Kévin Le Balc'h

GLOBAL NULL-CONTROLLABILITY AND NONNEGATIVE-CONTROLLABILITY OF SLIGHTLY SUPERLINEAR HEAT EQUATIONS

Global null-controllability and nonnegative-controllability of slightly superlinear heat equations

Kévin Le Balc'h

In [START_REF] Farid Ammar Khodja | Null-controllability of some reaction-diffusion systems with one control force[END_REF], y = y(t, .) : Ω → R is the state to be controlled and h = h(t, .) : Ω → R is the control input supported in ω, a nonempty open subset of Ω.

We assume that f satisfies (2) f (0) = 0.

In this case, y = 0 solves (1) with y 0 = 0 and h = 0.

In the following, we will also assume that f satisfies the restrictive growth condition

(3) ∃α > 0, f (s) |s| log α (1 + |s|) → 0 as |s| → +∞.

Under the hypothesis (3), blow-up may occur if h = 0 in [START_REF] Farid Ammar Khodja | Null-controllability of some reaction-diffusion systems with one control force[END_REF]. Take for example f (s) = -|s| log α (1 + |s|) with α > 1. The mathematical theory of blow-up for

   ∂ t y -∆y = |y| log α (1 + |y|) in (0, T ) × Ω, y = 0 on (0, T ) × ∂Ω, y(0, .) = y 0 in Ω, was established in [START_REF] Victor | Regional blow up in a semilinear heat equation with convergence to a Hamilton-Jacobi equation[END_REF] and [START_REF] Victor | Blow-up for quasilinear heat equations described by means of nonlinear Hamilton-Jacobi equations[END_REF]. It was shown that blow-up • occurs globally in the whole domain

Ω if α < 2,
• is of pointwise nature if α > 2,

• is 'regional', i.e., it occurs in an open subset of Ω if α = 2. See [26, Section 2 and Section 5] for a survey on this problem.

The goal of this paper is to analyze the null-controllability properties of [START_REF] Farid Ammar Khodja | Null-controllability of some reaction-diffusion systems with one control force[END_REF]. Let us define Q T := (0, T ) × Ω. We recall two classical definitions of nullcontrollability.

Definition 1.2. Let T > 0. The system (1) is

• globally null-controllable in time T if for every y 0 ∈ L ∞ (Ω), there exists h ∈ L ∞ (Q T ) such that the solution y of (1) satisfies y(T, .) = 0. • locally null-controllable in time T if there exists δ T > 0 such that for every y 0 ∈ L ∞ (Ω) verifying y 0 L ∞ (Ω) ≤ δ T , there exists h ∈ L ∞ (Q T ) such that the solution y of (1) satisfies y(T, .) = 0.

We have the following well-known local null-controllability result.

Theorem 1.3. For every T > 0, (1) is locally null-controllable in time T .

The proof of Theorem 1.3 is a consequence of the (global) null-controllability of the linear heat equation with a bounded potential (due to Andrei Fursikov and Oleg Imanuvilov, see [START_REF] Fursikov | Controllability of evolution equations[END_REF] or [START_REF] Fernández | Global Carleman inequalities for parabolic systems and applications to controllability[END_REF]Theorem 1.5]) and the small L ∞ perturbations method (see [START_REF] Anita | Null controllability for the dissipative semilinear heat equation[END_REF]Lemma 6] and [START_REF] Farid Ammar Khodja | Null-controllability of some reaction-diffusion systems with one control force[END_REF], [START_REF] Barbu | Local controllability of the phase field system[END_REF], [START_REF] Le Balc'h | Controllability of a 4 x 4 quadratic reaction-diffusion system[END_REF], [START_REF] Límaco | Null controllability of some reaction-diffusion systems with only one control force in moving domains[END_REF], [START_REF] Wang | Exact local controllability of a one-control reaction-diffusion system[END_REF] for other results in this direction).

The following global null-controllability (positive) result has been proved independently by Enrique Fernandez-Cara, Enrique Zuazua (see [START_REF] Fernández | Null and approximate controllability for weakly blowing up semilinear heat equations[END_REF]Theorem 1.2]) and Viorel Barbu under a sign condition (see [START_REF] Barbu | Exact controllability of the superlinear heat equation[END_REF]Theorem 2] or [START_REF] Barbu | Controllability and stabilization of parabolic equations[END_REF]Theorem 3.6]) for Dirichlet boundary conditions. It has been extended to semilinearities which can depend on the gradient of the state and to Robin boundary conditions (then to Neumann boundary conditions) by Enrique Fernandez-Cara, Manuel Gonzalez-Burgos, Sergio Guerrero and Jean-Pierre Puel in [START_REF] Fernández-Cara | Exact controllability to the trajectories of the heat equation with Fourier boundary conditions: the semilinear case[END_REF] (see also [START_REF] Doubova | On the controllability of parabolic systems with a nonlinear term involving the state and the gradient[END_REF] for the Dirichlet case).

Theorem 1.4. [19, Theorem 1] We assume that (3) holds for α ≤ 3/2. Then, for every T > 0, (1) is globally null-controllable in time T . Remark 1.5. Historically, the first global null-controllability (positive) result for [START_REF] Farid Ammar Khodja | Null-controllability of some reaction-diffusion systems with one control force[END_REF] with f satisfying (3) was proved by Enrique Fernandez-Cara in [START_REF] Fernández-Cara | Null controllability of the semilinear heat equation[END_REF] for α ≤ 1 and for Dirichlet boundary conditions.

The following global null-controllability (negative) result has been proved by Enrique Fernandez-Cara, Enrique Zuazua (see [START_REF] Fernández | Null and approximate controllability for weakly blowing up semilinear heat equations[END_REF]).

Theorem 1.6. [START_REF] Fernández | Null and approximate controllability for weakly blowing up semilinear heat equations[END_REF]Theorem 1.1] We set f (s) := |s| 0 log p (1 + σ)dσ with p > 2 and we assume that Ω \ ω = ∅. Then, for every T > 0, there exists an initial datum y 0 ∈ L ∞ (Ω) such that for every h ∈ L ∞ (Q T ), the maximal solution y of (1) blows-up in time T * < T . Remark 1.7. Such a function f does satisfy (3) for any α > p because |f (s)| ∼ |s| log p (1 + |s|) as |s| → +∞. Then, Theorem 1. [START_REF] Barbu | Controllability and stabilization of parabolic equations[END_REF] shows that (1) can fail to be nullcontrollable for every T > 0 under the hypothesis (3) with α > 2. Theorem 1.6 comes from a localized estimate in Ω \ ω that shows that the control cannot compensate the blow-up phenomena occurring in Ω \ ω (see [START_REF] Fernández | Null and approximate controllability for weakly blowing up semilinear heat equations[END_REF]Section 2]).

When the nonlinear term f is dissipative, i.e., sf (s) ≥ 0 for every s ∈ R, then blow-up cannot occur. Furthermore, such a nonlinearity produces energy decay for the uncontrolled equation, therefore naively one may be led to believe that it can help in steering the solution to zero in arbitrary short time. The results of Sebastian Anita and Daniel Tataru show that this is false, more precisely that for 'strongly' superlinear f one needs a sufficiently large time in order to bring the solution to zero. An intuitive explanation for this is that the nonlinearity is also damping the effect of the control as it expands from the controlled region into the uncontrolled region (see [START_REF] Anita | Null controllability for the dissipative semilinear heat equation[END_REF]).

Theorem 1.8. [3, Theorem 3]

We set f (s) := s log p (1 + |s|) with p > 2 and we assume that Ω \ ω = ∅. Then, there exist x 0 ∈ Ω \ ω, T 0 ∈ (0, 1) such that for every T ∈ (0, T 0 ), h ∈ L ∞ (Q T ), there exists y 0 ∈ L ∞ (Ω) such that the solution y to (1) satisfies y(T, x 0 ) < 0. Remark 1.9. In particular, for such a f as in Theorem 1.8, (1) is not globally nullcontrollable in small time T . Theorem 1.8 is due to pointwise upper bounds on the solution y of (1) which are independent of the control h (see [START_REF] Anita | Null controllability for the dissipative semilinear heat equation[END_REF]Section 3]).

Main results

2.1.

Small-time global nonnegative-controllability. We introduce a new concept of controllability. Definition 2.1. Let T > 0. The system (1) is globally nonnegative-controllable (respectively globally nonpositive-controllable) in time T if for every y 0 ∈ L ∞ (Ω), there exists h ∈ L ∞ (Q T ) such that the solution y of (1) satisfies [START_REF] Barbu | Local controllability of the phase field system[END_REF] y(T, .) ≥ 0 (respectively y(T, .) ≤ 0).

The first main result of this paper is a small-time global nonnegative-controllability result for (1). Theorem 2.2. We assume that (3) holds for α ≤ 2 and f (s) ≥ 0 for s ≥ 0 (respectively f (s) ≤ 0 for s ≤ 0). Then, for every T > 0, [START_REF] Farid Ammar Khodja | Null-controllability of some reaction-diffusion systems with one control force[END_REF] Theorem 2.5. We assume that (3) holds for α ≤ 2, f (s) > 0 for s > 0 or f (s) < 0 for s < 0 and 1/f ∈ L 1 ([1, +∞)). Then, there exists T sufficiently large such that (1) is globally null-controllable in time T . Remark 2.6. Theorem 2.5 proves that Theorem 1.6 is almost sharp. Indeed, let us take f (s) = |s| 0 log p (1 + σ)dσ with p < 2, then by Theorem 2.5, there exists T sufficiently large such that (1) is globally null-controllable in time T . In particular, one can find a localized control which prevents the blow-up from happening. The case f (s) = Remark 2.8. The small-time global null-controllability of ( 1) remains open when (3) holds for 3/2 < α ≤ 2.

2.3.

Proof strategy of the small-time global nonnegative-controllability. We will only prove the global nonnegative-controllability result. The nonpositivecontrollability result is an easy adaptation.

The proof strategy of Theorem 2.2 will follow Enrique Fernandez-Cara and Enrique Zuazua's proof of Theorem 1.4 (see [START_REF] Fernández | Null and approximate controllability for weakly blowing up semilinear heat equations[END_REF]).

The starting point is to get some precise observability estimates for the linear heat equation with a bounded potential a(t, x) for nonnegative initial data. More precisely, we show that observability holds with a sharp constant of the order exp C a 1/2 ∞ for nonnegative initial data (see Theorem 4.4 below). This is done thanks to a new Carleman estimate in L 1 (see Theorem 4.9 below). This leads to a nonnegativecontrollability result in L ∞ in the linear case with an estimate of the control cost of the order exp C a 1/2 ∞ which is the key point of the proof (see Theorem 4.1 below).

We end the proof of Theorem 2.2 by a Kakutani-Leray-Schauder's fixed-point strategy. The idea of taking short control times to avoid blow-up phenomena is the same as in [START_REF] Fernández | Null and approximate controllability for weakly blowing up semilinear heat equations[END_REF] and references therein. More precisely, the construction of the control follows two steps. The first step consists in steering the solution y of (1) to y(T * , .) ≥ 0 in time T * ≤ T with an appropriate choice of the control. Then, the two conditions: f (0) = 0 and the dissipativity of f in R + imply that the free solution y of (1) (with h = 0) defined in (T * , T ) stays nonnegative and bounded by using a comparison principle (see Section 5).

2.4.

Proof strategy of the large time global null-controllability. We will only treat the case where f (s) > 0 for s > 0. The other case, i.e., f (s) < 0 for s < 0 is an easy adaptation.

The proof strategy of Theorem 2.5 is divided into three steps. First, for every initial data y 0 ∈ L ∞ (Ω), one can steer the solution y of (1) in time T 1 := 1 (for instance) to a nonnegative state by using Theorem 2.2.

Secondly, we let evolve the system without control and we remark that

∀(t, x) ∈ [T 1 , +∞) × Ω, 0 ≤ y(t, x) ≤ G(t),
with G independent of y(T 1 , .) L ∞ (Ω) and G(t) → 0 when t → +∞. This kind of argument has already been used by Jean-Michel Coron in the context of the Burgers equation (see [START_REF] Coron | Some open problems on the control of nonlinear partial differential equations[END_REF]Theorem 8]). Finally, by using the second step, for T 2 sufficiently large, y(T 2 , .) belongs to a small ball of L ∞ (Ω) centered at 0, where the local null-controllability holds (see Theorem 1.3). Then, one can steer y(T 2 , .) to 0 with an appropriate choice of the control.

Parabolic equations: Well-posedness and regularity

The goal of this section is to state well-posedness results, dissipativity in time in L p -norm, maximum principle and L p -L q estimates for linear parabolic equations. We also give the definition of a solution to the semilinear heat equation [START_REF] Farid Ammar Khodja | Null-controllability of some reaction-diffusion systems with one control force[END_REF]. The references of these results only treat the case of Dirichlet boundary conditions but the proofs can be easily adapted to Neumann boundary conditions.

3.1. Well-posedness. We introduce the functional space (6)

W T := L 2 (0, T ; H 1 (Ω)) ∩ H 1 (0, T ; (H 1 (Ω)) ′ ),
which satisfies the following embedding (see [15, Section 5.9.2, Theorem 3])

W T ֒→ C([0, T ]; L 2 (Ω)).

3.1.1. Linear parabolic equations.

Definition 3.1. Let a ∈ L ∞ (Q T ), F ∈ L 2 (Q T ) and y 0 ∈ L 2 (Ω). A function y ∈ W T is a solution to (8)    ∂ t y -∆y + a(t, x)y = F in (0, T ) × Ω, ∂y ∂n = 0 on (0, T ) × ∂Ω, y(0, .) = y 0 in Ω, if for every w ∈ L 2 (0, T ; H 1 (Ω)), (9)
T 0

(∂ t y, w) ((H 1 (Ω)) ′ ,H 1 (Ω)) + Q T ∇y.∇w + Q T ayw = Q T F w, and (10)
y(0, .) = y 0 in L 2 (Ω).

The following well-posedness result in L 2 holds for linear parabolic equations.

Proposition 3.2. Let a ∈ L ∞ (Q T ), F ∈ L 2 (Q T ) and y 0 ∈ L 2 (Ω).
The Cauchy problem (8) admits a unique weak solution y ∈ W T . Moreover, there exists C = C(Ω) > 0 such that

(11) y W T ≤ C exp CT a L ∞ (Q T ) y 0 L 2 (Ω) + F L 2 (Q T ) .
The proof of Proposition 3.2 is based on Galerkin approximations, energy estimates and Gronwall's argument (see [START_REF] Evans | Partial differential equations[END_REF]Section 7.1.2]).

We also have the following classical L ∞ -estimate for [START_REF] Cazenave | An introduction to semilinear evolution equations[END_REF].

Proposition 3.3. Let a ∈ L ∞ (Q T ), F ∈ L ∞ (Q T ) and y 0 ∈ L ∞ (Ω).
Then the solution y of (8) belongs to L ∞ (Q T ) and there exists C = C(Ω) > 0 such that

(12) y L ∞ (Q T ) ≤ C exp CT a L ∞ (Q T ) y 0 L ∞ (Ω) + F L ∞ (Q T ) .
The proof of Proposition 3.3 is based on Stampacchia's method (see the proof of [28, Chapter 3, Paragraph 7, Theorem 7.1]).

Let us also mention the dissipativity in time of the L p -norm of the heat equation with a bounded potential.

Proposition 3.4. Let a ∈ L ∞ (Q T ), y 0 ∈ L 2 (Ω) and t 1 < t 2 ∈ [0, T ].
Then, there exists C = C(Ω) > 0 such that the solution y ∈ W T of (8) with F = 0, satisfies for every p ∈ [1, 2], [START_REF] Doubova | On the controllability of parabolic systems with a nonlinear term involving the state and the gradient[END_REF] y(t 2 , .)

L p (Ω) ≤ C exp CT a L ∞ (Q T ) y(t 1 , .) L p (Ω) .
The proof of Proposition 3.4 is based on the application of the variational formulation ( 9) with a cut-off of w = |y| p-2 y and a Gronwall's argument.

3.1.2. Nonlinear parabolic equations. We give the definition of a solution of (1).

Definition 3.5. Let y 0 ∈ L ∞ (Ω), h ∈ L ∞ (Q T ). A function y ∈ W T ∩ L ∞ (Q T ) is the solution of (1) if for every w ∈ L 2 (0, T ; H 1 (Ω)), T 0 (∂ t y, w) ((H 1 (Ω)) ′ ,H 1 (Ω)) + Q T ∇y.∇w + Q T ayw = Q T (f (y) + h1 ω )w, (14) and (15)
y(0, .) = y 0 in L ∞ (Ω).

The uniqueness of a solution to (1) is an easy consequence of the fact that f is locally Lipschitz because f ∈ C 1 (R; R).

Maximum principle.

We state the maximum principle for the heat equation.

Proposition 3.6. Let a ∈ L ∞ (Q T ), F ≤ G ∈ L 2 (Q T ) and y 0 ≤ z 0 ∈ L 2 (Ω).
Let y and z be the solutions to

(16)    ∂ t y -∆y + a(t, x)y = F, ∂y ∂n = 0, y(0, .) = y 0 ,    ∂ t z -∆z + a(t, x)z = G in (0, T ) × Ω, ∂z ∂n = 0 on (0, T ) × ∂Ω, z(0, .) = z 0 in Ω.
Then, we have the comparison principle

(17) ∀t ∈ [0, T ], a.e. x ∈ Ω, y(t, x) ≤ z(t, x).
The proof of Proposition 3.6 is based on the comparison principle for smooth solutions of [START_REF] Fellner | Global classical solutions to quadratic systems with mass conservation in arbitrary dimensions[END_REF] (see [START_REF] Wu | Elliptic & parabolic equations[END_REF]Theorem 8.1.6]) and a regularization argument.

We state a comparison principle for the semilinear heat equation (1) without control h. Proposition 3.7. Let y 0 ∈ L ∞ (Ω), h = 0. We assume that there exist a subsolution y and a supersolution y in L ∞ (Q T ) of (1), i.e., y (respectively y) satisfies (14), [START_REF] Evans | Partial differential equations[END_REF] replacing the equality = by the inequality ≤ (respectively by the inequality ≥). Moreover, we suppose that y and y are ordered in the following sense ∀t ∈ [0, T ], a.e. x ∈ Ω, y(t, x) ≤ y(t, x).

Then, there exists a (unique) solution y of (1). Moreover, y satisfies the comparison principle [START_REF] Fernández-Cara | Controllability of linear and semilinear non-diagonalizable parabolic systems[END_REF] ∀t ∈ [0, T ], a.e. x ∈ Ω, y(t, x) ≤ y(t, x) ≤ y(t, x).

For the proof of Proposition 3.7, see [START_REF] Wu | Elliptic & parabolic equations[END_REF]Corollary 12.1.1].

3.3. L p -L q estimates. We have the well-known regularizing effect of the heat semigroup.

Proposition 3.8. [8, Proposition 3.5.7] Let 1 ≤ q ≤ p ≤ +∞, y 0 ∈ L 2 (Ω) and y be the solution to [START_REF] Cazenave | An introduction to semilinear evolution equations[END_REF] with (a, F ) = (0, 0). Then, there exists C = C(Ω, p, q) > 0 such that for every t 1 < t 2 ∈ (0, T ), we have

(19) y(t 2 , .) L p (Ω) ≤ C(t 2 -t 1 ) -N 2 1 q -1 p y(t 1 , .) L q (Ω)
4. Global nonnegative-controllability of the linear heat equation with a bounded potential

4.1. Statement of the result. Let a ∈ L ∞ (Q T ).
We consider the heat equation with a bounded potential ( 20)

   ∂ t y -∆y + a(t, x)y = h1 ω in (0, T ) × Ω, ∂y ∂n = 0 on (0, T ) × ∂Ω, y(0, .) = y 0 in Ω,
and the following adjoint equation ( 21)

   -∂ t q -∆q + a(t, x)q = 0 in (0, T ) × Ω, ∂q ∂n = 0 on (0, T ) × ∂Ω, q(T, .) = q T in Ω.
The goal of this section is to prove the following theorem.

Theorem 4.1. For every T > 0, (20) is globally nonnegative-controllable in time T . More precisely, for every T > 0, there exists

C = C(Ω, ω, T, a) > 0, with C(Ω, ω, T, a) = exp C(Ω, ω) 1 + 1 T + T a L ∞ (Q T ) + a 1/2 L ∞ (Q T ) (22)
such that for every

y 0 ∈ L 2 (Ω), there exists h ∈ L ∞ (Q T ) such that (23) h L ∞ (Q T ) ≤ C(Ω, ω, T, a) y 0 L 2 (Ω) , and (24)
y(T, .) ≥ 0.

Remark 4.2. Actually, by looking carefully at the proof of Theorem 4.1 (see Section 4.5 below), we can see that the control h in Theorem 4.1 can be chosen constant in the time and the space variables.

Remark 4.3. It is well-known that ( 20) is globally nonnegative-controllable in time T because it is globally null-controllable in time T (see [START_REF] Fernández-Cara | Null controllability of the heat equation with boundary Fourier conditions: the linear case[END_REF]Theorem 2]) but the most interesting point is the cost of nonnegative-controllability given in Theorem 4.1. In particular, the exponent 1/2 of the term a

1/2 L ∞ (Q T )
will be the key point to prove Theorem 2.2 (see Section 5).

4.2.

A precise L 2 -L 1 observability inequality for the linear heat equation with bounded potential and nonnegative initial data. The proof of Theorem 4.1 is a consequence of this kind of observability inequality. Theorem 4.4. For every T > 0, there exists C = C(Ω, ω, T, a) > 0 of the form [START_REF] Fernández | Null and approximate controllability for weakly blowing up semilinear heat equations[END_REF] such that for every q T ∈ L 2 (Ω; R + ), the solution q to (21) satisfies [START_REF] Victor | Blow-up for quasilinear heat equations described by means of nonlinear Hamilton-Jacobi equations[END_REF] q(0, .

) 2 L 2 (Ω) ≤ C T 0 ω qdxdt 2 .
An immediate corollary of Theorem 4.4 is this observability inequality L 2 -L 2 that we state to discuss it below, but that will not be used in the present article.

Corollary 4.5. For every T > 0, there exists C = C(Ω, ω, T, a) > 0 of the form [START_REF] Fernández | Null and approximate controllability for weakly blowing up semilinear heat equations[END_REF] such that for every q T ∈ L 2 (Ω; R + ) the solution q to (21) satisfies [START_REF] Victor | The problem of blow-up in nonlinear parabolic equations[END_REF] q(0, .

) 2 L 2 (Ω) ≤ C T 0 ω q 2 dxdt .
It is well-known that null-controllability in L 2 is equivalent to an observability inequality in L 2 for every q T ∈ L 2 (Ω; R) (see [START_REF] Coron | of Mathematical Surveys and Monographs[END_REF]Theorem 2.44]). The main idea behind Corollary 4.5 is the fact that nonnegative-controllability in L 2 is a consequence of an observability inequality in L 2 for every q T ∈ L 2 (Ω; R + ) (see Section 4.5).

Remark 4.6. It is interesting to mention that [START_REF] Victor | The problem of blow-up in nonlinear parabolic equations[END_REF] holds with C of the form

C(Ω, ω, T, a) = exp C(Ω, ω) 1 + 1 T + T a L ∞ (Q T ) + a 2/3 L ∞ (Q T ) (27)
for every q T ∈ L 2 (Ω; R) (see [START_REF] Fernández-Cara | Null controllability of the heat equation with boundary Fourier conditions: the linear case[END_REF]Theorem 2]). The exponent 2/3 of the term

a 2/3 L ∞ (Q T )
is the key point to prove Theorem 1.4. Note that the optimality of the exponent 2/3 has been proved by Thomas Duyckaerts, Xu Zhang and Enrique Zuazua in the context of parabolic systems in even space dimensions N ≥ 2 and with Dirichlet boundary conditions (see [START_REF] Duyckaerts | On the optimality of the observability inequalities for parabolic and hyperbolic systems with potentials[END_REF]Theorem 1.1] and also [START_REF] Zuazua | Controllability and observability of partial differential equations: some results and open problems[END_REF]Theorem 5.2] for the main arguments of the proof). Corollary 4.5 shows that we can actually decrease the exponent 2/3 to the exponent 1/2 for nonnegative initial data. In some sense, we can make the connection between the recent preprint of Camille Laurent and Matthieu Léautaud who disprove the Miller's conjecture about the short-time observability constant of the heat equation in the general case and show that the conjecture holds true for nonnegative initial data by using Li-Yau estimates (see [START_REF] Laurent | Observability of the heat equation, geometric constants in control theory, and a conjecture of Luc Miller[END_REF] and [START_REF] Li | On the parabolic kernel of the Schrödinger operator[END_REF]).

Remark 4.7. In the context of the wave equation in one space dimension, the (optimal) constant of observability inequality for the linear wave equation with a bounded potential is actually exp

C 1 + a 1/2 L ∞ (Q T )
(see [START_REF] Zuazua | Exact controllability for semilinear wave equations in one space dimension[END_REF]Theorem 4]) which leads to the exact controllability of the semilinear wave equation in large time for semilinearities satisfying (3) with α < 2 (see [START_REF] Zuazua | Exact controllability for semilinear wave equations in one space dimension[END_REF]Theorem 1] and also [START_REF] Vincent | Unsolved problems in mathematical systems and control theory[END_REF]Problem 5.5] for the presentation of the related open problem in the multidimensional case). Roughly speaking, as an ordinary differential argument would indicate, this constant of observability inequality is very natural because the wave operator is of order two in the time and the space variables. Then, by analogy and by taking into account that the heat operator is of order one in the time variable and of order two in the space variable, one could rather expect a constant of obervability inequality of the order

exp C a L ∞ (Q T ) or exp C a 1/2 L ∞ (Q T ) which seem to be more intuitive than the term exp C a 2/3 L ∞ (Q T ) . 4.3. A new L 1 Carleman estimate.
The goal of this section is to establish a L 1 Carleman estimate for nonnegative initial data (see Theorem 4.9 below). First, we introduce some classical weight functions for proving Carleman inequalities.

∈ C 2 (Ω) such that η 0 > 0 in Ω, η 0 = 0 in ∂Ω, and |∇η 0 | > 0 in Ω \ ω 0 .
A proof of this lemma can be found in [START_REF] Coron | of Mathematical Surveys and Monographs[END_REF]Lemma 2.68]. Let ω 0 be a nonempty open set satisfying ω 0 ⊂⊂ ω and let us set [START_REF] Aleksandrovna | Linear and quasilinear equations of parabolic type[END_REF] α(t, x)

:= e 2λ η 0 ∞ -e λη 0 (x) t(T -t) , (29)
ξ(t, x) := e λη 0 (x) t(T -t) ,
for (t, x) ∈ Q T , where η 0 is the function provided by Lemma 4.8 for this ω 0 and λ ≥ 1 is a parameter. We have the following new L 1 Carleman estimate.

Theorem 4.9. There exist two constants C := C(Ω, ω) > 0 and

C 1 := C 1 (Ω, ω) > 0, such that, ( 30
) ∀λ ≥ 1, ∀s ≥ s 1 (λ) := C(Ω, ω)e 4λ η 0 ∞ T + T 2 + T 2 a 1/2 L ∞ (Q T )
, for every q T ∈ L 2 (Ω; R + ), the nonnegative solution q of (21) satisfies

Q T e -sα ξ 2 qdxdt ≤ C 1 (0,T )×ω e -sα ξ 2 qdxdt. ( 31
)
Proof. Unless otherwise specified, we denote by C various positive constants varying from line to line which may depend on Ω, ω but independent of the parameters λ and s.

We introduce other weights which are similar to α and ξ

(32) α(t, x) := e 2λ η 0 ∞ -e -λη 0 (x) t(T -t) , (33)
ξ(t, x) := e -λη 0 (x) t(T -t) .
The following estimates (34)

|∂ i α| = | -∂ i ξ| ≤ Cλξ, |∂ i α| = | -∂ i ξ| ≤ Cλ ξ, |∂ t α| ≤ 2T ξ 2 e 2λ η 0 ∞ , |∂ t α| ≤ 2T ξ 2 e 4λ η 0 ∞ , ξ(T /2) 2 ≥ 1, ξ(T /2) 2
≥ e -λ η 0 ∞ , will be very useful for the proof.

Let q T ∈ C ∞ 0 (Ω; R + ). The general case comes from an easy density argument by using the fact that C ∞ c (Ω; R + ) is dense in L 2 (Ω; R + ) for the L 2 (Ω; R) topology. The solution q of (21) is nonnegative by applying the maximum principle given in Proposition 3.6 with y = 0 and z(t, x) = q(t -T, x).

We define ψ := e -sα q and ψ := e -s α q. The proof is divided into five steps:

• Step 1: We integrate over (0, T ) × Ω an identity satisfied by ψ. [START_REF] Fursikov | Controllability of evolution equations[END_REF]Chapter 1] and also [START_REF] Fernández-Cara | Null controllability of the heat equation with boundary Fourier conditions: the linear case[END_REF]Appendix]).

Step 1: An identity satisfied by ψ. We readily obtain that

(35) M ψ = 0,
where

M ψ = -sλ 2 |∇η 0 | 2 ξψ -2sλξ∇η 0 .∇ψ + ∂ t ψ (36) + s 2 λ 2 |∇η 0 | 2 ξ 2 ψ + ∆ψ + sα t ψ -a(t, x)ψ -sλ∆η 0 ξψ.
Remark 4.11. The starting point, i.e., the identity [START_REF] Pao | Nonlinear parabolic and elliptic equations[END_REF] is the same as in the classical proof developed by Andrei Fursikov and Oleg Imanuvilov in [START_REF] Fursikov | Controllability of evolution equations[END_REF] (see also [21, Proof of Lemma 1.3] or [START_REF] Rousseau | On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations[END_REF]Section 7]). But, from now, the proof strategy of the L 1 -Carleman estimate is very different from the usual one of the L 2 -Carleman estimate. Indeed, we will focus on the fourth right hand side term of (36)

s 2 λ 2 |∇η 0 | 2 ξ 2 ψ.
It is nonnegative because ψ is nonnegative and it is of order two in the parameter s whereas the seventh right hand side term of ( 36)

a(t, x)ψ,
is of order 0 in the parameter s. This comparison suggests to integrate the identity [START_REF] Pao | Nonlinear parabolic and elliptic equations[END_REF] in order to obtain [START_REF] Rousseau | On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations[END_REF] for λ ≥ 1 and s ≥ s 1 (λ) as defined in [START_REF] Le Balc'h | Controllability of a 4 x 4 quadratic reaction-diffusion system[END_REF].

We integrate [START_REF] Pao | Nonlinear parabolic and elliptic equations[END_REF]

over (0, T ) × Ω (37) Q T s 2 λ 2 |∇η 0 | 2 ξ 2 ψ - Q T 2sλξ∇η 0 .∇ψ + Q T ∂ t ψ + Q T ∆ψ = Q T sλ 2 |∇η 0 | 2 ξψ - Q T sα t ψ + Q T a(t, x)ψ + Q T sλ∆η 0 ξψ.
Note that all the terms in (37) are well-defined. Indeed, by using q T ∈ C ∞ c (Ω) and the parabolic regularity in L 2 to (21

) (see [12, Theorem 2.1]), we deduce that q ∈ X 2 := L 2 (0, T ; H 2 (Ω)) ∩ H 1 (0, T ; L 2 (Ω)) then ψ ∈ X 2 .
Step 2: Estimates for ψ. As a consequence of the properties of η 0 (see Lemma 4.8), we have [START_REF] Simon | Compact sets in the space L p (0, T ; B)[END_REF] m := min |∇η 0 (x)| 2 ; x ∈ Ω \ ω 0 > 0, which yields

Q T s 2 λ 2 |∇η 0 | 2 ξ 2 ψ (39) ≥ (0,T )×(Ω\ω) s 2 λ 2 |∇η 0 | 2 ξ 2 ψ ≥ m Q T s 2 λ 2 ξ 2 ψ -m (0,T )×ω s 2 λ 2 ξ 2 ψ.
By combining [START_REF] Protter | Maximum principles in differential equations[END_REF] and [START_REF] Souplet | Global existence for reaction-diffusion systems with dissipation of mass and quadratic growth[END_REF], we have ( 40)

m Q T s 2 λ 2 ξ 2 ψ - Q T 2sλξ∇η 0 .∇ψ + Q T ∂ t ψ + Q T ∆ψ ≤ Q T sλ 2 |∇η 0 | 2 ξψ + Q T s|α t |ψ + Q T |a(t, x)|ψ + Q T sλ|∆η 0 |ξψ + m (0,T )×ω s 2 λ 2 ξ 2 ψ.
We have the following integration by parts

- Q T 2sλξ∇η 0 .∇ψ = Q T 2sλ ∇ξ.∇η 0 ψ + ξ∆η 0 ψ - Σ T 2sλξ ∂η 0 ∂n ψdσdt, (41) (42)
Q T ∂ t ψ = Ω (ψ(T, .) -ψ(0, .)) = 0, (43)
Q T ∆ψ = Σ T ∂ψ ∂n ,
where Σ T := (0, T ) × ∂Ω. From ( 40), ( 41), ( 42), ( 43), we have ( 44)

m Q T s 2 λ 2 ξ 2 ψ - Σ T 2sλξ ∂η 0 ∂n ψ + Σ T ∂ψ ∂n ≤ Q T sλ 2 |∇η 0 | 2 ξψ + Q T s|α t |ψ + Q T |a(t, x)|ψ + Q T 3sλ|∆η 0 |ξψ + Q T 2sλ|∇ξ||∇η 0 |ψ + m (0,T )×ω s 2 λ 2 ξ 2 ψ.
By using the first two lines of (34) and λ ≥ 1, we have

Q T sλ 2 |∇η 0 | 2 ξψ + Q T s|α t |ψ + Q T |a(t, x)|ψ + Q T 3sλ|∆η 0 |ξψ + Q T 2sλ|∇ξ||∇η 0 |ψ ≤ C Q T sλ 2 ξψ + Q T se 2λ η 0 ∞ T ξ 2 ψ + Q T |a(t, x)|ψ + Q T sλξψ ≤ C Q T sλ 2 ξψ + Q T se 2λ η 0 ∞ T ξ 2 ψ + Q T |a(t, x)|ψ . (45)
By combining ( 44) and (45), we get

(46) m Q T s 2 λ 2 ξ 2 ψ - Σ T 2sλξ ∂η 0 ∂n ψ + Σ T ∂ψ ∂n ≤ ≤ C Q T sλ 2 ξψ + Q T se 2λ η 0 ∞ T ξ 2 ψ + Q T |a(t, x)|ψ .
Absorption. The goal of this intermediate step is to absorb the right hand side of (46) by the first left hand side term of (46) by taking s sufficiently large. In order to do this, it is useful to keep in mind the fact that λ ≥ 1 and the third line of [START_REF] Lions | Exact controllability, stabilization and perturbations for distributed systems[END_REF] for the next estimates.

By taking s ≥ (T /2) 2 (4C/m), we have Csξ ≤ (m/4)(sξ) 2 and consequently (47)

C Q T sλ 2 ξψ ≤ m 4 Q T s 2 λ 2 ξ 2 ψ.
By taking s ≥ T e 2λ η 0 ∞ (4C/m), we have Cse 2λ η 0 ∞ T ξ 2 ≤ (m/4)(λsξ) 2 and consequently

(48) C Q T se 2λ η 0 ∞ T ξ 2 ψ ≤ m 4 Q T s 2 λ 2 ξ 2 ψ. By taking s ≥ (T /2) 2 a 1/2 L ∞ (Q T ) (4C/m) 1/2 , we have C a L ∞ (Q T ) ≤ (m/4)(λsξ) 2 and consequently (49) C Q T |a(t, x)|ψ ≤ m 4 Q T s 2 λ 2 ξ 2 ψ.
Therefore, by taking s ≥ s 1 (λ) as defined in [START_REF] Le Balc'h | Controllability of a 4 x 4 quadratic reaction-diffusion system[END_REF], we have from (47), ( 48) and ( 49) that

(50) C Q T sλ 2 ξψ + Q T se 2λ η 0 ∞ T ξ 2 ψ + Q T |a(t, x)|ψ ≤ 3m 4 Q T s 2 λ 2 ξ 2 ψ.
Then, from (46) and (50), for s ≥ s 1 (λ), we get

(51) m 4 Q T s 2 λ 2 ξ 2 ψ - Σ T 2sλξ ∂η 0 ∂n ψ + Σ T ∂ψ ∂n ≤ m (0,T )×ω s 2 λ 2 ξ 2 ψ.
Step 3: An identity satisfied by ψ. We readily obtain that

(52) M ψ = 0,
where

M ψ = -sλ 2 |∇η 0 | 2 ξ ψ + 2sλ ξ∇η 0 .∇ ψ + ∂ t ψ (53) + s 2 λ 2 |∇η 0 | 2 ξ 2 ψ + ∆ ψ + s α t ψ -a(t, x) ψ + sλ∆η 0 ξ ψ.
We integrate [START_REF] Pao | Nonlinear parabolic and elliptic equations[END_REF] over (0, T ) × Ω (54)

Q T s 2 λ 2 |∇η 0 | 2 ξ 2 ψ + Q T 2sλ ξ∇η 0 .∇ ψ + Q T ∂ t ψ + Q T ∆ ψ = Q T sλ 2 |∇η 0 | 2 ξ ψ - Q T s α t ψ + Q T a(t, x) ψ - Q T sλ∆η 0 ξ ψ.
Step 4: Estimates for ψ. By using [START_REF] Simon | Compact sets in the space L p (0, T ; B)[END_REF], we have

Q T s 2 λ 2 |∇η 0 | 2 ξ 2 ψ (55) ≥ (0,T )×(Ω\ω) s 2 λ 2 |∇η 0 | 2 ξ 2 ψ ≥ m Q T s 2 λ 2 ξ 2 ψ -m (0,T )×ω s 2 λ 2 ξ 2 ψ.
By combining (54) and (55), we have

(56) m Q T s 2 λ 2 ξ 2 ψ + Q T 2sλ ξ∇η 0 .∇ ψ + Q T ∂ t ψ + Q T ∆ ψ ≤ Q T sλ 2 |∇η 0 | 2 ξ ψ + Q T s| α t | ψ + Q T |a(t, x)| ψ + Q T sλ|∆η 0 | ξ ψ + m (0,T )×ω s 2 λ 2 ξ 2 ψ.
We have the following integration by parts

Q T 2sλ ξ∇η 0 .∇ ψ = - Q T 2sλ ∇ ξ.∇η 0 ψ + ξ∆η 0 ψ + Σ T 2sλ ξ ∂η 0 ∂n ψ, (57)
(58)

Q T ∂ t ψ = Ω ( ψ(T, .) -ψ(0, .)) = 0, (59)
Q T ∆ ψ = Σ T ∂ ψ ∂n .
From (56), (57), (58), (59), we have

(60) m Q T s 2 λ 2 ξ 2 ψ + Σ T 2sλ ξ ∂η 0 ∂n ψ + Σ T ∂ ψ ∂n ≤ Q T sλ 2 |∇η 0 | 2 ξ ψ + Q T s| α t | ψ + Q T |a(t, x)| ψ + Q T 3sλ|∆η 0 | ξ ψ + Q T 2sλ|∇ ξ||∇η 0 | ψ + m (0,T )×ω s 2 λ 2 ξ 2 ψ.
By using the first two lines of (34) and the fact that λ ≥ 1, we have

Q T sλ 2 |∇η 0 | 2 ξ ψ + Q T s| α t | ψ + Q T |a(t, x)| ψ + Q T 3sλ|∆η 0 | ξ ψ + Q T 2sλ|∇ ξ||∇η 0 | ψ ≤ C Q T sλ 2 ξ ψ + Q T se 4λ η 0 ∞ T ξ 2 ψ + Q T |a(t, x)| ψ + Q T sλ ξ ψ ≤ C Q T sλ 2 ξ ψ + Q T se 4λ η 0 ∞ T ξ 2 ψ + Q T |a(t, x)| ψ (61)
By combining (60) and (61), we get (62)

m Q T s 2 λ 2 ξ 2 ψ + Σ T 2sλ ξ ∂η 0 ∂n ψ + Σ T ∂ ψ ∂n ≤ C Q T sλ 2 ξ ψ + Q T se 4λ η 0 ∞ T ξ 2 ψ + Q T |a(t, x)| ψ
Absorption. Note that we will use the third line of (34) in the next four estimates.

By taking s ≥ e λ η 0 ∞ (T /2) 2 (4C/m), we have Cs ξ ≤ (m/4)(s ξ) 2 and consequently

(63) C Q T sλ 2 ξ ψ ≤ m 4 Q T s 2 λ 2 ξ 2 ψ.
By taking s ≥ T e 4λ η 0 ∞ (4C/m), we have Cse 2λ η 0 ∞ T ξ 2 ≤ (m/4)(λs ξ) 2 and consequently

(64) C Q T se 2λ η 0 ∞ T ξ 2 ψ ≤ m 4 Q T s 2 λ 2 ξ 2 ψ. By taking s ≥ e λ η 0 ∞ (T /2) 2 a 1/2 L ∞ (Q T ) (4C/m) 1/2 , we have C a L ∞ (Q T ) ≤ (m/4)(λs ξ) 2 and consequently (65) C Q T |a(t, x)| ψ ≤ m 4 Q T s 2 λ 2 ξ 2 ψ.
Therefore, by taking s ≥ s 1 (λ) as defined in [START_REF] Le Balc'h | Controllability of a 4 x 4 quadratic reaction-diffusion system[END_REF], we have from (47), ( 48) and ( 65) that

(66) C Q T sλ 2 ξ ψ + Q T se 4λ η 0 ∞ T ξ 2 ψ + Q T |a(t, x)| ψ ≤ 3m 4 Q T s 2 λ 2 ξ 2 ψ.
Then, from (62) and (66), for s ≥ s 1 (λ), we get

(67) m 4 Q T s 2 λ 2 ξ 2 ψ + Σ T 2sλ ξ ∂η 0 ∂n ψ + Σ T ∂ ψ ∂n ≤ m (0,T )×ω s 2 λ 2 ξ 2 ψ.
Step 5: Elimination of the boundary terms. From now, we take s ≥ s 1 (λ). By summing (51) and (67), we get

m 4 Q T s 2 λ 2 ξ 2 ψ - Σ T 2sλξ ∂η 0 ∂n ψ + Σ T ∂ψ ∂n + m 4 Q T s 2 λ 2 ξ 2 ψ + Σ T 2sλ ξ ∂η 0 ∂n ψ + Σ T ∂ ψ ∂n ≤ m (0,T )×ω s 2 λ 2 ξ 2 ψ + (0,T )×ω s 2 λ 2 ξ 2 ψ . (68)
Since η 0 = 0 on ∂Ω, we have

ξ = ξ, α = α and ψ = ψ on Σ T , which leads to (69) - Σ T 2sλξ ∂η 0 ∂n ψ + Σ T 2sλ ξ ∂η 0 ∂n ψ = 0.
Moreover, we have

∂ i ψ = e -sα (∂ i q + sλ∂ i η 0 ξq), ∂ i ψ = e -s α (∂ i q -sλ∂ i η 0 ξq),
whence by using ∂q ∂n = 0 on Σ T , we get ∂ψ ∂n = sλ ∂η 0 ∂n ξe -sα q, ∂ ψ ∂n = -sλ ∂η 0 ∂n ξe -s α q on Σ T .

This leads to (70)

Σ T ∂ψ ∂n + Σ T ∂ ψ ∂n = 0.
We get from (68), ( 69) and ( 70)

m 4 Q T s 2 λ 2 ξ 2 ψ + Q T s 2 λ 2 ξ 2 ψ ≤ C (0,T )×ω s 2 λ 2 ξ 2 ψ + (0,T )×ω s 2 λ 2 ξ 2 ψ . (71)
By using the fact that ξ ≤ ξ, e -s α ≤ e -sα in Q T , we get from (71) the Carleman estimate [START_REF] Rousseau | On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations[END_REF]. This concludes the proof of Theorem 4.9.

4.4.

Proof of the L 2 -L 1 observability inequality: Theorem 4.4. The goal of this subsection is to prove Theorem 4.4, which is a consequence of Theorem 4.9, L p -L q estimates and the dissipativity in time of the L p norm of (21).

Proof.

Step 1: L 1 -L 1 observability inequality. We fix λ = 1 and s = s 1 in Theorem 4.9 to get (72)

Q T t -2 (T -t) -2 e -sα qdxdt ≤ C 1 (Ω, ω) (0,T )×ω t -2 (T -t) -2 e -sα qdxdt.
First, we observe that in (T /4, 3T /4) × Ω,

t -2 (T -t) -2 e -sα ≥ C T 4 exp   - C(Ω, ω) T + T 2 + T 2 a 1/2 L ∞ (Q T ) T 2   ≥ C T 4 e -C(Ω,ω) 1+ 1 T + a 1/2 L ∞ (Q T ) . (73)
Secondly, from the fact that x 2 e -M x ≤ C/M 2 for every x, M ≥ 0 used with

x = t -1 (T -t) -1 and M = C(Ω, ω) T + T 2 + T 2 a 1/2 L ∞ (Q T ) , we remark that in (0, T )× ω, t -2 (T -t) -2 e -sα ≤ t -2 (T -t) -2 exp -C(Ω, ω) T + T 2 + T 2 a 1/2 L ∞ (Q T ) t -1 (T -t) -1 ≤ C C(Ω, ω) T + T 2 + T 2 a 1/2 L ∞ (Q T ) 2 ≤ C(Ω, ω) T 4 . (74)
Then, we get from (72), ( 73) and (74) (75)

(T /4,3T /4)×Ω qdxdt ≤ e C(Ω,ω) 1+ 1 T + a 1/2 L ∞ (Q T ) (0,T )×ω qdxdt.
On the other hand, we obtain by the dissipativity in time of the L 1 -norm (see Proposition 3.4 with p = 1)

(76) q(T /4, .) L 1 (Ω) ≤ 2C exp CT a L ∞ (Q T ) T 3T /4 T /4
q(t, .) L 1 (Ω) dt.

By using (75) and (76), we get (77) q(T /4, .) L 1 (Ω) ≤ C(Ω, ω, T, a)

(0,T )×ω qdxdt,
where C(Ω, ω, T, a) is defined in [START_REF] Fernández | Null and approximate controllability for weakly blowing up semilinear heat equations[END_REF].

From now, we denote by C(Ω, ω, T, a) various positive constants varying from line to line which are of the form [START_REF] Fernández | Null and approximate controllability for weakly blowing up semilinear heat equations[END_REF].

Step 2: Global L 2 -L 1 estimate. The goal of this step is to prove that (78) q(0, .) L 2 (Ω) ≤ C(Ω, ω, T, a) q(T /4, .) L 1 (Ω) .

To simplify the notations, we set q(t) := q(Tt) for t ∈ [0, T ]. Then, (78) rewrites as follows (79) q( T 2 , .)

L 2 (Ω)
≤ C(Ω, ω, T, a) q( T 1 , .)

L 1 (Ω)
.

with T 2 := T > T 1 := 3T /4. We introduce the following sequence (80)

r 0 := 1, ∀k ≥ 0, r k+1 := N r k N -r k if r k < N, 2r k if r k ≥ N.
We readily have from the definition (80) that

(81) ∀k ≥ 0, β k := N 2 1 r k - 1 r k+1 ≤ 1 2 < 1, and
(82) ∃l ≥ 1, r l ≥ 2.
We also introduce a sequence of times

(83) ∀k ∈ {0, . . . , l}, τ k := T 1 + k l ( T 2 -T 1 ).
Let us remark that (84) ∀k ∈ {0, . . . , l}, τ k+1 -

τ k = T 2 -T 1 l = T 2l .
By induction, we will show that (85) ∀k ∈ {0, . . . , l}, q(τ k , .) L r k (Ω) ≤ C(Ω, ω, T, a) q(τ 0 , .) L 1 (Ω) .

The case k = 0 is obvious (take C 0 = 1). Then, by denoting by S(t) = e t∆ the heat-semigroup with Neumann boundary conditions, we have for every k ≥ 0,

(86) q(τ k+1 ) = S(τ k+1 -τ k ) q(τ k ) + τ k+1 τ k
S(τ k+1s)(-a(s, .) q(s))ds, from the equation satisfied by q (see ( 21)). We assume that (85) holds for k ∈ {0, . . . , l}. From (86), (81) and the regularizing effect L r k -L r k+1 of the heat-semigroup (see Proposition 3.8), we have

q(τ k+1 ) L r k+1 (Ω) ≤ (τ k+1 -τ k ) -β k q(τ k ) L r k (Ω) + τ k+1 τ k (τ k+1 -s) -β k a L ∞ (Q T ) q(s) L r k (Ω) ds ≤ A 1,k + A 2,k , (87) where
(88) A 1,k := (τ k+1 -τ k ) -β k q(τ k ) L r k (Ω) , and
(89) A 2,k := τ k+1 τ k (τ k+1 -s) -β k a L ∞ (Q T ) q(s) L r k (Ω) ds.
From (88), ( 84), ( 81) and (85), we have (90) A 1,k ≤ CT -β k C(Ω, ω, T, a) q(τ 0 , .) L 1 (Ω) ≤ C(Ω, ω, T, a) q(τ 0 , .) L 1 (Ω) .

From (89), the dissipativity in time of the L r k -norm (see Proposition 3.4), the induction assumption (85), ( 81) and (84), we have (91)

A 2,k ≤ a ∞ τ k+1 τ k (τ k+1 -s) -β k Ce CT a ∞ q(τ k ) L r k (Ω) ds ≤ C a ∞ e CT a ∞ C(Ω, ω, T, a) q(τ 0 , .) L 1 (Ω) (τ k+1 -τ k ) -β k +1 ≤ C(Ω, ω, T, a) a ∞ T -β k +1 q(τ 0 , .) L 1 (Ω) ≤ C(Ω, ω, T, a) a ∞ (T + 1) q(τ 0 , .) L 1 (Ω) ≤ C(Ω, ω, T, a) e T a ∞ + 2e a 1/2 ∞ q(τ 0 , .) L 1 (Ω) ≤ C(Ω, ω, T, a) q(τ 0 , .) L 1 (Ω) .
The estimates (87), ( 90) and ( 91) prove (85) for (k + 1) and concludes the induction. Thus, (85) holds for k = l, which combined with (82) and (83), yields (79).

Step 3: By using (77) and (78), we prove (25) and consequently Theorem 4.4.

4.5.

Proof of the linear global nonnegative-controllability: Theorem 4.1. The goal of this section is to prove Theorem 4.1. The following proof is inspired by the so-called Hilbert Uniqueness method due to Jacques-Louis Lions (see [START_REF] Lions | Exact controllability, stabilization and perturbations for distributed systems[END_REF] and more precisely [43, Section 2.1]).

Proof. The proof is divided into two steps. First, we build a sequence of controls h ε ∈ L ∞ ((0, T ) × ω) with ε > 0 which provide the approximate nonnegativecontrollability of [START_REF] Fernández-Cara | Null controllability of the heat equation with boundary Fourier conditions: the linear case[END_REF]. Secondly, we pass to the limit when ε tends to 0.

Step 1. Let us fix T > 0, a ∈ L ∞ (Q T ) and y 0 ∈ L 2 (Ω). For any ε ∈ (0, 1), we consider the following functional: for every q T ∈ L 2 (Ω; R + ),

(92) J ε (q T ) = 1 2 (0,T )×ω qdxdt 2 + ε q T L 2 (Ω) + Ω q(0, x)y 0 (x)dx,
where q is the solution to [START_REF] Fernández | Global Carleman inequalities for parabolic systems and applications to controllability[END_REF]. The functional J ε is continuous, convex and coercive on the unbounded closed convex set L 2 (Ω; R + ). More precisely, we will show that (93) lim inf

q T L 2 (Ω) →+∞ J ε (q T ) q T L 2 (Ω) ≥ ε.
Indeed, given a sequence (q T,k ) k≥0 ∈ L 2 (Ω) with q T,k L 2 (Ω) → +∞, we normalize it: q T,k := q T,k q T,k L 2 (Ω) , and we denote by q k the solution to ( 21) associated to the initial data q T,k . We have

(94) J ε (q T,k ) q T,k L 2 (Ω) = q T,k L 2 (Ω) 2 (0,T )×ω q k dxdt 2 + ε + Ω q k (0, x)y 0 (x)dx.
We distinguish the following two cases. Case 1:

(95) lim inf k→+∞ (0,T )×ω q k dxdt > 0.

When (95) holds, we clearly have

lim inf k→+∞ J ε (q T,k ) q T,k L 2 (Ω) = +∞ ≥ ε
Case 2:

(96) lim inf k→+∞ (0,T )×ω

q k dxdt = 0.
In this case, by using the estimate (11) of Proposition 3.2, the embedding ( 7) and ( 96), extracting subsequences (that we denote by the index k to simplify the notation), we deduce that there exists q ∈ W T such that (97)

q k ⇀ q in W T , (98) q k (0, .) ⇀ q(0, .) in L 2 (Ω), (99)
(0,T )×ω q k dxdt → 0.

By using Aubin Lions' lemma (see [START_REF] Simon | Compact sets in the space L p (0, T ; B)[END_REF]Section 8,Corollary 4]) and ( 97), ( q k ) k∈N is relatively compact in L 2 (Q T ), then up to a subsequence we have (100)

q k → q in L 2 (Q T ; R + ).
In view of ( 99) and (100), we have (101) q = 0 in (0, T ) × ω.

Then, by using (101) and the observability inequality (25), we have (102) q(0, .) = 0.

Consequently, by combining (98) and (102), we have Ω q k (0, x)y 0 (x)dx → 0, which yields (93) thanks to (94). We deduce that J ε admits a minimum q ε,T ∈ L 2 (Ω; R + ). We take (103)

h ε := (0,T )×ω q ε 1 ω ,
and we denote by

y ε ∈ W T ∩ L ∞ (Q T ) the solution to (104)    ∂ t y ε -∆y ε + a(t, x)y ε = h ε 1 ω in (0, T ) × Ω, ∂yε ∂n = 0 on (0, T ) × ∂Ω, y ε (0, .) = y 0 in Ω.
We use the fact that J ε (q T,ε ) ≤ J ε (0) = 0 to get

(105) 1 2
(0,T )×ω

q ε 2 + ε q ε,T L 2 (Ω) ≤ - Ω q ε (0, x)y 0 (x)dx.
By using the observability inequality ( 25), ( 103), ( 105) and Young's inequality, we obtain the following bound on the sequence of controls

(106) h ε 2 L ∞ (Q T ) ≤ C(Ω, ω, T, a) y 0 2 L 2 (Ω) ,
where C(Ω, ω, T, a) is of the form [START_REF] Fernández | Null and approximate controllability for weakly blowing up semilinear heat equations[END_REF]. For λ > 0 and p T ∈ L 2 (Ω; R + ), we have (107) J ε (q ε,T ) ≤ J ε (q ε,T + λp T ).

Dividing the inequality (107) by λ and letting λ → 0 + , we easily obtain from (103),

-(y 0 , p(0, .)) L 2 (Ω) ≤ (0,T )×ω h ε p + ε lim inf λ→0 + q ε,T + λp T L 2 (Ω) -q ε,T L 2 (Ω) λ (108) ≤ (0,T )×ω h ε p + ε p T L 2 (Ω) ,
where p is the solution to [START_REF] Fernández | Global Carleman inequalities for parabolic systems and applications to controllability[END_REF] with initial data p T . Since systems [START_REF] Fernández-Cara | Null controllability of the heat equation with boundary Fourier conditions: the linear case[END_REF] and ( 21) are in duality, we have (109

) (0,T )×ω h ε p = (y ε (T, .), p T ) L 2 (Ω) -(y 0 , p(0, .)) L 2 (Ω) ,
which, combined with (108), yields

(110) (y ε (T, .), p T ) L 2 (Ω) ≥ -ε p T L 2 (Ω) , ∀p T ∈ L 2 (Ω; R + ).
Step 2. By using ( 106), (104), Proposition 3.2, Proposition 3.3 and the embedding ( 7), up to a subsequence, we get that there exist

h ∈ L ∞ (Q T ) and y ∈ W T ∩ L ∞ (Q T ) such that (111) h ε ⇀ * h in L ∞ (Q T ) as ε → 0, ( 112
) y ε ⇀ y in W T ⇒ y ε (0, .)⇀ y(0, .), y ε (T, .)⇀ y(T, .) in L 2 (Ω) as ε → 0.
Then, by using (104), ( 111) and (112), we obtain that y is the solution of (20) associated to the control h satisfying (23) (by letting ε goes to 0 in (106)) and

(113) (y(T, .), p T ) L 2 (Ω) ≥ 0, ∀p T ∈ L 2 (Ω; R + ).

Then, we deduce from (113) that y satisfies [START_REF] Victor | Regional blow up in a semilinear heat equation with convergence to a Hamilton-Jacobi equation[END_REF], which concludes the proof of Theorem 4.1.

A fixed-point argument to prove the small-time nonlinear global nonnegative controllability

The goal of this section is to prove Theorem 2.2. We assume that (3) holds for α ≤ 2 and f (s) ≥ 0 for s ≥ 0.

5.1.

A comparison principle. First, we begin with this lemma, which is a consequence of the comparison principle for subsolutions and supersolutions of (1) with control h = 0 stated in Proposition 3.7.

Lemma 5.1. Let T > 0, y 0 ∈ L ∞ (Ω). Assume that there exists T * ∈ (0, T ] and a control h * ∈ L ∞ (Q T * ) such that the solution y ∈ L ∞ (Q T * ) to (1) satisfies (5) (replacing T ← T * ). Then, if we set h(t, .) := h * (t, .) for t ∈ (0, T * ), 0 for t ∈ (T * , T ), the solution y of (1) belongs to L ∞ (Q T ) and satisfies [START_REF] Barbu | Local controllability of the phase field system[END_REF]. Moreover, there exists

C := C(Ω) > 0 such that (114) y L ∞ (Q T ) ≤ C y L ∞ (Q T * ) .
Proof. By using the fact that f (0) = 0, f (s) ≥ 0 for s ≥ 0 and the comparison principle (see Proposition 3.7), we have

(115) ∀t ∈ [T * , T ], a.e. x ∈ Ω, 0 ≤ y(t, x) ≤ y(t, x),
where y is the nonnegative solution to

(116)    ∂ t y -∆ y = 0 in (T * , T ) × Ω, ∂ y ∂n = 0 on (T * , T ) × ∂Ω, y(T * , .) = y(T * , .) in Ω.
Therefore, by using Proposition 3.3 for (116), we get that there exists

C := C(Ω) > 0 such that (117) y L ∞ ((T * ,T )×Ω) ≤ C y(T * , .) L ∞ (Ω) ≤ C y L ∞ (Q T * ) .
By using ( 115) and ( 117), we obtain that y ∈ L ∞ (Q T ), ( 5) and (114) hold.

5.2.

The fixed-point: definition of the application. We begin with some notations. Let us set

(118) g(s) = f (s) s if s = 0, f ′ (0) if s = 0.
The function g is continuous and by using the fact that f satisfies (3) with α ≤ 2, we deduce that for every ε > 0, there exists

C ε > 0 such that (119) ∀s ∈ R, |g(s)| 1/2 ≤ ε log(2 + |s|) + C ε .
The end of the section is devoted to the proof of Theorem 2.2.

Proof. Let T > 0, y 0 ∈ L ∞ (Ω).
Unless otherwise specified, we denote by C various positive constants varying from line to line which may depend on Ω, ω, T .

We will perform a Kakutani-Leray-Schauder's fixed-point argument in L ∞ (Q T ).

For each z ∈ L ∞ (Q T ), we consider the linear system (120)

   ∂ t y -∆y + g(z)y = h1 ω in (0, T ) × Ω, ∂y ∂n = 0 on (0, T ) × ∂Ω, y(0, .) = y 0 in Ω.
We set (121)

T * z := min T, g(z) -1/2 L ∞ (Q T ) . According to Theorem 4.1, there exists a control h z ∈ L ∞ (Q T * z ) satisfying (122) h z L ∞ (Q T * z ) ≤ exp C 1 + 1 T * z + T * z g(z) L ∞ (Q T ) + g(z) 1/2 L ∞ (Q T ) y 0 L 2 (Ω) ≤ exp C 1 + g(z) 1/2 L ∞ (Q T ) y 0 L 2 (Ω) ,
such that the solution y of (120) in (0, T * z ) × Ω with h = h z satisfies (123) y(T * z , .) ≥ 0. By extending by 0 the control h z in (T * z , T ), we get from (122)

(124) h z L ∞ (Q T ) ≤ exp C 1 + g(z) 1/2 L ∞ (Q T ) y 0 L 2 (Ω) .
For each z ∈ L ∞ (Q T ), we introduce the set of controls (125) H(z) := {h z ∈ L ∞ (Q T ) ; h z fulfills (124) and h z ≡ 0 in (T * z , T ) × Ω}. We have the following facts.

5.2. For every z ∈ L ∞ (Q T ), H(z) is compact for the weak-star topology of L ∞ (Q T ). Fact 5.3. Assume that z k → z in L ∞ (Q T ) and h k ∈ H(z k )⇀ * h in L ∞ (Q T ) as k → +∞. Then, we have h ∈ H(z).
We define the set-valued mapping Φ : L ∞ (Q T ) → P(L ∞ (Q T )) in the following way. For every z ∈ L ∞ (Q T ), Φ(z) is the set of y ∈ L ∞ (Q T ) such that for some h z ∈ H(z), y is the solution of (120) and this solution satisfies (123).

We recall the Kakutani-Leray-Schauder's fixed point theorem (see [27, Theorem 2.2, Theorem 2.4]).

Theorem 5.4 (Kakutani-Leray-Schauder's fixed point theorem). If

(1) Φ is a Kakutani map, that is to say for every

z ∈ L ∞ (Q T ), Φ(z) is a nonempty convex and closed subset of L ∞ (Q T ), (2) Φ is compact, that is to say for every bounded set B ⊂ L ∞ (Q T ), there exists a compact set K ⊂ L ∞ (Q T ) such that for every z ∈ B, Φ(z) ⊂ K, (3) Φ is upper semicontinuous in L ∞ (Q T ), that is to say for all closed subset A ⊂ L ∞ (Q T ), Φ -1 (A) = {z ∈ L ∞ (Q T ) ; Φ(z) ∩ A = ∅} is closed, (4) F := {y ∈ L ∞ (Q T ) ; ∃λ ∈ (0, 1), y ∈ λΦ(y)} is bounded in L ∞ (Q T ), hold.
Then Φ has a fixed point, i.e, there exists y ∈ L ∞ (Q T ) such that y ∈ Φ(y).

5.3.

Hypotheses of Kakutani-Leray-Schauder's fixed point theorem. We will check that the four hypotheses of Theorem 5.4 hold.

The point (1) holds. Indeed, for every z ∈ L ∞ (Q T ), we have seen that Φ(z) is nonempty. The convexity of Φ(z) comes from the fact that the inequality (123) is stable by convex combinations. Let us show that Φ(z) is closed. Let (y k ) k∈N be a sequence of elements in L ∞ (Q T ), such that for every k ∈ N, y k ∈ Φ(z) and y k → y in L ∞ (Q T ). Then, for every k ∈ N, there exists a control h k ∈ H(z) such that y k is the solution to (126)

   ∂ t y k -∆y k + g(z)y k = h k 1 ω in (0, T ) × Ω, ∂y k ∂n = 0 on (0, T ) × ∂Ω, y k (0, .) = y 0 in Ω,
and this solution satisfies (127) y k (T * z , .) ≥ 0. By using Fact 5.2, Proposition 3.2 and the embedding (7), we get that there exist a strictly increasing sequence (k l ) l∈N of integers and h ∈ H(z) such that (128)

h k l ⇀ * h in L ∞ (Q T ) as l → +∞, (129)
y k l ⇀ y in W T ⇒ y k l (0, .) ⇀ y 0 , y k l (T * z , .)⇀ y(T * z , .
) in L 2 (Ω) as l → +∞. By passing to the limit as l → +∞ in (126), (127) and by using (128) and (129), we get that y ∈ Φ(z). This concludes the proof of the point [START_REF] Farid Ammar Khodja | Null-controllability of some reaction-diffusion systems with one control force[END_REF].

The point (2) holds. Let B be a bounded set of L ∞ (Q T ). By using (124) and Proposition 3.3 applied to (120), we deduce that there exists R > 0 such that for every z ∈ B, for every y ∈ Φ(z) associated to a control h z ∈ H(z), we have

(130) z, y, h z ∈ B R := {ζ ∈ L ∞ (Q T ) ; ζ L ∞ (Q T ) ≤ R}. Let Y ∈ L ∞ (Q T ) be the solution to the Cauchy problem (131)    ∂ t Y -∆Y = 0 in (0, T ) × Ω, ∂Y ∂n = 0 on (0, T ) × ∂Ω, Y (0, .) = y 0 in Ω.
Let y * = y -Y , where y ∈ Φ(z), with z ∈ B, associated to a control h z ∈ H(z).

Then, y * is the solution to (132)

   ∂ t y * -∆y * + g(z)y = h z 1 ω in (0, T ) × Ω, ∂y * ∂n = 0 on (0, T ) × ∂Ω, y * (0, .) = 0 in Ω.
From (130), we have

(133) -g(z)y + h z 1 ω L ∞ (Q T ) ≤ C R .
From (133), a maximal parabolic regularity theorem in L p (see [START_REF] Denk | Optimal L p -L q -estimates for parabolic boundary value problems with inhomogeneous data[END_REF]Theorem 2.1]), with p = N + 2, applied to y * , solution of (132), we deduce that (134)

y * ∈ X p := W 1,p (0, T ; L p (Ω)) ∩ L p (0, T ; W 2,p (Ω)) and y * Xp ≤ C R . By the Sobolev embedding theorem X p ֒→ C β/2,β (Q T ) with β > 0 (see [41, Theorem 1.4.1]), we deduce that y * ∈ C 0 (Q T ) and (135) ∀(t, x) ∈ Q T , ∀(t ′ , x ′ ) ∈ Q T , |y * (t, x) -y * (t ′ , x ′ )| ≤ C R (|t -t ′ | β/2 + |x -x ′ | β ).
Let K * be the set of y * such that (135) holds. Then, we have

K := (Y + K * ) ∩ B R is a compact convex subset of L ∞ (Q T ) by Ascoli's theorem and ∀z ∈ B, Φ(z) ⊂ K.
This concludes the proof of the point [START_REF] Ammar-Khodja | Recent results on the controllability of linear coupled parabolic problems: a survey[END_REF].

The point (3) holds. Let A be a closed subset of

L ∞ (Q T ). Let (z k ) k∈N be a sequence of elements in L ∞ (Q T ), (y k ) k∈N be a sequence of elements in L ∞ (Q T ), and z ∈ L ∞ (Q T ) be such that (136) z k → z in L ∞ (Q T ) as k → +∞, (137) ∀k
∈ L ∞ (Q T ) such that (141) h k l ⇀ * h in L ∞ (Q T ) as l → +∞, (142)
y k l → y in L ∞ (Q T ) as l → +∞.
Since A is closed, (137) and (142) imply that y ∈ A. Hence, it suffices to check that (143) y ∈ Φ(z).

Letting l → +∞ in (139) and (140) and using (136), ( 141) and (142), we get that y satisfies (120) and (123). Hence, (143) holds. This concludes the proof of the point (3).

The point (4) holds. Let y ∈ F. Then, for some λ ∈ (0, 1) and h y ∈ H(y), we have    ∂ t y -∆y + f (y) = λh y 1 ω in (0, T ) × Ω, ∂y ∂n = 0 on (0, T ) × ∂Ω, y(0, .) = λy 0 in Ω. and y(T * y , .) ≥ 0. Therefore, by using Lemma 5.1 and Proposition 3.3, we have

(144) y L ∞ (Q T ) ≤ C y L ∞ (Q T * y ) ≤ C exp CT * y g(y) L ∞ (Q T ) y 0 L ∞ (Ω) + h y L ∞ (Q T ) .
Consequently, by taking into account the definition of T * y , i.e., (121) and using (124), (144), (119), we deduce that (145)

y L ∞ (Q T ) ≤ exp C 1 + g(y) 1/2 L ∞ (Q T ) y 0 L ∞ (Ω) ≤ exp C 1 + ε log 2 + y L ∞ (Q T ) + C ε y 0 L ∞ (Ω) ≤ exp (C ε ) 2 + y L ∞ (Q T ) εC y 0 L ∞ (Ω) .
Therefore, by taking ε sufficiently small such that εC = 1/2, we deduce from (145) that F is bounded in L ∞ (Q T ). This concludes the proof of the point (4). By Theorem 5.4, Φ has a fixed point y. We denote by h y the associated control. Then, by using Lemma 5.1, y is the solution to (1) with control h y such that [START_REF] Barbu | Local controllability of the phase field system[END_REF] holds. This concludes the proof of Theorem 2.2.

Application of the global nonnegative-controllability to the large time global null-controllability

In this section, we prove Theorem 2.5. We assume that (3) holds for α ∈ [3/2, 2], f (s) > 0 for s > 0 and 1/f ∈ L 1 ([1, +∞)).

Proof. Let y 0 ∈ L ∞ (Ω). The proof is divided into three steps.

Step 1: Steer the solution to a nonnegative state in time T 1 := 1. By using Theorem 2.2, there exists h 1 ∈ L ∞ (Q T 1 ) such that the solution y to (1) replacing T ← T 1 satisfies y T 1 := y(T 1 , .) ≥ 0.

Step 2: Dissipation of f on R + and comparison to an ordinary differential equation. We set h 2 (t, .) := 0, for t ∈ [T 1 , T 2 ], with T 2 which will be determined later.

Then, by using the comparison principle given in Proposition 3.7, we deduce that the solution y to

   ∂ t y -∆y = -f (y) in (T 1 , T 2 ) × Ω, ∂y ∂n = 0 on (T 1 , T 2 ) × ∂Ω, y(T 1 , .) = y T 1 in Ω, satisfies (146) ∀t ∈ [T 1 , T 2 ], a.e. x ∈ Ω, 0 ≤ y(t, x) ≤ v(t),
where v is the (global) nonnegative solution to the ordinary differential equation

(147) v(t) = -f (v(t)) in (T 1 , +∞), v(T 1 ) = y T 1 L ∞ (Ω) + 1 . A straightforward calculation leads to (148) ∀t ∈ [T 1 , +∞), v(t) > 0 and F (v(t)) -F (v(T 1 )) = t -T 1 ,
where F is defined as follows

(149) ∀s > 0, F (s) = s +∞ -1 f (σ) dσ = +∞ s 1 f (σ) dσ.
Note that F is well-defined because f (σ) > 0 for every σ > 0 and 1/f ∈ L 1 ([1, +∞)) by hypothesis. We check that F is a C 1 strictly decreasing function. Moreover, we have 1/f / ∈ L 1 ((0, 1]) because f ∈ C 1 (R; R) and f (0) = 0. Hence, we have by ( 149 Therefore, we deduce that F : (0, +∞) → (0, +∞) is a C 1 -diffeomorphism. We denote by F -1 : (0, +∞) → (0, +∞) its inverse, which is strictly decreasing. Then, by (148), we have

(151) ∀t ∈ [T 1 , +∞), v(t) = F -1 (t -T 1 + F (v(T 1 )) ≤ F -1 (t -T 1 ).
The estimate (151) is the key point because it states that we can upperbound v by a function independent of the size of v(T 1 ) and we also have

(152) F -1 (t -T 1 ) → 0 as t → +∞,
by using (150).

Let δ > 0 be such that the null-controllability of (1) holds in B L ∞ (Ω) (0, δ) in time T = 1. The existence of δ is given by Theorem 1. [START_REF] Anita | Null controllability for the dissipative semilinear heat equation[END_REF].

By (152), we deduce that there exists T 2 sufficiently large such that (153)

F -1 (T 2 -T 1 ) ≤ δ.
Consequently, by using ( 146), ( 151), (153), we have (154) a.e. x ∈ Ω, 0 ≤ y(T 2 , x) ≤ δ.

Step 3: Local null-controllability. By using Theorem 1.3 with T = 1, we deduce from (154) that there exists a control h 3 ∈ L ∞ ((T 2 , T 3 )×Ω) with T 3 := T 2 +1 such that the solution y of (1) replacing (0, T ) ← (T 2 , T 3 ) satisfies y(T 3 , .) = 0.

To sum up, the control h(t, .) :=    h 1 (t, .) for t ∈ (0, T 1 ), h 2 (t, .) for t ∈ (T 1 , T 2 ), h 3 (t, .) for t ∈ (T 2 , T 3 ), steers the initial data y 0 ∈ L ∞ (Ω) to 0. It is worth mentioning that the final time of control T 3 does not depend on y 0 . This concludes the proof of Theorem 2.5.

Dirichlet boundary conditions

Theorem 2.2 and Theorem 2.5 remain valid for Dirichlet boundary conditions, as to say for (155)

   ∂ t y -∆y + f (y) = h1 ω in (0, T ) × Ω, y = 0 on (0, T ) × ∂Ω, y(0, .) = y 0 in Ω.
The main point is to establish a L 1 -Carleman estimate similar to Theorem 4.9 for (156)

   -∂ t q -∆q + a(t, x)q = 0 in (0, T ) × Ω, q = 0 on (0, T ) × ∂Ω, q(T, .) = q T in Ω.
We keep the notations of Section 4.3.

Theorem 7.1. There exists two constants C = C(Ω, ω) > 0 and C

1 := C 1 (Ω, ω) > 0, such that, (157) ∀λ ≥ 1, ∀s ≥ s 1 (λ) := C(Ω, ω) e 2λ η 0 ∞ T + T 2 + T 2 a 1/2 L ∞ (Q T ) , for every q T ∈ L 2 (Ω; R + ), the nonnegative solution q of (156) satisfies λ Q T e -sα sξ 2 η 0 q + Q T e -sα ξq ≤ C 1 λ (0,T )×ω e -sα sξ 2 qdxdt. ( 158
)
Proof. The proof follows the one of Theorem 4.9. This is why we omit some details. We multiply the identity (35) by η 0 and we integrate over (0, T ) × Ω

(159) QT s 2 λ 2 |∇η 0 | 2 ξ 2 ψη 0 - QT 2sλξ(∇η 0 .∇ψ)η 0 + QT (∂ t ψ)η 0 + QT (∆ψ)η 0 = QT sλ 2 |∇η 0 | 2 ξψη 0 - QT sα t ψη 0 + QT a(t, x)ψη 0 + QT sλ∆η 0 ξψη 0 .
By the properties of η 0 , we have (160)

Q T s 2 λ 2 |∇η 0 | 2 ξ 2 ψη 0 ≥ m Q T s 2 λ 2 ξ 2 ψη 0 -m (0,T )×ω s 2 λ 2 ξ 2 ψη 0 ,
where m is defined in [START_REF] Simon | Compact sets in the space L p (0, T ; B)[END_REF].

By combining (159) and (160), we have

(161) m QT s 2 λ 2 ξ 2 ψη 0 - QT 2sλξ(∇η 0 .∇ψ)η 0 + QT (∂ t ψ)η 0 + QT (∆ψ)η 0 ≤ QT sλ 2 |∇η 0 | 2 ξψη 0 + QT s|α t |ψη 0 + QT |a(t, x)|ψη 0 + QT sλ|∆η 0 |ξψη 0 + m (0,T )×ω s 2 λ 2 ξ 2 ψη 0 .
We have the following integration by parts From (161), (162), (163), (164) and the properties of η 0 , we have

- QT 2sλξ(∇η 0 .∇ψ)η 0 = QT 2sλ   (∇ξ.∇η 0 )η 0 ψ + ξ(∆η 0 )η 0 ψ + ξ|∇η 0 | 2 ψ ≥0    . (162) (163) QT (∂ t ψ)η 0 = Ω η 0 (.)(ψ(T, .) -ψ(0, .)) = 0, (164)
(165) m QT s 2 λ 2 ξ 2 ψη 0 + 2m QT sλξψ ≤ QT sλ 2 |∇η 0 | 2 ξψη 0 + QT s|α t |ψη 0 + QT |a(t, x)|ψη 0 +3 QT sλ|∆η 0 |ξψη 0 + 2 QT sλ|∇ξ||∇η 0 |ψη 0 + QT ψ|∆η 0 | +m (0,T )×ω s 2 λ 2 ξ 2 ψη 0 + 2m (0,T )×ω sλξψ.
The first five right hand side terms of (165) can be absorbed by the first left hand side term provided s ≥ s 1 (λ) as defined in (157) (see 'Step 2, Absorption' of the proof of Theorem 4.9 for details: it is exactly the same mechanism as in the proof for the Neumann case). The sixth right hand side term of (165) can be absorbed by the second left hand side term provided s ≥ C(Ω, ω)T 2 . The two last right hand side terms of (165) are smaller than (0,T )×ω s 2 λ 2 ξ 2 ψ provided s ≥ C(Ω, ω)T 2 . This leads to

Q T s 2 λ 2 ξ 2 ψη 0 + Q T sλξψ ≤ C (0,T )×ω s 2 λ 2 ξ 2 ψ,
which yields (158) by dividing by sλ. From Theorem 7.1, we deduce a precise L 2 -L 1 observability inequality as in Theorem 4.4 by using the second left hand side term of (158). It is an easy adaptation of Section 4.4.

The proof of the linear global nonnegative-controllability result as Theorem 4.1 and the fixed-point argument (see Section 5) remain unchanged. This leads to the small-time global nonnegative controllability for (155). The proof of the large time global null-controllability result for (155) follows the same lines as Section 6. In particular, the comparison principle between the free solution and the solution to the ordinary differential equation, i.e., (146) stays valid because v(t) > 0 on (T 1 , T 2 ) × ∂Ω.

Comments

8.1. Nonlinearities depending on the gradient of the state. We do not treat semilinearties F (y, ∇y) as considered in [START_REF] Fernández-Cara | Exact controllability to the trajectories of the heat equation with Fourier boundary conditions: the semilinear case[END_REF] (see also [START_REF] Doubova | On the controllability of parabolic systems with a nonlinear term involving the state and the gradient[END_REF]) because the left hand side of the L 1 -Carleman estimate [START_REF] Rousseau | On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations[END_REF] established in Theorem 4.9 does not provide estimates on the gradient of the state. 8.2. Nonlinear reaction-diffusion systems. We may wonder to what extent our main results, i.e., Theorem 2.2 and Theorem 2.5 for (1), can be adapted to the m×m semilinear reaction-diffusion system

(166) ∀1 ≤ i ≤ m,    ∂ t u i -d i ∆u i = f i (u 1 , . . . , u m ) + h i 1 ω in (0, T ) × Ω, ∂u i ∂n = 0 on (0, T ) × ∂Ω, u i (0, .) = u i,0 in Ω, with (d 1 , . . . , d m ) ∈ (0, +∞) m and (f 1 , . . . , f m ) ∈ C 1 (R m ; R) m satisfying (167) ∀i ∈ {1, . . . , m}, f i (0, . . . , 0) = 0.
We assume that the nonlinearity is strongly quasi-positive, i.e.,

∈ R m , ∀i = j ∈ {1, . . . , m}, ∂f i ∂u j (u 1 , . . . , u m ) ≥ 0. (168) ∀u
and satisfies a 'mass-control structure'

(169) ∀u ∈ [0, +∞) m , m i=1 f i (u) ≤ C 1 + m i=1 u i .
Lots of systems come naturally with the two properties (168) and (169) in applications (see [START_REF] Pierre | Global existence in reaction-diffusion systems with control of mass: a survey[END_REF]Section 2]).

We have the following global-nonnegative controllability result in small time.

Theorem 8.1. For each f i , we assume that (3) holds for α ≤ 2. For every T > 0, the system (166) is globally nonnegative-controllable in time T .

Application 8.2. Let α ∈ (0, 2). The system (170)

       ∂ t u -∆u = -u log α (2 + |u|) + h 1 1 ω in (0, T ) × Ω, ∂ t v -∆v = u log α (2 + |u|) + h 2 1 ω in (0, T ) × Ω,
∂u ∂n = ∂v ∂n = 0 on (0, T ) × ∂Ω, (u, v)(0, .) = (u 0 , v 0 ) in Ω, is globally nonnegative-controllable for every time T > 0.

Proof. As the proof is very similar to that of Theorem 2.2, we limit ourselves to pointing out only the differences. Difference 1: A L 1 -Carleman estimate for a linear parabolic system. Let A ∈ L ∞ (Q T ; R m×m ) be such that (171) ∀i = j ∈ {1, . . . , m}, a.e. (t, x) ∈ Q T , A i,j (t, x) ≥ 0.

Remark 8.3. The condition (171) is satisfied by the linearized system of (166) around (0, 0) thanks to (168).

We consider the adjoint system In order to prove (173), we first remark that the nonnegativity of ζ comes from (171) (see [START_REF] Protter | Maximum principles in differential equations[END_REF]Chapter 3,Theorem 13]). Then, by applying the same proof strategy to each line of (172) as performed in Theorem 4.9 and by forgetting for the moment the terms involving A i,j (t, x)ζ j , we get ∞ T + T 2 . We conclude the proof of (173) by absorbing the first right hand side term of (174) provided s ≥ C(Ω, ω)T 2 A 1/2 L ∞ (Q T ) . Difference 2: Without control, the free solution associated to a nonnegative initial data of (166) stays nonnegative and remains bounded. An adaptation of Lemma 5.1 to the system (166) holds true. But, the reason is different. It comes from [START_REF] Fellner | Global classical solutions to quadratic systems with mass conservation in arbitrary dimensions[END_REF]Theorem 1.1] which ensures global existence of classical solutions associated to nonnegative initial data for nonlinear reaction-diffusion systems with semilinearities satisfying (168), (169) and a (super)-quadratic growth (see also [START_REF] Souplet | Global existence for reaction-diffusion systems with dissipation of mass and quadratic growth[END_REF] under an additional structure assumption, the so-called dissipation of entropy).

Remark 8.4. It is worth mentioning that if the nonlinearities of (166) are bounded in L 1 (Q T ) for all T > 0 (which is the case of (170) for instance), then the solutions exist globally because the growth of the semilinearity (f i ) 1≤i≤m is less than |u| N+2 N (see [START_REF] Pierre | Global existence in reaction-diffusion systems with control of mass: a survey[END_REF]Section 1]). This concludes the proof of Theorem 8.1.

In the following result, we give a sufficient condition to ensure the global nullcontrollability of (166). Proof. As the proof is very similar to that of Theorem 2.2, we omit the details. The first step consists in steering the initial data to a nonnegative state in time T 1 := 1. This is possible thanks to Theorem 8.1. After that, we use the following comparison principle between u, the solution to This comes from the quasi-monotone nondecreasing of (f i ) 1≤i≤m which is a consequence of (168) (see [START_REF] Wu | Elliptic & parabolic equations[END_REF]Theorem 12.2.1] or also [START_REF] Pao | Nonlinear parabolic and elliptic equations[END_REF]Chapter 8,Theorem 3.1]). Then, by using (175), ( 176), (177) and the arguments of the step 2 of the proof of Theorem 2.5, we readily get ∀i ∈ {1, . . . , m}, a.e. x ∈ Ω, 0 ≤ u i (T 2 , x) ≤ δ, where T 2 is chosen sufficiently large and δ > 0 is the radius of the ball of L ∞ (Ω) m centered at 0 where the local null-controllability of (166) holds in time T = 1 (see for instance [START_REF] Fernández-Cara | Controllability of linear and semilinear non-diagonalizable parabolic systems[END_REF]Theorem 1.1] and the small L ∞ perturbations method).

∀1 ≤ i ≤ m,    ∂ t u i -d i ∆u i = f i (u 1 , . . . , u m ) in (T 1 , T 2 ) × Ω,
Then, one can steer u(T 2 , .) to 0 with an appropriate choice of the control.

Another interesting problem could be to determine if Theorem 8.1 and Theorem 8.5 can be generalized with fewer controls than equations in (166). The usual strategy of Luz de Teresa to 'eliminate controls' in a linear parabolic system (see [START_REF] De | Insensitizing controls for a semilinear heat equation[END_REF] or [2, Theorem 4.1]) seems to be difficult to implement because the Carleman inequality in L 1 (see Theorem 4.9) only provide estimates on the function (and not on its partial derivatives in time and space).

∂

  results 2.1. Small-time global nonnegative-controllability 2.2. Large time global null-controllability 2.3. Proof strategy of the small-time global nonnegative-controllability 2.4. Proof strategy of the large time global null-controllability 3. Parabolic equations: Well-posedness and regularity 3.1. Well-posedness 3.2. Maximum principle 3.3. L p -L q estimates 4. Global nonnegative-controllability of the linear heat equation with a bounded potential 4.1. Statement of the result 4.2. A precise L 2 -L 1 observability inequality for the linear heat equation with bounded potential and nonnegative initial data 4.3. A new L 1 Carleman estimate 4.4. Proof of the L 2 -L 1 observability inequality: Theorem 4.4 4.5. Proof of the linear global nonnegative-controllability: Theorem 4.1 5. A fixed-point argument to prove the small-time nonlinear global nonnegative controllability 5.1. A comparison principle 5.2. The fixed-point: definition of the application 5.3. Hypotheses of Kakutani-Leray-Schauder's fixed point theorem 1. Introduction Let T > 0, N ∈ N * , Ω be a bounded, connected, open subset of R N of class C 2 and n be the outer unit normal vector to ∂Ω. We consider the semilinear heat equation with Neumann boundary conditions: t y -∆y + f (y) = h1 ω in (0, T ) × Ω, ∂y ∂n = 0 on (0, T ) × ∂Ω, y(0, .) = y 0 in Ω, where f ∈ C 1 (R; R). Remark 1.1. All our results stay valid for Dirichlet boundary conditions (see Section 7).

|s| 0 log 2

 2 (1 + σ)dσ is open. Remark 2.7. Theorem 2.5 does not treat the case f (s) = -s log p (1+|s|) with p < 2 because of the sign condition.

Lemma 4 . 8 .

 48 Let ω 0 ⊂⊂ ω be a nonempty open subset. Then there exists η 0

Fact

  ∈ N, y k ∈ A, (138) ∀k ∈ N, y k ∈ Φ(z k ).By (138) and (124), for every k ∈ N, there exists a control h k ∈ H(z k ) such that y k is the solution to(139)    ∂ t y k -∆y k + g(z k )y k = h k 1 ω in (0, T ) × Ω, ∂y k ∂n = 0 on (0, T ) × ∂Ω, y k (0, .) = y 0 in Ω,and this solution satisfies (140) y k (T * z k , .) ≥ 0. By (136), Fact 5.3 and the point (2) of Theorem 5.4, we get that there exist a strictly increasing sequence (k l ) l∈N of integers, h ∈ H(z) and y

F

  (s) = +∞ and lim s→+∞ F (s) = 0.

∞T + T 2 + T 2 A 1 / 2 L

 2212 ζ -∆ζ = A(t, x)ζ in (0, T ) × Ω, ∂ζ ∂n = 0 on (0, T ) × ∂Ω, ζ(T, .) = ζ T in Ω.Our goal is to establish this L 1 -Carleman inequality: for every ζ T ∈ L 2 (Ω; R + ) m , the nonnegative solution ζ of (172) satisfiesm i=1 Q T e -sα ξ 2 ζ i dxdt ≤ C(Ω, ω) m i=1 (0,T )×ω e -sα ξ 2 ζ i dxdt ,(173)for any λ ≥ 1, s ≥ s 1 (λ) := C(Ω, ω)e 4λ η 0 ∞ (Q T ;R m×m ) .

  e -sα λ 2 (sξ) 2 ζ i dxdt ≤ C(Ω, ω) A L ∞ (Q T ) Q T e -sα |ζ|dxdt (174) + m i=1 (0,T )×ω e -sα λ 2 (sξ) 2 ζ i dxdt , for λ ≥ 1, s ≥ C(Ω, ω)e 4λ η 0

Theorem 8 . 5 .Application 8 . 6 .

 8586 Let α ∈ (1, 2). For each f i , we assume that (3) holds with α and (175) ∃C > 0, ∀r ∈ [0, +∞) m , exists T sufficiently large such that (166) is globally null-controllable in time T . Let α ∈ (1, 2). There exists T > 0 such that the system       ∂ t u -∆u = -u log α (2 + |u| + |v|) + h 1 1 ω in (0, T ) × Ω, ∂ t v -∆v = -v log α (2 + |u| + |v|) + h 2 1 ω in (0, T ) × Ω, ∂u ∂n = ∂v ∂n = 0 on (0, T ) × ∂Ω, (u, v)(0, .) = (u 0 , v 0 )in Ω, is globally null-controllable in time T > 0.

1 , T 2 )

 12 × ∂Ω, u i (T 1 , .) = u i,T 1 in Ω,and v, the nonnegative (global) solution to the ordinary differential system (176)∀1 ≤ i ≤ m, vi (t) = -f i (v(t)) in (T 1 , +∞), v i (T 1 ) = u i,T 1 L ∞ (Ω) + 1 , that is to say (177) ∀i ∈ {1, . . . , m}, ∀t ∈ [T 1 , T 2 ], a.e. x ∈ Ω, 0 ≤ u i (t, x) ≤ v i (t).

  is globally nonnegativecontrollable (respectively globally nonpositive-controllable) in time T . Remark 2.3. Theorem 2.2 is almost sharp because it does not hold for α > 2 according to Theorem 1.8. The case where |f (s)| ∼ |s| log 2 (1 + |s|) as |s| → +∞ is open. Remark 2.4. Theorem 2.2 does not treat the case f (s) = -s log p (1+|s|) with p < 2 because of the sign condition. 2.2. Large time global null-controllability. The second main result of this paper is the following one.

  We repeat the step 1 for ψ. • Step 4: We repeat the step 2 for ψ. • Step 5: We sum the estimates of the step 2 and the step 4 to get rid of the boundary terms. Remark 4.10. The 'trick' of the proof to get rid of the boundary terms is inspired by the proof of the usual L 2 Carleman estimate for Neumann boundary conditions due to Andrei Fursikov and Oleg Imanuvilov (see

• Step 2: We get an estimate which looks like to (31) up to some boundary terms. • Step 3:

Acknowledgments. I would like to very much thank Karine Beauchard and Michel Pierre (Ecole Normale Supérieure de Rennes) for many fruitful, stimulating discussions, helpful advice. I am also grateful to Frédéric Marbach (Ecole Normale Supérieure de Rennes) for suggesting me to look at [10, Lemma 9] and to Sylvain Ervedoza (Institut de Mathématiques de Toulouse) for reading a preliminary draft of this paper and for pointing me the two references [29] and [32].


Y A Bioud
email: y.bioud@usherbrooke.ca

A Boucherif
email: a.boucherif@usherbrooke.ca

E Paradis

A Soltani

D Drouin

R Arès

DEFECT-FREE, HETERO-EPITAXY THROUGH ELECTROCHEMICAL ETCHING OF GERMANIUM FOR MULTIJUNCTION SOLAR CELLS APPLICATIONS

A low-cost method to reduce the threading dislocations density (TDD) in relaxed germanium (Ge) epilayers grown on silicon (Si) substrates is presented. Ge/Si substrate was treated with post epitaxial process to create a region with a high density of nano-voids. The nano-voids are formed by a selective electrochemical porosification through dislocations followed by thermal annealing which act as a free surface for inhibiting dislocation propagation. The TD density is reduced significantly which will be crucial in cutting manufacturing costs of III-V multi-junction solar cells, if the bulk Ge substrates can be replaced with our engineered substrate.

INTRODUCTION

Multi-junction III-V concentrator cells have the most important energy conversion efficiency, competing with any other solar cell technology. Germanium appears to be the material of choice for the bottom cell to cover the IR part of the solar spectrum. A high efficiency of 40.7% was measured for a metamorphic three junctions GaInP/GaInAs/Ge cell at 240 suns. This architecture has achieved system efficiency of 30%. A road map based on the integration of Ge on Si wafers can be a solution to replace bulk Ge substrates and would result in a significant cost reduction. However, due to the important 4.2% lattice mismatch between Si and Ge, strain is introduced in epitaxial Ge layers on Si which leads to the generation of a high density of misfit dislocations. Dislocations act as recombination centers for carriers and strongly degrade the solar cell performance. Several techniques of dislocation reduction were reported in recent years such as graded SiGe buffer layers, cyclic annealing, aspect-ratio-trapping (ART), selective area depositions (SAD) and deep substrate patterning (DSP) [START_REF] Ye | Germanium epitaxy on silicon[END_REF]. However, the treading dislocations density (TDD) in the Ge layer remains higher than 10 6 cm -1 , reducing its usefulness for many device applications. In this talk, we will discuss a new strategy for making cost effective Ge/Si virtual substrate which is based on a simple and cost effective technique that uses nano-voids as a barrier for dislocation propagation. Then, we will present detailed optimization of the porosification and the annealing of Ge on Si and study the effect of different process parameters on TDD and on surface roughness.

EXPERIMENTAL RESULTS AND DISCUSSIONS

Approach:

As sketched in Fig 1, Ge/Si substrate was treated with post epitaxial process to create a region with a high density of nano-voids within the Ge layer on Si is expected to act as a barrier for threading dislocations propagation. Nano-voids are formed by electrochemical porosification followed by thermal annealing. The process is completed by CMP (chemical mechanical planarization) step to obtain an epi-ready virtual substrate [START_REF] Bioud | Low cost Ge/Si virtual substrate through dislocation trapping by nanovoids[END_REF].

Electrochemical etching of dislocated Ge/Si layer:

The porous morphologies obtained from the electrochemical etching of Ge/Si layer are quite different from that of the bulk Ge. For bulk Ge, the layer structure shows uniformly distributed meso-pores with controllable nanostructure and size [3]- [START_REF] Bioud | MESOPOROUS GERMANIUM BY HIGH FREQUENCY BIPOLAR ELECTROCHEMICAL ETCHING FOR OPTICAL SENSOR APPLICATIONS[END_REF]. However, for dislocated Ge/Si, non-uniformly distributed meso-pores are obtained (see Fig 2a and4b). The porous layer consists of two zones: (i) a thin PGe layer formed at the surface, (ii) a thick PGe layer with weakly branched pores that trend to follow threading dislocation cores. Calibration of etching rates has been done by varying the current density in the electrochemical process. Fig 2c illustrates the dependence of the etching rate on the current density of bulk and dislocated Ge. The etch rate ratio between dislocated and bulk regions shows very high values up to 6. The preferential etching via dislocations can be explained by their high electrical conduction and their role in the transport properties with the electrolyte.

Nanovoid layer formation:

The morphological change of porous Ge during annealing is based on thermally activated jumps of surface atoms to stable positions following to the mass transport mechanism, for which atoms have a maximum of covalent bonds until their outer energy level is full. In terms of geometry, atoms diffuse from the larger curvature surface toward the lower curvature surface down a gradient potential. Therefore, the pores in the volume are enlarged and the pores on the surface are closed. This reveals the formation of a crystalline region in the top surface and a voided region in the volume.

Dislocation-void interactions

Fig. 3 revealed that threading dislocations in relaxed Ge epilayer grown on Si were pinned and bent towards the nanovoid area. Similar effect was observed by Myers et al. he suggests that this overlap is due to an attractive force created by a population of nano-voids with a binding energy that depends up-on the surface tension of the void and the relative positions of the interacting dislocation [START_REF] Myers | Forces between cavities and dislocations and their influence on semiconductor microstructures[END_REF]. Also, during thermal annealing, TDs were stopped moving at the void probably because their gliding distance to the voids side-walls were very short. Therefore, the formation of surface-free regions located between the top surface of Ge and Ge/Si interface could be an effective tool to reduce significantly the TDD in Ge/Si layer. The TD free top layers offer a unique potential and a technologically attractive path for overgrowth of high quality Ge based optical devices.

Figure1: Schematic illustration of dislocation trapping approach, which starts by the electrochemical porosification of the Ge/Si layer, followed by thermal treatment to form nano-voids.

Figure2:

  Figure2: Etching rate evolution versus the current density of bulk Ge and dislocated Ge/Si layer (a) and their cross-section SEM view (b,c) showing the preferential etching via dislocations.

Figure3:

  Figure3: TEM images shows gliding dislocations pinned by nano-voids.

CONCLUSIONS

In this work, we proposed a new architecture of low cost virtual substrate engineering. This concept is promising for the fabrication of dislocation-free Ge on Si substrate. The resulting epi-ready surface will be used as the starting point for the deposition of the III-V junctions in the 3J PV cell. Using a simple, inexpensive process through electrochemical etching and thermal annealing will be crucial in cutting manufacturing costs and enabling market penetration of Ge based devices.


A S Caro-Bretelle

F Bernardeau

Romain Léger

Perrin Didier

J C C Benezet

P Ienny

A S Caro

D Perrin

R Leger

Computational modelling of void growth in Phenolic Molding Compounds

Keywords: Phenolic molding compounds, Damage, Volume change, Finite element

HAL is

Introduction

As most thermoset materials, phenolic molding compound (PMC) wastes generate environmental problems and very few recycling solutions have been proposed for this type of material. In a previous paper, authors consider the possibility to use PMC as fillers in a polymer matrix [START_REF] Bernardeau | Development of a recycling solution for waste thermoset material: waste source study, comminution scheme and filler characterization[END_REF], [START_REF] Bernardeau | Valorization of waste thermoset material as a filler in thermoplastic: Mechanical properties of phenolic molding compound waste-filled PP composites[END_REF]. Mineral fillers such as calcium carbonate, talc, silica, clay, etc., are usually used to reduce costs, to give some flame-retardancy and/or in some cases to enhance mechanical material properties. The point of this article is to explore the mechanical enhancement possibilities of PMC incorporated into PolyPropylene (PP) matrix. From authors' knowledge, there are very few studies about its incorporation as a fillers into polymers [START_REF] Żuchowska | Some physical properties of polypropylene-phenolic microsphere blends[END_REF], [START_REF] Chiang | Modification of polypropylene, blending with resole type phenol-formaldehyde resins[END_REF]. The major point is the weak interaction between PMC and PP (due to their chemical discrepancies) leading to poor mechanical properties. PP/PMC composites were compatibilized with maleic anhydride-grafted PP (PP-g-MA) to solve this drawback [START_REF] Zhang | Dynamically cured polypropylene/Novolac blends compatibilized with maleic anhydride-g-polypropylene[END_REF].

The present study aims at improving the comprehension of the micromechanics of deformation involved during a tensile test, especially the inelastic volume variation.

To better understand the key role of the fillers on the material response, a predictive modeling is constructed from mechanical tests performed for each of composite phase components. The Gurson-Tvergaard-Needleman (GTN) model ( [START_REF] Gurson | Continuum Theory of Ductile Rupture by Void Nucleation and Growth: Part I-Yield Criteria and Flow Rules for Porous Ductile Media[END_REF]- [START_REF] Tvergaard | Analysis of the cup-cone fracture in a round tensile bar[END_REF]) has been used to describe the elasto-visco-plastic and damageable behavior of PP matrix. Actually, in references [START_REF] Boisot | Experimental investigations and modeling of volume change induced by void growth in polyamide 11[END_REF]- [START_REF] Selles | Voiding mechanisms in semicrystalline polyamide 6 during creep tests assessed by damage based constitutive relationships and finite elements calculations[END_REF], authors used the same strategy to describe volume change in polyamide 11. A Kachanov elastic damage model is used to characterize the brittle behavior of bakelite [START_REF] Kachanov | Introduction to Continuum Damage Mechanics[END_REF]. Knowing the microstructure, a predictive Finite Element Analysis (FEA) was conducted as an useful tool to understand local phenomena leading to composite breakage.

The paper is structured as follows: section 2 presents materials and experiments; section 3 highlights modelling; section 4 discusses about results and section 5 brings a conclusion.

Materials and experiments

Materials

Bakelite

Phenolic Molding Compounds known as Bakelite TM used in this study was recovered from WEEE (Waste Electrical and Electronic Equipment) casing. This resin is based on phenol and formaldehyde condensation. They were identified via Fourier Transform Infrared Spectroscopy (FTIR) analysis and manually sorted from other plastic parts. It was found in a previous research that this resin was reinforced with various fillers and additives such as calcium carbonate, silica, magnesium carbonate and wood. Bakelite densities is between 1.31 and 1.39 g/L [START_REF] Bernardeau | Valorization of waste thermoset material as a filler in thermoplastic: Mechanical properties of phenolic molding compound waste-filled PP composites[END_REF].

PMC waste is therefore transformed into usable fillers of appropriate size by comminution. The polydispersity of PMC filler size distribution was quantified by laser diffraction Particle Sizing Technique (PSD) (Figure 1). The procedure is fully described in [START_REF] Bernardeau | Valorization of waste thermoset material as a filler in thermoplastic: Mechanical properties of phenolic molding compound waste-filled PP composites[END_REF].

16kg). A

Polypropylene grafted with maleic anhydride (PP-g-MA) Orevac CA100 from Arkema was also used in this study as compatibiliser.

Composite

Particulates composite were prepared by extrusion. All materials were dried at 80°C during 24h in an air circulating oven prior to compounding. A Clextral BC21 corotating twin screw extruder was used. The temperature was set to 220°C at the primary feeding port, and to 210°C at the extrusion head. A rotation speed of 250 rpm was used. An overall extrusion rate of 5kg/h was set for all composite. PMC fillers were fed in a port located in the middle of the barrel. PP-g-MA was dry-mixed with polypropylene before processing. The extruded material was cooled in a water bath and granulated. Compounds with 20 % in weight of fillers were produced.

After compounding, extruded materials were dried at 80°C during 24h in an air circulating oven. The pellets were then injection molded with a Krauss Maffei KM50-T180CX. Dog bone samples (ISO 1A according to ISO 527-2) were injected for mechanical testing.

The crystallinity was measured by differential scanning calorimetry (DSC). For pure polypropylene, a crystallinity of 48% was measured. The crystallinity of composites is around 46%.

In order to improve the adhesion between the filler and the matrix, 3% by weight of PP-g-MA was added into the composite.

Three different formulations (PP, PP with PMC with or without PP-g-MA), presented in Table 1, were thus prepared for this study. In the following, PP, PP/PMC, PP/PMC/gMA denote respectively the neat PP, PP with 20% in weight of Bakelite TM and PP with 20% in weight of Bakelite TM and 3% by weight of PP-g-MA. The nominal stress is given by the following expression (eq. 1):

(

) 1
where F is the recorded load and S 0 the initial sample section surface.

The optical extensometer involves a high resolution Charge-Coupled Device (CCD) camera (Redlake Megaplus II, 1920 x 1080 contiguous and square pixels, coded in 256 grey levels), set in front of the specimen, which records images during the test (Figure 2). The optical axis of the camera remains perpendicular to the in-plane surface of the specimen during the test. The images acquisition is commanded by a LabVIEW ® software which allows the simultaneous acquisition of the images and the data from the testing machine (such as load and crosshead displacement). According to the test speed used, images are recorded every 2s. The scale factor is fixed to 42 µm per pixel.

The in-plane strains were deduced from digital image correlation (DIC) following a method well described in [START_REF] Christmann | Mechanical behaviour at large strain of polycarbonate nanocomposites during uniaxial tensile test[END_REF]. A transverse isotropic assumption is considered and was validated in a previous study on polycarbonate nanocomposite specimen submitted to uniaxial tensile test [START_REF] Christmann | Mechanical behaviour at large strain of polycarbonate nanocomposites during uniaxial tensile test[END_REF].

The volumetric strain is defined as (eq. 2):

(

From (eq. 2) the part of elastic volumetric strain can be computed (eq. 3):

(

) 3
where E and  are the elastic material parameters (Young modulus and Poisson's ratio). 2). The ultimate strain is lowered by the incorporation of fillers since they act as defects which initiate the composite breakage. The presence of PP-g-MA produces the opposite effect (see Figure 5 (a) for a lower imposed speed). The measured volumetric strain is higher for the composites in comparison of the neat polymer (Figure 6 (b)). For example, at 5% of strain, volumetric strain value is 5 times (respectively 2 times) higher for the composite with PMC (with both PMC and PP-g-MA). The fillers increase the composite volumetric variation. The presence of PP-g-MA into the composite reduces this variation. The anhydride maleic could maintain cohesion at the interface matrix/fillers and limits the global volumetric variation. This latest assumption will be checked with an in-situ

tensile

Uniaxial tensile tests with environmental scanning electron microscopy (ESEM)

Tensile tests were carried out by ESEM Quanta FEG 200 on rectangular shaped samples (50*10*4 mm 3 ) with a 10 mm gauge length and repeated three times. Two notches were performed symmetrically on each side of samples (corresponding to a minimal section of 6*4 mm 2 ) to initiate the fracture in the observation field (Figure 7).

The device is equipped with a 5 kN load cell and the test speed is set at 5 mm/min.

One image is recorded every ten seconds and force and displacement are recorded at a frequency of 2 Hz. such as decohesion at the interfaces clearly appear around the apparition of the maximum stress. At the same level of stress, the interface for the PP/PMC/gMA composite seems to be stronger, since the previous observed decohesion is replaced by a breakage inside the Bakelite TM (Figure 7).

Modelling

Behavior

The literature shows an important number of constitutive models, namely "damaged model", able to account mechanical degradation in ductile materials. Usually, these models include new internal variables and evolution laws coupled to plasticity problems. Relevant approaches are, for example, the works of Gurson [START_REF] Gurson | Continuum Theory of Ductile Rupture by Void Nucleation and Growth: Part I-Yield Criteria and Flow Rules for Porous Ductile Media[END_REF], Lemaitre [START_REF] Lemaitre | A Continuous Damage Mechanics Model for Ductile Fracture[END_REF] and Rousselier [START_REF] Rousselier | Ductile fracture models and their potential in local approach of fracture[END_REF]. The present work focuses on Gurson-type (for PP) and

Lemaitre-type (for Bakelite TM ) material models.

PP: modeling from a Gurson-type damage model

Gurson-type approach is based on mechanics of porous media. The damage indicator, related to the degradation of the material, is the void volume fraction (i.e.

the porosity) which progressively down-scale the yield surface. In the early eighties Tvergaard and Needleman [START_REF] Tvergaard | Analysis of the Cup-Cone Round Tensile Fracture[END_REF], [START_REF] Tvergaard | Influence of voids on shear band instabilities under plane strain conditions[END_REF] extended the Gurson approach, namely GNT model, including material hardening, multiple voids and void coalescence. More recently, this model has been written within the thermodynamical framework of Lemaitre and Chaboche [START_REF] Hughes | A simultaneous SAXS/WAXS and stress-strain study of polyethylene deformation at high strain rates[END_REF].

The volume strain, usually used to quantify the porosity, is measured by some authors with SEM, wide-angle X-ray scattering (WAXS) or small-angle X-ray scattering (SAXS) [START_REF] Boisot | Experimental investigations and modeling of volume change induced by void growth in polyamide 11[END_REF], [START_REF] Addiego | Characterization of volume strain at large deformation under uniaxial tension in high-density polyethylene[END_REF]- [START_REF] Castagnet | Cavitation in strained polyvinylidene fluoride: Mechanical and X-ray experimental studies[END_REF]. Even if initial porosity has not been observed for the neat PP, this approach remains consistent as observed volume change is simultaneous to void nucleation and growth (see results of section 2.3).

In this damage formulation the yield criterion uses hydrostatic pressure and porosity:

the criterion of plasticity and the plastic potential, depend on a macroscopic stress and an effective porosity fraction in the following way (eq. 4):

where is the stress tensor, , and are model parameters.

Effective stress is implicitly evaluated from eq. 4.

The plastic flow potential is written as (eq. 5):

(

) 5
where p is the effective plastic strain and are material parameters.

The viscoplastic strain is written with a Norton law: [START_REF] Gurson | Continuum Theory of Ductile Rupture by Void Nucleation and Growth: Part I-Yield Criteria and Flow Rules for Porous Ductile Media[END_REF] where are material parameters.

The volumetric change can therefore be divided in two parts:

(7)

Bakelite: modeling from continuum damage mechanics

Lemaitre-type approach is based on continuum damage mechanics. A macroscopic damage variable d is introduced to describe the progressive degradation of the material [START_REF] Kachanov | Introduction to Continuum Damage Mechanics[END_REF]. In Bakelite TM , we suppose that the damage accumulation occurs under elastic deformation as Bakelite TM is an elastic brittle material. Then the stress can be expressed as follows: [START_REF] Tvergaard | Analysis of the cup-cone fracture in a round tensile bar[END_REF] where are material parameters and is the elastic part of the strain tensor.

is the elasticity tensor depending on Young modulus and Poisson's ratio.

Discussion

The non-linear numerical simulation has been carried out using the FE software Zebulon, which has been developed at Mines Paristech [START_REF] Besson | Object-Oriented Programming Applied to the Finite Element Method Part II. Application to Material Behaviors[END_REF].

Identification of material parameters

Material parameters were identified by using an inverse method of optimization of ZeBuLON software.

PP matrix

The method used a minimization algorithm based on that of Levenberg Marquardt leading to a set of parameters (given in Table 3) that conduct to a good accordance between simulated and experimental stress/strain (Figure 8 (a)). The porosity associated to cavitational volumetric strain is deduced from eq. 7 and compared to the experimental one in Figure 8 (b) for both imposed strain rates. Young's modulus and Poisson's ratio are deduced from optical measurements at low strain. Bakelite mechanical properties are summarized in

Predictive modelling

For sake of simplicity, a plane stress assumption is assumed for the numerical computations. This strong hypothesis is very far from reality; nevertheless finite elements evaluations are faster in this case and a 2D microstructure enables, in a first approach, to understand micro mechanisms of deformation. The interface between matrix and fillers is assumed to be perfect and phase properties are described through the set of parameters of Table 3. As the PP crystallinity is not such impacted by the fillers (see section 2.3.1), we assume that the properties of PP matrix into the composite are similar to those of the neat PP.

An example of microstructure (generated via DIGIMAT software) is given in Figure 9, the granulometry of the fillers size is the one mentioned in Figure 1. Mesh microstructure based on triangular elements and boundary conditions are given in this table. We assume the structure symmetry (on the line of symmetries components of displacements are zero). A prescribed displacement (10 mm, corresponding to 10% of strain) is applied on the top of the sample. The size of this representative volume element (RVE), namely box length in Figure 9 and the number of elements correspond to minimal values leading to a convergent stress/strain numerical data. This RVE model is thus useful to predict the mechanical properties of PP/Bakelite composites. The major advantage of numerical simulations is the access to microstructural information. For example, the average stresses in both matrix and fillers and the damage area in the Bakelite TM can be computed. This latest value corresponds to the relative surface of fillers that have reached breakage (d=1 in eq. 8). Average stress for PP in PP/PMC/g-MA composite is plotted vs. strain in Figure 11 (a) for both imposed speeds. On average, the higher the imposed speed, the higher the average stress in the matrix, unlike bakelite with an average behavior almost insensitive to the applied speed. The fillers do not impact the matrix behavior from this point of view.

Regarding the Bakelite TM (Figure 11 More interesting, a higher average porosity is observed in the PP matrix as the fillers are added and as the velocity decreases (see Figure 12a). This is not in contradiction with the assessment that the average volumetric strain of composites increases with the imposed speed, as the volumetric strain in Bakelite follows an inverse trend (see Figure 12a). This is a consequence of this higher solicitation of bakelite as the tensile test speed increases. Beside these average variations, the histogram representing the frequencies of porosity levels in elements of the meshed structure reveals higher level of porosity for the speed 50 mm/min (Figure 12b). For these two-phase composites whose softer matrix is associated with viscoplastic behavior, the matrix accommodating capacity is annihilated at high solicitation rates.

This leads to greater damage in particulate reinforcements when the interface is cohesive. The particulate reinforcement acts as a fuse, thus delaying the cavitation damage of the matrix.

Conclusion

The mechanical properties of a polypropylene based composites have been studied in this work. Instrumented (photomechanics) experiments (uniaxial tensile tests) were carried out on PP and PP filled with 20% by weight of phenolic molding compound waste powder to highlight the damage process. PP-g-MA was added as a compatibiliser. At first, global measured values (as volume variation and load during the test) lead, for the neat PP, to a set of mechanical parameters describing the GTN model that well reproduce the global stress/strain curve. Knowing the fillers mechanical properties, PP/PMC/g-MA is considered as a two phases composite as the adhesion between matrix and fillers is supposed to be perfect. A FE code is used both to predict composites behavior and to understand the damage micromechanisms involved during any during any structuraltest. It was found that bakelite can be a good candidate to reinforce polymer. As usual fillers, it improves Young's modulus and strength. Tensile tests performed with a higher strain rates velocities lead to both delay the cavitation damage of the matrix and increase the damage level of fillers. The charges concentrate then the damage limiting the cavitation which usually leads to a premature rupture of the composite.

Figure 1 :

 1 Figure 1: Recycling of Bakelite, from WEEE to its incorporation in PP matrix with a controlled size distribution

Figure 2

 2 Figure 2: Testing apparatus

Figure 3 :

 3 Figure 3: Stress vs. strain (a), Volumetric Strain (b) in uniaxial tensile tests at several imposed strain rates for PP matrix

Figure 4 :

 4 Figure 4: Bakelite TM samples for mechanical testing

Figure 5 :

 5 Figure 5: Stress vs. strain (a), Volumetric Strain (b) in uniaxial tensile tests at 5 mm/min for PP matrix and composites with and without PP-gMA

Figure 6 :

 6 Figure 6: ESEM micrographs of PP with 20 % w of PMC, scale factors 150 (a), 250 (b), 1000 (c), 2500 (d)

Figure 7 :

 7 Figure 7: Tensile tests on PP with 20 wt% of PMC (with and without PP-g-MA), ESEM observation near to the notch

Figure 8 :

 8 Figure 8: Modelling of PP at several imposed speed, stress vs. strain (a) cavitation volumetric strain vs. time (b)

Figure 9 :

 9 Figure 9: Finite Element Modelling: structure and boundary conditions

Figure 10 :

 10 Figure 10: Stress vs. strain (a) volumetric strain vs. strain (b) for PP/PMC/gMA composite at several imposed speeds, experiments and modelling

  (b)) the initiation of damage occurs at the same time, independently of the imposed speed, whereas its development is faster as the velocity increases. Tests at high speeds are therefore suitable to discriminate interfacial properties in a composite as the particles involve a faster damage.

Figure 11 :

 11 Figure 11: Average stresses in PP and Bakelite phases (a) and damaged area of bakelite (b) in PP/PMC/g-MA composite vs. strain for several imposed speed

Figure 12 :

 12 Figure 12: Average porosity of PP, average volumetric strain in Bakelite versus strain (a) and porosity repartition into PP phase

Table 1 :

 1 Materials under study

	Acronym	PP	20%w PMC	3%w PP-g-MA
	PP	×

allows the recording of time, load and elongation. Let's define by x the tensile direction, y perpendicular to x and in the planar surface and z the out of plane direction.

Table 2 :

 2 test with ESEM equipment (see the next part 2.3: microscopic observations).

		PP	PP/PMC	PP/PMC/gMA
	E (MPa)	1701	2197	2839
		0.38	0.38	0.38
		25	22	30
		0.15	0.06	0.1

Elastic and ultimate properties deduced from experiments for PP, PP/PMC and PP/PMc/gMA composites

Table 2 .

 2 Ultimate stress ( ) and strain ( ) are recorded in this table.

		Bakelite TM		PP
	E (MPa)	7800	E (MPa)	1701
		0.38		0.38

Table 3 :

 3 Material parameters for PP and Bakelite TM