Boumediene Abdellaoui

Kheireddine Biroud

El-Haj Laamri
email: el-haj.laamri@univ-lorraine.fr.

E.-H Laamri

EXISTENCE AND NONEXISTENCE OF POSITIVE SOLUTIONS TO A FRACTIONAL PARABOLIC PROBLEM WITH SINGULAR WEIGHT AT THE BOUNDARY

Keywords: fractional Nonlinear parabolic problems, Singular Hardy potential, complete blow-up results. Mathematics Subject Classification, 2010: 35B05, 35K15, 35B40, 35K55, 35K65

In this work, we consider a nonlocal semilinear parabolic problem related to a fractional Hardy inequality with singular weight at the boundary. More precisely, we consider the problem

where 0 < s < 1, Ω ⊂ IR N is a bounded regular domain, d(x) = d(x, ∂Ω), p > 0 and λ > 0 is a positive constant. The initial data u 0 0 is a nonnegative function in a suitable Lebesgue space that we make precise later.

The main goal of this work is to analyze the interaction between the parameters s, p and λ in order to show the existence or the nonexistence of solution to problem (P ) in a suitable sense. We will show that our results have a significative difference with respect to the local case s = 1.

Introduction and main results

Recently, much attention has been devoted to the fractional Laplacian (-∆) s , which for 0 < s < 1 is defined on smooth functions as follows (1.1) (-∆) s u(x) := a N,s P.V.

R N u(x) -u(y) |x -y| N +2s dy, where P.V stands for the Cauchy principal value and a N,s is a dimensional constant given by (see e.g [21, section 2]),

a N,s := s2 2s Γ( N +2s 2 ) π N 2 Γ(1 -s)
where Γ denotes the Gamma function. The choice of the constant a N,s is motivated, among others, by the following proposition (see [START_REF] Di Nezza | Hitchhiker's guide to the fractional Sobolev spaces[END_REF]Proposition 4.4]) : Proposition 1.1. For all φ ∈ C ∞ 0 (R N ), the following statements hold : (1.2) lim s→0 + (-∆) s φ = φ and lim s→1 -(-∆) s φ = -∆φ where ∆ is the classical Laplacian.

Of course, there are other ways to define the fractional Laplacian. For this, we refer the interested reader to [START_REF] Di Nezza | Hitchhiker's guide to the fractional Sobolev spaces[END_REF][START_REF] Bisci | Variational Methods For Nonlocal Fractional Problems[END_REF][START_REF] Caffarelli | An extension problem related to the fractional Laplacian[END_REF] and the references therein. Note that the fractional Laplacian is a nonlocal operator and that this raises several technical difficulties. However, great attention has recently been focused on the study of nonlocal operators because they play a crucial role in describing several phenomena, as, for instance, the thin obstacle problem, anomalous diffusion, quasigeostrophic flows, ... In spirit of this nonlocal setting, the aim of this paper is the study of the following nonlocal parabolic problem :

(1.3)      u t + (-∆) s u = λ u p d 2s in Ω T = Ω × (0, T ), u(x, 0) = u 0 (x) in Ω, u = 0 in (IR N \ Ω) × (0, T ),
where Ω is a bounded regular domain of IR N , d(x) = dist (x, ∂Ω), p > 0, λ > 0, u 0 0 is a nonnegative measurable function, 0 < s < 1 and (-∆) s is the fractional Laplacian .

More precisely, our goal here is to get natural conditions on the parameters s, p, λ and initial data in order to prove the existence of positive solutions to the problem (1.3). By solution, we mean energy solution or weak solution (for more specifics, see Definition 2.4 and Definition 2.5).

In order to give precise statements of the results obtained, let us introduce first the following Hardy's inequality (for a proof, see [START_REF] Dyda | A fractional order Hardy inequality[END_REF] and [START_REF] Filippas | Sharp Trace Hardy Sobolev Mazýa Inequalities and the Fractional Laplacian[END_REF]Theorem 5]) under the form that we will use.

Theorem 1.2. Assume that s ∈ (0, 1), N ≥ 2s and let Ω be a bounded regular domain (C 1,1 regularity is sufficient). Then, there exists a positive constant Λ s (Ω) such that for all φ ∈ C ∞ 0 (Ω)

(1.4) Λ s (Ω) Ω φ 2 (x) d 2s (x) dx ≤ a N,s 2 DΩ |φ(x) -φ(y)| 2 |x -y| N +2s dxdy,
where [START_REF] Abatangelo | Large Solutions for Fractional Laplacian Operators[END_REF] and Ω is a convex domain, then the constant Λ s (Ω) is independent of Ω and it is given by

D Ω := IR N × IR N \ Ω C × Ω C . If s ∈ [ 1 2 ,
(1.5) Λ s (Ω) = Λ * s := Γ 2 (s + 1 2 ) π .
Remark 1.3.

• In all cases, we always have Λ s (Ω) ≤ Λ * s , see [START_REF] Filippas | Sharp Trace Hardy Sobolev Mazýa Inequalities and the Fractional Laplacian[END_REF]Theorem 5]. • Let us point out that Λ s (Ω) is achieved if and only if Λ s (Ω) < Λ * s . We refer to [START_REF] Bogdan | The best constant in a fractional Hardy inequality[END_REF] for more details.

Before going further, let us mention some previous works concerning the local case, i.e. s = 1 in order to compare it with the nonlocal case. First, Problem (1.3) has been recently treated in [START_REF] Abdellaoui | A semilinear parabolic problem with singular term at the boundary[END_REF] (see also [START_REF] Abdellaoui | Nonlinear elliptic problem related to the Hardy inequality with singular term at the boundary[END_REF] for the elliptic case). In their work, the authors have proved a strong nonexistence result if p < 1. However, for p > 1 and for adequate assumptions on u 0 , they proved the existence of a positive solution using suitable sub and supersolution and iteration arguments.

If p = 1 and Ω is a bounded convex domain, then the authors proved in [START_REF] Cabré | Existence versus explosion instantanée pour des équations de la chaleur linéaires avec potentiel singulier[END_REF] a strong nonexistence result for λ > Λ * 1 . As a consequence, a complete and instantaneous blow-up result was proved for approximating problems. Second, in the case where the weight d -2 is replaced by |x| -2 with 0 ∈ Ω, the best constant in the Hardy's inequality is Λ * * = N -2 2 2 , see for instance [START_REF] Garcia Azorero | Hardy inequalities and some critical elliptic and parabolic problems[END_REF] and the references therein. Moreover, if p = 1, the authors proved in [START_REF] Baras | The heat equation with a singular potential[END_REF] that the existence of solutions holds if and only if λ ≤ Λ * * . For p > 1, a strong nonexistence result was obtained in [START_REF] Brezis | Some simple nonlinear PDE's without solution[END_REF].

In the nonlocal case (i.e s < 1), a different phenomenon appears in booth cases p < 1 and p = 1.

The main results of this paper can be summarized in the following points. • If p < 1, we will show that Problem (1.3) has a nonnegative solution, in a suitable sense, for all u 0 ∈ L 1 (Ω). This makes a significative difference in comparing with the local case s = 1, where a strong nonexistence result is proved in [START_REF] Abdellaoui | A semilinear parabolic problem with singular term at the boundary[END_REF]. Here we will prove the following result.

Theorem 1.4. Assume 0 < s < 1 and 0 < p < 1. Then, for all 0 ≤ u 0 ∈ L 1 (Ω), Problem (1.3) has a positive distributional solution. Moreover, if s > 1 2 and 0 < p < 2s -1, then u is a weak solution in the sense of Definition 2.5 given below.

• In the linear case p = 1, we will show that the range of the parameter s, together with the validity of the Hardy's inequality with optimal constant stated in Theorem 1.2 and Remark 1.3, will be decisive for existence and nonexistence of positive solutions for λ large. The main result in this case is the following. Theorem 1.5. Assume that s ∈ (0, 1 2 ). Then, for all λ > 0, Problem (1.3) has a positive weak solution. If s ∈ [ 1 2 , 1) and Λ s (Ω) = Λ * s (that includes the case of convex domain), then the existence of a nonnegative solution holds if and only if 0 < λ ≤ Λ * s . • Finally, in the superlinear case p > 1, we are able to show the existence of a nonnegative solution for a suitable u 0 ∈ L ∞ (Ω). More precisely, we will prove the next theorem.

Theorem 1.6. Assume that s ∈ (0, 1) and p > 1. Then, there exists u 0 ∈ L ∞ (Ω) such that Problem (1.3) has a positive bounded solution.

As far as we know, all the results presented here are new. For the sake of completeness and for the reader's convenience, we have tried to write this paper in almost self contained form. Moreover, we give precise references for all points that are not detailed in this work.

To end this section, our paper is organized as follows. In Section 2, we give some auxiliary results related to fractional Sobolev spaces and some functional inequalities. We also define the two notions of solution that we will use systematically along the paper : energy solutions and weak solutions. The case 0 < p < 1 is treated in Section 3. Using monotone arguments and suitable test functions, we are able to prove the existence and uniqueness of a global weak solution that is in a suitable Sobolev space. This seems to be surprising since in the local case a strong nonexistence result holds with complete blow-up for the sequence of solutions to the approximating problems. In section 4, we treat the linear case p = 1. According to the value of s, we will show the existence and the nonexistence of solution for λ large. In particular, if s ∈ (0, 1 2 ), we are able to prove the existence and the uniqueness of a positive weak solution for all λ > 0. However, if s ∈ [ 1 2 , 1) and Λ * s (Ω) = Λ * s (that holds for example if Ω is convex), then the existence of solution holds if and only if λ ≤ Λ * s . The superlinear case p > 1 is treated in Section 5. Under suitable assumptions on the initial data and using monotone arguments, we are able to get the existence of a bounded positive solution. Finally, in the last section we give some extension to the case of the spectral fractional Laplacian.

2. The functional setting and tools 2.1. Functional framework. As already announced in the introduction, we are working in a bounded domain Ω with homogeneous Dirichlet boundary conditions u = 0 in IR \ Ω and not simply u = 0 in ∂Ω. This makes the classical fractional Sobolev space approach not sufficient for studying our problem. Then, we need to set a natural functional framework that is inspired by (but not equivalent to) the fractional Sobolev spaces which will allow us to deal with the Dirichlet boundary condition.

Of course, this subsection can be skipped by readers already familiar with them.

Let Ω ⊂ IR N and s ∈ (0, 1). In the sequel, we denote

D Ω = R 2N \(Ω C × Ω C
). In the following, we denote by H s (Ω) the classical fractional Sobolev space defined as

H s (Ω) =: u ∈ L 2 (Ω) ; Ω Ω |u(x) -u(y)| 2
|x -y| N +2s dxdy < +∞ , endowed with the norm

(2.1) u H s (Ω) = ||u|| L 2 (Ω) + a N,s 2 Ω Ω |u(x) -u(y)| 2 |x -y| N +2s dxdy 1 2 .
Now, we introduce the space H s 0 (Ω) as the completion of C ∞ 0 (Ω) with respect to the norm of H s (IR N ). Notice that if u ∈ H s 0 (Ω), we have u = 0 a.e. in IR N \ Ω. It is clear that for u ∈ H s 0 (Ω), we have

I R N I R N |u(x) -u(y)| 2 |x -y| N +2s dxdy = DΩ |u(x) -u(y)| 2 |x -y| N +2s dxdy.
If Ω is a bounded domain, then using Poincaré's inequality, we can endowed H s 0 (Ω) with the equivalent norm

||u|| 2 H s 0 (Ω) = a N,s 2 DΩ |u(x) -u(y)| 2 |x -y| N +2s dxdy,
The pair (H s 0 (Ω), ||.|| H s 0 (Ω) ) is a Hilbert space (see for instance [START_REF] Servadei | Valdinoci Mountain pass solutions for non-local elliptic operators[END_REF]Lemma 7] or [START_REF] Bisci | Variational Methods For Nonlocal Fractional Problems[END_REF]Lemma 1.29]). The dual space of H s 0 (Ω) will be denoted by H -s (Ω). For more properties of the previous spaces, we refer to [START_REF] Di Nezza | Hitchhiker's guide to the fractional Sobolev spaces[END_REF], [START_REF] Adams | Sobolev spaces[END_REF] and [10, subsection 1.5].

2.2. Some useful functional inequalities. In this subsection, we collect and present some functional inequalities under the form in which they will be exploited.

• The following Sobolev's inequality is proved in [START_REF] Di Nezza | Hitchhiker's guide to the fractional Sobolev spaces[END_REF] (see also [START_REF] Ponce | Elliptic PDEs, Measures and Capacities[END_REF] for a simple proof).

Theorem 2.1. Assume that 0 < s < 1 with 2s < N . There exists a positive constant S = S(N, s) such that for all u ∈ H s 0 (Ω), we have

S Ω |u(x)| 2 * s dx 2 2 * s ≤ a N,s 2 DΩ |u(x) -u(y)| 2 |x -y| N +2s dx dy where 2 * s := 2N N -2s
is the so-called critical Sobolev exponent.

• In order to treat the case λ = Λ * s :=

Γ 2 (s + 1 2 ) π
, we need the following improved Hardy-Sobolev inequality obtained in [START_REF] Filippas | Sharp Trace Hardy Sobolev Mazýa Inequalities and the Fractional Laplacian[END_REF]. For this purpose, let us first state the definition of inner radius of a domain Ω : the inner radius of a domain Ω is defined as R in := sup x∈Ω d(x). We say that the domain Ω has a finite inner radius whenever R in < ∞.

Theorem 2.2. (Improved Hardy-Sobolev inequality) Assume that 1 2 ≤ s < 1 and Ω ⊂ IR N (N ≥ 2) is a uniformly Lipschitz and convex domain with finite inner radius. Then, there exists a positive constant C ≡ C(Ω, s) such that

(2.2) C Ω |u(x)| 2 * s dx 2 2 * s ≤ a N,s 2 DΩ |u(x) -u(y)| 2 |x -y| N +2s dx dy -Λ * s Ω u 2 d 2s dx where 2 * s = 2N N -2s .
• The Kato type inequality below will be useful in this paper.

Theorem 2.3. Let Φ ∈ C 2 (IR) be a convex function. Assume that u and Φ(u) are such that (-∆) s u and (-∆) s Φ(u) exist. Then

(2.3) (-∆) s Φ(u) ≤ Φ (u)(-∆) s (u).
2.3. Notion of solutions. Now, let us begin by making precise the sense that solutions are defined.

Definition 2.4. Let (g, u 0 ) ∈ L 2 (Ω T )×L 2 (Ω).
We say that u is an energy solution to the problem

(2.4)    u t + (-∆) s u = g(x, t) in Ω T = Ω × (0, T ), u(x, t) = 0 in (IR N \ Ω) × (0, T ) u(x, 0) = u 0 (x) in Ω, if u ∈ L 2 ((0, T ), H s 0 (Ω)) ∩ C([0, T ], L 2 (Ω)) with u t ∈ L 2 ((0, T ), L 2 (Ω)) and for all w ∈ L 2 ((0, T ), H s 0 (Ω)), we have (2.5) T 0 Ω u t w dx dt+ T 0 DΩ (u(x) -u(y))(w(x) -w(y)) |x -y| N +2s dx dy dt = T 0 Ω gw dx dt, for all w ∈ L 2 ((0, T ), H s 0 (Ω)).
Since we will consider problems with general datum in L 1 , we have to specify the meaning of weak solutions. We first need to define the space of test functions

P (Ω T ) := {φ : R N × [0, T ] → R, s.t. -φ t + (-∆) s φ = ϕ, ϕ ∈ L ∞ (Ω T ) φ(x, t) = 0 in (IR N \ Ω) × (0, T ], φ(x, T ) = 0 in Ω}.
Notice that if φ ∈ P (Ω T ), then φ ∈ L ∞ (Ω T ) (see for instance [START_REF] Leonori | Basic estimates for solutions of a class of nonlocal elliptic and parabolic equations[END_REF]). Then we have the subsequent definition.

Definition 2.5. Let (g, u 0 ) ∈ L 1 (Ω T ) × L 1 (Ω). We say that u ∈ C([0, T ], L 1 (Ω)) is a weak solution to (2.6)    u t + (-∆) s u = g(x, t) in Ω T u(x, t) = 0 in (IR N \ Ω) × (0, T ), u(x, 0) = u 0 (x)
in Ω, if for all φ ∈ P (Ω T ), we have

Ω T u(-φ t + (-∆) s φ) dx dt = Ω T u ϕ dx dt = Ω T gφ dx dt + Ω u 0 φ(x, 0) dx.
The following existence and regularity result is obtained in [START_REF] Leonori | Basic estimates for solutions of a class of nonlocal elliptic and parabolic equations[END_REF].

Theorem 2.6 ( See [26, Theorem 28]). Assume that (g, u 0 ) ∈ L 1 (Ω T ) × L 1 (Ω).
Then, Problem (2.6) has a unique weak solution u, that is obtained as a limit of approximations, such that ∀k ≥ 0,

T k (u) ∈ L 2 ((0, T ), H s 0 (Ω)), u ∈ L q (Ω T ) for all q ∈ [1, N +2s N ) and |(-∆) s 2 u| ∈ L r (Ω T ) for all r ∈ [1, N +2s N +s ).
Here, T k (σ) := max{-k, min{k, σ}} for k > 0, Taking into consideration the singular weight d -2s at the boundary of Ω and in order to prove a priori estimates for approximate problems, we will use this existence result obtained in [START_REF] Abatangelo | Large Solutions for Fractional Laplacian Operators[END_REF].

Theorem 2.7 (See [1, Proposition 1.10]). theorem 2.7 Assume that s ∈ (0, 1) and β ∈ (0, 2s). Then, the following problem

(2.7)    (-∆) s ξ = 1 d β (x) in Ω, φ = 0 in IR N \Ω, has a unique bounded distributional solution such that (i) if β < s, then ξ d s , (ii) if β = s, then ξ d s log( D d(x) ) where D >> diam(Ω), (iii) if β ∈ (s, 2s), then ξ d 2s-β .
Note that, if f and g are nonnegative functions, then by f g, we mean that there exist two positive constants

C 1 and C 2 such that C 1 f ≤ g ≤ C 2 f .
Finally, we conclude this section with the next Picone's inequality that the proof can be found in [START_REF] Leonori | Basic estimates for solutions of a class of nonlocal elliptic and parabolic equations[END_REF] and [START_REF] Abdellaoui | Nonlinear fractional elliptic problem with singular term at the boundary[END_REF].

Proposition 2.8. Consider u ∈ H s 0 (Ω) such that u > 0 in Ω. Then, for all φ ∈ C ∞ 0 (Ω), we have (2.8) Ω (-∆) s u u φ 2 dx ≤ a N,s 2 DΩ |φ(x) -φ(y)| 2 |x -y| N +2s dx dy.
3. The sublinear case p < 1 : Existence results.

In this section, we are interested in the sub-linear case, namely 0 < p < 1. Contrary to the local case studied in [START_REF] Abdellaoui | A semilinear parabolic problem with singular term at the boundary[END_REF], we are able here to show the existence of a solution, at least, in a weak sense.

In order to make easier the calculations and using the homogeneity, we can assume that λ = 1.

The main existence result of this section is the following.

Theorem 3.1. Assume that 0 < s < 1 and 0 < p < 1. Then, for all 0

u 0 ∈ L 1 (Ω), Problem (1.
3) has a positive distributional solution. Moreover, if s > 1 2 and 0 < p < 2s -1, we get the existence of a weak solution in the sense of Definition 2.5.

Proof. : We proceed by iterations. Let n ≥ 1 and define u n to be the unique positive solution to

(3.1)        u nt + (-∆) s u n = u p n-1 (d(x) + 1 n ) 2s in Ω T u n (x, t) = 0 in (IR N \ Ω) × (0, T ), u n (x, 0) = u 0n (x)
in Ω, where u 0n := min(n, u 0 ) and u 00 := 0. The existence and the uniqueness of u n follow using classical arguments for monotone operators. Moreover u n ≥ 0. Now, taking into consideration that u 0n ↑ u 0 as n → ∞, then by the comparison principle it holds that u n ≤ u n+1 for all n.

Let ξ be the solution to Problem (2.7) with β ∈ (s, min{2s, 1}), then β < 1. By Theorem 2.7, it follows that ξ d 2s-β . Using ξ as a test function in (3.1) and taking into consideration the estimate on ξ given in Theorem 2.7, it holds

d dt Ω u n ξ dx + Ω u n d β dx = Ω u p n-1 ξ (d(x) + 1 n ) 2s dx ≤ C Ω u p n d β dx.
Since p < 1, by Young's inequality, we obtain

(3.2) d dt Ω u n ξ dx + (1 -ε) Ω u n d β dx ≤ C(ε) Ω 1 d β dx. Since β < 1, then Ω 1 d β dx < ∞. Choosing ε small enough and integrating (3.2) in time, we get sup t∈[0,T ] Ω u n (x, t) ξ dx + (1 -ε) T 0 Ω u n d β dx dt ≤ T CC(ε) + Ω u 0 ξ dx < ∞.
Thus, there exists a measurable function

u ∈ L 1 (Ω T ) ∩ L ∞ ((0, T ), L 1 loc (Ω)) such that u n ↑ u strongly in L 1 (Ω T ) and in L ∞ ((0, T ), L 1 loc (Ω)) as n → ∞. Moreover u p n (d(x) + 1 n ) 2s ↑ u p d 2s strongly in L 1 loc (Ω T ).
Hence u is, at least, a distributional solution to problem (1.3) with

u d β ∈ L 1 (Ω T ).
Let us assume now that s > 1 2 and that p < 2s -1. Taking into consideration that u ∈ L 1 (Ω T ) and using Hölder's inequality, we can show that u p d 2s ∈ L 1 (Ω T ). Finally, by Theorem 2.6, we easily obtain the desired result. Remark 3.2. • In the local case (i.e s = 1), the authors proved in [START_REF] Abdellaoui | A semilinear parabolic problem with singular term at the boundary[END_REF] a strong nonexistence result to problem (1.3) for all p < 1. As a consequence, they get a complete and instantaneous blow-up for the approximating problems. Hence, our existence result in Theorem 3.1 shows a significative difference between the local and the nonlocal cases.

• In the linear case, as it was established in [START_REF] Dipierro | All functions are locally sharmonic up to a small error[END_REF] a closely phenomenon occurs. More precisely, the authors proved that all functions are locally s-harmonic up to a small error. This produces more solutions in the fractional case than the local case (that disappear when letting s → 1).

The linear case : Existence and nonexistence results.

In this section, we consider the linear problem (4.1)

     u t + (-∆) s u = λ u d 2s in Ω T , u = 0 in (IR N \ Ω) × (0, T ), u(x, 0) = u 0 (x) in Ω,
where λ > 0 and u 0 is a measurable positive function satisfying some assumptions that we will specify later. We will show that the rang of the parameter s will be the key point in order to show the existence or the nonexistence of solution for large value of λ.

4.1.

The case s ∈ (0, 1 2 ) : existence result. The main existence result of this subsection is the following. Theorem 4.1. Assume that Ω is a bounded regular domain of IR N with N > 2s, s ∈ (0, 1 2 ) and 0 u 0 ∈ L 1 (Ω). Then, for all λ > 0, Problem (4.1) has a weak solution u ∈ C((0, T ), L 1 (Ω)) such that

1) ∀k ≥ 0, T k (u) ∈ L 2 ((0, T ), H s 0 (Ω)) ; 2) u ∈ L q (Ω T ) for all q ∈ [1, N +2s N ) ; 3) |(-∆) s 2 u| ∈ L r (Ω T ) for all r ∈ [1, N +2s N +s ). Remark 4.2.
• The above existence result holds for all bounded domains and in particular for convex domains.

• The existence result in Theorem 4.1 (at least for convex domains) is another case where local and nonlocal problems have significant difference. In fact, if s = 1 and if Ω is a convex domain, existence holds only if λ ≤ Λ * 1 .

Before starting the proof of Theorem 4.1, we shall recall the following existence result obtained in [5, Theorem 3.1] which will have a key role in our demonstration. Theorem 4.3. Assume that 0 < s < 1 2 and fix α > 0 be such that 2s + 2α < 1. Then, the problem

(4.2)      (-∆) s w = 1 d 2s+α in Ω, w > 0 in Ω, w = 0 in IR N \ Ω,
has a unique weak solution w obtained as a limit of approximate problems with w ∈ H s 0 (Ω). Moreover (4.3) w(x) ≥ Cd s (x) in Ω.

Proof. Let 0 < s < 1 2 and choosing 0 < α < 1-2s 2 , then d -2(s+α) ∈ L 1 (Ω). Therefore defining w n as the unique solution to the problem (4.4)

       (-∆) s w n = 1 d 2s+α + 1 n in Ω, w n > 0 in Ω, w n = 0 in IR N \ Ω,
Thanks to Theorem 2.6, w n ↑ w strongly in L 1 (Ω) where w is the unique weak solution to Problem (4.3). Now, using w n as a test function in (4.4) and applying Hölder's inequality, we get

a N,s 2 DΩ |w n (x) -w n (y)| 2 |x -y| N +2s dx dy ≤ C(Ω) Ω w 2 n (x) d 2s (x) dx
     u nt + (-∆) s u n = λ u n-1 (d(x) + 1 n ) 2s in Ω T u n (x, t) = 0 in (IR N \ Ω) × (0, T ), u n (x, 0) = u 0n (x)
in Ω, where u 0n := T n (u 0 ) and u 00 := 0.

In what follows, we denote by C any positive constant that can change from one line to the other and it is independent of n.

Fix α ∈ (0, 1-2s 2 ) and consider w ∈ H s 0 (Ω) the unique weak solution to Problem (4.2) given in Theorem 4.3.

We claim that,

(4.6) ∀ε > 0, ∀λ > 0, ∃C = C(λ, ε) > 0 such that λ < C d 2s (x) + ε d α (x) ∀x ∈ Ω.
It is clear that (4.6) holds trivially in a neighborhood of the boundary. Now, far from the boundary we use the fact that for every compact set K ⊂⊂ Ω, d 2s > C(K). Thus, by choosing C large in (4.6), the claim follows.

Taking w (the solution to Problem (4.2)) as test function in (4.5), we get

d dt Ω u n w dx + Ω w(-∆) s u n dx ≤ λ Ω u n w d 2s dx. Thus (4.7) d dt Ω u n w dx + Ω u n d 2s+α dx ≤ λ Ω u n w d 2s dx.
For k > 0, we consider the set A k := {x ∈ Ω ; w(x) ≥ k}. Then, we get the existence of k > 0 such that A k ⊂ B k := {x ∈ Ω ; d(x) ≥ k}. Hence, by using (4.6) we have

λ Ω u n w d 2s dx = λ A k u n w d 2s dx + λ Ω\A k u n w d 2s dx ≤ λ B k u n w d 2s dx + ε Ω\A k u n w d 2s+α dx + C(ε) Ω\A k u n w dx ≤ (λ + k-2s + C(ε)) Ω u n w dx + εk Ω\A k u n d 2s+α dx.
Going back to (4.7) and choosing ε such that εk << 1, we obtain

(4.8) d dt Ω u n w dx + (1 -ε k) Ω u n d 2s+α dx < C Ω u n w dx. Thus d dt Ω u n w dx ≤ C Ω u n w dx.
By Gronwall's Lemma and integrating in time, it holds that sup

t∈[0,T ] Ω u n (x, t)w dx + (1 -ε) T 0 Ω u n d 2s+α dx dt ≤ C(T ) Ω u 0 w dx. Thus sup t∈[0,T ] Ω u n (x, t)w dx ≤ C(T ) and T 0 Ω u n d 2s dx dt ≤ C(T ).
Taking into consideration that the sequence {u n } n is increasing in n, we get the existence of a measurable function u such that (1)

u n → u strongly in L 1 loc (Ω T ), (2) u n d 2s → u d 2s strongly in L 1 (Ω T ).
Moreover, thanks to the above estimates, we can show that u ∈ C([0, T ], L 1 (Ω)). Thus, u is a weak solution to problem (4.1) in the sense of Definition 2.5. Since u is obtained as a limit of approximate problems with monotone behavior, then by a simple comparison principle, it follows that u is the minimal solution to Problem (4.1).

To prove the uniqueness of the solution of Problem (4.1), we argue by contradiction. Assume that v is another weak solution to Problem (4.1). It is clear that v ≥ u.

Define U = v -u, then U d 2s ∈ L 1 (Ω T ) and U satisfies (4.9)      U t + (-∆) s U = λ U d 2s in Ω T , U = 0 in (IR N \ Ω) × (0, T ), U(x, 0) = 0 in Ω.
Taking w (the solution to Problem (4.2)) as a test function in (4.9), then thanks to (4.6), we obtain

d dt Ω U w dx + (1 -ε) Ω U d 2s dx < C Ω Uw dx,
where ε << 1. Moreover U, w ≥ 0 in Ω T , then

d dt Ω U w dx ≤ C Ω U w dx.
Using Gronwall's lemma, we reach that U(t, x)w(x) = 0 a.e. in Ω for all t > 0. Thus U(t, x) ≡ 0 for all t ≥ 0 and the uniqueness follows.

In the case where the data u 0 has more regularity, we can improve the regularity of the solution u. More precisely, we have : Theorem 4.4. Assume that Ω is a bounded regular domain with N > 2s, s ∈ (0, 1 2 ) and λ > 0. Assume that 0 u 0 ∈ L σ (Ω, w dx) with σ > 1 where w is the solution to Problem (4.2). Then, the unique solution u to Problem (4.1) satisfies u ∈ L ∞ ((0, T ), L σ (Ω, w dx)).

Proof. As in the proof of Theorem 4.1, let us consider u n the unique solution to the approximate Problem (4.5). Since σ > 1, then we use the Kato type inequality (??) to obtain (4.10)

u σ nt + (-∆) s u σ n ≤ σu σ-1 n u nt + σu σ-1 n (-∆) s u n .
By using u σ-1 n w as a test function in (4.5) and taking into consideration estimates (4.6) and (4.10), we get

1 σ d dt Ω u σ n w dx + 1 σ Ω u σ n d 2s+α dx ≤ d dt Ω u σ-1 n u nt w dx + Ω wu σ-1 n (-∆) s u n dx ≤ λ Ω u σ n w d 2s ≤ C Ω u σ n w dx + ε Ω u σ n d 2s+α dx. Therefore 1 σ d dt Ω u σ n w dx + 1 σ -ε Ω u σ n d 2s+α dx ≤ C Ω u σ n w dx.
Hence, by choosing ε << 1 σ , we reach that

d dt Ω u σ n w dx ≤ C Ω u σ n w dx.
Gronwall's inequality allows us to conclude that

Ω u σ n (x, t)w dx ≤   Ω u σ 0 w dx   e Ct , ∀t ≥ 0.
Using the monotone convergence Theorem, we get the existence of a solution u such that u ∈ L σ (Ω T , w dx dt) and

T 0 Ω u σ d 2s+α dx dt ≤ C(T ).
In the case where λ ≤ Λ s (Ω), then according to the attainability of Λ s (Ω), we are able to improve the regularity of the solution. More precisely, we have :

Theorem 4.5. Let Ω ⊂ IR N be a bounded regular domain with N ≥ 2s and s ∈ ( 1 2 , 1). Assume that 0 ≤ u 0 ∈ L 2 (Ω). 1) If Λ s (Ω) is attained, then for all λ ≤ Λ * s (Ω), Problem (4.1) has a unique positive u such that u ∈ L 2 (0, T, H s 0 (Ω)) if λ < Λ s (Ω) and u ∈ L 2 (Ω T ) if λ = Λ s (Ω). 2) If Λ s (Ω) = Λ * s := Γ 2 (s + 1 2 ) π
(recall that in this case, Λ s (Ω) is not attained), then :

• if λ < Λ *
s , then Problem (4.1) has a unique positive finite energy solution u ∈ L 2 ((0, T ), H s 0 (Ω)) ;

• if λ = Λ * s , then problem (4.1) has a distributional solution u ∈ L 2 (Ω T ).
Proof. We proceed by approximation. Let n ∈ IN * and define u n to be the unique nonnegative solution to the following approximate problem, (4.11)

     u nt + (-∆) s u n = λ u n-1 (d(x) + 1 n ) 2s in Ω T , u n (x, t) = 0 in (IR N \ Ω) × (0, T ), u n (x, 0) = u 0n (x)
in Ω, where u 0n := T n (u 0 ) and u 00 := 0. The existence of u n follows using classical arguments for monotone operators. It is clear that {u n } n is monotone in n.

We have to distinguish two cases : First case Λ s (Ω) < Λ * s . In this case, Λ s (Ω) is attainted. Using u n as a test function in (4.11) and by Hardy's inequality, we get,

1 2 Ω u 2 n (x, T ) dx + a N,s 2 1 - λ Λ s (Ω) T 0 DΩ |u n (x, t) -u n (y, t)| 2 |x -y| N +2s dx dy dt ≤ 1 2 Ω u 2 0 dx.
• If λ < Λ s (Ω), we deduce the existence of a measurable function u ∈ L 2 ((0, T ), H s 0 (Ω)) such that u n u weakly in L 2 ((0, T ), H s 0 (Ω)), u n → u a.e in Ω T , and

u 2 n d 2s → u 2 d 2s strongly in L 1 (Ω T ). It is not difficult to show that u n → u strongly in L 2 ((0, T ), H s 0 (Ω))
. Thus u is an energy solution to Problem (4.1).

• If λ = Λ s (Ω), we have max t∈[0,T ] Ω u 2 n (x, t) dx ≤ C.
Then we conclude that the sequence {u n } n is bounded in L 2 (Ω T ) and by monotonicity of u n we get u n → u in L 2 (Ω T ) and u is a distributional solution to (4.1).

Second case Λ s (Ω) = Λ * s . If λ < Λ *
s , then as in the first case, the existence of an energy solution follows directly by using the same a priori estimates. Now, let us deal with the case λ = Λ * s . As above, using u n as test function in (4.11), we get

1 2 Ω u 2 n (x, t) dx + a N,s 2 t 0 DΩ |u n (x, σ) -u n (y, σ)| 2 |x -y| N +2s dx dy dσ ≤ C s t 0 Ω u 2 n d 2s dx dσ + 1 2 Ω u 2 0 dx.
Now using the improved Hardy-Sobolev inequality (2.2) (see Theorem 2.2), it holds that

1 2 Ω u 2 n (x, t) dx + k(N, s) t 0 Ω u 2 * s n (x, σ) dxdσ 2 2 * s ≤ 1 2 Ω u 2 0 dx.
By Sobolev's inequality, we get

1 2 Ω u 2 n (x, t) dx + C t 0 Ω u 2 n (x, σ) dxdσ ≤ 1 2 Ω u 2 0 dx. Hence sup t∈[0,T ] Ω u 2 n (x, t) dx ≤ Ω u 2 0 dx.
Therefore, we conclude that the sequence {u n } n is bounded in L 2 (Ω T ). As above by monotonicity of u n we get u n → u in L 2 (Ω T ) and u is a distributional solution of (4.1).

Related to the asymptotic behavior of the above solution, we have the following proposition.

Proposition 4.6. Assume that λ ∈ (0, Λ s (Ω)) if Λ s (Ω) < Λ * s or λ ∈ (0, Λ * s ] if Λ s (Ω) = Λ * s . If u is the solution to problem (4.1) obtained above, then u(x, t) → 0 in L 2 (Ω) as t → +∞.
Proof. The first case follows easily by using Gronwall's lemma. Suppose that λ = Λ * s , then 1 2

d dt Ω u 2 dx ≤ - a N,s 2 DΩ |u(x, t) -u(y, t)| 2 |x -y| N +2s dx dy + Λ * s Ω u 2 d 2s dx.
Thus, by improved Hardy-Sobolev inequality (2.2), we obtain, 1 2

d dt Ω u 2 dx ≤ -C Ω u 2 * s dx 2 2 * s .
Thus, Hölder's inequality allows us to get 1 2

d dt Ω u 2 dx ≤ -C 0 Ω u 2 dx.
Thanks to Gronwall's lemma, we obtain

||u(t)|| L 2 (Ω) ≤ e -C0t ||u 0 || L 2 (Ω) .
This ends up the proof.

4.2. The case s ∈ [ 1 2 , 1) : Nonexistence result. Let us recall that, in this case and if Ω is a regular convex domain, then Hardy-Sobolev inequality in Theorem 1.2 holds with the optimal constant Λ s (Ω) = Λ * s . As in [START_REF] Cabré | Existence versus explosion instantanée pour des équations de la chaleur linéaires avec potentiel singulier[END_REF], if Ω is a convex domain, we are able to show the next nonexistence result.

Theorem 4.7. Assume that 1 2 ≤ s < 1, N ≥ 2, Ω ⊂ IR N is a bounded regular domain such Λ s (Ω) = Λ *
s (that includes the case of convex domains). If λ > Λ * s , then Problem (4.1) does not have any weak positive solution.

In order to prove Theorem 4.7, we need to introduce two useful propositions. The first one is the next version of the maximum principle. Proposition 4.8. Assume that v 0 ∈ L 1 (Ω) and consider v the unique solution to the problem (4.12)

   v t + (-∆) s v = 0 in Ω T , v(x, t) = 0 in (IR N \ Ω) × (0, T ), v(x, 0) = v 0 (x) in Ω.
Then, for all t > 0, there exist two positive constants C 1 (t) and C 2 (t) such that for all x ∈ Ω (4.13)

C 1 (t)d s (x) ≤ v(x, t) ≤ C 2 (t)d s (x).
Proof. The proof follows closely the argument used in [START_REF] Martel | Complete blow up and global behaviour of solution of ut -∆u = g(u)[END_REF] for the local case. However, for the reader's convenience, we include here some details. Define to be the unique solution to the problem

(4.14) (-∆) s = 1 in Ω, = 0 in IR N \ Ω.
Then by [START_REF] Capella | Regularity of radial extremal solutions for some non-local semilinear equations[END_REF], we have c

1 d s (x) ≤ (x) ≤ c 2 d s (x) with c 1 , c 2 > 0.
From [START_REF] Barrios | Strong maximum principles for fractional elliptic and parabolic problems with mixed boundary conditions[END_REF], we reach that, if t > 0, then v(x, t) ≥ C(t) . Hence, the left hand side inequality in (4.13) follows.

Let us now prove the right hand side inequality in (4.13). For this, we will use the regularizing effect of the fractional heat semi-group. Fix t > 0, then

|v(x, t)| ≤ C t N 2s ||v 0 || L 1 (Ω) .
Consider V to be the unique solution to this problem (4.15)

   V t + (-∆) s V = 0 in Ω T , V(x, t) = 0 in (IR N \ Ω) × (0, T ), V(x, 0) = 1 in Ω.
By using Hopf's lemma (see [START_REF] Fernandez-Real | Boundary regularity for the fractional heat equation. Revista de la Real Academia de Ciencias Exactas, Físicas[END_REF]), it follows that for all t > 0

V(•, t) d s L ∞ (Ω) ≤ C(t).
Denoting by S the corresponding semi-group, then for simplicity of typing we set

V(•, t) = S(t)v 0 (•).
Thanks to the semi-group properties, if v solves (4.12), then

||d -s v(•, t)|| L ∞ (Ω) = d -s S( t 3 ) S( 2t 3 )v 0 (•, ) L ∞ (Ω) ≤ C(t)||S( 2t 3 )v 0 (•)|| L ∞ (Ω) , and
S( 2t 3 )v 0 (•) L ∞ (Ω) ≤ C 2 (t)||S( t 3 )v 0 (•)|| L 1 (Ω) .
Now, taking into consideration that

S( t 3 )v 0 (•, ) L 1 (Ω) = Ω v 0 (x) d s (x) S( t 3 )χ Ω d s (x) dx ≤ C(Ω) Ω v 0 (x) d s (x) V(x, t 3 ) d s (x) dx ≤ C(t)||d s u 0 || L 1 (Ω) ,
and combining with the above estimates, it follows that

d -s v(•, t) ≤ C(t)||d s v 0 || L 1 (Ω) .
Hence we conclude.

We need also the next result that extends the one obtained in [START_REF] Brezis | Hardy's inequalities revisited[END_REF] in the local case.

Proposition 4.9. Let Ω be a bounded regular domain such that Λ s (Ω) = Λ * s . Define

Λ = inf φ∈C ∞ 0 (Ω),φ =0 a N,s 2 DΩ |φ(x) -φ(y)| 2 |x -y| N +2s dxdy -C Ω φ 2 d 2s dx Ω φ 2 dx , where C > Λ * s , then Λ = -∞.
Proof. We argue by contradiction. Assume that | Λ| < ∞, then for all φ ∈ C ∞ 0 (Ω), we have

a N,s 2 DΩ |φ(x) -φ(y)| 2 |x -y| N +2s dxdy -C Ω φ 2 d 2s dx ≥ Λ Ω φ 2 dx.
Thus (4.16)

J( Λ) := inf φ∈C ∞ 0 (Ω),φ =0 a N,s 2 DΩ |φ(x) -φ(y)| 2 |x -y| N +2s dxdy + | Λ| Ω φ 2 dx Ω φ 2 d 2s dx ≥ C > Λ * s .
For β > 0, we define the set

Ω β := x ∈ Ω ; dist(x, ∂Ω) < β , then J( Λ) ≤ inf φ∈C ∞ 0 (Ω β ),φ =0 a N,s 2
DΩ β |φ(x) -φ(y)| 2 |x -y| N +2s dxdy + | Λ| Ω β φ 2 dx Ω β φ 2 d 2s dx
.

By using Hardy's inequality (1.4), we obtain

Ω β φ 2 dx = Ω β φ 2 d 2s d 2s dx ≤ β 2s Ω β φ 2 d 2s dx ≤ β 2s Λ * s a N,s 2
DΩ β |φ(x) -φ(y)| 2 |x -y| N +2s dx dy. Hence, for φ ∈ C ∞ 0 (Ω β ), a N,s 2 DΩ β |φ(x) -φ(y)| 2 |x -y| N +2s dxdy + | Λ| Ω β φ 2 dx ≤ a N,s 2 1 + β 2s Λ * s | Λ| DΩ β |φ(x) -φ(y)| 2 |x -y| N +2s dxdy.
Therefore we conclude that

J( Λ) ≤ 1 + β 2s Λ * s | Λ| inf φ∈C ∞ 0 (Ω β ) a N,s 2 DΩ β |φ(x) -φ(y)| 2 |x -y| N +2s dxdy Ω β φ 2 d 2s dx ≤ 1 + β 2s Λ * s | Λ| Λ * s (Ω β ). Recall that Λ * s (Ω) = Λ * s , then since Λ * s (Ω) ≤ Λ * (Ω β ) ≤ Λ * s , it follows that Λ * (Ω β ) = Λ *
s . Now, using the fact that C > Λ * s , we can choose β small enough such that

1 + β 2s Λ * s | Λ| Λ * s < C,
wich gives a contradiction with (4.16). Thus Λ = -∞.

We are now ready to prove our main nonexistence result.

Proof of Theorem 4.7.

Without loss of generality, we suppose that 0 u 0 ∈ L ∞ (Ω). We argue by contradiction and suppose that problem (4.1) has a positive weak solution u. Using an approximating argument, we get the existence of a minimal solution to Problem (4.1) obtained as a limit of approximating problems. We denote this minimal solution by u. Thus u = lim n→+∞ u n where u n is the unique positive solution to (4.17)

   u nt + (-∆) s u n = λa n (x)u n in Ω T , u n (x, t) = 0 in (IR N \ Ω) × (0, T ), u n (x, 0) = u 0 (x)
in Ω, with a n (x) = min{n,

1 d 2s }, λ > Λ * s . It is clear that {u n } n is increasing in n and u n → u strongly in C([0, T ], L 1 (Ω)).
Notice that u(x, t) > 0 for all (x, t) ∈ Ω × (0, T ). By applying Proposition 4.8, it follows that for all (x, t) ∈ Ω × [t 1 , t 2 ] with t 1 > 0, we have

u(x, t) ≥ C(t 1 , t 2 , Ω)d s (x). Let φ ∈ C ∞ 0 (Ω),
Ω a n (x)φ 2 dx ≤ a N,s 2 DΩ |φ(x) -φ(y)| 2 |x -y| N +2s dx dy + 1 t 2 -t 1 Ω ln u n (t 2 ) u n (t 1 ) φ 2 dx. Notice that ln u n (•, t 2 ) u n (•, t 1 ) = ln u n (•, t 2 )d(•) u n (•, t 1 )d(•) .
Since u n ↑ u strongly in L 1 (Ω T ) and using the fact that u

n (•, t) ≥ C d(•) for all n ≥ 1 and uniformly for t ∈ [t 1 , t 2 ], it holds that ln (u n (•, t 2 )d(•)) L σ (Ω) ≤ C for all σ > 1. Thus ln u n (•, t 2 ) u n (•, t 1 ) L σ (Ω) ≤ ln (u n (•, t 2 )d(•)) L σ (Ω) + ln (u n (•, t 1 )d(•)) L σ (Ω) ≤ C.
Hence for all ε > 0, we get the existence of C(ε) such that

1 t 2 -t 1 Ω ln u n (t 2 ) u n (t 1 ) φ 2 dx ≤ ε a N,s 2 Ω |φ(x) -φ(y)| 2 |x -y| N +2s dx dy + C(ε) Ω φ 2 dx.
Letting n → ∞, it follows that for all φ ∈ C ∞ 0 (Ω) with φ = 0, we have

(1 + ε) a N,s 2 DΩ |φ(x) -φ(y)| 2 |x -y| N +2s dx dy -λ Ω φ 2 d 2s dx Ω φ 2 dx ≥ -C(ε).
Since λ > Λ * s , we can choose ε small enough such that

λ 1 + ε > Λ * s . Then, we obtain inf φ∈C ∞ 0 (Ω) a N,s 2 DΩ |φ(x) -φ(y)| 2 |x -y| N +2s dxdy - λ 1 + ε Ω φ 2 d 2s dx Ω φ 2 dx ≥ - C(ε) 1 + ε > -∞,
Hence, we reach a contradiction with the result of Proposition 4.9. This ends up the proof.

5. The superlinear case p > 1 : Existence results.

In this section, we are interested in the super-linear case, namely p > 1. The main idea to get the existence result is to find a suitable supersolution and then we proceed by iteration. The main existence result of this section is the following. Theorem 5.1. Assume that Ω is a bounded regular domain of IR N with N > 2s and p > 1. Then, there exists u 0 ∈ L ∞ (Ω) such that the problem

(5.1)      u t + (-∆) s u = u p d 2s in Ω T , u(x, t) = 0 in (IR N \ Ω) × (0, T ), u ( x, 0) = u 0 (x) in Ω, has a minimal positive solution u ∈ L 2 ((0, T ), H s 0 (Ω)) ∩ L ∞ (Ω T ).
Proof. In order to prove Theorem 5.

ξ t + (-∆) s ξ ≥ ξ p d 2s .
It is clear that if u 0 ≤ ξ, then ξ is supersolution to (5.1). Consider now ξ, the unique solution to the following problem (5.2)

   v t + (-∆) s v = 0 in Ω T , v = 0 in (IR N \ Ω) × (0, T ), v(x, 0) = u 0 (x) in Ω. Then ξ ∈ L 2 (0, T, H s 0 (Ω)) ∩ L ∞ (Ω T )
and ξ is a subsolution to (5.1) with ξ ≤ ξ.

We define now the sequence {u k } k by u 0 := ξ, and for k ≥ 1, u k is the unique positive solution to the problem

(5.3)        (u k ) t + (-∆) s u k = (u k-1 ) p d 2s + 1 k in Ω T , u k = 0 in (IR N \ Ω) × (0, T ), u k (x, 0) = u 0 (x)
in Ω.

Applying the comparison principle to obtain ξ ≤ u k ≤ u k+1 ≤ ξ for all k ≥ 1.

Thus, we get the existence of u ≤ ξ such that u k ↑ u strongly in L σ (Ω T ) for all σ ≥ 1 and u k → u strongly in L 2 ((0, T )), H s 0 (Ω)). Then u is a bounded solution to Problem (5.1) with u p d 2s ∈ L 1 (Ω T ). This concludes the proof.

Remark 5.2. It seems to be interesting to find natural class of initial data u 0 such that Problem (5.1) has a weak solution in the sense that u p d 2s ∈ L 1 (Ω T ).

Extension and open problems.

In this section, we state some extension of the previous results to the case of the spectral Laplacian. First let us define the space In addition, Observe that H s (Ω) is the interpolation space (H s 0 (Ω), L 2 (Ω)) [1-s] , see [START_REF] Adams | Sobolev spaces[END_REF][START_REF] Bisci | Variational Methods For Nonlocal Fractional Problems[END_REF]27]. Therefore, we obtain where k s is a normalization constant. As consequence, for all u ∈ H s (Ω), we have Then, ζ ∈ C α ( Ω) for all α ∈ (0, min{2s, 1}). Moreover, there exists a positive constant C = C(Ω, s) such that (i) ζ ≤ Cd 2s if s < 1 2 , (ii) ζ ≤ Cd s if 1 2 ≤ s < 1. Set ψ := ζ a where a ∈ (0, 1). By the Kato type inequality 2.3, we get

H s (Ω) =    H s (Ω) if 0 < s < 1 2 , H 1 2 00 (Ω) if s = 1 2 , H s 0 (Ω) if 1 2 < s < 1, where
A s ψ ≥ C ζ 1-a .
As in Theorem 3.1, we have the next existence result where an additional hypothesis is needed in the case where s < 1 2 . Theorem 6.2. Let 0 < s < 1 and consider the problem where 0 u 0 ∈ L 1 (Ω). Then (1) if s ∈ (0, 1 2 ), Problem (6.8) has a positive distributional solution for all p ∈ (0, 1) ;

(2) if s ∈ [ 1 2 , 1), Problem (6.8) has a positive distributional solution for all p ∈ (0, 1 -s).

1 2 .

 12 Hence by the Hardy-Sobolev inequality (1.4), we conclude thata N,s 2 DΩ |w n (x) -w n (y)| 2 |x -y| N +2s dx dy ≤ C(Ω) for all n.Thus w ∈ H s 0 (Ω) and the result follows.Now we are ready to show Theorem 4.1. Proof of Theorem 4.1. We proceed by approximation. Let n ∈ IN * and u n ∈ L 2 ((0, T ), H s 0 (Ω)) ∩ L ∞ (Ω T ) be the positive solution to the approximate problem(4.5)

(6. 1 )i=1 λ s i a 2 iλ s i a 2 i

 122 H s (Ω) := {u = +∞ i=1 a i φ i ∈ L 2 (Ω) with ∞ < +∞},where (λ i , φ i ) are the eigenvalues and the eigenfunctions of (-∆) with Dirichlet boundary conditions. The space H s (Ω) endowed with the norm||u|| 2 H s (Ω) = +∞ i=1is a Hilbert space.

  a i φ i .As it was proved by Caffarelli-Silvestre in[START_REF] Caffarelli | An extension problem related to the fractional Laplacian[END_REF] by adding a new variable, it is possible to consider the spectral fractional Laplacian operator in a bounded domain. This extension allows us to write nonlocal problems in a local way.More precisely, consider the problem (6.3)A s u = f (x, u) in Ω, u = 0 on ∂Ω.Then, defineC Ω = Ω × (0, ∞) ⊂ R N +1+ and denote by (x, y) a point in C Ω . For u ∈ H s (Ω), we define the s-harmonic extension w = E s (u) in C Ω as the solution to the y 1-2s ∇w) = 0 inC Ω , w = 0 in ∂ L Ω, w = u on Ω × {0}, with ∂ L Ω = ∂Ω × (0, ∞).It is obvious that w belongs to the following spaceX s 0 (C Ω ) := C ∞ 0 (C Ω ) ||.|| X s 0 (C Ω ) with ||w|| 2 X s 0 (CΩ) = k s CΩ y 1-2s |∇w| 2 dx dy,

(6. 5 )Theorem 6 . 1 .

 561 ||E s (u)|| X s 0 (CΩ) = ||u|| H s (Ω) .Going back to problem (6.4), we get(∂y = A s w(x, 0) = A s u(x).Now, let us recall the next regularity result obtained in [18, Lemma 2.9]. Assume that s ∈ (0, 1) and consider ζ to be the unique solution to problem (6.7)A s ζ = 1 in Ω, ζ = 0 in ∂Ω.

  t + A s (u) = λ u p d 2s in Ω T = Ω × (0, T ), u(x, 0) = u 0 (x) in Ω, u = 0 in ∂Ω) × (0, T ),

  1, we use a monotony argument. Fix β > 0 be such that β ∈ (0, s) if p ≥ 2 and β < (2 -p)s if p ∈ (1, 2). Let ξ be the solution to Problem (2.7), then by Theorem 2.7, it holds that ξ d s . Since p > 1, by setting ξ = Cξ, we can choose C small enough such that

Acknowledgments :

Part of this work was realized while the first and the second authors were visiting the "Institut Elie Cartan", Université de Lorraine. They would like to thank the Institute for the warm hospitality.

by project MTM2016-80474-P, MINECO,

Proof. As in the proof of Theorem 3.1, we proceed by approximation. Define u n to be the unique solution to the problem (6.9)

in Ω, with u 0n := min(n, u 0 ) and u 00 := 0. It is clear that u n ≥ 0 and u n ≤ u n+1 for all n.

• Let us begin by the case s ∈ (0, 1 2 ). Define ψ = ζ a with a ∈ (0, 1) where ζ is the unique solution to Problem (6.7). Using ψ as a test function in (6.9), we reach that

Since p < 1, by Young's inequality, we obtain (6.10)

Hence the rest of the proof follows as in the proof of Theorem 3.1.

• We deal now with the case s ∈ [ 1 2 , 1). In this case, φ ≤ Cd s . Since p < 1 -s, we can choose a ∈ (0, 1) such that p < (1 -s 1-s(1-a) ), then as above we have (6.11)

Then we conclude. Remark 6.3. In the case s > 1 2 , we believe that the condition p < 1 -s is technical and general existence result holds for all p < 1.


Gridsada Phanomchoeng

Ali Zemouche

Woongsun Jeon

Frédéric Rajesh Rajamani

Keywords: Descriptor systems, LMIs, analysis, nonlinear systems

I. INTRODUCTION

State observer design for nonlinear systems has attracted the attention of many researchers in the field of automatic control. The topic is a subject of constant evolution [START_REF] Wang | Observer design for differentiable lipschitz nonlinear systems with time-varying parameters[END_REF], [START_REF] Zemouche | On LMI conditions to design observers for Lipschitz nonlinear systems[END_REF], [START_REF] Oueder | A high gain observer with updated gain for a class of MIMO non-triangular systems[END_REF], [START_REF] Andrieu | High gain observers with updated gain and homogeneous correction terms[END_REF], [START_REF] Darouach | Observers for Lipschitz nonlinear descriptor systems: Application to unknown inputs systems[END_REF], [START_REF] Tsinias | Time-varying observers for a class of nonlinear systems[END_REF], [START_REF] Alessandri | Design of observers for Lipschitz nonlinear systems using LMI[END_REF], [START_REF] Gauthier | A simple observer for nonlinear systems, applications to bioreactors[END_REF] and is motivated by several applications such as disturbance input estimation, fault diagnosis, control system design with unknown partial dynamics or synchronization and encryption in chaotic communication systems. One of the well-established problems that uses the theory of descriptor systems is the simultaneous state and unknown input estimation problem [START_REF] Alma | Adaptive observers design for a class of linear descriptor systems[END_REF], [START_REF] Ha | State and input simultaneous estimation for a class of nonlinear systems[END_REF], [START_REF] Zemouche | Sobolev norms-based state estimation and input recovery for a class of nonlinear systems. Design and experimental results[END_REF], [START_REF] Zemouche | Nonlinear observer-based ℋ synchronization and unknown input recovery[END_REF]. Observer design for the class of descriptor systems with nonlinearities in both the dynamics of the system and the output measurements is an important unsolved problem.

While the problem of state observer design for nonlinear systems has been widely investigated in the recent literature and several research results have been established, the study becomes complicated when we face systems in descriptor/singular structures [START_REF] Darouach | Reduced-order observer design for descriptor systems with unknown inputs[END_REF]. Indeed, very few results have been investigated or established in the case of descriptor systems, especially when the output measurements contain nonlinear terms.

Motivated by the lack of results in the field of observer design for descriptor nonlinear systems in the presence of nonlinearities in the measured outputs, we propose in this paper a new LMI design technique aimed at meeting a ℋ performance criterion. The presence of nonlinear terms in the output measurements renders the design of the observer gain parameters to be particularly complicated. To overcome this obstacle, we propose a new LMI technique, where the method requires a certain assumption on system parameters concerning the numbers of outputs, unknown inputs, states and numbers of nonlinear functions in the measurement equations. Moreover, the proposed LMI technique is less conservative compared to the existing results in the literature. Indeed, the nonlinearities are handled in a less conservative way. The Young's inequality is used in a convenient manner that leads to dilated LMIs [START_REF] Zemouche | ℋ circle criterion observer design for Lipschitz nonlinear systems with enhanced LMI conditions[END_REF]. The contributions of this paper can be summarized as follows:

 The technique is able to estimate unknown inputs with only a minimum number of measurements. Indeed, the rank condition requires a necessary number of outputs.

 The contribution can be viewed as an extension of the work in [START_REF] Zemouche | ℋ circle criterion observer design for Lipschitz nonlinear systems with enhanced LMI conditions[END_REF] to a more general class of nonlinear systems: descriptor structure of the system and presence of nonlinearities in the outputs. The rest of this paper is organized as follows. In Section II we introduce the problem formulation and the motivations of the proposed work. Section III is devoted to the proposed LMI design method based on a rank condition based assumption. Then the application to unknown input estimation is presented in Section IV. Finally, a conclusion and some perspectives are presented in Section V.

Notation:

The following notation will be used throughout this paper.

 ‖•‖ is the usual Euclidean norm;  ( * ) is used for the blocks induced by symmetry;  represents the transposed matrix of ;  represents the identity matrix of dimension ;  For a square matrix , > 0 ( < 0) means that this matrix is positive definite (negative definite);

 The notation ‖ ‖ ℒ = ∫ ‖ ( )‖
is the ℒ ℋ Observer for Descriptor Nonlinear Systems with Nonlinear Output Equations G. Phanomchoeng1,2 , A. Zemouche 3,4 , W. Jeon 5 , R. Rajamani 5 , and F. Mazenc 3 norm of the vector ∈ ℝ . The set ℒ is define by

ℒ = ∈ ℝ ∶ ‖ ‖ ℒ < +∞ and then ℒ , ‖•‖ ℒ is called the Lebesgue space;  ( ) = 0, … , 0, 1 ⏞ , 0, … , 0 ∈ ℝ , ≥ 1 is a
vector of the canonical basis of ℝ ;

II. PROBLEM FORMULATION AND MOTIVATIONS

A. System description The class of system to be investigated in this paper is defined by the following equations:

⎩ ⎨ ⎧ ( ) ̇= ( ) + ( ( ), ( )) + ( ), ( ) + ( ) ( ) = ( ) + ℎ ( ), ( ) + ( ) (1 )
where ∈ ℝ ; ∈ ℝ , and ∈ ℝ are respectively the state vector of the system, the known control input vector, and the output measurements vector. The vector ∈ ℝ represents the unknown disturbances affecting the system dynamics and the measured variables. The matrices

∈ ℝ × , ∈ ℝ × , ∈ ℝ × , ∈ ℝ × , ∈ ℝ × , ∈ ℝ × , ∈ ℝ ×
, and ∈ ℝ × are known and constant. The functions ∶ ℝ × ℝ → ℝ and ℎ ∶ ℝ × ℝ → ℝ are globally Lipschitz 1 with respect to the variable , uniformly on , with Lipschitz constants and , respectively. Assume also that the pair , satisfies the following condition:

= .

(

) 2
The aim consists in finding an observer to estimate asymptotically or exponentially the state . Contrary to regular systems 2 for which a simple Luenberger observer with one constant gain can be used to investigate the ℋ criterion, the problem of state observer for the descriptor system (1) turns out to be difficult in some cases. In this paper, we will first consider a well-known two-stage structure of the observer. The results we will propose can then be extended to more complicated structures [START_REF] Darouach | Observers for Lipschitz nonlinear descriptor systems: Application to unknown inputs systems[END_REF], [START_REF] Darouach | Observers for a class of nonlinear singular systems[END_REF]. Although the structure of the state observer is simple, the synthesis of the observer gain parameters has complicated obstacles that we need to overcome in order to have tractable sufficient LMI conditions.

B. The two-stage observer structure

The two-stage state observer we consider in this note is described by the following equations: 1 If and ℎ are only locally Lipschitz, we may consider their saturated versions and ℎ , respectively, on an invariant compact set in which and ℎ satisfy the global Lipschitz property [START_REF] Shim | Semi-global observer for multi-output nonlinear systems[END_REF]. 2 This means that the matrix is invertible, i.e. = .

⎩ ⎪ ⎨ ⎪ ⎧ ( ) = ( ) + + ( ( ), ( )) + ( ), ( ) ( )
= ( ) + ( ) ( ) = ( ) -( ), = 1,2 (3)
where ( ) is the estimate of ( ). The auxiliary variables ( ), ( ) and the matrices , , , , and are the observer parameters to be determined so that the estimation error ( ) = ( ) -( ) converges towards zero in sense of a pre-defined performance criterion, namely ℋ or , criteria.. By using (1) and (3) ( ) is expressed as:

( ) = ( ) + - ( ) + ℎ ( ), ( ) -( ) + ( ). (4)
In the case ( ) = ( ), the problem becomes easy and the results are well known in the literature, because with (. ) ≡ 0, equation ( 4) reduces to

( ) = ( ) + - ( ). (5)
Hence, from the condition (2), there exist two matrices and such that + = .

(6) Therefore, we can write the dynamics of the estimation error as:

̇( ) = ( ) - ( ) ̇ (7)
and then we can exploit the singular equation in (1) and the dynamics of ( ) in (3) to lead to an appropriate and standard estimation error dynamics after some matrix manipulations [START_REF] Darouach | Observers for Lipschitz nonlinear descriptor systems: Application to unknown inputs systems[END_REF], [START_REF] Ha | State and input simultaneous estimation for a class of nonlinear systems[END_REF], [START_REF] Darouach | Reduced-order observer design for descriptor systems with unknown inputs[END_REF]. However, the presence of ℎ(( ( ), ( )) and ( ) in the output measurements renders the problem more complicated. Indeed, the computation of the parameters , , , , , and the choice of the performance criterion depend on additional assumptions on the matrices , on the disturbance ( ), and on the nonlinear function ℎ( , ). In particular, the presence of nonlinearities in the output measurements renders the problem difficult and original from a LMI point of view. Depending on the distribution of the nonlinearity ℎ(. , . ) in ( ) (this is related to the structure of the matrix ) and its time derivative, a different LMI technique can be provided in each case.

C. ℋ criterion

This subsection is devoted to some definitions related to the ℋ performance criterion. The aim consists in finding ( ), ( ) and the matrices , , , , and so that the estimation error ( ) satisfies the criterion:

( ) ℒ ≤ ‖ ‖ ℒ + (8)
where > 0 is the disturbance attenuation level and > 0 is to be determined. In fact, √ is the gain from to .

Department of Mechanical Engineering, Faculty of Engineering, Chulalongkorn University, Bangkok 10330, Thailand.

Smart Wireless Communication Ecosystem Research Group, Chulalongkorn University, Bangkok 10330, Thailand.

EPI Inria DISCO, Laboratoire des Signaux et Syst`emes, CNRSCentraleSup´elec, 91192 Gif-sur-Yvette, France.

University of Lorraine, 186, rue de Lorraine, CRAN UMR CNRS 7039,

Cosnes et Romain, France.[START_REF] Darouach | Observers for Lipschitz nonlinear descriptor systems: Application to unknown inputs systems[END_REF] Laboratory for Innovations in Sensing, Estimation, and Control, Department of Mechanical Engineering, University of Minnesota, Minneapolis, USA.

Usually we use Lyapunov functions to get checkable conditions guaranteeing [START_REF] Gauthier | A simple observer for nonlinear systems, applications to bioreactors[END_REF]. In the LMI framework, we take a quadratic Lyapunov function ( ), such that

The objective is to develop some LMI conditions under which the inequality (9) holds.

III. OBSERVER DESIGN METHODOLOGY

In this section we will introduce a new LMI method to deal with the problem of state observer design for the described descriptor nonlinear systems.

To avoid the nonlinear term ℎ ( ), ( ) -( ) in ( 4), an obvious solution consists in choosing so that = 0. However, this solution requires the following additional rank condition:

which implies implicitly + ≥ + . This means that the nonlinear function ℎ is considered as unknown input, which requires additional measurements to satisfy [START_REF] Ha | State and input simultaneous estimation for a class of nonlinear systems[END_REF]. However, for practical reasons, it is often difficult to measure additional variables because of their expensive cost or unavailability at any cost.

To avoid the above constraint, we will introduce a new assumption on the nonlinear function ℎ, which will not affect the number of measurements required to achieve the procedure.

Assumption 3.1: There exist two matrices , and nonlinear function : ℝ × ℝ → ℝ so that the time derivative of the nonlinear output function ℎ( ( ), ( )) satisfies the following conditions: ℎ ( ), ( ) = ( ) + ( ), ( ) ,

A. A new two-stage observer structure Let us consider the two-stages observer (3) by specifying the variables , = 1,2. That is, we use the following observer structure:

From the rank condition (2), there exist two matrices and so that [START_REF] Zemouche | ℋ circle criterion observer design for Lipschitz nonlinear systems with enhanced LMI conditions[END_REF] which leads to [START_REF] Tsinias | Time-varying observers for a class of nonlinear systems[END_REF]. Then, the dynamics equation ( 4) becomes

+ ℎ ( ), ( ) -( ) . where ( ) = ( ) -( ).

(16) The new variable ( ) is introduced in order to avoid the derivative of the disturbance ( ). For some details on this issue, we refer the reader to [START_REF] Hassan | Robust unknown input observers for nonlinear time-delay systems[END_REF].

From Assumption 3.1, after developing the calculations, we get the following dynamics of ( ):

) Since , ℎ, and are globally Lipschitz, then from [START_REF] Zemouche | On LMI conditions to design observers for Lipschitz nonlinear systems[END_REF] there exist functions

The dynamics of the estimation error is now rewritten under the detailed form:

(25)

B. New LMI design conditions By calculating the derivative of the Lyapunov function

along the trajectory of (25), we get ( ) ≤ 0 if the following inequality holds: Using the Young's relation [START_REF] Zemouche | ℋ circle criterion observer design for Lipschitz nonlinear systems with enhanced LMI conditions[END_REF], we deduce that for all symmetric positive definite matrices , and ℕ we have

Finally, by following the methodology in [START_REF] Zemouche | ℋ circle criterion observer design for Lipschitz nonlinear systems with enhanced LMI conditions[END_REF], we get the following theorem.

Theorem 1: Assume that there exist symmetric positive

and a matrix ℛ ∈ ℝ × such that the following convex optimization problem is solvable:

53d) Then, the ℋ criterion ( 8) is satisfied with = (ℙ). The observer gain parameter is then computed by = ℙ ℛ .

(54)

IV. APPLICATION TO UNKNOWN INPUT ESTIMATION

This section is devoted to unknown input estimation for nonlinear systems with nonlinear outputs. The aim is to use the results of the previous section to estimate, in the ℋ sense, unknown inputs occurring in the system. The class of systems concerned by this study is described by the following nonlinear equations:

where ( ) ∈ ℝ is the system state and ( ) ∈ ℝ is the unknown input vector. The nonlinearities (. ) and (. )

satisfy the global Lipschitz assumption as in the previous section.

We will propose some schemes to estimate simultaneously the system state ( ) and the unknown input ( ). In each estimation scheme, we rewrite system (55) under the descriptor form (1) under specific assumptions on the parameters of (55), namely the matrices , , , , and the nonlinear function ( , ). There are several ways to organize the presentation of all the possible scenarios due to different assumptions required in each scenario. In this initial conference paper, we will only investigate the specific case of: =

For this specific case, LMI conditions ensuring the simultaneous asymptotic estimation of the states and the unknown inputs will be provided. System (55) can be rewritten below

( , ) = ( , , ), ℎ( , ) = ( , , ), ( , ) = ( , ).

Since the matrix is full column rank then it follows that the condition (2) holds. Indeed,

Hence the LMI design techniques given in Section III can be applied under additional assumptions.

Under Assumption 3.1, the observer based filter (13) provides a simultaneous estimation of the state ( ) and the unknown input ( ) in the ℋ sense defined in [START_REF] Gauthier | A simple observer for nonlinear systems, applications to bioreactors[END_REF] provided that the filter parameters are obtained by solving the convex optimization problem (31) in Theorem 1 according to the definitions (57).

V. CONCLUSION

In this note we have proposed a new LMI design technique that satisfies a ℋ performance criterion for a class of descriptor nonlinear systems. The presence of nonlinearities in the output measurements renders the synthesis problem particularly complicated. To overcome this obstacle, we have proposed a different LMI method, where the method requires a certain assumption on system parameters involving the numbers of measurements, unknown inputs and nonlinear functions in the output equations. We have shown that the introduction of these new assumptions allows handling the problem of observer synthesis. New and less conservative LMI conditions for observer design are proposed.

As future work, we aim to apply this technique to complicated real-word models, such as the tripped rollover problem and radar vehicle tracking problems in automotive applications.