Serge Nicaise

Cristina Pignotti

Exponential stability of abstract evolution equations with time delay

We consider abstract semilinear evolution equations with a time delay feedback. We show that, if the C 0 -semigroup describing the linear part of the model is exponentially stable, then the whole system retains this good property when a suitable smallness condition on the time delay feedback is satisfied. Some examples illustrating our abstract approach are also given.

Introduction

Let H be a fixed Hilbert space with norm • , and consider an operator A from H into itself that generates a C 0 -semigroup (S(t)) t≥0 that is exponentially stable, i.e., there exist two positive constants M and ω such that

S(t) L(H) ≤ Me -ωt , ∀t ≥ 0, (1.1)
where, as usual, L(H) denotes the space of bounded linear operators from H into itself. For a fixed delay parameter τ , a fixed bounded operator B from H into itself and for a real parameter k, we consider the evolution equation U t (t) = AU(t) + F (U(t)) + kBU(t -τ ) in (0, +∞) U(0) = U 0 , BU(t -τ ) = f (t), ∀t ∈ (0, τ ), (1.2) where F : H → H satisfies some Lipschitz conditions, the initial datum U 0 belongs to H and f ∈ C([0, τ ]; H). Time delay effects often appear in many applications and physical problems. On the other hand, it is well-known (cfr. [START_REF] Bátkai | Semigroups for delay equations[END_REF][START_REF] Datko | Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks[END_REF][START_REF] Datko | An example on the effect of time delays in boundary feedback stabilization of wave equations[END_REF][START_REF] Nicaise | Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks[END_REF][START_REF] Xu | Stabilization of wave systems with input delay in the boundary control[END_REF]) that they can induce some instability. Hence we are interested in giving an exponential stability result for such a problem under a suitable condition between the constant k and the constants M, ω, τ, the norm of B and the nonlinear term F. For some particular examples (see e.g. [START_REF] Ammari | Feedback boundary stabilization of wave equations with interior delay Systems and[END_REF][START_REF] Bátkai | Semigroups for delay equations[END_REF][START_REF] Pignotti | A note on stabilization of locally damped wave equations with time delay[END_REF][START_REF] Guesmia | Well-posedness and exponential stability of an abstract evolution equation with infinite memory and time delay[END_REF][START_REF] Said-Houari | Stability result of the Timoshenko system with delay and boundary feedback[END_REF][START_REF] Alabau-Boussouira | Exponential stability of the wave equation with memory and time delay[END_REF]) we know that the above problem, under certain smallness conditions on the delay feedback kB, is exponentially stable, the proof being from time to time quite technical because some observability inequalities or perturbation methods are used. Hence our main goal is to furnish a direct proof of this stability result by using the so-called Duhamel's formula (or variation of parameters formula).

Observe that our proof is simpler with respect to the ones used so far for particular models. Moreover, we emphasize its generality. Indeed, it applies to every model in the form (1.2) when the operator A generates an exponentially stable semigroup.

In the same spirit, we want to prove existence and exponential stability results when the operator B is unbounded (and F = 0). In that case they are proved using Duhamel's formula but under some admissibility conditions.

Note also that previous papers deal with linear models and B bounded, while here we include a nonlinear term F or B unbounded.

The paper is organized as follows. In section 2 we study the case with bounded feedback operator B and nonlinear term F Lipschitz, giving a well-posedness result and an exponential decay estimate. The analysis is then extended to a more general linear term F in section 3, under more restrictive assumptions. In section 4, we consider, only for the linear model, unbounded delay feedback operators B and prove a well-posedness and an exponential stability result. Finally, in section 5, some illustrative examples with B unbounded are given.

The case F globally Lipschitz

In this section, we assume that F is globally Lipschitz continuous, namely

∃γ > 0 such that F (U 1 ) -F (U 2 ) H ≤ γ U 1 -U 2 H , ∀ U 1 , U 2 ∈ H . (2.1)
Moreover, we assume that F (0) = 0.

The following well-posedness result holds.

Proposition 2.1 For any initial datum U 0 ∈ H and f ∈ C([0, τ ]; H), there exists a unique (mild) solution U ∈ C([0, +∞), H) of problem (1.2). Moreover,

U(t) = S(t)U 0 + t 0 S(t -s)[F (U(s)) + kBU(s -τ )] ds. (2.2)
Proof. We use an iterative argument. Namely in the interval (0, τ ), problem (1.2) can be seen as an inhomogeneous evolution problem

U t (t) = AU(t) + F (U(t)) + g 0 (t) in (0, τ ) U(0) = U 0 , (2.3)
where g 0 (t) = kf (t). This problem has a unique solution U ∈ C([0, τ ], H) (see Th. 1.2, Ch. 6 of [START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF]) satisfying

U(t) = S(t)U 0 + t 0 S(t -s)[F (U(s)) + g 0 (s)] ds.
This yields U(t), for t ∈ [0, τ ]. Therefore on (τ, 2τ ), problem (1.2) can be seen as an inhomogeneous evolution problem

U t (t) = AU(t) + F (U(t)) + g 1 (t) in (τ, 2τ ) U(τ ) = U(τ -), (2.4)
where g 1 (t) = kBU(t -τ ). Hence, this problem has a unique solution

U ∈ C([τ, 2τ ], H)
given by

U(t) = S(t -τ )U(τ -) + t τ S(t -s)[F (U(s)) + g 1 (s)] ds, ∀t ∈ [τ, 2τ ].
By iteration, we obtain a global solution U satisfying (2.2). Now we will prove the following exponential stability result.

Theorem 2.2 Let M, ω, γ as in (1.1) and (2.1). There is a positive constant k 0 such that for k satisfying

|k| < k 0 := e τ ω -1 τ B L(H) Me τ ω , (2.5)
and for γ < γ(|k|), where γ(|k|) is a suitable constant depending on |k|, then there exist

ω ′ > 0 and M ′ > 0 such that the solution U ∈ C([0, +∞), H) of problem (1.2), with U 0 ∈ H and f ∈ C([0, τ ]; H), satisfies U(t) H ≤ M ′ e -ω ′ t ( U 0 H + τ 0 e ωs f (s) H ds), ∀t ≥ τ. (2.6)
From its definition the constant k 0 depends only on M, ω, τ and the norm of B.

Proof. First assume F ≡ 0. We use again an iterative argument and Duhamel's formula but here on the whole R + , namely we can write

U(t) = S(t)U 0 + k t 0 S(t -s)BU(s -τ ) ds, ∀t > 0. (2.7)
Then,

U(t) H ≤ Me -ωt ( U 0 H + |k| t 0 e ωs BU(s -τ ) H ds), ∀t > 0.
Let us show that this implies that

e ωt U(t) H ≤ M( U 0 H + |k|α)(1 + |k|τ BMe ωτ ) n , t ∈ [0, (n + 1)τ ], n ∈ N, (2.8)
where for shortness we have set α := τ 0 e ωs f (s) H ds and

B = B L(H) . Now if we set σ = τ -1 ln(1 + |k|τ BMe ωτ ), (2.9)
then we see that (2.8) gives

U(t) H ≤ Me -(ω-σ)t ( U 0 H + |k|α), ∀t > 0.
Hence U(t) H will decay exponentially if σ -ω is negative or equivalently if

1 + |k|τ BMe ωτ < e τ ω ,
which is nothing else than (2.5).

Under this constraint, we deduce that the estimate (2.6) holds with ω ′ = ω -σ.

Hence we are reduced to prove (2.8). First in (0, τ ), (2.2) and the initial condition from (1.2) yield

U(t) H ≤ Me -ωt ( U 0 H + |k| t 0 e ωs f (s) H ds), ∀t ∈ (0, τ ).
Then, e ωt U(t) H ≤ M( U 0 H + |k|α), ∀t ∈ (0, τ ), which is nothing else than (2.8) for n = 0. Second for any m ∈ N * , we assume that (2.8) holds for all n ≤ m -1 and prove that it holds for m. Indeed by (2.2), we have for all t ∈ (mτ, (m + 1)τ )

U(t) H ≤ Me -ωt U 0 H + |k|α + |k| m-1 ℓ=0 (ℓ+2)τ (ℓ+1)τ
e ωs BU(s -τ ) H ds . Now for s ∈ ((ℓ + 1)τ, (ℓ + 2)τ ), with ℓ = 0, . . . , m -1, we notice that s -τ belongs to (ℓτ, (ℓ + 1)τ ), and using our iterative assumption, we get

U(t) H ≤ Me -ωt U 0 H + |k|α + |k| m-1 ℓ=0 (ℓ+2)τ (ℓ+1)τ e ωs BMe -ω(s-τ ) ( U 0 H + |k|α)(1 + |k|τ BMe ωτ ) ℓ ds ≤ Me -ωt U 0 H + |k|α + |k|τ BM( U 0 H + |k|α)e ωτ m-1 ℓ=0 (1 + |k|τ BMe ωτ ) ℓ
Hence we have obtained that

U(t) H ≤ Me -ωt ( U 0 H + |k|α) 1 + |k|τ BMe ωτ m-1 ℓ=0 (1 + |k|τ BMe ωτ ) ℓ .
Because one readily checks that

1 + |k|τ BMe ωτ m-1 ℓ=0 (1 + |k|τ BMe ωτ ) ℓ = (1 + |k|τ BMe ωτ ) m ,
we obtain

e ωt U(t) H ≤ M( U 0 H + |k|α)(1 + |k|τ BMe ωτ ) m , t ∈ [mτ, (m + 1)τ ].
This estimate and the recurrence assumption, as

(1 + |k|τ BMe ωτ ) n ≤ (1 + |k|τ BMe ωτ ) m for all n ≤ m -1,
imply that (2.8) holds for m. So the result is proved in the linear case.

In order to extend it to the nonlinear model, let us introduce (cfr. [START_REF] Nicaise | Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks[END_REF]) the new variable

Z(t, ρ) := BU(t -τ ρ), ρ ∈ (0, 1), t > 0 .
Then problem (1.2) may be rewritten as

       U t (t) = AU(t) + F (U(t)) + kZ(t, 1) in (0, +∞) Z t (t, ρ) = -τ -1 Z ρ (t, ρ) in (0, +∞) × (0, 1) Z(t, 0) = BU(t) U(0) = U 0 , Z(0, ρ) = f ((1 -ρ)τ ), ∀ρ ∈ (0, 1).
(2.10) Therefore, if we set V := (U, Z) T , the linear part of (2.10), namely

       U t (t) = AU(t) + kZ(t, 1) in (0, +∞) Z t (t, ρ) = -τ -1 Z ρ (t, ρ) in (0, +∞) × (0, 1) Z(t, 0) = BU(t) U(0) = U 0 , Z(0, ρ) = f ((1 -ρ)τ ), ∀ρ ∈ (0, 1), becomes V t = ÃV V (0) = (U(0), Z(0, •)) T .
It is easy to see that à generates a strongly continuous semigroup (T (t)) t≥0 in the Hilbert space H := H × L 2 (0, 1; H). Moreover, (T (t)) t≥0 is exponentially stable. Indeed, we clearly have

Z(t, ρ) 2 L 2 (0,1;H) = 1 0 BU(t -τ ρ) 2 H dρ ≤ B 2 1 0 U(t -τ ρ) 2 H dρ .
Then, the exponential estimate for U gives, for t ≥ 2τ,

Z(t, ρ) 2 L 2 (0,1;H) ≤ B 2 M ′ 2 h 2 e -2ω ′ t U 0 H + τ 0 e ωs f (s) H ds 2 with h 2 := 1 τ τ 0 e 2ω ′ s
ds. Thus, there exists a positive constant M , depending on M, ω, τ, |k| and the norm of B, such that

T (t) H ≤ M e -ω ′ t , t > 0.
(2.11)

Coming back to (2.10) and using Duhamel's formula, V := (U, Z) T can be written as

V (t) = T (t)V 0 + t 0 T (t -s) F (V (s))ds,
where F (V (s)) = (F (U(s)), 0) T . Therefore,

V (t) H ≤ M e -ω ′ t V 0 H + Me -ω ′ t t 0 e ωs γ V (s) Hds
and the exponential stability estimate follows from Gronwall's lemma if ω ′ -γ M < 0.

Remark 2.3 From our proof we see that for F ≡ 0 the explicit decay of U(t) H is

U(t) H ≤ Ce (σ-ω)t ( U 0 H + α), (2.12)
for some C > 0.

Remark 2.4 Note that Theorem 2.2 is very general. Indeed, it gives stability results, when the delay feedback parameter k is sufficiently small, for every model in the form (1.2) if the semigroup (S(t)) t≥0 generated by the linear operator A is exponentially stable. For instance, it furnishes stability results for previously studied models for wave type equations (cfr. [START_REF] Ammari | Feedback boundary stabilization of wave equations with interior delay Systems and[END_REF][START_REF] Pignotti | A note on stabilization of locally damped wave equations with time delay[END_REF][START_REF] Nicaise | Stabilization of second-order evolution equations with time delay[END_REF]), Timoshenko models (cfr. [START_REF] Said-Houari | Stability result of the Timoshenko system with delay and boundary feedback[END_REF]). Also, it includes recent stability results for problems with viscoelastic damping and time delay (cfr. [START_REF] Guesmia | Well-posedness and exponential stability of an abstract evolution equation with infinite memory and time delay[END_REF][START_REF] Alabau-Boussouira | Exponential stability of the wave equation with memory and time delay[END_REF]).

3 More general nonlinearities

Abstract existence and stability results

Here we consider a more general class of nonlinearities. More precisely, we assume that for every constant c there exists a positive constant L(c) such that

F (U 1 ) -F (U 2 ) H ≤ L(c) U 1 -U 2 H , (3.13)
for all U 1 , U 2 ∈ H with U 1 H ≤ c, U 2 H ≤ c.
Moreover, we assume that there exists an increasing continuous function χ : [0, +∞) → [0, +∞), with χ(0) = 0, such that

F (U) H ≤ χ( U H ) U H , ∀ U ∈ H . (3.14)
Now, the nonlinear term introduces additional difficulties. We can give an exponential stability result under a well-posedness assumption for small initial data. Then, we will show that this assumption is satisfied for a quite large class of examples. Theorem 3.1 Let M, ω ′ be as in (2.11). Suppose that for |k| sufficiently small

∃ ρ 0 > 0 and C ρ 0 > 0 such that ∀ U 0 ∈ H, f ∈ C([0, τ ]; H) with ( U 0 2 H + τ 0 |k| f (s) 2
H ds) 1/2 < ρ 0 , there exists a unique global solution

U ∈ C([0, +∞, H) to (1.2) with U(t) H ≤ C ρ 0 < χ -1 ω ′ M , ∀ t > 0.
(3.15)

Then there exists k > 0 such that if |k| < k, for every U 0 ∈ H and f ∈ C([0, τ ]; H) satisfying the assumption from (3.15), the solution U of problem (1.2) satisfies the exponential decay estimate

U(t) H ≤ M * e -ωt ( U 0 H + τ 0 e ωs f (s) H ds), ∀t ≥ τ, (3.16)
for suitable constants M * , ω.

Proof. We can simply repeat the previous proof of Theorem 2.2 with χ(C ρ 0 ) instead of γ.

Examples

We now give some examples for wich assumption (3.15) is satisfied.

Let H be a real Hilbert space, with norm • H , and let A 1 : D(A 1 ) → H, a positive self-adjoint operator with a compact inverse in H. Denote by V := D(A

1 2
1 ) the domain of A 1 2

1 . Further, for i=1,2, let W i be a real Hilbert space (which will be identified to its dual space) and let C ∈ L(W 1 , H), B ∈ L(W 2 , H). Assume that, for some constant µ > 0

B * u 2 W 2 ≤ µ C * u 2 W 1 , ∀ u ∈ V . (3.17)
Let be given a functional G : V → IR such that G is Gâteaux differentiable at any x ∈ V . We further assume (cfr. [START_REF] Alabau-Boussouira | Decay estimates for second order evolution equations with memory[END_REF]) that a) For any u ∈ V there exists a positive constant c(u) such that

|DG(u)(v)| ≤ c(u) v H , ∀v ∈ V,
where DG(u) is the Gâteaux derivative of G at u. Consequently DG(u) can be extended in the whole H and we will denote by ∇G(u) the unique element in H such that

(∇G(u), v) H = DG(u)(v), ∀v ∈ H.
b) For all c > 0, there exists L(c) > 0 such that

∇G(u) -∇G(v) H ≤ L(c) A 1 2 1 (u -v) H for all u, v ∈ V such that A 1 2 1 u H ≤ c and A 1 2
1 v H ≤ c. c) There exists a suitable increasing continuous function ψ satisfying ψ(0) = 0 such that

∇G(u) H ≤ ψ( A 1 2 1 u ) A 1 2 1 u 2 H , ∀u ∈ V.
In this setting let us consider the second order evolution equation

u tt + A 1 u + CC * u t = ∇G(u) + kBB * u t (t -τ ), t > 0, u(0) = u 0 , u t (0) = u 1 , B * u t (t -τ ) = g(t), t ∈ (0, τ ), (3.18)
with (u 0 , u 1 ) ∈ V × H. Denoting v := u t and U := (u, v) T , this problem may be rewritten in the form (1.2) with

A := 0 1 -A 1 -CC * , F (U) := (0, ∇G(u)) T , BU := (0, BB * v) T .
The above assumptions on G imply that F satisfies (3.13) and (3.14) in H := V × H, with χ = ψ. We define the energy of solutions of problem (3.18) as

E(t) := E(t, u(•)) = 1 2 u t 2 H + 1 2 A 1 2 1 u 2 H -G(u) + 1 2 t t-τ |k| B * u t (s) 2 W 2 ds . (3.19)
We will show that for the above model Theorem 3.1 holds.

First of all note that

E ′ (t) = -C * u t (t) 2 W 1 + k B * u t (t), B * u t (t -τ ) + |k| 2 B * u t (t) 2 W 2 - |k| 2 B * u t (t -τ ) 2 W 2 ≤ -C * u t (t) 2 W 1 + |k| B * u t (t) 2 W 2
Then, if |k| < 1 µ , the energy is not increasing. We can prove the following well-posedness result for sufficiently small data. Proof. Note that the condition |k| < 1/µ guarantees that the energy is not increasing.

First of all, on [0, τ ] the abstract system may be rewritten in the form (2.3) with g 0 (t) = (0, kBg(t)).

Then, from classical theory for nonlinear evolution equation (see Th. 1.4, Ch. 6 of [START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF]), there exists a unique mild local solution U defined in a maximal time interval [0, σ) with 0 < σ ≤ τ. We will show that σ = τ .

We argue similarly to [START_REF] Alabau-Boussouira | Decay estimates for second order evolution equations with memory[END_REF]. Note that if ψ( A

1 2 1 u 0 H ) < 1 4 , then E(0) ≥ 1 2 u 1 2 H + 1 2 A 1 2 1 u 0 2 H -G(u 0 ) ≥ 1 2 u 1 2 H + 1 4 A 1 2 1 u 0 2 H ≥ 0 .
We first show that if

ψ( A 1 2 1 u 0 H ) < 1 4
and ψ(2(E(0))

1 2 ) < 1 4 , (3.20)
then

E(t) ≥ 1 2 u t (t) 2 H + 1 4 A 1 2 1 u(t) 2 H , ∀t ∈ [0, σ). (3.21)
Let r := sup{s ∈ [0, σ) such that (3.21) holds for every t ∈ [0, s]}. Suppose that r < σ, then,

E(r) ≥ 1 2 u t (r) 2 H + 1 4 A 1 2 1 u(r) 2 H ≥ 0 . (3.22)
Thus, from (3.22), we have

ψ( A 1 2 1 u(r) H ) ≤ ψ(2(E(r)) 1 2 ) < ψ(2(E(0)) 1 2 ) < 1 4 .
This gives

E(r) ≥ 1 2 u t (r) 2 H + 1 2 A 1 2 1 u(r) 2 H -G(u(r)) > 1 2 u t (r) 2 H + 1 4 A 1 2 1 u(r) 2 H ,
which contradicts the maximality of r. This implies r = σ. Now, let us set

ρ 0 = min 1 2 ψ -1 1 4 , 1 2 √ 2 ψ -1 ω ′ M > 0.
In a second step we show that (3.20) holds for all u 0 ∈ D(A

1 2 1 ), u 1 ∈ H, g ∈ C([0, τ ], W 2 ), satisfying A 1 2 1 u 0 2 H + u 1 2 H + τ 0 |k| g(s) 2 W 2 ds 1 2 < ρ 0 . (3.23)
Indeed, as this assumption implies that A 1 2

1 u 0 H < ρ 0 , then one has

ψ( A 1 2 1 u 0 H ) < ψ(ρ 0 ) = ψ( 1 2 ψ -1 ( 1 4 )) < 1 4 .
Hence by the assumption c) on G, we deduce that

E(0) ≤ 3 4 A 1 2 1 u 0 2 H + 1 2 u 1 2 H + 1 2 τ 0 |k| g(s) 2 W 2 ds < ρ 2 0 ,
and, by definition of ρ 0 , we conclude that

ψ(2(E(0)) 1 2 ) < ψ(ψ -1 ( 1 4 )) = 1 4 .
In conclusion under the assumption (3.23), the estimate (3.20) holds, implying in particular that

0 ≤ 1 2 u t (t) 2 H + 1 4 A 1 2 1 u(t) 2 H ≤ E(t) ≤ E(0) ≤ ρ 2 0 , ∀t ∈ [0, σ].
Then again by [11, Th. 1.4, Ch. 6]), σ = τ. Now we can consider the interval [τ, 2τ ) and we can rewrite the problem in the form (2.3) with g 1 (s) = (0, kBu t (t -τ )). As before, there exists a local solution and arguing as on [0, τ ] we obtain a solution on [0, 2τ ] under the assumption (3.23).

By repeating this argument we prove that, if (3.23) holds, then the solution exists on [0, +∞) and

u t (t) 2 H + A 1 2 1 u(t) 2 H < 4ρ 2 0 ≤ 1 2 ψ -1 ω ′ M 2 .
This proves (3.15).

If A generates an exponentially stable continuous semigroup on H, then the exponential estimate (3.16) holds for k small enough, for small initial data.

Remark 3.3

The abstract model (3.18) includes semilinear versions of previously analyzed concrete models for wave-type equations (cfr. [START_REF] Nicaise | Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks[END_REF]); see the example below. Of course, due to the presence of the nonlinearity we obtain the stability result (for small initial data) under a more restrictive assumption on the size of the delay feedback parameter k. Observe also that models with viscoelastic damping could be considered but with also an extra not-delayed damping necessary to avoid blow-up of solutions, at least for small data.

Γ 0 = { x ∈ ∂Ω : m(x) • ν(x) > 0 }. (3.24)
Moreover, let ω 2 be any set satisfying ω 2 ⊆ ω 1 . Let us consider the initial boundary value problem

u tt (x, t) -∆u(x, t) + aχ ω 1 u t (x, t) + kχ ω 2 u t (x, t -τ ) = |u(x, t)| β u(x, t) in Ω × (0, +∞) (3.25) u(x, t) = 0 on ∂Ω × (0, +∞) (3.26) u(x, 0) = u 0 (x) and u t (x, 0) = u 1 (x) in Ω (3.27) u t (x, t -τ ) = f (x, t), in ω 2 × (0, τ ), (3.28) with initial data (u 0 , u 1 , f ) ∈ H 1 0 (Ω) × L 2 (Ω) × L 2 (ω 2 × (0, τ ))
, and a, k real constants, a > |k|. The constant β > 0 satisfies a suitable restriction to be specified below.

This problem enters into our previous framework, if we take H = L 2 (Ω) and the operator A 1 defined by

A 1 : D(A 1 ) → H : u → -∆u,
where D(A 1 ) = H 1 0 (Ω) ∩ H 2 (Ω). The operator A 1 is a self-adjoint and positive operator with a compact inverse in H and is such that V = D(A 1/2 ) = H 1 0 (Ω). We then define W i = L 2 (ω i ) and the operators B, C as

B : W 2 → H : v → √ kṽχ ω 2 , C : W 1 → H : v → √ aṽχ ω 1 ,
where ṽ ∈ L 2 (Ω) is the extension of v by zero outside ω i . It is easy to verify that

B * (ϕ) = √ kϕ |ω 2 for ϕ ∈ H,
and thus BB * (ϕ) = kϕχ ω 2 , for ϕ ∈ H. Analogously,

C * (ϕ) = √ aϕ |ω 1 for ϕ ∈ H,
and CC * (ϕ) = aϕχ ω 1 , for ϕ ∈ H. Moreover, since ω 2 ⊆ ω 1 and a > |k| the inequality (3.17) holds. Next, consider the functional

G(u) := 1 β + 2 Ω |u(x)| β+2 dx, u ∈ H 1 0 (Ω),
which, for 0 < β ≤ 4 n-2 , is well-defined by Sobolev's embedding theorem. Note that G is Gâteaux differentiable at any u ∈ H 1 0 (Ω) and its Gâteaux derivative is given by

DG(u)(v) = Ω |u(x)| β u(x)v(x)dx, v ∈ H 1 0 (Ω).
As proved in [START_REF] Alabau-Boussouira | Decay estimates for second order evolution equations with memory[END_REF], if we assume that 0 < β < 2 n-2 , then G satisfies the previous assumptions a), b), c). Therefore problem (3.25)-(3.28) enters in the abstract framework (3.18) and so the previous stability result holds for small initial data if the delay parameter |k| is sufficiently small.

The case B unbounded

In this case we need more assumptions on B, indeed we assume that B = CC * , with C * ∈ L(D(A), U) and hence C ∈ L(U, H -1 ), where U is a complex Hilbert space (which is identified with its dual space) and H -1 = D(A * ) ′ is the dual space of D(A * ) with respect to the pivot space H (see [15, section 2.10]). In such a setting, for all t ≥ 0 we can define Φ t ∈ L(L 2 (0, τ ; U), H -1 ) by

Φ t v = t 0 S t-σ Cv(σ) dσ, for all v ∈ L 2 (0, τ ; U).
We further need the following assumptions that are satisfied by different examples (see below). (H1) For all v ∈ L 2 (0, τ ; U), one has Φ t v ∈ C([0, τ ], H) and there exists C 1 > 0 such that

Φ τ v H ≤ C 1 v L 2 (0,τ ;U ) . (4.1)
By Remark 4.2.3 of [START_REF] Tucsnak | Observation and control for operator semigroups[END_REF] we see that this property (H1) implies that C is an admissible control operator for the semigroup generated by A in the sense of Definition 4.2.1 of [START_REF] Tucsnak | Observation and control for operator semigroups[END_REF]. Note further that the estimate (4.2.5) of [START_REF] Tucsnak | Observation and control for operator semigroups[END_REF] implies that

Φ t v H ≤ C 1 v L 2 (0,τ ;U ) , ∀t ∈ [0, τ ]. (4.2)
(H2) For all v ∈ L 2 (0, τ ; U), one has C * Φ t v ∈ L 2 (0, τ ; U) and there exists

C 3 > 0 such that C * Φ t v L 2 (0,τ ;U ) ≤ C 3 v L 2 (0,τ ;U ) . (4.3)
(H3) For all z 0 ∈ H, C * S(•)z 0 ∈ L 2 (0, τ ; U) and C * S(•)z 0 L 2 (0,τ ;U ) ≤ C 2 z 0 H . (4.4)
Note that this last condition directly implies that for all ℓ ∈ N,

C * S(•)z 0 L 2 (ℓτ,(ℓ+1)τ ;U ) ≤ C 2 Me -ℓτ ω z 0 H . (4.5)
Indeed for any t ∈ (ℓτ, (ℓ + 1)τ ), we can write S(t)z 0 = S(t -ℓτ )S(ℓτ )z 0 , and therefore by (4.4)

C * S(•)z 0 L 2 (ℓτ,(ℓ+1)τ ;U ) ≤ C 2 S(ℓτ )z 0 H ,
which leads to (4.5) owing to our assumption (1.1).

We are first able to prove the next well-posedness result.

Proposition 4.1 Under the previous assumptions on C, then for any initial datum U 0 ∈ H and f ∈ L 2 (0, τ ; U), there exists a unique (mild

) solution U ∈ C([0, +∞), H) of problem U t (t) = AU(t) + kCC * U(t -τ ) in (0, +∞) U(0) = U 0 , C * U(t -τ ) = f (t), ∀t ∈ (0, τ ). (4.6)
Proof. We use an iterative argument. Namely in the interval (0, τ ), problem (4.6) can be seen as an inhomogeneous evolution problem

U t (t) = AU(t) + kCf (t) in (0, τ ) U(0) = U 0 . (4.7)
Hence by the hypothesis (H1), this problem has a unique solution U ∈ C([0, τ ], H) given by

U(t) = S(t)U 0 + k t 0 S(t -s)Cf (s) ds. (4.8)
This yields U on (0, τ ) and therefore on (τ, 2τ ), problem (4.6) can be seen as an inhomogeneous evolution problem

U t (t) = AU(t) + kCv(t) in (τ, 2τ ) U(τ ) = U(τ -), (4.9)
where

v(t) = C * U(t -τ ) = C * S(t -τ )U 0 + kC * Φ t-τ f
. But owing to the hypotheses ((H2) and (H3), v belongs to L 2 (τ, 2τ ; U). Hence by the hypothesis (H1), this problem has a unique solution U ∈ C([τ, 2τ ], H) given by

U(t) = S(t -τ )U(τ -) + k t τ S(t -s)Cv(s) ds, ∀t ∈ [τ, 2τ ]. (4.10)
By iteration, we obtain a global solution.

Similarly we will prove the following exponential stability result.

Theorem 4.2 Let the assumptions (H1) to (H3) be satisfied. Set

k 0 := e τ ω -1 M ′2 C 1 C 4 e 2ωτ ,
where

M ′ = max{M, 1}, C 4 = max{C 2 , C 3 M ′ C 1 }. Then for any k satisfying |k| < k 0 , (4.11)
there exist ω ′ > 0 and M ′′ > 0 such that the solution

U ∈ C([0, +∞), H) of problem (4.6) with U 0 ∈ H and f ∈ L 2 (0, τ ; U) satisfies U(t) H ≤ M ′′ e -ω ′ t ( U 0 H + |k|M ′ C 1 e 2ωτ f L 2 (0,τ,U ) ), ∀t > 0. (4.12)
From its definition the constant k 0 depends only on M, ω, τ and the constants appearing in the assumptions (H1) to (H3).

Proof. We use again an iterative argument and the estimates (4.1) to (4.4). First on (0, τ ) using (4.8), the assumptions (1.1) and (4.1), we see that

U(t) H ≤ Me -ωt U 0 H + |k|C 1 f L 2 (0,τ,U ) , ∀t ∈ (0, τ ),
that directly leads to

U(t) H ≤ e -ωt (M U 0 H + |k|C 1 f L 2 (0,τ,U ) e ωτ ), ∀t ∈ (0, τ ). (4.13)
Now coming back to (4.8) and using (4.3) and (4.4), we get

C * U L 2 (0,τ,U ) ≤ C 2 U 0 H + |k|C 3 f L 2 (0,τ,U ) . (4.14)
Let us now prove by iteration that for all ℓ ∈ N, we have

U(t) H ≤ K 1 (ℓ)e -ωt , ∀t ∈ (ℓτ, (ℓ + 1)τ ), (4.15)
as well as

C * U L 2 (ℓτ,(ℓ+1)τ ;U ) ≤ K 2 (ℓ)e -ℓτ ω , (4.16)
where

K 1 (ℓ) ≤ M ′ ( U 0 H + δα)(1 + δC 4 M ′ ) ℓ , (4.17)
and

K 2 (ℓ) ≤ C 4 M ′ ( U 0 H + δα)(1 + δC 4 M ′ ) ℓ , (4.18)
with δ = |k|C 1 M ′ e 2ωτ and α = f L 2 (0,τ,U ) . Note that (4.15) and (4.16) hold for ℓ = 0 due to (4.13) and (4.14) since simple calculations yield

M U 0 H + |k|C 1 f L 2 (0,τ,U ) e ωτ ≤ K 1 (0), C 2 U 0 H + |k|C 3 f L 2 (0,τ,U ) ≤ K 2 (0).
Let us now prove that if (4.15)-(4.18) hold up to ℓ then they hold for ℓ + 1. Indeed for t ∈ ((ℓ + 1)τ, (ℓ + 2)τ ), we have This identity can be equivalently written

U(t) = S(t)U 0 + k ℓ j=1 (
U(t) = S(t)U 0 + k ℓ j=1 S(t -(j + 1)τ )Φ τ C * U((j -1)τ + •) (4.19) + kΦ t-(ℓ+1)τ C * U(ℓτ + •) + kS(t -τ )Φ τ f.
Hence by our assumptions (1.1) and (4.1), we deduce that

U(t) H ≤ Me -ωt U 0 H + |k|MC 1 ℓ j=1 e -(t-(j+1)τ )ω C * U((j -1)τ + •) L 2 (0,τ,U ) + |k|C 1 C * U(ℓτ + •) L 2 (0,τ,U ) + |k|Me -(t-τ )ω C 1 α.
Hence by our iterative assumption, the estimate (4.16) for all j ≤ ℓ yields

U(t) H ≤ M ′ e -ωt U 0 H + |k|C 1 e 2τ ω ℓ+1 j=1 K 2 (j -1) + |k|e τ ω C 1 α .
By setting K 2 (-1) = α, we have found that

U(t) H ≤ M ′ e -ωt U 0 H + |k|C 1 e 2τ ω ℓ+1 j=0 K 2 (j -1) .
This proves (4.15) for ℓ + 1 with

K 1 (ℓ + 1) = M ′ U 0 H + |k|C 1 e 2τ ω ℓ+1 j=0 K 2 (j -1) . (4.20)
Now we come back to (4.19) and applying C * to this identity (meaningful due to our assumptions (H2) and (H3)), we get

C * U L 2 ((ℓ+1)τ,(ℓ+2)τ,U ) ≤ C 2 Me -(ℓ+1)τ )ω U 0 H + |k|C 2 C 1 M ℓ j=1 e -(ℓ-j)τ ω C * U((j -1)τ + •) L 2 (0,τ,U ) + |k|C 3 C * U L 2 (ℓτ,(τ +1)τ,U ) + |k|C 2 C 1 e -ℓτ ω α.
As our iterative assumption means that (4.16) holds for all j ≤ ℓ, we get

C * U L 2 ((ℓ+1)τ,(ℓ+2)τ,U ) ≤ C 2 Me -(ℓ+1)τ ω U 0 H + |k|C 2 C 1 M ′ e -ℓτ ω e τ ω ℓ j=0 K 2 (j -1) + |k|C 3 e -ℓτ ω K 2 (ℓ). As C 2 ≤ C 4 and C 3 ≤ C 1 M ′ C 4 , we deduce that C * U L 2 ((ℓ+1)τ,(ℓ+2)τ,U ) ≤ C 4 M ′ e -(ℓ+1)τ ω U 0 H + δ ℓ j=-1 K 2 (j) .
This proves (4.16) for ℓ + 1 with

K 2 (ℓ + 1) = C 4 M ′ ( U 0 H + δ ℓ j=-1 K 2 (j)). ( 4

.21)

Let us now show that K 2 (ℓ) given by (4.21) satisfies (4.18). Indeed it holds for ℓ = 0 and then we again prove (4.18) by induction. If it holds up ℓ then by (4.21) we will have

K 2 (ℓ + 1) ≤ C 4 M ′ U 0 H + δα + δ ℓ j=0 C 4 M ′ ( U 0 H + δα)(1 + δC 4 M ′ ) j ≤ C 4 M ′ ( U 0 H + δα) 1 + δC 4 M ′ ℓ j=0 (1 + δC 4 M ′ ) j ≤ C 4 M ′ ( U 0 H + δα) 1 + δC 4 M ′ (1 + δC 4 M ′ ) ℓ+1 -1 δC 4 M ′ ≤ C 4 M ′ ( U 0 H + δα)(1 + δC 4 M ′ ) ℓ+1 .
This proves (4.18) for ℓ + 1.

Once (4.18) holds for all ℓ, we come back to (4.20) and get

K 1 (ℓ + 1) ≤ M ′ ( U 0 H + |k|C 1 e 2τ ω (α + ℓ j=1 C 4 M ′ ( U 0 H + δα)(1 + δC 4 M ′ ) j ) ≤ M ′ ( U 0 H + δ(α + δ -1 ( U 0 H + δα)((1 + δC 4 M ′ ) ℓ+1 -1) ≤ M ′ ( U 0 H + (αδ + ( U 0 H + δα)((1 + δC 4 M ′ ) ℓ+1 -1) ≤ M ′ U 0 H + ( U 0 H + δα)(1 + δC 4 M ′ ) ℓ+1 -U 0 H ≤ M ′ ( U 0 H + δα)(1 + δC 4 M ′ ) ℓ+1 .
This proves (4.18) for all ℓ + 1.

We end up the proof by combining (4.15) and (4.17) to get

U(t) H ≤ M ′ ( U 0 H + δα)(1 + δC 4 M ′ ) ℓ e -ωt , ∀t ∈ (ℓτ, (ℓ + 1)τ ).
Hence setting

σ = τ -1 ln(1 + δC 4 M ′ ) = τ -1 ln(1 + |k|C 4 M ′2 C 1 e 2ωτ ), (4.22)
we conclude as in Theorem 2.2 that

U(t) H ≤ M ′ ( U 0 H + δα)e (σ-ω)t , ∀t > 0. (4.23)
Therefore U(t) H will decay exponentially if σ -ω is negative or equivalently if

1 + |k|C 4 M ′2 C 1 e 2ωτ < e τ ω ,
which is nothing else than (4.11).

Consider an extension ṽ of v by taking an odd extension of v to (τ, 2τ ) and by taking ṽ = 0 outside (0, 2τ ). Then such an extension satisfies IR ṽ(t) dt = 0, and

IR |ṽ(t)| 2 dt ≤ 2 τ 0 |v(t)| 2 dt.
(5.31)

Then we can consider the solution w of

w tt (x, t) -w xx (x, t) = 0 in Ω × (0, ∞), w x (1, t) = -w t (1, t) -aw(1, t
) on (0, +∞), w x (0, t) = -ṽ(t) on (0, +∞), w(x, 0) = 0 and w t (x, 0) = 0 in Ω.

But since the corresponding operator A generates a strongly continuous semigroup, this solution w coincides with u in (0, τ ). Furthermore we can extend w by zero in (0, 1) × (-∞, 0) that then satisfies w tt (x, t) -w xx (x, t) = 0 in Ω × IR, w x (1, t) = -w t (1, t) -aw(1, t) on IR, w x (0, t) = -ṽ(t) on IR.

Taking Fourier transform in time, we deduce that for all ξ ∈ IR, ŵ(•, ξ) satisfies Recalling that w coincides with u in (0, τ ) and using the estimate (5.31), we have proved that As before it suffices to check the assumption (H1) for v ∈ D(0, τ ). For such a v consider the (strong) solution u = Φ t v, 0 < t < τ of (5.30). Then we consider its energy

E(t) = 1 2 ( 1 0 (|u t | 2 + |u x | 2 ) dx + a|u(1, t)| 2 ).
Differentiating and integrating by parts we have

E ′ (t) = -|u t (1, t)| 2 + v(t)u t (0, t).
Integrating this identity between 0 and t ∈ (0, τ ] and using Cauchy-Schwarz's inequality we find that E(t) ≤ v L 2 (0,τ ) u t (0, t) L 2 (0,τ ) .

Hence using the estimate (5.32) we arrive at (4.1). The continuous property is proved similarly by integrating between t ∈ (0, τ ] and t ′ ∈ (0, τ ].

In conclusion, as the system (5.26)-(5.29) with k = 0 is exponentially stable and the assumptions (H1) to (H3) hold, system (5.26)-(5.29) remains exponentially stable if k is small enough. Remark 5.1 Our approach cannot be used for the wave equation in IR d , with d ≥ 2 since according to the results from [START_REF] Tataru | On the regularity of boundary traces for the wave equation[END_REF] (see for instance Theorem 3 in [START_REF] Tataru | On the regularity of boundary traces for the wave equation[END_REF] and the comments before), the assumption (H2) is wrong once d ≥ 2.

The wave equation with boundary and internal unbounded feedbacks in 1d

Here we want to consider the following problem: For a fixed a ∈ (0, 1) consider the solution of u tt (x, t) -u xx (x, t) = 0 in (0, a) ∪ (a, 1) × (0, +∞) (5.33)

Proposition 3 . 2

 32 The assumption (3.15) is satisfied for |k| < 1/µ.

Example 3 . 4

 34 As an explicit example of system (3.18) let us consider the wave equation with local internal damping and internal delay. More precisely, let Ω ⊂ IR n , n ≥ 3, be an open bounded domain with a boundary ∂Ω of class C 2 . Denoting by m the standard multiplier m(x) = x-x 0 , x 0 ∈ IR n , let ω 1 be the intersection of Ω with an open neighborhood of the subset of ∂Ω

k τ 0 S

 0 j+1)τ jτ S(t -s)CC * U(s -τ ) ds + k t (ℓ+1)τ S(t -s)CC * U(s -τ ) ds + (t -s)Cf (s) ds.

ξ 2 ŵ

 2 + ŵxx = 0 in Ω, ŵx (1) = -(a + iξ) ŵ(1), ŵx (0) = -ṽ(ξ).Hence easy calculations showthat ŵ(x, ξ) = ṽ(ξ) iξ (e -iξx + c(ξ) cos(ξx)), ∀x ∈ (0, 1), with c(ξ) = ae -iξ ξ sin ξ -(a + iξ) cos ξ .This identity implies thatiξ ŵ(0, ξ) = ṽ(ξ)(1 + c(ξ))and since one can show that there exists a positive constant C depending on a such that|c(ξ)| ≤ C, ∀ξ ∈ IR,we deduce that |iξ ŵ(0, ξ)| ≤ (1 + C)| ṽ(ξ)|, ∀ξ ∈ IR. By Parseval's identity we find that IR |w t (0, t)| 2 dt ≤ (1 + C) IR |ṽ(t)| 2 dt.

  (H2) holds reminding thatC * (u, u t ) = B * 2 u t = u t (0, •).

Examples in the case B unbounded

Most of our examples are second order evolution equations with damping. Namely they are in the following form. Let H be a complex Hilbert space and let A : D(A) → H be a positive self-adjoint operator with a compact inverse in H. Denote by V := D(A 1 2 ) the domain of A 1 2 . Moreover, for i = 1, 2, let U i be complex Hilbert spaces with norm and inner product denoted respectively by • U i and •, • U i and let B i : U i → V ′ be linear operators. In this setting we consider the problem

where the constant τ > 0 is the time delay and k is a real parameter.

We transform this problem into a first order system by using the standard reduction of order: setting

it satisfies formally

In such a setting, we easily check that the adjoint A * of A is given by

In other words, if we introduce the unitary mapping

Consequently the semigroup (S * (t)) t≥0 generated by A * will be given by

To apply our stability results from section 4 to our system (5.24)-( 5.25) we need to check the assumptions (H1) to (H3) for the operators A and B = CC * . But in this case, (H1) implies (H3) since by Remark 4.2.4 of [START_REF] Tucsnak | Observation and control for operator semigroups[END_REF], (H1) implies that C is an admissible control for the semigroup S(t) and by Theorem 4.4.3 of [15] this is equivalent to the fact that C * is an admissible operator for the semigroup S * . As C * S * = OC * SO, we deduce that (H3) holds owing to Proposition 4.4.1 of [START_REF] Tucsnak | Observation and control for operator semigroups[END_REF].

The wave equation with boundary feedbacks in 1d

Our first application concerns the wave equation with boundary feedbacks in dimension 1. More precisely let Ω = (0, 1) ⊂ IR be the unit interval.

Given a positive constant a, let us consider the initial boundary value problem

with initial data (u 0 , u 1 ) ∈ H 1 (Ω) ×L 2 (Ω). This problem enters in the abstract framework (5.24)-( 5.25) if we take H = L 2 (Ω) with its standard norm and V = H 1 (Ω) with the norm

The operator A is defined by

where D(A) := {u ∈ H 2 (Ω) ∩ H 1 (Ω) : u x (0) = 0 and u x (1) + au(1) = 0}.

We then define U 1 = U 2 := IR and for i = 1 or 2, the operator

Finally we need to take

Hence in such a situation it remains to check the hypotheses (H1) and (H2).

To check the assumption (H2), as D(0, τ ) is dense in L 2 (0, τ ), it suffices to check it for v ∈ D(0, τ ). For such a v consider u = Φ t v, 0 < t < τ , that is the (strong) solution of

on (0, τ ), u(x, 0) = 0 and u t (x, 0) = 0 in Ω.

(5.30) u(0, t) = 0 on (0, +∞)

(5.36) u(x, 0) = u 0 (x) and u t (x, 0) = u 1 (x) in (0, 1)

(5.37) with initial data (u 0 , u 1 ) ∈ {w ∈ H 1 (0, 1) : w(0) = 0} × L 2 (0, 1), and [u](a) means the jump of u at the point a, i.e., [u](a) = u(a+) -u(a-). This problem corresponds to the case where a standard dissipative law (cfr. [START_REF] Komornik | Exact controllability and stabilization, the multiplier method[END_REF]) is acting at 1, while a dissipation with delay appears at the interior point a.

As in subsection 5.1, we only need to check the assumption (H2) (since as before one can show that (H2) implies (H1)), that is proved exactly as before by using an extension method and Fourier transform in time to get the system ξ 2 ŵ -ŵxx = 0 in (0, a) ∪ (a, 1), ŵ(0) = 0, ŵx (1) = -iξ ŵ( 1 In conclusion, for k small enough, system (5.33)-(5.37) is exponentially stable since it is for k = 0.

The wave equation with a bounded internal feedback and a boundary unbounded feedback in 1d

Arguing as before we can consider the following problem

(5.40) u(x, 0) = u 0 (x) and u t (x, 0) = u 1 (x) in (0, 1)

(5.41) with α > 0 and initial data (u 0 , u 1 ) ∈ {w ∈ H 1 (0, 1) : w(0) = 0} × L 2 (0, 1). This problem corresponds to the case where a standard dissipative law is acting on the whole domain, while a dissipation with delay appear at the boundary point 1. As this system with k = 0 is exponentially stable and the assumptions (H1) to (H3) are valid, system (5.38)-(5.41) remains exponentially stable if k is small enough (cfr. [START_REF] Datko | An example on the effect of time delays in boundary feedback stabilization of wave equations[END_REF]).


Serge Piero D'ancona

Roland Nicaise
email: serge.nicaise@univ-valenciennes.fr

Schnaubelt
email: schnaubelt@kit.edu

P D'ancona
email: dancona@mat.uniroma1.it

S Nicaise

R Schnaubelt

BLOW-UP FOR NONLINEAR MAXWELL EQUATIONS

Keywords: 2000 Mathematics Subject Classification. 35L60, 35Q61 Quasilinear Maxwell system, blow-up, shock, Kerr nonlinearity

We construct classical solutions to the nonlinear Maxwell system with periodic boundary conditions which blow up in H(curl). A similar result is shown on the full space. Our construction is based on an analysis of a shock wave in one space dimension.

Introduction and main results

The Maxwell system

∂ t D(t) = curl H(t), ∂ t H(t) = -curl E(t), (1.1)
div

D(t) = 0, div B(t) = 0 (1.2)
is one of the fundamental equations of physics (which is still poorly understood analytically in the nonlinear case). One has to complement (1.1) and (1.2) by material laws that connect the electric fields E and D, as well as the magnetic ones B and H. We focus on materials without magnetic response (as appearing in optics) and look at the case H = B. Moreover, we only treat instantaneous material relations where D is given as a pointwise function of E. This class of problems fits to the theory of quasilinear hyperbolic systems, cf. [START_REF] Benzoni-Gavage | Multidimensional Hyperbolic Partial Differential Equations[END_REF], [START_REF] Majda | Compressible Fluid Flow and Systems of Conservation Laws in Several Space Dimensions[END_REF] or [START_REF] Serre | Hyperbolicity of the nonlinear models of Maxwell's equations[END_REF]. There is a great variety of other material laws which are retarded in time, see [START_REF] Babin | Nonlinear Maxwell equations in inhomogeneous media[END_REF] and [START_REF] Roach | Mathematical Analysis of Deterministic and Stochastic Problems in Complex Media Electromagnetics[END_REF], or which are given by additional coupled ordinary or partial differential equations, see [START_REF] Ammari | Global existence and regularity of solutions to a system of nonlinear Maxwell equations[END_REF], [START_REF] Dumas | Cauchy problem and quasi-stationary limit for the Maxwell-Landau-Lifschitz and Maxwell-Bloch equations[END_REF], [START_REF] Jochmann | Asymptotic behavior of the electromagnetic field for a micromagnetism equation without exchange energy[END_REF] and [START_REF] Joly | Global solutions to Maxwell equations in a ferromagnetic medium[END_REF], for instance. For the physical background we refer to, e.g., [START_REF] Busch | Periodic nanostructures for photonics[END_REF] or [START_REF] Moloney | Nonlinear Optics[END_REF]. In our analysis we concentrate on conservative systems such as (1.1) and (1.2) without conductivity and given currents or charges.

For rather general instantaneous nonlinear material laws, the Cauchy problem for (1.1) and (1.2) on R 3 is locally wellposed due to standard results on quasilinear hyperbolic systems if one works in H 3 (R 3 ) (or H s (R 3 ) for s > 5/2), see [START_REF] Bahouri | Fourier Analysis and Nonlinear Partial Differential Equations[END_REF], [START_REF] Benzoni-Gavage | Multidimensional Hyperbolic Partial Differential Equations[END_REF] or [START_REF] Majda | Compressible Fluid Flow and Systems of Conservation Laws in Several Space Dimensions[END_REF]. Nothing seems to be known in lower regularity. The situation on domains is more delicate since one typically has characteristic boundary conditions. Here the general theory only yields result in even higher regularity, see e.g. [START_REF] Guès | Problème mixte hyperbolique quasi-linéaire caractéristique[END_REF]. For the Maxwell equations itself partial results were established in [START_REF] Picard | Local existence of solutions of impedance initialboundary value problem for non-linear Maxwell equations[END_REF]. Again on R 3 one knows at least for certain nonlinearities of Kerr type (1.5) that small initial data in H k (R 3 ) (for large k ∈ N) lead to global smooth solutions, see [START_REF] Racke | Lectures on Nonlinear Evolution Equations. Initial Value Problems[END_REF]. On the other hand, shock solutions on R 3 are present for the Kerr nonlinearity, see [START_REF] Aregba-Driollet | Kerr-Debye relaxation shock profiles for Kerr equations[END_REF]. Here the first derivative blows up only pointwise and these solutions do not belong to L 2 (R 3 ). As a result, global existence in H(curl) × H(curl) is not ruled out so far.

In this paper, we construct a classical solution (E 0 , B 0 ) whose curl blows up in L 2 for the Maxwell system on a cube with periodic or certain mixed boundary conditions. This solution is essentially given by a shock solution for a one-dimensional subproblem whose compactly supported initial data were chosen in such way that the first derivative of the solution explodes in L 2 (R) in finite time. To transfer this result to R 3 , one has to localize (E 0 , B 0 ) since these fields only depend on one space variable. The analysis of the localized solution requires a local uniqueness result for (1.1) and (1.2) which does not seem to be available in H(curl) × H(curl). As a result, we obtain blow-up on R 3 only within a smaller class of solutions, see Theorem 1.4. Our results apply to a large class of nonlinearities described below, where we impose no, respectively very mild, assumptions on their behavior at infinity.

To make use of subproblems in one space dimension, we look at scalar type material laws D = Φ(E)E for a given function Φ : R 3 → R and thus on the nonlinear Maxwell system

∂ t [Φ(E(t))E(t)] = curl B(t), ∂ t B(t) = -curl E(t), (1.3
)

div[Φ(E(t))E(t)] = 0, div B(t) = 0 (1.4)
An important special case is the Kerr model with

Φ(x) = 1 + a |x| 2 (1.5)
for some a > 0, see [START_REF] Busch | Periodic nanostructures for photonics[END_REF] or [START_REF] Moloney | Nonlinear Optics[END_REF]. In Examples 2.1 and 3.1 we treat more general versions of (1.5). The one-dimensional version of (1.3) reads as

∂ t b(u) = ∂ x v, ∂ t v = ∂ x u, (u(0, x), v(0, x)) = (u 0 (x), v 0 (x)) (1.6)
In (1.6) the function b has to satisfy the following hypotheses.

Assumption 1.1. The map b belongs to C 2 (R, R), there exist numbers w -< 0 < w 0 < w + such that b > 0 on J := (w -, w + ), the function q ∈ C(J, R) given by

q(s) := b (s) 2b (s) 3/2 , s ∈ J,
has a global maximum at s = w 0 , q is C 1 near w 0 , and q(s) > 0 for 0 < s ≤ w 0 .

We point out that there are no assumptions on the behavior of b at infinity. In particular, b could become constant or linear. We now state our basic blow-up result about (1.6), which we prove in Section 2.

Theorem 1.2 (1D case). Let Assumption 1.1 be true. Then there exist compactly supported initial data

(u 0 , v 0 ) ∈ C 1 (R, R 2 ) and a C 1 solution (u, v) to Problem (1.6) on [0, t * ) × R for some t * ∈ (0, ∞) which is compactly supported in x for each t and satisfies ∂ x u(t, •) L 2 (R) → +∞ as t → t - * .
In the proof one actually sees that this blow-up occurs for a large class of initial functions. We mainly require that the slope of the 'electric part' attains its positive maximum at x = 0, see (2.10) and (2.12). The blow-up solutions in the following two results are modifications of those from Theorem 1.2.

In three space dimensions we first look at the cube

Q M = [-M, M ] 3
for some M > 0 with outer unit normal ν. Besides periodic boundary condition we treat the mixed conditions (BC) given as (i) E • ν = 0 and B × ν = 0 for x 3 = ±M ;

(ii) E × ν = 0 and B • ν = 0 on the rest of the boundary.

Here (ii) are the usual conditions for perfectly conducting boundaries, while (i) corresponds to a perfect magnetic conductor. Though there is no real material with the behavior in (i), it is used as a symmetry boundary condition in numerical computations.

Theorem 1.3 (3D boundary value problem).

Assume that b(s) := Φ(s, 0, 0)s for s ∈ R fulfills Assumption 1.1. Consider the Cauchy problem for (1.3) and (1.4) on the cube Q M with the boundary conditions (BC) or with periodic boundary conditions, where M > 0 is sufficiently large. Then there exist initial data

(E 0 0 , B 0 0 ) in C 1 (Q M , R 6 ) satisfying (1.4
) and the respective boundary conditions and a corresponding

C 1 solution (E 0 , B 0 ) on [0, t * ) × Q M for some t * ∈ (0, ∞) such that curl E 0 (t, •) L 2 (Q M ) → +∞ as t → t - * .
This and the following theorem are shown in Section 3. For our result on R 3 , we also assume that Φ(E) = β(|E|) for a function β ∈ C 2 ([0, ∞), R) and let b(s) := sβ(|s|) for s ∈ R. We define the symmetric matrix

A 0 (y) = β(|y|)I 3×3 + β (|y|)|y| -1 yy (1.7)
for y ∈ R 3 \ {0} and A 0 (0) = β(0)I 3×3 . Our asumptions will imply that this matrix is positive definite with a uniform lower bound. For C 1 solutions the equation (1.3) can be equivalently rewritten as the symmetric hyperbolic system

A 0 (E)∂ t E = curl B, ∂ t B = -curl E. (1.8)
The space of functions

ϕ ∈ L 2 (R 3 , R 3 ) such that curl ϕ ∈ L 2 (R 3 , R 3 ) is called H(curl).
It is a Hilbert space when endowed with the natural norm.

Theorem 1.4 (3D problem on R 3 ). Let Φ(E) = β(|E|) for a function β in C 2 ([0, ∞), R) such that b(s) := β(|s|)s for s ∈ R fulfills Assumption 1.1, b (s) ≥ κ and sβ (s) ≤ cβ(s)
for some c, κ > 0 and all s ≥ 0. Consider the Cauchy problem for (1.3) and (1.4) on R 3 . There exist compactly supported initial data

(E c 0 , B c 0 ) in C 1 (R 3 , R 6 ) satisfying (1.4) such that the corresponding local C 1 solution (E c , B c ) can not be continued to a global solution of (1.8) in C([0, ∞), H(curl) × H(curl)) such that curl B is bounded on [0, T ] × R 3 for each T > 0.
As we will see in Example 2.1 and 3.1 the Kerr model (1.5) fulfills the assumptions of the above theorem, even if we modify β(s) = 1 + as 2 to a constant function for large s > 0.

Remark 1.5. Straightforward modifications of our proofs yield the following generalizations. First, if we assume in addition that b belongs to C ∞ (R, R), then one can replace in the above theorems C 1 by C ∞ . Second, Theorem 1.4 is also true for functions Φ as in Theorem 1.3 such that the map Ψ : R 3 → R 3 ; Ψ(x) = Φ(x)x, is an diffeomorphism and its derivative DΨ(x) is symmetric and uniformly positive definite for x ∈ R 3 . The extra conditions on b and β in Theorem 1.4 just ensure these properties of Ψ.

Remark 1.6. Let Φ(E) = β(|E|)E with a function β ∈ C 2 (R, R) as in Theorem 1.4. For Q = R 3 or Q = [-M, M ] 3 as in Theorem 1.3 we define h(s) = s 0 β( √ r)
dr for s ≥ 0 and the 'energy'

E(E, B) = Q 1 2 |B| 2 + β(|E|)|E| 2 -1 2 h(|E| 2 ) dx
for functions on Q such that each summand is integrable. Take maps (E, B) in

C 1 ([0, T ] × Q, R 6 ) with sufficient decay at ∞ if Q = R 3 ,
E(E, B) = Q 1 2 |E| 2 + 3 4 a |E| 2 + 1 2 |B| 2 dx.
But, these conserved quantities are not strong enough to prevent the blow-up in H(curl) stated in the theorems.

2. The one dimensional case, proof of Theorem 1.2

Let b satisfy Assumption 1.1. For C 1 solutions taking values in J × R, we can rewrite the system (1.6) as

∂ t u v + A(u, v)∂ x u v = 0 with A(u, v) = 0 -b (u) -1 -1 0 (2.1)
on R. For (u, v) ∈ J × R, the matrix A(u, v) has the eigenvalues and eigenvectors

λ 1,2 (u, v) = ±b (u) -1 2 , η 1,2 (u, v) = ∓1 b (u) 1 2
.

These observations are a special case of the analysis in Section 3 of [START_REF] Aregba-Driollet | Kerr-Debye relaxation shock profiles for Kerr equations[END_REF]. In the following we take λ = λ 1 and η = η 1 and drop the index 1. Using the construction in Section 1.4 of [START_REF] Majda | Compressible Fluid Flow and Systems of Conservation Laws in Several Space Dimensions[END_REF], we first construct a bounded C 1 solution of (2.1) whose first derivative has a finite time blow-up in the sup-norm. The main step is to show that it even blows up in L 2 if one chooses the initial functions in the right way.

Fix (α, β) ∈ (w 0 , w + ) × R such that q(s) > 0 for 0 < s ≤ α.

Observe that the interval α -J = (α -w

+ , α -w -) contains [0, α]. The C 2 function φ : α -J → J × R φ 1 (s) = α -s, φ 2 (s) = β + s 0 b (α -τ ) 1/2 dτ,
solves the ordinary differential equation

φ (s) = η(φ(s)), s ∈ α -J, φ(0) = (α, β). (2.2)
For later use, we note the identities

∇λ(φ(s)) • φ (s) = ∇λ(φ(s)) • η(φ(s)) = q(α -s), s ∈ α -J. (2.3)
Let σ 0 : R → [0, α] be C 1 and equal to α outside a compact set. There is a unique C 1 solution σ of the scalar partial differential equation

∂ t σ(t, x) + λ(φ(σ(t, x)))∂ x σ(t, x) = 0, t ≥ 0, x ∈ R, σ(0, x) = σ 0 (x), x ∈ R, (2.4)
on a sufficiently small (bounded) time interval [0, t), where σ takes values in α -J. See e.g. Theorems 2.1 and 2.2 of [START_REF] Majda | Compressible Fluid Flow and Systems of Conservation Laws in Several Space Dimensions[END_REF]. We now define

u(t, x) v(t, x) = φ(σ(t, x)).
It is easy to check that (u, v) is a C 1 solution of (2.1) on [0, t) × R. We observe that

∂ x u = φ 1 (σ)∂ x σ = -∂ x σ. (2.5)
The methods of characteristics yields the implicit formula

σ(t, x) = σ 0 (x -tλ(φ(σ(t, x)))) = σ 0 (y(t, x)), (2.6)
where

y(t, x) := x -tλ(φ(σ(t, x)) = x -t • b (α -σ(t, x)) -1/2 , (2.7) as long as 1+t∇λ(φ(σ(t, x))) • η(φ(σ(t, x)))σ 0 (x -tλ(φ(σ(t, x)))) = 1 + tσ 0 (x -tλ(φ(σ(t, x))))q(α -σ(t, x)) > 0,
(2.8) see e.g. p.114 of [START_REF] Evans | Partial Differential Equations[END_REF], as well as (2.3). We now set

γ(t) := inf x∈R σ 0 (y(t, x))q(α -σ(t, x)) for t ∈ [0, t).
Let t 0 ∈ R + be the supremum of t ∈ [0, t) such that τ γ(τ ) > -1 for all τ ∈ [0, t]. In the following, we take t ∈ [0, t 0 ) so that the inequality (2.8) is valid for all x ∈ R. Equations (2.6) and (2.3) then imply

∂ x σ(t, x) = σ 0 (x -tλ(φ(σ(t, x)))) 1 -tq(α -σ(t, x))∂ x σ(t, x) , ∂ x σ(t, x) = σ 0 (x -tλ(φ(σ(t, x)))) 1 + tq(α-σ(t, x))σ 0 (x -tλ(φ(σ(t, x)))) = σ 0 (y(t, x)) 1 + tq(α-σ(t, x))σ 0 (y(t, x))
.

In particular, σ and ∂ x σ are bounded on [0, t 0 -δ] × R for each δ ∈ (0, t 0 ]. The blow-up condition in Theorem 2.2 Annex of [START_REF] Majda | Compressible Fluid Flow and Systems of Conservation Laws in Several Space Dimensions[END_REF] thus yields t = t 0 . From formula (2.6) we further deduce ∂ x σ(t, x) = σ 0 (y(t, x))∂ x y(t, x) and hence

∂ x y(t, x) = 1 1 + tq(α -σ(t, x))σ 0 (y(t, x)) > 0.
(2.9) (In the case σ 0 (y(t, x)) = 0, we have ∂ x σ(t, x) = 0 and the identity ∂ x y(t, x) = 1 > 0 follows from (2.7).) Using also (2.7), we see that the map x → y(t, x) is a bijection from R to R; let y -1 t : R → R be its inverse. This fact and (2.6) lead to the equation

γ(t) = inf z∈R σ 0 (z)q(α -σ 0 (z)) =: γ 0 for all t < t 0 .
We now fix a C 1 function σ 0 : R → [0, α] which is equal to α outside some compact set and satisfies

σ 0 (0) = α -w 0 , σ 0 (0) = min z∈R σ 0 (z) < 0. (2.10)
In view of Assumption 1.1, we can determine γ 0 = σ 0 (0)q(w 0 ) and

t 0 = - 1 γ 0 . (2.11)
Substituting z = y(t, x) and employing (2.9), we deduce from formula (2.6) the identities

∂ x σ(t, •) 2 2 = R |∂ x σ(t, x)| 2 dx = R |σ 0 (y(t, x))∂ x y(t, x)| 2 dx = R |σ 0 (z)| 2 1 + tq(α -σ 0 (z)) σ 0 (z) dz.
We shall employ the expansions

σ 0 (z) = α -w 0 + O(z), σ 0 (z) = σ 0 (0) + o + (z), q(w) = q(w 0 ) -o + (w -w 0 )
where o + (z) denotes any nonnegative function with the property o + (z)/z → 0 as z → 0. Here we used the assumptions that q has a global maximum at w 0 while σ 0 has a global minimum at 0. Hence, (2.11) yields

1 + tq(α -σ 0 (z)) σ 0 (z) = 1 + tγ 0 + t[q(w 0 )o + (z) + o + (z) |σ 0 (0)| -o + (z 2 )] = 1 + tγ 0 + to + (z).
By means of this equality, we arrive at

∂ x σ(t, •) 2 2 = R |σ 0 (z)| 2 1 + tγ 0 + to + (z) dz.
Fix a number δ 0 > 0 such that

|σ 0 (z)| 2 ≥ |σ 0 (0)| 2 2 =: c 0 for all |z| ≤ δ 0 .
For all > 0 there exists a radius δ ∈ (0, δ 0 ) such that 0 ≤ o + (z) ≤ δ for all |z| < δ.

E(t, x) = 1 β(b -1 (|D(t, x)|)) D(t, x) =: ψ(|D(t, x)|) D(t, x),
where ψ : [0, b(w

+ )] → [0, w + ] is C 2 . Let D 0 0 (x) = Φ(E 0 0 (x))E 0 0 (x) = β(|u 0 (x 2 )|)(u 0 (x 2
), 0, 0). We now introduce the functions D c 0 , B c 0 : R 3 → R 3 by ) for all t because of (1.3) and div D c 0 = div B c 0 = 0. We suppose that Theorem 1.4 was wrong. Then (E c , B c ) can actually be extended to a global solution of (1.8) such that the functions E c (t, •) and B c (t, •) belong to H(curl) and curl B c (t, •) to L ∞ (R 3 ) for each t ≥ 0, and they are locally bounded in the respective spaces. We introduce the maps

D c 0 (x) =   β(|u 0 (x 2 )|)u 0 (x 2 )χ(x 1 )χ(x 3 ) 0 -β(|u 0 (x 2 )|)u 0 (x 2 )χ (x 1 )X(x 3 )   , B c 0 (x) =   -v 0 (x 2 )X(x 1 )χ (x 3 ) 0 v 0 (x 2 )χ(x 1 )χ(x 3 )   . Note that D c 0 (x) and B c 0 (x) vanish if x ∈ [-r 1 , r 1 ] 3 . Moreover, for x ∈ [-r, r]
E = E 0 -E c and B = B 0 -B c on [0, t 0 ) × R 3 . (3.3)
Our assumptions yield that b (s) = β(s) + sβ (s) ≥ κ > 0 and β(s) ≥ κ > 0 for all s ≥ 0. Therefore the matrix A 0 (y) from (1.7) has the inverse

A 0 (y) -1 = 1 β(|y|) I 3×3 - β (|y|) |y| β(|y|) 2 + |y| 2 β(|y|)β (|y|) yy (3.4)
for each y ∈ R 3 \ {0}, and A 0 (0) -1 = β(0) -1 I 3×3 . These inverses are uniformly bounded since β(s) ≥ κ and sβ (s) ≤ cβ(s) for all s ≥ 0. As a result, the positive definite symmetric matrices a 0 (t, x) := A 0 (E 0 (t, x)) and a c (t, x) (1.8) and that E c (0, •) belongs to L ∞ (R 3 ), we deduce the boundedness of the functions ∂ t E c and E c on [0, t 0 ] × R 3 . It follows that a c and a 0 are bounded on [0, t 0 ) × R 3 , and ∂ t a 0 (t, x) on [0, t 0 -δ] × R 3 for each δ ∈ (0, t 0 ).

:= A 0 (E c (t, x)) have a uniform lower bound η ∈ (0, 1] for (t, x) ∈ [0, t 0 ) × R 3 . Using further that ∂ t E c = a -1 c curl B c by
In view of (1.8), the function (E , B ) satisfies the equation

a 0 ∂ t E = curl B + (a c -a 0 )a -1 c curl B c = curl B + ZE , ∂ t B = -curl E , (3.5)
for the zero order term ZE := Formula (1.7) implies that the derivative DA 0 (y), y = 0, is bounded on bounded subsets. This fact and the above observations show that M is a uniformly bounded matrix function on [0, t 0 ) × R 3 .

Recall that E (0, •) = B (0, •) = 0 on [-r, r] 3 . Take now r ≥ √ 3M + √ 6t 0 /η. Due to the properties of a 0 and Z, we can apply local uniqueness results for the linear system (3.5), cf. Theorem 4.11 in [START_REF] Bahouri | Fourier Analysis and Nonlinear Partial Differential Equations[END_REF] and its proof, deducing that E = B = 0 and hence E c = E 0 on the truncated cone C = {(t, x) | 0 ≤ t < t 0 , |x| ≤ r -√ 6t/η}. Since the cuboid [0, t 0 ) × [-M, M ] 3 is contained in C, we see that curl E c (t, •) blows up in L 2 (R 3 ) as t → t 0 . This fact contradicts the assumption made before (3.3), and hence Theorem 1.4 is true.

Example 3.1. As in Example 2.1, let γ > 2 and a > 0. Set β 1 (s) = 1 + as γ-1 for s ≥ 0. Alternatively, we modify β 1 to a function β 2 ∈ C 2 ([0, ∞), R) with β 2 ≥ 0 which is constant on [w 2 , ∞) for some w 2 > w 0 , where w 0 > 0 is given by Example 2.1. Define b j (s) = sβ j (s) for s ≥ 0. These functions satisfy the assumptions of Theorem 1.4.

r 1 >

 1 r > 0 with r 1 > M , and has the additional property that∞ -∞ χ(s) ds = 0. The support of X(s) := s -r1 χ(σ) dσ, s ∈ R, is thus contained in [-r 1 , r 1 ].We next define the new fieldD(t, x) = Φ(E(t, x))E(t, x) = β(|E(t, x)|)E(t, x)(3.2)for any given E so that (1.4) reduces to div D = 0, div B = 0. Since b(0) = 0 and b (s) ≥ κ > 0 for s ≥ 0, we have b(s) = sβ(s) ≥ κs and so β(s) ≥ κ for all s ≥ 0. The formula (3.2) then yields |E(t, x)| = b -1 (|D(t, x)|) for all (t, x) ∈ [0, t 0 ) × R. Hence, the transformation in (3.2) has the inverse given by

3

 3 the identities D c 0 (x) = D 0 0 (x) and B c 0 (x) = B 0 0 (x) follow from χ = 1 and χ = 0 on [-a, a]. In addition, on R 3 we can easily compute div D c 0 = div B c 0 = 0. We then define the initial field E c 0 : R 3 → R 3 E c 0 (x) = ψ(|D c 0 (x)|)D c 0 (x). whose support is also contained in [-r 1 , r 1 ] 3 and which coincides with E 0 0 on [-r, r] 3 . Theorem 2.1 of [12] yields a local in time C 1 solution (E c , B c ) of (1.8) having the initial values (E c 0 , B c 0 ). It satisfies the divergence conditions (1.4

1 0

 1 DA 0 (E 0 + s(E c -E 0 ))[E , a -1 c curl B c ] ds.

  say, which solve(1.3) and satisfy (BC) or periodic boundary conditions ifQ = [-M, M ] 3 . It is then easy to check that E(E(t), B(t)) is constant for t ∈ [0, T ]. (See[START_REF] Serre | Hyperbolicity of the nonlinear models of Maxwell's equations[END_REF] for more general material laws.) For the Kerr model one has

We gratefully acknowledge financial support by the Deutsche Forschungsgemeinschaft (DFG) through CRC 1173.

We can then estimate

Because of t 0 = -1/γ 0 in (2.11), it follows lim inf

Since > 0 is arbitrary, equation (2.5) finally implies that lim inf

is compactly supported by construction. On the other hand, we have

for all sufficiently large |x|. If we choose

the function v 0 is then compactly supported as well. This concludes the proof of Theorem 1.2.

Example 2.1. Let γ > 2 and a > 0. Set 3. The three dimensional case, proofs of Theorems 1.3 and 1.4

We begin with the proof of Theorem 1.3. We use the solution (u, v) of the one dimensional problem (1.6) constructed in the previous section. Equations (2.6) and (2.8) imply that the supports of u(t, •) and v(t, •) are contained in an interval (-M, M ) for all times 0 ≤ t < t 0 with t 0 from (2.11), provided M > 0 is chosen large enough. We then define (E 0 (t, x), B 0 (t, x)) = (u(t, x 2 ), 0, 0, 0, 0, v(t, x 2 )).

(3.1)

It is easy to check that these functions solve (1.3) and (1.4) on Q M = [-M, M ] 3 and that they satisfy the boundary conditions (BC) as well as the periodic ones. We set (E 0 0 (x), B 0 0 (x)) = (u 0 (x 2 ), 0, 0, 0, 0, v 0 (x 2 )). Since we have curl

) , the conclusion of Theorem 1.3 follows from Theorem 1.2.

We pass now to the proof of Theorem 1.4. The functions (E 0 , B 0 ) defined in (3.1) solve (1.3) and (1.4) on the whole strip [0, t 0 ) × R 3 ; and the initial data have compact support in the variable x 2 , but not in x 1 and x 3 . We now modify the initial funtions outside a compact set so that they become compactly supported and still satisfy the divergence conditions (1.4). To this end, let χ be a test functions which is equal to 1 on an interval [-r, r], vanishes outside the interval [-r 1 , r 1 ] for some


Oumaima Laraj
email: oumaima.laraj@um5r.ac.ma

Noha El Khattabi

Alain Rapaport

Mathematical model of anaerobic digestion with leachate recirculation

Keywords: Anaerobic digestion, Mathematical modeling, Biogas, Leachate recirculation, Stability analysis, Non-hyperbolic equilibrium

Landfilled biodegradable materials are subject to biological evolution, namely biodegradation or digestion, under the action of aerobic bacteria and further anaerobic bacteria. In the absence of special provisions, the liquid flowing through the mass of waste produces leachate, being charged with chemical or biological substances. Therefore, the mathematical modeling is of great interest for making decisions in the field of waste management since it makes possible to predict the behavior of the various complex systems underlying the management of waste. The analysis of the dynamics of the model considered in our work describes the process of anaerobic biodegradation with leachate reinjection, and allows us to predict the amount of biogas produced in the long term. The system admits an infinity of non hyperbolic equilibria but we are able to describe the asymptotic behavior of the solutions depending on the initial condition. Thanks to this model, we can highlight the influence of the leachate recirculation on the production of biogas, with the help of numerical simulations. This recirculation could be a part of the sustainable solutions adapted by recent technologies for the treatment of waste, not only to optimize the total production of biogas, but also to obtain a good quality of the outgoing liquid to meet the discharge standards.

I INTRODUCTION

The anaerobic digestion process is characterized by a succession of complex reactions. In the literature, there exists different multi-step models but whose mathematical analysis is often difficult or merely not possible [START_REF] Batstone | The IWA Anaerobic Digestion Model No 1 (ADM1)[END_REF][START_REF] Hassam | A generic and systematic procedure to derive a simplified model from the Anaerobic Digestion Model No. 1 (ADM1)[END_REF].

In the Rouez's model [START_REF] Rouez | Dégradation anaérobie de déchets solides: Caractérisation, facteurs d'influence et modélisations[END_REF], a simplification has been proposed in which the process is reduced to two fundamental steps which are the hydrolysis/acidogenesis and the methanogenesis. This model has been extended in [START_REF] Ouchtout | Analysis of an anaerobic digestion model in landfill with mortality term[END_REF] by considering that a part of the dead microorganisms returns to the slowly biodegradable material. On the other hand, in [START_REF] Rapaport | Dynamical modelling and optimal control of landfills[END_REF] the organic matter is decomposed into two types of substrat, soluble and insoluble. Our model gathers these features, that are 1. the two-steps process, 2. the consideration of both soluble and insoluble substrates, 3. the distribution of the dead microbial matter into a part returning (slowly) to degradable material and another one transformed directly into carbon dioxide.

Our objective is to investigate the influence of leachate recirculation on the performances of biogas production.

The biological scheme underlining the different interactions of the process is depicted on Figure 1.

Figure 1: Biological diagram of the interactions of the anaerobic digestion process of organic matter (where

f 1 = f 1 i + f 1s ).

II MODEL AND ASSUMPTIONS 2.1 Model equations

The process described in Figure 1 is modeled by the following dynamical system:

                 Ẋ = -δX + αmB Ṡi = f 1 i δX -h(u)g(S i ) Ṡs = f 1s δX + h(u)g(S i ) -1 Y µ(S s )B Ḃ = (µ(S s ) -m)B Ċ O 2 = (1 -f 1 )δX + (1 -f 2 ) 1-Y Y µ(S s )B + (1 -α)mB Ċ H 4 = f 2 1-Y Y µ(S s )B (1)
where we posit

f 1 = f 1 i + f 1s .
The state variables are the concentrations : X(t) of complex organic matter at time t, S i (t) of insoluble substrate at time t, S s (t) of soluble substrate at time t, B(t) of methanogenic bacteria at time t, and the output variables are the amounts produced per unit of volume : CO 2 (t) of carbon dioxide at time t, CH 4 (t) of methane at time t. The function µ(•) is the specific growth rate of the biological reaction, while the function (u, S i ) → h(u)g(S i ) describes the conversion rate of insoluble substrate into soluble one, where u is the recirculation rate. As the conversion rate of insoluble substrate into soluble one is a function of two variables that is equal to 0 in absence of insoluble substrate or null recirculation, we have assumed that it can be written as a product of two functions h(•), g(•) of each variable (equal to 0 when their argument is zero).

δ > 0, 0 < α ≤ 1 and 0 < u ≤ u max < +∞, 0 < m < max Ss>0 µ(S s ), 0 < Y < 1, 0 < f 1 i , f 1s , f 2 < 1 (with f 1 i + f 1s = f 1 ).

III MATHEMATICAL RESULTS

In this section, we present some results, their proofs are given in the appendix.

Existence and uniqueness of solutions

Proposition 1. The vector field associated to the system of equations ( 1) is C 1 in R 6 . Therefore, for any non-negative initial condition, the Cauchy problem admits a unique solution defined for any time (see for instance [START_REF] Walter | Ordinary Differential Equations[END_REF]). Moreover, the solution remains non-negative for any positive time.

Asymptotic behavior

Theorem 1. Under hypotheses H 1 , H 2 , H 3 , for any non-negative initial condition, the solution of (1) is bounded. In addition, one has

lim t→+∞ X(t) = lim t→+∞ B(t) = 0, lim t→+∞ S i (t) = 0, lim t→+∞ S s (t) := S ⋆ s ≥ 0, with S ⋆ s ≤ X(0) αY + (S i (0) + S s (0)) + B(0) Y .
Moreover, if X(0), S i (0) or S s (0) is non-null, then S ⋆ s > 0.

Attractor

Let us define the set

E := {S s ∈ R + : µ(S s ) ≤ m}.
Proposition 2. Under hypotheses H 1 , H 2 , H 3 , for any non-negative initial condition, one has

S ⋆ s = lim t→+∞ S s (t) ∈ E.
The system (1) admits a continum of equilibria which are all non hyperbolic. Therefore, one cannot conclude about their stability by studying the single linearization of the dynamics (and the Center Manifold Theorem is not enough informative). We use the same technique as in [START_REF] Ouchtout | Analysis of an anaerobic digestion model in landfill with mortality term[END_REF] which consists in considering a (non-linear) change of coordinates parameterized by S ⋆ s ∈ E, such that the dynamics admits a unique equilibrium in these coordinates (see Annex 2). For each S ⋆ s ∈ E, this equilibrium is a hyperbolic saddle point, with a stable manifold of dimension three that matches all trajectories of (1) with lim t→+∞ S s (t) = S ⋆ s . Then, we obtain the following result. Proposition 3. Under the same hypotheses than before,

A := {0} × {0} × E × {0}
is a forward attractor of any non-negative initial condition of (1).

Asymptotic production of biogas

Let us denote by G(t) the total amount of biogas per unit of volume that is produced at time t, that is

G(t) = CO 2 (t) + CH 4 (t).
Proposition 4. Under the same hypotheses than before, for any non-negative initial condition, the solution of the system (1) verifies

lim t→+∞ G(t) = G(0) + X(0) + aB(0) + S i (0) + S s (0) -S ⋆ s where a := 1 -αf 1 Y 1 -f 1 Y > 0.

IV NUMERICAL SIMULATIONS

For the growth function µ, we have chosen the Haldane law [START_REF] Andrews | A mathematical model for the continuous culture of microorganisms utilizing inhibitory substrates[END_REF] which modeled a growth inhibition under large concentration of substrate

µ(S s ) = μ S s K s + K i + S 2 s K i
where μ is the maximum growth rate, K s is the half-saturation constant and K i the inhibition constant, and the simplest functions for g and h:

g(S i ) = S i , h(u) = u.
The values of parameters are inspired from [START_REF] Ouchtout | Analysis of an anaerobic digestion model in landfill with mortality term[END_REF] and given in Table 1. Initial condition and parameter u for the recirculation are given in Table 2.

μ K s K i δ m Y α f 1 i f 1s f 2 0.3 h -1 160 mg.l -1 10 mg.l -1 0.176 h -1 0.02 h -1 0.05 0.9 0.4 0.3 0.76

X(0)

S i (0) S s (0) B(0) u 300 mg.l -1 0 0 2 mg.l -1 0.3 The time evolution of the variables and the production of biogas are depicted on Figure 2. To have insights on the impacts of changing operating conditions, we have performed different simulations:

1. Figure 3: soluble substrate for different load of organic matter. As expected, the asymptotic value of S s never reaches the interval (14.09, 125.90).

2. Figure 4: effect of absence of recirculation. When u = 0, the concentration S i is carrying on increasing all the time, differently to the case u > 0 for which we have proved that it converges asymptotically to 0. 3. Figure 5: effect of leachate recirculation and initial load of organic matter on the biogas production. This shows that a minimum of recirculation is necessary to obtain good performances. Moreover, the inhibition in the growth (which implies that the set E is not connected) induces a discontinuity in the biogas production as a function of X(0). There exists a threshold on the initial load of organic matter above which the biogas production falls down drastically. V CONCLUSIONS

• The theoretical results are confirmed by the numerical simulations.

• With Haldane growth function, the biogas production is a non monotonic function of the initial load of organic matter X(0) (it increases and then decreases with a discontinuity at the maximal value). • Future perspectives of this work concern the maximization of the biogas production:

1. for large amounts of organic matter to be treated, the optimal supply of organic matter over time, 2. the optimal control of the recirculation rate u.

A ANNEX 1. PROOF OF PROPOSITION 1

Under hypotheses H 1 , H 2 , H 3 , for any non-negative initial condition (X 0 , S 0 i , S 0 s , B 0 , CO 0 2 , CH 0 4 ) we have

Ḃ(t) = (µ(S s ) -m)B(t), then B(t) = B 0 exp( t 0 (µ(S s (z)) -m)dz), therefore B(t) ≥ 0, ∀t ≥ 0.
Moreover, Ẋ(t) = -δX(t) + αmB(t). Since B(t) ≥ 0, ∀t ≥ 0, one has Ẋ(t) ≥ -δX(t), ∀t ≥ 0, and, by Grönwall Lemma [START_REF] Walter | Ordinary Differential Equations[END_REF], X(t) ≥ X 0 . exp(-δt), ∀t ≥ 0. As X 0 ≥ 0, then X(t) ≥ 0, ∀t ≥ 0.

In addition, Ṡi (t) = f 1i δX(t) -h(u)g(S i ). Since X(t) ≥ 0, ∀t ≥ 0, then Ṡi (t) ≥ -h(u)g(S i ) ∀t ≥ 0. Considering the following Cauchy problem:

Ṡi (t) = -h(u)g( Si ) Si (0) = S 0 i . (2)
By Comparison Lemma [START_REF] Walter | Ordinary Differential Equations[END_REF], one has Si (t) ≤ S i (t), ∀t ≥ 0. Then, it suffices to show that 0 ≤ Si (t), ∀t ≥ 0. ⋆ if S 0 i = 0, as g(0) = 0 , it's clear that 0 is a solution of the Cauchy problem (2). ⋆ if S 0 i > 0, therefore Si (t) > 0, for all t ≥ 0, indeed by uniqueness of the solution of the Cauchy problem (2), Si (.) can't cross the axis Si = 0. We deduce that Si (t) ≥ 0, ∀t ≥ 0.

In the same way, we show that S s (t) ∈ R + , ∀t > 0.

Let us now prove that CO 2 (t) ∈ R + , ∀t ≥ 0, and CH 4 (t) ∈ R + , ∀t ≥ 0.

By integrating between 0 and t the following equations

Ċ O 2 (t) = (1 -f 1 )δX(t) + (1 -f 2 ) • 1 -Y Y µ(S s )B(t) + (1 -α)mB(t) (3)
ĊH 4 (t) = f 2 • 1 -Y Y µ(S s )B(t) (4)
and since X(t), B(t) and µ(S s ) are non negative for all t ≥ 0, our result is proved.

B ANNEX 2. PROOF OF THEOREM 1

To show the boundedness of the solution of the system (1) we introduce the following function:

F (t) = X(t) + αY (S i (t) + S s (t)) + αB(t), ∀t ≥ 0. ( 5
)
From the equations of the system (1) we obtain:

Ḟ (t) = (αf 1 Y -1)δX(t). (6)
Since αf 1 Y < 1, and X(.) is non negative, then F (.) is non-increasing, so that F (t) ≤ F (0), ∀t ≥ 0. And in addition, by the non-negativity of the variables X(.), S i (.), S s (.) and B(.), one has:

0 ≤ F (t) ≤ X 0 + αY (S 0 i + S 0 s ) + αB 0 , ∀t ≥ 0.
Therefore, F (.) is bounded, which implies by the non-negativity of the variables that X(t), S i (t), S s (t) and B(t) are bounded. By integrating the equation ( 6) between 0 and t, we obtain for all non-negative t,

F (t) = F (0) + (αf 1 Y -1)δ t 0 X(s)ds. (7)
As F (.) is bounded then

t 0 X(s)ds < +∞. (⋆)
Moreover, from the first equation of (1), we have

X(t) = X(0) -δ t 0 X(s)ds + αm t 0 B(s)ds. (8)
Therefore, with (⋆) we get

t 0 B(s)ds < +∞. (⋆⋆)
Then, from the fourth equation of ( 1), we obtain:

B(t) = B(0) + t 0 µ(S s (s))B(s)ds -m t 0 B(s)ds. (9)
As B(.) is bounded and with (⋆⋆), we deduce that

t 0 µ(S s (s))B(s)ds < +∞. (⋆ ⋆ ⋆)
In addition, the integration of the last two equations of the system (1) between 0 and t leads to the following formulas

CO 2 (t) = CO 2 (0) +(1-f 1 )δ t 0 X(s)ds+ (1-f 2 ) 1 -Y Y t 0 µ(S s (s))B(s)ds+ (1-α)m t 0 B(s)ds (10) CH 4 (t) = CH 4 (0) + f 2 1 -Y Y t 0 µ(S s (s))B(s)ds. (11)
Then we get the boundedness of the variables CO 2 (t) and CH 4 (t) taking into account (⋆), (⋆⋆) and (⋆ ⋆ ⋆).

Let us show the convergence of X(.), S i (.), S s (.) and B(.).

The function F : [0, +∞[→ [0, +∞[ is differentiable, non-increasing and bounded by 0, therefore, F (.) converges to a limit denoted by F lim when t tends to +∞. Moreover, one has:

F (t) = (αf 1 Y -1)δ(-δX(t) + αmB(t)). (12)
From (12) and since X(.) and B(.) are bounded, therefore, F is bounded and thus Ḟ is uniformly continuous on R + . We get by Barbalat's Lemma [START_REF] Barbalat | Systèmes d' équations diférentielles d'oscillations non linéaires[END_REF], lim t→+∞ Ḟ (t) = 0. Then from (6), we obtain lim t→+∞ X(t) = 0.

In the same way, as X(.), B(.) and µ(.) are bounded, then Ẍ is bounded and thus Ẋ is uniformly continuous on R + and by Barbalat's Lemma, one has

lim t→+∞ B(t) = 0.
Similarly, Si is bounded wich implies that Ṡi is uniformly continuous on ]0, +∞[, and by Barbalat's lemma, we get lim t→+∞ S i (t) = 0.

In addition, as F (t) converges to F lim , we deduce that

∃S ⋆ s ∈ R + , lim t→+∞ S s (t) = S ⋆ s = F lim αy .
Moreover, from (5) and as F (.) is non increasing, by tending t to ∞, we conclude that

S ⋆ s ≤ X 0 αY + (S 0 i + S 0 s ) + B 0 Y .
Now let us show that S ⋆ s ̸ = 0, when X 0 , S 0 i or S 0 s is non null. Assuming that S ⋆ s = 0, as µ(.) is continuous, one has lim t→+∞ µ(S s (t)) = 0. For η = (f 1i + f 1s )αmY > 0, one has ∃T > 0, ∀t > T : |µ(S s (t))| ≤ η therefore, from the equations of the system (1), we obtain:

(f 1i + f 1s ) Ẋ(t) + Ṡi (t) + Ṡs (t) ≥ 0, ∀t > T.
When (f 1i + f 1s )X(0) + S i (0) + S s (0) > 0, the variables X(.), S i (.) or S s (.) cannot reaches 0 in finite time. > 0 one gets

Then (f 1i + f 1s )X(t) + S i (t) + S s (t) ≥ (f 1i + f 1s )X(T ) + S i (T ) + S s (T ) > 0,
∃T > 0, ∀t > T : (µ(S s (t)) -m) > ϵ.
Consequently, and since B(.) is non negative we obtain

Ḃ(t) > ϵB(t), ∀t > T.
Since B(.) is bounded and by a standard comparison theorem [START_REF] Walter | Ordinary Differential Equations[END_REF], we deduce that lim t→+∞ B(t) = +∞ wich is in contradiction with the fact that B(.) converges asmptotically to 0. Therefore, we conclude that S s (.) converges to S ⋆ s with S ⋆ s ∈ E . Now let us fix e = (0, 0, S ⋆ s , 0) with S ⋆ s ∈ int E. We consider the variable R(.) defined by:

R(t) = (f 1i + f 1s )X(t) + S i (t) + (S s (t) -S ⋆ s ) B(t) + c 1 , ∀t ≥ 0 where c 1 = µ(S ⋆ s )/Y -α(f 1i + f 1s )m µ(S ⋆ s ) -m By a direct computation, we get Ṙ(t) = -r 1 (µ(S s ) -µ(S ⋆ s )) -(µ(S s ) -m)R(t) with r 1 = (1 -α(f 1i + f 1s )Y )m Y (m -µ(S ⋆ s )) > 0
We consider the domain:

K = {(X, S i , R, B) ∈ R 2 + × R × R ⋆ + /S ⋆ s -S i -(f 1i + f 1s )X + (R -c 1 )B ≥ 0}
and the application p(.) defined by:

p : R 4 + → K Z → p(Z) = (X, S i , (f1 i +f1 s )X+Si+(Ss-S ⋆ s ) B + c 1 , B) with Z = (X, S i , S s , B) ∈ R 4
+ . p is a homeomorphism from R 4 + to K. Then, the system (1) can be written equivalently on K, as follows:

       Ẋ = -δX + αmB Ṡi = f 1i δX -h(u)g(S i ) Ṙ = -r 1 (k(X, S i , R, B) -µ(S ⋆ s )) -(k(X, S i , R, B) -m)R Ḃ = (k(X, S i , R, B) -m)B (13) with k(X, S i , R, B) = µ(S ⋆ s -(f 1i + f 1s )X + (R -c 1 )B)
The systems (1) and ( 13) are locally topologically equivalent. However, trajectories with B(t) = 0, ∀t ≥ 0 are not considered by the homeomorphism p, therefore, we consider the domain K 0 defined by:

K 0 = K ∪ I × R × {0} with I = {(X, S i ) ∈ R 2 + /S ⋆ s -S i -(f 1i + f 1s )X ≥ 0}
Since S ⋆ s ∈ int E, the origin (0, 0, 0, 0) is the unique equilibrium point of (13) in K 0 . The linearization of (13) near the origin is based on the following jacobian matrix:

J(0, 0, 0, 0) =     -δ 0 0 δm f 1i δ -h(u)g ′ (0) 0 0 r 1 (f 1i + f 1s )µ ′ (S ⋆ s ) r 1 µ ′ (S ⋆ s ) m -µ(S ⋆ s ) r 1 c 1 µ ′ (S ⋆ s ) 0 0 0 µ(S ⋆ s ) -m    
Its eigenvalues are λ 1 = -δ < 0, λ 2 = -h(u)g ′ (0) < 0, λ 3 = -(µ(S ⋆ s ) -m) > 0 and λ 4 = (µ(S ⋆ s ) -m) < 0. Since the origin is an hyperbolic equilibrium of (13) in K 0 , by the Stable Manifold Theorem [START_REF] Perko | Differential Equations and Dynamical Systems[END_REF], there exists a stable manifold W s (0) and an instable manifold W u (0), such that: dim(W s (0)) = 3 dim(W u (0)) = 1

It's clear that W u (0) = {0} × {0} × R × {0}. Now let us show that any point (X, S i , R, B) ∈ W s (0) \ (0, 0, 0, 0) has to be such that B ̸ = 0. If B 0 = 0, any solution of (13) verifies B(t) = 0, ∀t > 0, since lim t→+∞ X(t) = 0 and lim t→+∞ S i (t) = 0, then, as µ(.) is C 1 on R + , one has lim t→+∞ k(X(t), S i (t), R(t), B(t)) = lim t→+∞ µ(S ⋆ s -S 1 (t) -(f 1 + f 2 )X(t)) = µ(S ⋆ s ),

For η = m-µ(S ⋆ s ) 2 > 0, one has

∃T > 0 : ∀t > T : m -µ(S ⋆ s -S i (t) -(f 1i + f 1s )X(t)) > η
From the equation of the system (13), one has Ṙ(t) > ηR(t) -r 1 (µ(S ⋆ s -S i (t) -(f 1i + f 1s )X(t)) -µ(S ⋆ s )) , ∀t > T.

Then lim t-→+∞ R(t) = +∞. K is positively invariant by the dynamics (13), therefore, any trajectory (X(t), S i (t), R(t), B(t)) in S = W s (0) ∩ K is homeomorphic to a trajectory (X(t), S i (t), S s (t), B(t)) of (1) and converges asymptotically to (0, 0, S ⋆ s , 0). Finally and with the results of proposition 2 and proposition 3, we conclude that the set A := {0} × {0} × E × {0} is a forward attractor of (1).

Figure 2 :

 2 Figure 2: Time evolution of the state variables (left) and total biogas production (right).

Figure 3 :

 3 Figure 3: Time evolution of the soluble substrate for different values of X(0).

Figure 4 :

 4 Figure 4: Insoluble substrate (left) and biogas production (right) with and without recirculation.

Figure 5 :

 5 Figure 5: Total production of biogas as a function of the recirculation (left) and of the initial load of organic matter X(0) (right).

  The parameters of the model are :α: the proportion of nutrient recycling, u: the leachate recirculation rate (that can be a manipulating variable), δ: the degradation rate of the organic matter, m: the mortality rate of the biomass, Y : the conversion rate of the substrate, f 1 i , f 1s , f 2 : the stochiometric coefficients. The function µ(•) is C 1 on R + , non-negative and bounded with µ(0) = 0. H 2 . The functions g(•) and h(•) are C 1 on R + , increasing with bounded derivative and equal to 0 at zero. H 3 . The parameters α, u, m, ,Y , f 1 i , f 1s , f 2 satisfy the following properties:

	2.2 Hypotheses
	H 1 .

Table 1 :

 1 Parameters valuesFor these values, the set E of attracting values for S s is computed numerically as E ≃ [0, 14.09] ∪ [125.90, +∞).

Table 2 :

 2 Operating conditions

  ∀t > T wich is in contradiction with the fact that lim t→+∞ (f 1i + f 1s )X(t) + S i (t) + S s (t) = 0.

	C ANNEX 2. PROOFS OF PROPOSITION 2 AND PROPOSITION 3
	If S ⋆ s / ∈ E, one has lim t→+∞ µ(S s (t)) = µ(S ⋆ s ), therefore by choosing ϵ =	µ(S ⋆ s )-m 2

D ANNEX 3. PROOF OF PROPOSITION 4

From the formulas (8) and [START_REF] Barbalat | Systèmes d' équations diférentielles d'oscillations non linéaires[END_REF] and by tending t to ∞, one has:

Moreover, by integrating the second and third equation of the system (1) between t = 0 and t = +∞ we obtain

From a simple calculus, we get the following expressions

where

C . Therefore, from the formulas ( 16), ( 17) and (18) we deduce the asymptotic value of the total biogas G(t) provided in Proposition 4.