equations, formation of singularities[END_REF]) and then to obtain by a different method, the break-down of solutions to the Jeffrey's model. We look for a solution ϕ(t, x) to (2.19) in the form of a polarized traveling profile

ϕ(t, x) = ψ(x -ct) t (λ 1 , λ 2 , λ 3 ).
Take a = 0, c = 0, +1, -1, and ψ(ξ) = a ξ * -ξ for some fixed value ξ * . The function

(2.20) ϕ(t, x) = ψ(x -ct) t (± c a , ∓ c + 1 a , c(c + 1) a(c -1) ) is a solution to (2.19) out of the set {(x, t), x -ct = ξ * }.
To yield an actual solution to (2.19) corresponding to a given Cauchy datum p(0, x) = p 0 (x), we use the finite speed of propagation. Choose for instance

(2.21)    c > 2, ξ * = -2, p 0 ∈ C ∞ 0 (] -2, +2[), p 0 (x) = ψ(x)λ for |x| ≤ 1.
. Then, the unique solution p(t, x) to (2.19) with initial condition p 0 is such that

p(t, x) = ϕ(t, x) for t ≥ 0, t ≤ 1 + x, t ≤ 1 -x, and t ≤ x+2 c . Therefore, p blows up at T * ≤ T := 1 c-1 . Next, we construct u 0 ∈ (C ∞ 0 (] -2, +2[)) 3 such that (2.22) ∂ x u 0 = p 01 r 1 (u 0 ) + p 02 r 2 (u 0 ) + p 03 r 3 (u 0 ).
With this object we can modify

p 0 on ] -2, 2[\[-1, 1] preserving (2.21) such that R (p 01 (ξ) + p 02 (ξ))dξ = 0 so that u 02 (x) = x -∞ (p 01 (ξ) + p 02 (ξ))dξ ∈ C ∞ 0 (R).
We fix now p 01 and p 03 such that

R (cosh u 02 (ξ)p 01 (ξ) + sinh u 02 (ξ)p 03 (ξ))dξ = 0, and R (sinh u 02 (ξ)p 01 (ξ) + cosh u 02 (ξ)p 03 (ξ))dξ = 0, so that u 0 defined integrating (2.22) is compactely supported in [0, 2].
Hence, the smooth solution of the Cauchy problem for (2.7) with A(u) given by (2.16), blows up in T * ≤ T . Let us notice however that, by this argument, we do not know if also u blows up in T * . Then, the problem of the semilinear behavior for the Jeffrey's model stays unsolved.

2.3. The Suliciu model. The original model proposed by Suliciu in [START_REF] Suliciu | On modelling phase transition by means of rate-type constitutive equations, shock wave structure[END_REF], was a semilinear approximation to the following one dimensional p-system arising in elasticity

   ∂ t u 1 -∂ x u 2 = 0, ∂ t u 2 -∂ x (p(u 1 )) = 0, p ′ (u 1 ) > 0.
The viscoelastic Suliciu approximation is

           ∂ t u 1 -∂ x u 2 = 0, ∂ t u 2 -∂ x v = 0, ∂ t v -µ∂ x u 2 = 1 ε (p(u 1 ) -v)
, where µ is a positive constant and ε << 1 is the relaxation parameter. Many authors have investigated the convergence properties of this model when ε → 0, see [START_REF] Carbou | Relaxation approximation of some initial-boundary value problem for p-systems[END_REF] and references therein. More recently a similar model has been proposed for the approximation of the system of isentropic gas dynamics in Eulerian coordinates

(2.23)    ∂ t ρ + ∂ x (ρu) = 0, ∂ t (ρu) + ∂ x (ρu 2 + p(ρ)) = 0.
Here, ρ ≥ 0 is the density of the gas, u its fluid velocity and the pressure function p = p(ρ) satisfies p ′ (ρ) > 0. In [START_REF] Bouchut | Nonlinear stability of finite volume methods for hyperbolic conservation laws and well-balanced schemes for sources[END_REF] and [START_REF] Chalons | Navier-Stokes equations with several independent pressure laws and explicit predictor-corrector schemes[END_REF], the following Suliciu relaxation approximation was introduced:

(2.24)

           ∂ t ρ + ∂ x (ρu) = 0, ∂ t (ρu) + ∂ x (ρu 2 + π) = 0, ∂ t (ρπ) + ∂ x (ρuπ + c 2 u) = 1 ε ρ(p(ρ) -π)
, for a constant c > 0 to be fixed later.

Set U = (ρ, u, π) and Ω = {U ∈ R 3 ; ρ > 0}. It is easy to see that the system (2.24) can be written in the standard form (2.25)

∂ t U + A(U )∂ x U = F (U ),
with A defined in Ω and given by (2.26)

A(U ) =       u ρ 0 0 u 1/ρ 0 c 2 /ρ u       and (2.27) F (U ) = t (0, 0, 1 ε (p(ρ) -π)).
Notice that, when the source term for (2.24) is completely general, namely it is given by

f (U ) = t (f 1 (U ), f 2 (U ), f 3 (U )),
the interaction in (2.25) reads

F (U ) = t (f 1 (U ), 1 ρ (f 2 (U ) -uf 1 (U )), 1 ρ (f 3 (U )) -πf 1 (U )).
This system (2.25) is strictly hyperbolic and TLD in Ω. Indeed, its eigenvalues are given by (2.28)

λ 1 (U ) = u - c ρ < λ 2 (U ) = u < λ 3 (U ) = u + c ρ ,
and the right and left eigenvectors are given, respectively, by

(2.29) r 1 (U ) = t (1, - c ρ 2 , c 2 ρ 2 ), r 2 (U ) = t (1, 0, 0), r 3 (U ) = t (1, c ρ 2 , c 2 ρ 2 ) and (2.30) l 1 (U ) = t (0, - ρ 2 2c , ρ 2 2c 2 ), l 2 (U ) = t (1, 0, - ρ 2 c 2 ), l 3 (U ) = t (0, ρ 2 2c , ρ 2 2c 2 ). Let G(U, p) = 3 k=1 p k F ′ (U )r k (U ) -r ′ k (U )F (U ) . The John's decomposition of system (2.26) is (2.31)              ∂ t p 1 + λ 1 (U )∂ x p 1 = -3c ρ 2 p 1 p 2 -6c ρ 2 p 1 p 3 + t l 1 (U )G, ∂ t p 2 + λ 2 (U )∂ x p 2 = c ρ 2 (p 1 p 2 + p 2 p 3 ) + t l 2 (U )G, ∂ t p 3 + λ 3 (U )∂ x p 3 = 6c ρ 2 p 1 p 3 + 3c ρ 2 p 2 p 3 + t l 3 (U )G.
It is not easy to deduce the semilinear behavior directly from (2.31). To show this property we are going to use in Section 3 the definition of richness of a systems. According to [START_REF] Sévennec | Géométrie des systèmes de lois de conservation[END_REF], see also [START_REF] Serre | Systems of conservation laws. 2. Geometric structures, oscillations, and initial-boundary value problems[END_REF], a strictly hyperbolic system is rich if it has a conservative form and it is diagonalizable along its Riemann invariants.

Proposition 2.1. The Suliciu model (2.25), (2.26) is a rich system in Ω.

Proof. The Riemann invariants for the Suliciu model are (2.32)

w 1 = π -cu, w 2 = 1 ρ + π c 2 , w 3 = π + cu.
We have also

w ′ 1 r 1 = 2c 2 ρ 2 , w ′ 1 r 2 = 0, w ′ 1 r 3 = 0; w ′ 2 r 1 = 0, w ′ 2 r 2 = -1 ρ 2 , w ′ 2 r 3 = 0; w ′ 3 r 1 = 0, w ′ 3 r 2 = 0, w ′ 3 r 3 = 2c 2 ρ 2 .
Therefore w 1 is a 2,3-Riemann invariant, w 2 is a 1,3-Riemann invariant, w 3 is a 1,2-Riemann invariant. The map (2.32) defines a diffeomorphism from Ω to Ω 1 := {W ∈ R 3 ; 2c 2 w 2 -w 1 -w 3 > 0}, and it holds

(2.33) ρ = 2c 2 (2c 2 w 2 -w 1 -w 3 ) -1 , u = 1 2c (w 3 -w 1 ), π = 1 2 (w 1 + w 3 ).
In the new unknown W = (w 1 , w 2 , w 3 ), the system has the diagonal form (2.34)

∂ t W + Λ(W )∂ x W = F 1 (W ),
where

(2.35) Λ(W ) =       λ 1 (W ) 0 0 0 λ 2 (W ) 0 0 0 λ 3 (W )       =       1 c (w 3 -c 2 w 2 ) 0 0 0 1 2c (w 3 -w 1 ) 0 0 0 -1 c (w 1 -c 2 w 3 )       .
The system is strictly hyperbolic in Ω 1 and, thanks to (2.34), also rich.

2.4.

The Kerr-Debye model. The Kerr-Debye model is a relaxation approximation of the Kerr model in nonlinear optics [START_REF] Ziolkowski | The incorporation of microscopic material models into FDTD approach for ultrafast optical pulses simulations[END_REF]. Recall that the propagation of the electromagnetic waves is described by the Maxwell equations

           ∂ t D -curl H = 0, ∂ t B + curl E = 0, div D = div B = 0.
The Kerr model describes an instantaneous response of the medium, where the constitutive relations read

B = µ 0 H, D = ε 0 (1 + ε r |E| 2 )E.
The Kerr-Debye model describes a delayed response of the medium, by the constitutive relations

B = µ 0 H, D = ε 0 (1 + χ)E,
where χ solves the equation

∂ t χ = 1 τ (ε r |E| 2 -χ),
the constant τ > 0 being a delay time. The analytical convergence of the Kerr-Debye to the Kerr model has been investigated in [START_REF] Hanouzet | Approximation par relaxation d'un système de Maxwell non linéaire[END_REF] for the Cauchy problem and in [START_REF] Carbou | Relaxation approximation of some nonlinear Maxwell initial-boundary value problem[END_REF][START_REF] Carbou | Comportement semi-linéaire d'un système hyperbolique quasi-linéaire : le modèle de Kerr Debye[END_REF][START_REF] Carbou | Relaxation approximation of the Kerr Model for the three dimensional initial-boundary value problem[END_REF] for the initial-boundary value problem. Following [START_REF] Carbou | Relaxation approximation of some nonlinear Maxwell initial-boundary value problem[END_REF], we deal with the following one dimensional version of the Kerr-Debye model (2.36)

             ∂ t d + ∂ x h = 0, ∂ t h + ∂ x e = 0, ∂ t χ = 1 ε (e 2 -χ), with d = (1 + χ)e. If the initial condition (2.37) (d, h, χ)(0, x) = (d 0 , h 0 , χ 0 )(x)
is such that χ 0 ≥ 0, then for every positive time, where the solution is defined, we still have χ(t, x) ≥ 0, and we can replace e by (1 + χ) -1 d in system (2.36). So, setting u = (d, h, χ) and Ω = {u ∈ R 3 ; χ ≥ -1}, we can rewrite the system (2.36) as

(2.38)

∂ t u + A(u)∂ x u = F (u),
where A(u) is given in Ω by

(2.39)

A(u) =       0 1 0 (1 + χ) -1 0 -(1 + χ) -2 d 0 0 0      

and

(2.40)

F (u) = t (0, 0, 1 ε ((1 + χ) -2 d -χ)).
This system is strictly hyperbolic and TLD since its eigenvalues are given by (2.41)

λ 1 (u) = -(1 + χ) -1 2 < λ 2 (u) = 0 < λ 3 (u) = (1 + χ) -1 2 ,
and the right and left eigenvectors are given, respectively, by (2.42)

r 1 (u) = t (1, -(1 + χ) -1 2 , 0), r 2 (u) = t ((1 + χ) -1 d, 0, 1), r 3 (u) = t (1, (1 + χ) -1 2 , 0),

and

(2.43)

l 1 (u) = 1 2 t (1, -(1 + χ) 1/2 , -(1 + χ) -1 d), l 2 (u) = t (0, 0, 1), l 3 (u) = 1 2 t (1, (1 + χ) 1/2 , -(1 + χ) -1 d).
Let G be given by

G(U, p) = 3 k=1 p k F ′ (U )r k (U ) -r ′ k (U )F (U ) .
The John's formula for the Kerr-Debye model is given by (2.44)

           ∂ t p 1 + λ 1 (U )∂ x p 1 = -5 4 (1 + χ) -3 2 p 1 p 2 + 1 4 (1 + χ) -3 2 p 2 p 3 + t l 1 (U )G, ∂ t p 2 = t l 2 (U )G, ∂ t p 3 + λ 3 (U )∂ x p 3 = -1 4 (1 + χ) -3 2 p 1 p 2 + 5 4 (1 + χ) -3 2 p 2 p 3 + t l 3 (U )G.
Remark 2.1. Unlike the Suliciu model, the Kerr-Debye model is not a rich system. There are no functions which are at the same time the Riemann invariants for the first two characteristic fields. In Section 4, we establish the semilinear behavior for a generalized version of the Kerr-Debye system, using energy estimates. Moreover, for the original Kerr-Debye system, we are also able to prove global existence for smooth initial data.

3. The Suliciu model 3.1. Semilinear behavior of rich systems. Consider the N × N system (1.1) and assume it is strictly hyperbolic and rich. So, such a system can be written in a diagonal form as

(3.1) ∂ t W + Λ(W )∂ x W = G(W ),
where Λ(W ) = diag(λ 1 (W ), . . . , λ N (W )), and

λ 1 (W ) < λ 2 (W ) < • • • < λ N (W ).
Moreover, according to [START_REF] Sévennec | Géométrie des systèmes de lois de conservation[END_REF][START_REF] Serre | Systems of conservation laws. 2. Geometric structures, oscillations, and initial-boundary value problems[END_REF], the following relations are always verified (set

∂ i := ∂ w i ): (3.2) ∂ k ∂ i λ j λ i -λ j = ∂ i ∂ k λ j λ k -λ j , i = j = k.
From these relations there exist N smooth functions α j (W ) such that:

(3.3) ∂ i α j = ∂ i λ j λ i -λ j , i = j.
If moreover we assume that the system is TLD, then, in the diagonal form (3.1) we have

(3.4) ∂ i λ i = 0, i = 1, . . . , N.
For this kind of systems the John's decomposition is specially simple and effective. Thanks to (3.3), we can prove the following result.

Theorem 3.1. A strictly hyperbolic system with source term, which is both rich and TLD, has the semilinear behavior.

Proof. We consider the Cauchy problem for the system (3.1), with a smooth initial condition W 0 , which is bounded in C 1 . We assume that the local smooth solution is defined and bounded on [0, T [×R:

(3.5) there exists C > 0 such that:

|W (t, x)| ≤ C, 0 ≤ t < T, x ∈ R.
To show the semilinear behavior it suffices to show that p = ∂ x W is also bounded on the same strip [0, T [×R. Since the system is TLD, using (3.4), we can show that p = t (p 1 , . . . , p N ) = t (∂ x w 1 , . . . , ∂ x w N ) is a solution to the Cauchy problem (3.6)

∂ t p i + λ i (W )∂ x p i +   j =i ∂ j λ i (W )p j   p i = k ∂ k G i (W )p k , i = 1, . . . , N, (3.7
) p(0, x) = ∂ x W 0 (x).
Now, for j = i, we have

p j = λ i -λ j ) -1 (∂ t w j + λ i (W )∂ x w j -G j (W ) .
So, using (3.3) we obtain

  j =i ∂ j λ i (W )p j   = - j =i ∂ j α i (W )(∂ t w j + λ i (W )∂ x w j -G j (W ))) = -(∂ t + λ i (W )∂ x )α i (W ) + j ∂ j α i (W )G j (W ).
Inserting this equality in (3.6), we obtain (3.8)

(∂ t + λ i (W )∂ x )p i -[(∂ t + λ i (W )∂ x )α i (W )]p i = -   j ∂ j α i (W )G j (W )   p i + k ∂ k G i (W )p k .
So, setting q i := e -α i (W ) p i , the function q is the solution to the Cauchy problem (3.9)

∂ t q i + λ i (W )∂ x q i = -   j ∂ j α i (W )G j (W )   q i +e -α i (W ) k ∂ k G i (W )e α k (W ) q k , i = 1, . . . , N, (3.10)
q i (0, x) = e -α i (W 0 ) ∂ x w 0i (x), i = 1, . . . , N.
The system (3.9) is a diagonal linear system with smooth and bounded coefficients on the strip [0, T [×R. Therefore the function q, and then also p = ∂ x W , is bounded on the same strip as required.

Remark 3.1. For N = 2, we recover the results in [START_REF] Rozhdestvenskii | Impossibility of the "gradient catastrophe" for slightly non-linear systems[END_REF], since in this case the conditions (3.2) are empty and so trivially verified. On the other hand, for N ≥ 3, these conditions appear to be quite restrictive.

Remark 3.2. In Theorem 3.1, the assumptions are taken globally on R N . However, in many cases, it should be possible to restrict the analysis to an open bounded domain of R N .

3.2.

Semilinear behavior of the Suliciu model. Let us focus now on the system (2.24), but for a generic source term. In the unknown U = (ρ, u, π), we have (2.25), where A is given by (2.26) and F is a smooth interaction term. We write the system for the Riemann invariants W = t (w 1 , w 2 , w 3 ) given by (2.32), which is

(3.11) ∂ t W + Λ(W )∂ x W = G(W ),
where Λ(W ) is given by (2.35). This system is strictly hyperbolic in Ω 1 = {W ∈ R 3 ; 2c 2 w 2 -w 1 -w 3 > 0}, since we have

λ 2 -λ 1 = λ 3 -λ 2 = c ρ = 1 2c (2c 2 w 2 -w 1 -w 3 ).
It is easy to see that the system is rich and TLD in the same domain and the functions α j , for j = 1, 2, 3 are explicitly given by (3.12)

α 1 = α 2 = α 3 = -log(2c 2 w 2 -w 1 -w 3 ).
Therefore, we can write the system (3.9) and come back to the original variables if

(3.13) 2c 2 w 2 -w 1 -w 3 = 2c 2 ρ > 0.
Let us also observe that, for the original variables, the matrix A(U ) is just defined and strictly hyperbolic in the domain Ω = {U ∈ R 3 ; ρ > 0}. Now, following [START_REF] Majda | Compressible fluid flow and systems of conservation laws in several space variables[END_REF], the loss of regularity for the local smooth solutions can be stated in the following more precise form:

If T * (U 0 ) < +∞, then i) sup 0≤t<T * ∂ t U L ∞ (R) + ∂ x U L ∞ (R) = +∞ or ii) for every compact set K ⊂⊂ Ω, U (t) escapes from K as t ր T * .
To establish the semilinear behavior, we have to avoid the singularity coming out from the vanishing of the density ρ. Hence, we are going to assume that our local smooth solutions of Cauchy problem for equation (2.25), is defined and bounded on the strip [0, T [×R and moreover

(3.14) ∃ρ > 0 such that ρ(t, x) ≥ ρ, 0 ≤ t < T, x ∈ R.
Therefore, the function W given by (2.32), satisfies (3.5); actually W (t, x) is in a compact set of Ω 1 for 0 ≤ t < T, x ∈ R. Then, thanks to (3.9), ∂ x W is bounded in [0, T [×R, and the same is true for ∂ x U . So, we have proved the following result.

Theorem 3.2. For the Suliciu model (2.25), with A(U ) given by (2.27) and F being a smooth source term, let U 0 be an initial data such that T * (U 0 ) < +∞. Assume that there exists ρ > 0 such that ρ(t, x) ≥ ρ in [0, T * [×R.

Then

sup 0≤t<T * U (t, •) L ∞ (R) = +∞.
According to Theorem 3.2, the smooth solutions to Suliciu model cannot develop shocks, as far as the density ρ is strictly positive. However, it is still possible to wonder about the existence of global solutions, and in particular when the source term is given by the relaxation term (2.27) and, for some fixed interval I in R, the subcharacteristic condition

(3.15) ∀ξ ∈ I, ξ 2 p ′ (ξ) < c 2 ,
holds. This condition has been introduced in [START_REF] Bouchut | Entropy satisfying flux vector splittings and kinetic BGK models[END_REF] to guarantee the dissipativity of the system (see also the next Subsection 3.3). Nevertheless, even under these conditions, it is possible to show the blow-up of smooth solutions in finite time.

More precisely, if we take the pressure law

(3.16) p(ρ) = - c 2 0 ρ ,
with the interaction term given by (2.27), the subcharacteristic condition (3.15) holds on I =]0, +∞[ for c 2 0 < c 2 . Following an example given by F. Bouchut [START_REF] Bouchut | [END_REF], we are going to show the existence of a smooth solution for this system, such that ρ(t, x) ≥ ρ > 0, and which nevertheless blows up in finite time. let us rewrite system (2.24) in the Lagrangian coordinates, still denoted by (t, x). We have (3.17)

             ∂ t ( 1 ρ ) -∂ x u = 0, ∂ t u + ∂ x π = 0, ∂ t π + c 2 ∂ x u = 1 ε (p(ρ) -π),
When p(ρ) is given by (3.16), this system is linear in the unknowns ( 1 ρ , u, π). We are going to find a solution to (3.17) such that 1 ρ is bounded, but vanishes in finite time, which then implies that ρ → +∞.

Let us look for a solution of the form

(3.18) t ( 1 ρ , u, π) = e ikx ϕ(t) + t ( 1 ρ 0 , 0, - c 2 0 ρ 0 ).
This yields

(3.19) ϕ ′ +       0 -ik 0 0 0 ik c 2 0 ε ic 2 k 1 ε       ϕ = 0. Choosing c 0 = c 3 , k 2 c 2 = 1 3ε 2 , λ = 1 3ε
is a triple eigenvalue of the matrix of the system (3.19). Therefore, for every fixed constant β > 0, the vector

(3.20)                      1 ρ(t, x) = 1 ρ 0 + iβke ikx e -t 3ε t(1 + t 3ε ), u(t, x) = βe ikx e -t 3ε (1 + t 3ε - t 2 9ε 2 ), π(t, x) = - c 2 0 ρ 0 + iβc 2 ke ikx e -t 3ε t(-1 + t 9ε ),
is a solution to (3.17) for the initial condition t ( 1 ρ 0 , βe ikx , -c 2 0 ρ 0 ). Fixing ρ 0 > 0, it is possible to take β large enough to obtain that the real part of 1 ρ vanishes in finite time.

3.3.

Other properties of the Suliciu model. Next we aim to show that the Suliciu model (2.24) fits in the general framework of the partially dissipative hyperbolic systems with a strictly convex entropy function, which have been recently investigated in [START_REF] Hanouzet | Global existence of smooth solutions for partially dissipative hyperbolic systems with a convex entropy[END_REF] and [START_REF] Bianchini | Asymptotic behavior of smooth solutions for partially dissipative hyperbolic systems with a convex entropy[END_REF]. Therefore, following [START_REF] Chalons | Navier-Stokes equations with several independent pressure laws and explicit predictor-corrector schemes[END_REF], we modify the source term (assuming that the subcharacteristic condition (3.15) is always verified) in order to construct a regular entropy function.

We have that the function h(ξ) = p(ξ) + c 2 ξ is invertible for ξ ∈ I and so we can set, for π + c 2 ρ ∈ h(I),

(3.21) ρ = h -1 (π + c 2 ρ ).
Notice that, ρ and ρ are connected by the relation

(3.22) π + c 2 ρ = p(ρ) + c 2 ρ .
In (2.24), we replace the source term 1 ε ρ(p(ρ) -π) by 1 ε ρ(p(ρ) -π), so that the system (2.25) reads now

(3.23) ∂ t U + A(U )∂ x U = F (U ),
with A(U ) given by (2.26) and

(3.24) F (U ) = t (0, 0, 1 ε (p(ρ) -π)).
This new system has the same properties we have studied in Subsection 3.2, and shares with the original Suliciu model also the equilibrium manifold. This follows by considering that the equilibrium manifold is given by F (U ) = 0, which is equivalent to p(ρ) = π. Therefore, using (3.22), we have c 2 ρ = c 2 ρ , which implies ρ = ρ.

Fixing the relaxation parameter ε, we can show the global existence of smooth solutions at least for initial data which are small perturbations of constant equilibrium states. Theorem 3.3. Let Ū = (ρ > 0, ū, π = p(ρ)) a constant state belonging to the equilibrium manifold of the system (3.23), with A(U ) given by (2.26) and F (U ) by (3.24). Let U 0 be a smooth perturbation of Ū . There exists δ > 0 such that, if U 0 -Ū H 2 (R) ≤ δ, then there exists a global smooth solution U to system (3.23) corresponding to the initial condition U 0 and

U -Ū ∈ C 0 ([0, +∞[; H 2 (R)) ∩ C 1 ([0, +∞[; H 1 (R)).
Proof. The Theorem follows from Theorem 1 in [START_REF] Hanouzet | Global existence of smooth solutions for partially dissipative hyperbolic systems with a convex entropy[END_REF], by proving that the system (3.23) has i) a strictly dissipative entropy according to Definition 2 in [START_REF] Hanouzet | Global existence of smooth solutions for partially dissipative hyperbolic systems with a convex entropy[END_REF] and ii) the Shizuta-Kawashima condition [START_REF] Shizuta | Systems of equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equation[END_REF] holds.

i) Let us rewrite (3.23) for the conservative variables U : (ρ, v, w) := (ρ, ρu, ρπ). Let ϕ be the function defined by

(3.25) ϕ ′ (Y ) = - p(ρ(Y )) c 2 .
First we show that the function E, given by

(3.26) E(U ) = v 2 2ρ + w 2 2c 2 ρ + ρϕ c 2 + w ρ ,
is a strictly convex entropy for (

∂ t E(U ) + ∂ x (ρuE(U ) + πu) = - ρ εc 2 (π -p(ρ)) 2 . Set X = u 2 2 + π 2c 2 and Y = c 2 2 + π. Then E(U ) = ρ(X + ϕ(Y )) and we have ∂ t E(U ) + ∂ x (ρuE(U ) + πu) = - ρ εc 2 (π -p(ρ))(π + c 2 ϕ ′ (Y ) 3.23) and (3.27)
), Then (3.25) implies (3.27). Concerning the strict convexity, let us notice that, if the function ϕ is convex in Y , then E is also strictly convex. Now, thanks to the subcharacteristic condition (3.15), we find that

ϕ ′′ (Y ) = - p ′ (ρ(Y )) c 2 p ′ (ρ(Y )) -c 2 (ρ(Y )) 2 > 0.
We also have that the entropy function E is dissipative, since

(E ′ (ρ, v, w) -E ′ (ρ, v, w)) F (ρ, v, w) = E ′ (ρ, v, w)F (ρ, v, w) = -ρ εc 2 (π -p(ρ)) 2 ≤ 0. (3.28)
Finally, we show that E is a strictly dissipative entropy. Following [START_REF] Hanouzet | Global existence of smooth solutions for partially dissipative hyperbolic systems with a convex entropy[END_REF], let us introduce the entropy variable

(3.29) W := E ′ (ρ, v, w) = (U 1 , U 2 , V ).
So the condition holds, since

(3.30) F (W ) = t (0, 0, - c 2 ε V ).
ii) To check the Shizuta-Kawashima condition, we have to verify that the eigenvalues of A( Ū ) are not belonging to the kernel of F ′ ( Ū ), where F is given by (3.24).

The right eigenvectors r i are given by (2.29), so, from

F ′ ( Ū )r 1 = F ′ ( Ū )r 3 = t (0, 0, - c 2 ερ 2 ) = 0,
and

F ′ ( Ū )r 2 = t (0, 0, - c 2 ερ 2 p ′ (ρ)ρ ′ (ȳ))
= 0, we deduce that the Shizuta-Kawashima condition holds.

Remark 3.3. Thanks to Theorem 1 in [START_REF] Hanouzet | Global existence of smooth solutions for partially dissipative hyperbolic systems with a convex entropy[END_REF], we can estimate the entropy variable (3.29) as follows (for ε = 1):

(3.31) sup 0≤t<+∞ U -Ū 2 H 2 (R) + ∞ 0 U 1 (τ ) 2 H 1 (R) + U 2 (τ ) 2 H 1 (R) + V (τ ) 2 H 2 (R) dτ ≤ C U 0 -Ū0 2 H 2 (R) ,
for some positive constant C. Also, it is possible to describe the asymptotic behavior for large times of the smooth global solutions, see [START_REF] Bianchini | Asymptotic behavior of smooth solutions for partially dissipative hyperbolic systems with a convex entropy[END_REF] for more details.

4. The Kerr-Debye Model 4.1. Semilinear behavior for a generalized Kerr-Debye model. In [START_REF] Carbou | Comportement semi-linéaire d'un système hyperbolique quasi-linéaire : le modèle de Kerr Debye[END_REF] we proved the semilinear behavior for the Kerr-Debye system in the one-dimensional case. The proof is based on a careful choice of variables: we rewrite Kerr-Debye system in the variables U = (e, h, χ), and we obtain (4.1)

             (1 + χ)∂ t e + ∂ x h = -e∂ t χ = -1 ε e(e 2 -χ), ∂ t h + ∂ x e = 0, ∂ t χ = 1 ε (e 2 -χ).
Here, we study the following generalization of the Kerr-Debye system (4.2)

   A 0 (χ)∂ t u + A 1 ∂ x u = ϕ(v), ∂ t χ = ψ(v),
with the initial condition (4.3) (u(0, x), χ(0, x)) = (u 0 (x), χ 0 (x)) := v 0 (x).

We make the following assumptions. The unknown v = (u, χ) takes its values in R n-r × R r . A 0 ∈ C ∞ (R r ; M n-r (R)), and for all χ ∈ R r , A 0 (χ) is a symmetric positive definite (n -r) × (n -r)-matrix such that

(4.4) ∃α > 0, ∀χ ∈ R r , ∀ξ ∈ R n-r , A 0 (χ)ξ • ξ ≥ α ξ 2 .
A 1 is a symmetric invertible (n -r) × (n -r)-matrix. The function ϕ ∈ C ∞ (R n ; R n-r ) with ϕ(0) = 0 and ψ ∈ C ∞ (R n ; R r ) with ψ(0) = 0. We denote Φ = (ϕ, ψ). The initial condition v 0 ∈ H 2 (R).

Remark 4.1. As the original Kerr-Debye system, the system (4.2) is totally linearly degenerate.

We have the following result.

Theorem 4.1. The system (4.2) has the semilinear behavior.

Proof. Let v be the regular maximal solution of the Cauchy problem (4.2).

Let us assume that its lifespan T * is finite. From the continuation principle in [START_REF] Majda | Compressible fluid flow and systems of conservation laws in several space variables[END_REF], we know that

(4.5) v L ∞ ([0,T * [×R) + ∂ t v L ∞ ([0,T * [×R) + ∂ x v L ∞ ([0,T * [×R) = +∞.
We will prove that if

(4.6) v L ∞ ([0,T * [×R) ≤ K, then ∂ t u L ∞ ([0,T * [×R) + ∂ x u L ∞ ([0,T * [×R)
< +∞, which contradicts Majda's result and shows the semilinear behavior.

The L 2 estimate. Taking the inner product of (4.2) with v we obtain 1 2

d dt R (A 0 (χ)u • u + |χ| 2 )dx = R Φ(v) • vdx + 1 2 R ∂ t (A 0 (χ))u • udx.
Since Φ(0) = 0, with (4.6), there exists a constant K such that

R Φ(v) • vdx ≤ K v 2 L 2 (R) .
In addition, since ∂ t χ = ψ(v), there exists a constant K such that (4.7)

∂ t χ L ∞ (R) ≤ K.
As ∂ t (A 0 (χ)) = A ′ 0 (χ)(∂ t χ), using (4.6) we obtain (4.8)

∂ t (A 0 (χ)) L ∞ ([0,T * [×R) ≤ K.
Therefore, there exists K such that

R ∂ t (A 0 (χ))u • udx ≤ K v 2 L 2 (R) ,
and we obtain 1 2

d dt R (A 0 (χ)u • u + |χ| 2 )dx ≤ K v 2 L 2 (R) .
Integrating in time and using (4.4) we conclude by Gronwall's Lemma that there exists a constant C such that

(4.9) v L ∞ ([0,T * [;L 2 (R)) ≤ C.
The H 1 -estimate. We differentiate the system (4.2) with respect to t and we obtain (4.10)

  A 0 (χ)∂ tt u ∂ tt χ   +   A 1 ∂ x ∂ t u 0   = -   (∂ t (A 0 (χ))∂ t u 0   + Φ ′ (v)∂ t v,
with the initial data obtained by the equation (4.2)

∂ t u(0, x) = (A 0 (χ 0 )) -1 (ϕ(v 0 ) -A 1 ∂ x u 0 ), ∂ t χ(0, x) = ψ(v 0 ).
Taking the inner product of (4.10) with ∂ t v we find 1 2

d dt R (A 0 (χ)∂ t u • ∂ t u + |∂ t χ| 2 )dx = - 1 2 R ∂ t (A 0 (χ))∂ t u • ∂ t udx + R Φ ′ (v)(∂ t v) • ∂ t v.
Using (4.8), (4.6) and Gronwall's Lemma, we can see there exists a constant C such that

(4.11) ∂ t v L ∞ ([0,T * [;L 2 (R)) ≤ C.
Next, from (4.2), we have

∂ x u = (A 1 ) -1 (ϕ(v) -A 0 (χ)∂ t u).
Hence, from (4.11), there exists C such that (4.12)

∂ x u L ∞ ([0,T * [;L 2 (R)) ≤ C.
In addition, derivating (4.2) 2 with respect to x we have

∂ t ∂ x χ = ∂ 2 ψ(v)(∂ x χ) + ∂ 1 ψ(v)(∂ x u).
Integrating in time from 0 to t, we obtain

∂ x χ(t, x) = ∂ x χ 0 (x) + t 0 ∂ 2 ψ(v)(∂ x χ) + ∂ 1 ψ(v)(∂ x u).
Then we deduce

∂ x χ(t, .) L 2 (R) ≤ ∂ x χ 0 L 2 (R) + t 0 ∂ 2 ψ(v) L ∞ (R) ∂ x χ L 2 (R) + ∂ 1 ψ(v) L ∞ (R) ∂ x u L 2 (R) ,
and so, using (4.6), (4.11), (4.12) and Gronwall's Lemma, we find

∂ x χ L ∞ ([0,T * [;L 2 (R)) ≤ C.
Therefore, we have obtained that there exists a constant C such that

(4.13) v L ∞ ([0,T * [;H 1 (R)) + ∂ t v L ∞ ([0,T * [;L 2 (R)) ≤ C.
The H 2 -estimate. We first remark that

∂ tt χ = ψ ′ (v)(∂ t v)
so, using (4.6) and (4.13), we can see that there exists C such that

(4.14) ∂ tt χ L ∞ ([0,T * [;L 2 (R)) ≤ C.
We differentiate (4.10) with respect to t and we obtain (4.15)

  A 0 (χ)∂ ttt u ∂ ttt χ   +   A 1 ∂ x ∂ tt u 0   = -2   (∂ t (A 0 (χ))∂ tt u 0   +   (∂ tt (A 0 (χ))∂ t u 0   + Φ ′′ (v)(∂ t v, ∂ t v) + Φ ′ (v)(∂ tt v)
Taking the inner product of this equation with ∂ tt v we have 1 2

d dt R (A 0 (χ)∂ tt u • ∂ tt u + |∂ tt χ| 2 )dx = I 1 + . . . + I 5 ,
where

I 1 = - 3 2 R ∂ t (A 0 (χ))∂ tt u • ∂ tt u dx, I 2 = - R A ′′ 0 (χ)(∂ t χ, ∂ t χ)(∂ t u) • ∂ tt u dx, I 3 = - R A ′ 0 (χ)(∂ tt χ)(∂ t u) • ∂ tt u dx, I 4 = R Φ ′′ (v)(∂ t v, ∂ t v) • ∂ tt v dx, I 5 = R Φ ′ (v)(∂ tt v) • ∂ tt v dx.
Now, using (4.8) we find

|I 1 | ≤ K ∂ t u 2 L 2 (R) .
Next, from (4.6) and (4.7), we have

|I 2 | ≤ K ∂ t u L 2 (R) ∂ tt u L 2 (R) ,
and from (4.6) and (4.14), we have

|I 3 | ≤ K ∂ tt u L 2 (R) . By (4.13), we know that v is bounded in L ∞ ((0, T * ) × R). So |I 4 | ≤ K ∂ t v L ∞ (R) ∂ t v L 2 (R) ∂ tt v L 2 (R) . Since v is bounded in L ∞ ((0, T * ) × R), we obtain |I 5 | ≤ K ∂ tt v 2 L 2 (R) . From (4.10) we have (4.16) ∂ x ∂ t u = (A 1 ) -1 (-A 0 (χ)∂ tt u -∂ t (A 0 (χ))∂ t u + ϕ ′ (v)(∂ t v)).
By (4.8) and (4.11) we find, adding up the previous estimates,

d dt R (A 0 (χ)∂ tt u • ∂ tt u + |∂ tt χ| 2 )dx ≤ C(1 + ∂ tt v 2 L 2 (R) ),
and by Gronwall's Lemma we deduce (4.17)

∂ tt v L ∞ (0,T * ;L 2 (R)) ≤ C.
Next, using (4.16), we find

∂ t u L ∞ (0,T * ;H 1 (R)) ≤ C and, since ∂ x ∂ t χ = ψ ′ (v)(∂ x v), we have ∂ t χ L ∞ (0,T * ;H 1 (R)) ≤ C. Now, differentiating (4.
2) with respect to x, we obtain

∂ xx u = (A 1 ) -1 (ϕ ′ (v)(∂ x v) -A ′ 0 (χ)(∂ x χ)∂ t u)
and the following estimate follows

u L ∞ (0,T * ;H 2 (R)) ≤ C.
In addition

∂ t ∂ xx χ = ∂ 1 ψ(v)∂ xx χ + 2∂ 2 12 ψ(v)(∂ x u, ∂ x χ) +∂ 2 1 ψ(v)(∂ x u, ∂ x u) + ∂ 2 2 ψ(v)(∂ x χ, ∂ x χ).
Integrating this equation, we obtain by Gronwall's Lemma,

∂ xx χ L ∞ (0,T * ;L 2 (R)) ≤ C.
Therefore we have proved that, provided (4.6), there exists a constant C such that

v L ∞ (0,T * ;H 2 (R)) + ∂ t v L ∞ (0,T * ;H 1 (R)) + ∂ tt v L ∞ (0,T * ;L 2 (R)) ≤ C
and, by Sobolev inequalities, we have

∂ x v L ∞ ([0,T * [×R) + ∂ t v L ∞ ([0,T * [×R) ≤ C,
which contradicts (4.5).

For the Kerr-Debye system (4.1), we have u = (e, h), v = U = (e, h, χ) and

A 0 (χ) =   1 + χ 0 0 1   .
We assume that the initial data χ 0 ≥ 0. Then from the last equation in (4.1), χ remains positive, and the condition (4.4) is satisfied on the domain under consideration. So we can adapt the proof of Theorem 4.1 and we obtain the following result.

Corollary 4.1. The Kerr-Debye system (4.1) has the semilinear behavior.

4.2.

Global solutions for the Kerr-Debye system. In the previous section, in order to obtain the semilinear behavior for the Kerr-Debye system (4.1), we only use the structure of the system (4.2); we did not make use of the fact that the system (4.1) admits a strictly convex partially dissipative entropy given by (4. We rewrite the Kerr-Debye system using the variables U = (e, h, χ):

(4.20)             
(1 + χ)∂ t e + e∂ t χ + ∂ x h = 0,

∂ t h + ∂ x e = 0, ∂ t χ = 1 ε (e 2 -χ).
with the initial condition U (0, x) = U 0 ∈ H 2 (R). ): if the initial data U 0 satisfies that U 0 -U is in H 2 (R) with χ 0 + χ ≥ 0, then the solution of the Kerr-Debye system with initial data U 0 is global in time and we have

U -U ∈ C 0 (R + ; H 2 (R)) ∩ C 1 (R + ; H 1 (R)).
So for this result, the smallness condition on U 0 -U H 2 (R) in [START_REF] Hanouzet | Global existence of smooth solutions for partially dissipative hyperbolic systems with a convex entropy[END_REF] is relaxed.

Actually, more general stability results can be obtained in the framework of [START_REF] Hanouzet | Global existence of smooth solutions for partially dissipative hyperbolic systems with a convex entropy[END_REF][START_REF] Bianchini | Asymptotic behavior of smooth solutions for partially dissipative hyperbolic systems with a convex entropy[END_REF]: the function E given by (4.18), is a strictly dissipative entropy as in [START_REF] Hanouzet | Global existence of smooth solutions for partially dissipative hyperbolic systems with a convex entropy[END_REF] and the Shizuta-Kawashima condition holds if and only if ē = 0. So in this case, we obtain the estimates similar to (3.31). However, if ē = 0, the stability problem remains open.

Theorem 4 . 2 .

 42 The Cauchy problem(4.20) with the initial data U 0 , such that χ 0 ≥ 0, has a global smooth solution.Proof. let us assume that the lifespan T * of the regular solution U is finite. Then, from Corollary 4.1, we know that U L ∞ ([0,T * [×R) = +∞. On the other side, we will obtain by variational estimates that U = (e, h, χ) is bounded in L ∞ ([0, T * [×R), so proving the result by contradiction.

  The H 1 estimate. We differentiate(4.20) with respect to t and we have(1 + χ)∂ tt e + 2∂ t e∂ t χ + e∂ tt χ + ∂ x ∂ t h = 0, ∂ tt h + ∂ x ∂ t e = 0, ∂ tt χ = 1 ε (2e∂ t e -∂ t χ),where the initial data on ∂ t U (0, x) is given by (4.20).Taking the inner product of (4.22) 1 with ∂ t e and of (4.22) 2 with ∂ t h weIn the same way, from (4.20) 1 , we have∂ x h = -(1 + χ)∂ t e -e∂ t χ,Stability for a constant equilibrium state. We can generalize Theorem 4.2 for a perturbation U of a constant equilibrium state U = (ē, h, χ = (ē)2

	The L 2 estimate. Integrating (4.19) on R we obtain 1 2 d dt R ((1 + χ)e 2 + h 2 + 1 2 χ 2 )dx + ε 2 R |∂ t χ| 2 dx = 0. Then there exists a constant C such that (4.21) U L ∞ (0,T (4.22)              obtain that 1 2 d R ∂ t χ(∂ t e) 2 dx = 1 ε R e 2 (∂ t e) 2 dx -1 ε R χ(∂ t e) 2 dx as well as from (4.22) 3 we have R ∂ tt χe∂ t e = ε 2 R |∂ tt χ| 2 dx + 1 4 d dt R |∂ t χ| 2 dx. Consequently we deduce 1 2 d dt R (1 + χ)(∂ t e) 2 + (∂ t h) 2 + 1 2 |∂ t χ| 2 dx + 3 2ε R e 2 (∂ t e) 2 dx + ε 2 R |∂ tt χ| 2 dx = 3 2ε R χ(∂ t e) 2 dx and thus, by Gronwall's Lemma, we obtain that there exists a constant C such that (4.23) ∂ Then, by Sobolev inequalities, we conclude that (4.25) e L ∞ ([0,T Next, solving (4.20) 3 we have χ(t, x) = χ 0 (x) exp(-t ε ) + t 0 exp(-t -s ε )(e(s, x)) 2 ds. Therefore, from (4.25), we obtain (4.26) χ L ∞ ([0,T so using (4.24), (4.26) and (4.23), we find (4.27) h L ∞ (0,T * ;H 1 (R)) ≤ C, and again by Sobolev inequalities dt From (4.20) 3 we have (4.28) h L ∞ ([0,T

* ;L 2 (R)) ≤ C. R ((1 + χ)(∂ t e) 2 + (∂ t h) 2 )dx + 3 2 R ∂ t χ(∂ t e) 2 dx + R ∂ tt χe∂ t e = 0. t U L ∞ (0,T * ;L 2 (R)) ≤ C.

Now, from (4.20) 2 we have ∂ x e = -∂ t h and so there exists a constant C such that

(4.24) e L ∞ (0,T * ;H 1 (R)) ≤ C. * [×R) ≤ C. * [×R) ≤ C. * [×R) ≤ C.

Therefore we have proved that there exists a constant C such that

U L ∞ ([0,T * [×R) ≤ C,

and we obtain a contradiction. Hence T * = +∞.

4.3.

Acknowledgements. The authors would like to thank C. Berthon and F. Bouchut for fruitfull discussions about the Suliciu model.