Huyuan Chen
email: chenhuyuan@yeah.net

Laurent Véron
email: laurent.veron@lmpt.univ-tours.fr

Ying Wang
email: yingwang00@126.com

Fractional heat equations with subcritical absorption having a measure as initial data

Keywords: Fractional heat equation, Radon measure, Dirac mass, Self-similar solution, Very singular solution MSC2010: 35R06, 35K05, 35R11

We study existence and uniqueness of weak solutions to (F)

, where N ≥ 2, the operator (-∆) α is the fractional Laplacian with α ∈ (0, 1), ν is a bounded Radon measure and h : (0, ∞) × R → R is a continuous function satisfying a subcritical integrability condition.

In particular, if h(t, u) = t β u p with β > -1 and 0 < p < p * β := 1 + 2α(1+β) N , we prove that there exists a unique weak solution u k to (F) with ν = kδ 0 , where δ 0 is the Dirac mass at the origin. We obtain that u k → ∞ in (0, ∞) × R N as k → ∞ for p ∈ (0, 1] and the limit of u k exists as k → ∞ when 1 < p < p * β , we denote it by u ∞ . When 1 + 2α(1+β)

N +2α := p * * β < p < p * β , u ∞ is the minimal self-similar solution of (F ) ∞ ∂ t u + (-∆) α u + t β u p = 0 in (0, ∞) × R N with the initial condition u(0, •) = 0 in R N \ {0} and it satisfies u ∞ (0, x) = 0 for x = 0. While if 1 < p < p * * β , then u ∞ ≡ U p , where U p is the maximal solution of the differential equation y ′ + t β y p = 0 on R + .

Introduction

Let h : (0, ∞) × R → R be a continuous function and Q ∞ = (0, ∞) × R N with N ≥ 2. The first object of this paper is to consider existence and uniqueness of weak solutions to fractional heat equations

∂ t u + (-∆) α u + h(t, u) = 0 in Q ∞ , u(0, •) = ν in R N , (1.1)
where ν belongs to the space M b (R N ) of bounded Radon measures in R N and (-∆) α (0 < α < 1) is the fractional Laplacian defined by (-∆) α u(t, x) = lim ǫ→0 + (-∆) α ǫ u(t, x),

where, for ǫ > 0,

(-∆) α ǫ u(t, x) = R N u(t, x) -u(t, z) |z -x| N +2α χ ǫ (|x -z|)dz and χ ǫ (r) = 0 if r ∈ [0, ǫ], 1 if r > ǫ.
In a pioneering work, Brezis and Friedman [START_REF] Brezis | Nonlinear parabolic equations involving measures as initial conditions[END_REF] have studied the semilinear heat equation with measure as initial data

∂ t u -∆u + u p = 0 in Q ∞ , u(0, •) = kδ 0 in R N , (1.2)
where k > 0 and δ 0 is the Dirac mass at the origin. They proved that if 1 < p < (N + 2)/N , then for every k > 0 there exists a unique solution u k to (1.2). When p ≥ (N + 2)/N , problem (1.2) has no solution and even more, they proved that no nontrivial solution of the above equation vanishing on R N \{0} at t = 0 exists. When 1 < p < 1 + 2 N , Brezis, Peletier and Terman used a dynamical system technique in [START_REF] Brezis | A very singular solution of the heat equation with absorption[END_REF] to prove the existence of a very singular solution u s to

∂ t u -∆u + u p = 0 in Q ∞ , (1.3)
vanishing at t = 0 on R N \ {0}. This function u s is self-similar, i.e. expressed under the form

u s (t, x) = t -1 p-1 f |x| √ t , (1.4)
and f is uniquely determined by the following conditions

f ′′ + N -1 η + 1 2 η f ′ + 1 p-1 f -f p = 0 on R + f > 0 and f is smooth on R + f ′ (0) = 0 and lim η→∞ η 2 p-1 f (η) = 0.
(1.5) Furthermore, it satisfies

f (η) = c 1 e -η 2 η 2 p-1 -N {1 -O(|x| -2 )} as η → ∞
for some c 1 > 0. Later on, Kamin and Peletier in [START_REF] Kamin | Singular solutions of the heat equation with absorption[END_REF] proved that the sequence of weak solutions u k converges to the very singular solution u s as k → ∞. After that, Marcus and Véron in [START_REF] Marcus | Initial trace of positive solutions of some nonlinear parabolic equations[END_REF] studied the equation in the framework of the initial trace theory. They pointed out the role of the very singular solution of (1.3) in the study of the singular set of the initial trace, showing in particular that it is the unique positive solution of (1.3) satisfying lim t→0 Bǫ u(t, x)dx = ∞, ∀ǫ > 0, B ǫ = B ǫ (0), (1.6) and lim t→0 K u(t, x)dx = 0 , ∀K ⊂ R N \ {0}, K compact.

(1.7)

If one replaces u p by t β u p with p ∈ (1, 1 + 2(1+β) N ), these results were extended by Marcus and Véron (β ≥ 0) in [START_REF] Marcus | Initial trace of positive solutions to semilinear parabolic inequalities[END_REF] and then Al Sayed and Véron (β > -1) in [START_REF] Wang | Initial trace of solutions of semilinear heat equations with absorption[END_REF]. The initial data problem with measure and general absorption term ∂ t u -∆u + h(t, x, u) = 0 in (0, T) × Ω, u = 0 in (0, T ) × ∂Ω, u(0, •) = ν in Ω, (1.8) in a bounded domain Ω of R N , has been studied by Marcus and Véron in [START_REF] Marcus | Initial trace of positive solutions to semilinear parabolic inequalities[END_REF] in the framework of the initial trace theory. They proved that the following general integrability condition on h 0 ≤| h(t, x, r) |≤ h(t)f (|r|)

, ∀(x, t, r)

∈ Ω × R + × R T 0 h(t)f (σt N 2 )t -N 2 dt < ∞ , ∀σ > 0
either h(t) = t α with α ≥ 0 or f is convex, (1.9) in order that the problem has a unique solution for any bounded measure. In the particular case with h(t, x, r) = t β |u| p-1 u, it is fulfilled if 1 < p < 1 + 2(1+β) N and β > -1, and the very singular solution exists in this range of values.

Motivated by a growing number of applications in physics and by important links on the theory of Lévy process, semilinear fractional equations has been attracted much interest in last few years, (see e.g. [START_REF] Caffarelli | Regularity theory for nonlinear integral operators[END_REF][START_REF] Caffarelli | Regularity of solutions to the parabolic fractional obstacle problem[END_REF][START_REF] Lara | Regularity for solutions of non local parabolic equations[END_REF][START_REF] Chen | Large solution to elliptic equations involving fractional Laplacian[END_REF][START_REF] Chen | Semilinear fractional elliptic equations involving measures[END_REF][START_REF] Chen | Global heat kernel estimates for fractional laplacians in unbounded open sets[END_REF][START_REF] Droniou | Fractal first-order partial differential equations[END_REF][START_REF] Felmer | Radial symmetry of positive solutions to equations involving the fractional laplacian[END_REF]). Recently, in [START_REF] Chen | Semilinear fractional elliptic equations with gradient nonlinearity involving measures[END_REF] we obtained the existence and uniqueness of a weak solution to semilinear fractional elliptic equation

(-∆) α u + f (u) = ν in Ω, u = 0 in Ω c , (1.10)
when ν is a Radon measure and f satisfies a subcritical integrability condition. In [START_REF] Chen | Semilinear fractional elliptic equations involving measures[END_REF] we studied the the different types of isolated singularities when f (u) = u p where 1 < p < N N -2α . In particular, assuming that 0 ∈ Ω, we proved that the sequence of solutions {u k } (k ∈ N) of (1.10), with ν = kδ 0 converges to infinity when k → ∞, if p ∈ (0, 1 + 2α N ) and it converges to a solution with a strong singularity at 0 if p ∈ (1 + 2α N , N N -2α ). One purpose of this paper is to study the existence and uniqueness of weak solutions to semilinear fractional heat equation (1.1) in a measure framework. We first make precise the notion of weak solution of (1.1) that we will use in this note. Definition 1.1 We say that u is a weak solution of (1.1), if for any T > 0, u ∈ L 1 (Q T ), h(t, u) ∈ L 1 (Q T ) and

Q T (u(t, x)[-∂ t ξ(t, x) + (-∆) α ξ(t, x)] + h(t, u)ξ(t, x)) dxdt = R N ξ(0, x)dν - R N ξ(T, x)u(T, x)dx ∀ξ ∈ Y α,T , (1.11)
where Q T = (0, T ) × R N and Y α,T is a space of functions ξ : [0, T ] × R N → R satisfying

(i) ξ L 1 (Q T ) + ξ L ∞ (Q T ) + ∂ t ξ L ∞ (Q T ) + (-∆) α ξ L ∞ (Q T ) < +∞;
(ii) for t ∈ (0, T ), there exist M > 0 and ǫ 0 > 0 such that for all ǫ ∈ (0,

ǫ 0 ], (-∆) α ǫ ξ(t, •) L ∞ (R N ) ≤ M.
Before stating our main theorems, we introduce the subcritical integrability condition for the nonlinearity h, that is,

(H) (i) The function h : (0, ∞) × R → R is continuous and for any t ∈ (0, ∞), h(t, 0) = 0 and h(t, r 1 ) ≥ h(t, r 2 ) if r 1 ≥ r 2 .
(ii) There exist β > -1 and a continuous, nondecreasing function g :

R + → R + such that |h(t, r)| ≤ t β g(|r|) ∀(t, r) ∈ (0, ∞) × R and +∞ 1 g(s)s -1-p * β ds < +∞, (1.12)
where

p * β = 1 + 2α(1 + β) N . (1.13)
We denote by

H α : (0, ∞) × R N × R N → R + the heat kernel for (-∆) α in (0, ∞) × R N , by H α [ν] the associated heat potential of ν ∈ M b (R N ), defined by H α [ν](t, x) = R N H α (t, x, y)dν(y)
and by H α [µ] the Duhamel operator defined for (t, x) ∈ Q T and any µ ∈ L 1 (Q T ) by

H α [µ](t, x) = t 0 H α [µ(s, .)](t -s, x)ds = t 0 R N H α (t -s, x, y)µ(s, y)dyds.
Now we state our first theorem as follows.

Theorem 1.1 Assume that ν ∈ M b (R N ) and the function h satisfies (H). Then problem (1.1) admits a unique weak solution u ν such that

H α [ν] -H α [h(., H α [ν + ])] ≤ u ν ≤ H α [ν] -H α [h(., -H α [ν -])] in Q ∞ , (1.14)
where ν + and ν -are respectively the positive and negative part in the Jordan decomposition of ν. Furthermore,

(i) if ν is nonnegative, so is u ν ;
(ii) the mapping: ν → u ν is increasing and stable in the sense that if {ν n } is a sequence of positive bounded Radon measures converging to ν in the weak sense of measures, then {u νn } converges to u ν locally uniformly in Q ∞ .

According to Theorem 1.1, there exists a unique positive weak solution u k to

∂ t u + (-∆) α u + t β u p = 0 in Q ∞ , u(0, •) = kδ 0 in R N , (1.15)
where β > -1, k > 0 and p ∈ (0, p * β ). We observe that u k → ∞ in (0, ∞) × R N as k → ∞ for p ∈ (0, 1], see Proposition 4.2 for details. Our next interest in this paper is to study the limit of u k as k → ∞ for p ∈ (1, p * β ), which exists since {u k } k is an increasing sequence of functions, bounded by 1+β p-1

1 p-1 t -1+β
p-1 , and we set

u ∞ = lim k→∞ u k in Q ∞ . (1.16)
Actually, u ∞ and {u k } k are classical solutions to equation

∂ t u + (-∆) α u + t β u p = 0 in Q ∞ , (1.17)
see Proposition 4.3 for details.

Definition 1.2 (i) A solution u of (1.17) is called a self-similar solution if u(t, x) = t -1+β p-1 u(1, t -1 2α x) (t, x) ∈ Q ∞ .
(ii) A solution u of (1.17) is called a very singular solution if it vanishes on R N \{0} at t = 0 and

lim t→0 + u(t, 0) Γ α (t, 0) = +∞,
where

Γ α := H α [δ 0 ] is the fundamental solution of ∂ t u + (-∆) α u = 0 in Q ∞ , u(0, •) = δ 0 in R N . (1.18)
We remark that for p ∈ (1, p * β ), a self-similar solution u of (1.17) is also a very singular solution, since lim

t→0 + Γ α (t, 0)t N 2α = c 2 , (1.19)
for some c 2 > 0. For any self-similar solution u of (1.17

), v(η) := u(1, t -1 2α x) with η = t -1 2α
x is a solution of the self-similar equation

(-∆) α v - 1 2α ∇v • η - 1 + β p -1 v + v p = 0 in R N . (1.20) Since 1+β p-1 1
p-1 is a constant nonzero solution of (1.20), the function

U p (t) := 1 + β p -1 1 p-1 t -1+β p-1 t > 0 (1.21)
is a flat self-similar solution of (1.17). It is actually the maximal solution of the ODE y ′ + t β y p = 0 defined on R + . Our next goal in this paper is to study non-flat self-similar solutions of (1.17).

Theorem 1.2 Assume that β > -1, u ∞ is defined by (1.16) and

p * * β < p < p * β ,
where

p * * β = 1 + 2α(1+β) N +2α
. Then u ∞ is a very singular self-similar solution of (1.17) in Q ∞ . Moreover, there exists c 3 > 1 such that

c -1 3 1 + |x| N +2α ≤ u ∞ (1, x) ≤ c 3 ln(2 + |x|) 1 + |x| N +2α x ∈ R N . (1.22)
When p * * β < p < p * β with β > -1, we observe that u ∞ and U p are self-similar solutions of (1.17) and u ∞ is non-flat. Now we are ready to consider the uniqueness of non-flat self-similar solution of (1.17) with decay at infinity, precisely, we study the uniqueness of self-similar solution to

∂ t u + (-∆) α u + t β u p = 0 in Q ∞ , lim |x|→∞ u(1, x) = 0.
(1.23)

We remark that if u is self-similar, then the assumption lim |x|→∞ u(1, x) = 0 is equivalent to lim |x|→∞ u(t, x) = 0 for any t > 0. Finally, we state the properties of u ∞ when 1 < p ≤ p * * β as follows.

Theorem 1.3 (i) Assume 1 < p < p * * β and u ∞ is defined by (1.16). Then u ∞ = U p , where U p is given by (1.21).

(ii) Assume p = p * * β and u ∞ is defined by (1.16). Then u ∞ is a self-similar solution of (1.17) such that

u ∞ (t, x) ≥ c 4 t -N+2α 2α 1 + |t -1 2α x| N +2α (t, x) ∈ (0, 1) × R N , (1.24)
for some c 4 > 0.

We note that Theorem 1.3 indicates that there exists no self-similar solution of (1.17) with an initial data u(0,

•) vanishing in R N \ {0} if p ∈ (1, p * * β )
, since u ∞ is the least self-similar solution. In Theorem 1.3 part (ii), we do not know if the self-similar solution is flat or not. From the above theorems, we have the following result. (1.25)

Furthermore, It is worth comparing the above theorems with the results obtained by Nguyen and Véron [START_REF] Nguyen-Phuoc | Local and global properties of solutions of heat equation with superlinear absorption[END_REF] concerning the limit, when k → ∞ of the solutions u = u k of

c -1 3 1 + |η| N +2α ≤ v ∞ (η) ≤ c 3 ln(2 + |η|) 1 + |η| N +2α ∀η ∈ R N (1.26) (
∂ t u -∆u + u(ln(u + 1))) α = 0 in Q ∞ , u(0, .) = kδ 0 in R N , (1.27)
where α > 0. Note that u k > 0 and the sequence {u k } is increasing. In this problem, they proved that the diffusion is dominating if 0 < α ≤ 1 and the limit of the u k is infinite. If 1 < α ≤ 2 the absorption dominates, but the limit of the u k is the maximal solution of the associated ODE, y ′ + y(ln(y + 1))) α = 0 on R + . Finally, if α > 2 the limit of the u k is a solution with a strong isolated singularity at (0, 0), which could be called a very singular solution, although it is not self-similar.

This paper is organized as follows. In Section 2 we introduce some properties of Marcinkiewicz spaces and Kato's type inequality for non-homogeneous problems. In Section 3 we prove Theorem 1.1. Section 4 is devoted to investigate the properties of solutions to (1.15). In Section 5 we give the proof of Theorem 1.2 and Theorem 1.3. Finally, we prove Theorem 1.4.

Linear estimates 2.1 The Marcinkiewicz spaces

We recall the definition and basic properties of the Marcinkiewicz spaces. Definition 2.1 Let Θ ⊂ R N +1 be an open domain and µ be a positive Borel measure in Θ. For κ > 1, κ ′ = κ/(κ -1) and u ∈ L 1 loc (Θ, dµ), we set

u M κ (Θ,dµ) = inf c ∈ [0, ∞] : E |u|dµ ≤ c E dµ 1 κ ′ , ∀E ⊂ Θ, E Borel set (2.1) and M κ (Θ, dµ) = {u ∈ L 1 loc (Θ, dµ) : u M κ (Θ,dµ) < ∞}. (2.2)
M κ (Θ, dµ) is called the Marcinkiewicz space of exponent κ or weak L κ space and . M κ (Θ,dµ) is a quasi-norm. The following property holds.

Proposition 2.1 [START_REF] Ph | A semilinear elliptic equation in L 1 (R N )[END_REF][START_REF] Chen | Semilinear fractional elliptic equations with gradient nonlinearity involving measures[END_REF] Assume that 1 ≤ q < κ < ∞ and u ∈ L 1 loc (Θ, dµ). Then there exists c 5 > 0 dependent of q, κ such that

E |u| q dµ ≤ c 5 u M κ (Θ,dµ) E dµ 1-q/κ , for any Borel set E of Θ. Remark 2.1 If Ω is a smooth domain of R N , we denote by H Ω α : (0, ∞) × Ω × Ω → R + the heat kernel for (-∆) α and, if ν ∈ M b (Ω), by H Ω α [ν]
the corresponding heat potential of ν defined by

H Ω α [ν](t, x) = Ω H Ω α (t, x, y)dν(y).
When Ω = R N , by Fourier transform, it is clear that

H α (t, x, y) = 1 (2π) N/2 R N e i(x-y)•ζ-t|ζ| 2α dζ = H α (t, x -y, 0).
Furthermore, H α (t, ., 0) L 1 is independent of t. This implies

H Ω α [ν](t, .) L p ≤ ν L p , ∀1 ≤ p ≤ ∞ , ∀ν ∈ L p (R N ). (2.3) Since H Ω α [ν](t + s, .) = H Ω α [H Ω α [ν](s, .)](t, .
) for all t, s > 0 (semigroup property) and

ν ≥ 0 =⇒ H Ω α [ν](t, .) ≥ 0 the semigroup {H Ω α [.](t, .
)} t≥0 is sub-Markovian. Furthermore, since the operator (-∆) α is symmetric in L 2 (R N ), the above semigroup is analytic in L p (R N ) for all 1 ≤ p < ∞: if 1 < p < ∞ it follows from a general result of Stein [START_REF] Stein | Harmonic Analysis: Real-variable methods, orthogonality and oscillatory integrals[END_REF]) and for p = 1 it is a consequence of regularity result from fractional powers of operators theory (see e.g. [START_REF] Komatsu | Fractional powers of operaors[END_REF]). For 1 ≤ p < ∞ the generator A p of the semigroup in L p (R N ) is the operator -(-∆) α with domain

D(A p ) := {ν ∈ L p (R N ) : (-∆) α ν ∈ L p (R N )}.
(2.4) and D(A p ) is dense since it contains C ∞ 0 (R N ). If p = ∞, the natural space is the space C 0 (R N ) of continuous functions in R N tending to 0 at infinity. The domain of the corresponding operator A c 0 is

D(A c 0 ) := {ν ∈ C 0 (R N ) : (-∆) α ν ∈ C 0 (R N )}. (2.5)
This operator is densely defined in C 0 (R N ). In order to avoid confusion,

C c (R N ) (resp. C ∞ c (R N )) denotes the space of continuous (resp. C ∞ ) functions in R N with compact support. It is a dense subset of C 0 (R N ).
The following regularizing effect L p (R N ) → L q (R N ) (1 ≤ p ≤ q ≤ ∞) is valid for any submarkovian semigroup of contractions in all L p (R N )-spaces which has a self-adjoint generator in L 2 (R N ) (see e.g. [START_REF] Stein | Topics in Harmonic Analysis Related to the Littlewood-Paley Theory[END_REF]).

Proposition 2.2 Assume 1 ≤ p ≤ q ≤ ∞, p = ∞. Then for any ν ∈ L p (R N ), H α [ν](t, .) ∈ L q (R N )∩D(A q
) for all t > 0 and there holds, for some positive constant c = c(α, N, p, q),

H α [ν](t, .) L q (R N ) ≤ c t N 2α ( 1 p -1 q ) ν L q (R N ) . (2.6)
Note also that the function

(t, x) → H α [ν](t, x) is C ∞ in Q ∞ as a result of the analyticity on the semigroup {H α [.](t)} t>0 .
Proposition 2.3 For any β > -1 and T > 0, there exists c 6 > 0 dependent of N, α, β such that for ν ∈ M b (Ω),

H Ω α [|ν|] M p * β (Q Ω T ,t β dxdt) ≤ c 6 ν M b (Ω) , (2.7)
where p * β is defined by (1.13) and

Q Ω T = (0, T ) × Ω.
In order to prove this proposition, we introduce some notations. For λ > 0 and y ∈ Ω, let us denote

A Ω λ (y) = {(t, x) ∈ Q Ω T : H Ω α (t, x, y) > λ} and m Ω λ (y) = A Ω λ (y) t β dxdt. We also set A R N λ = A λ and m R N λ = m λ .
Lemma 2.1 There exists c 7 > 0 such that for any λ > 1,

A λ (y) ⊂ (0, c 7 λ -2α N ] × B c 7 λ -1 N (y), (2.8)
where B r (y) is the ball with radius r and center y in R N .

Proof. We observe that

H α (t, x, y) = t -N 2α Γ α (1, (x -y)t -1 2α
), where Γ α is the fundamental solution of (1.18). From [START_REF] Blumenthal | Some theorems on stable processes[END_REF] (see also [START_REF] Chen | Heat kernel estimates for the Dirichlet fractional Laplacian[END_REF] for an analytic proof), there exists

c 8 > 0 such that Γ α (1, z) ≤ c 8 1 + |z| N +2α .
This implies in particular

H α (t, x, y) ≤ c 8 t -N 2α 1 + t -1 2α |x -y| N +2α .
(2.9)

On the one hand, for (t, x) ∈ A λ (y), we have that

t -N 2α Γ α (1, 0) ≥ t -N 2α Γ α (1, (x -y)t -1 2α ) > λ, which implies t < Γ 2α N α (1, 0)λ -2α N . (2.10)
On the other hand, letting r = |x -y|,

c 8 t t 1+ N 2α + r N +2α ≥ t -N 2α Γ α (1, (x -y)t -1 2α ) > λ, then r ≤ (c 8 tλ -1 ) 1 N+2α , (2.11)
which, together with (2.10), implies

r ≤ c 9 λ -1 N ,
for some c 9 > 0.

Proof of Proposition 2.3. By Lemma 2.1, there exists c 10 > 0 such that

m λ (y) ≤ c 10 λ -1-2α(1+β) N . Clearly H Ω α (t, x, y) ≤ H α (t, x, y), (2.12)
then for any Borel set E ⊂ Q Ω T and y ∈ Ω, we have that

E H Ω α (t, x, y)t β dxdt ≤ λ E t β dxdt + A λ (y) H α (t, x, y)t β dxdt and A λ (y) H α (t, x, y)t β dxdt = - +∞ λ sdm s (y) = λm λ (y) + +∞ λ m s (y)ds ≤ c 10 λ -2α(1+β) N + c 10 +∞ λ s -1-2α(1+β) N ds ≤ c 11 λ -2α(1+β) N ,
where

c 11 = c 10 1 + N 2α(1+β) . As a consequence, it follows E H Ω α (t, x, y)t β dxdt ≤ λ E t β dxdt + c 11 λ -2α(1+β) N . Taking λ = ( E t β dxdt) - N N+2α(1+β) , we obtain that E H Ω α (t, x, y)t β dxdt ≤ (c 11 + 1)( E t β dxdt) 2α(1+β) N+2α(1+β) . (2.13)
Since, by Fubini's theorem,

E H Ω α [|ν|](t, x)t β dxdt = E Ω H Ω α (t, x, y)d|ν(y)|t β dxdt = Ω E H Ω α (t, x, y)t β dxdtd|ν(y)|,
together with (2.13), it yields

E H Ω α [|ν|](t, x)t β dxdt ≤ (c 11 + 1) ν M b (Ω) E t β dxdt 2α(1+β) N+2α(1+β) .
Thus,

H Ω α [|ν|] M 1+ 2α(1+β) N (Q Ω T ,t β dxdt) ≤ (c 11 + 1) ν M b (Ω) ,
which ends the proof.

The non-homogeneous problem

In this section we consider the linear non-homogeneous problem

∂ t u + (-∆) α u = µ in Q T , u(0, •) = ν in R N . (2.14) If µ ∈ L 1 (Q T ) and ν ∈ L 1 (R N ) a function u defined in Q T is an integral solution of (2.14) in Q T if it is expressed by Duhamel's formula, that is u(t, x) = H α [ν](t, x) + H α [µ](t, x) a.e. in Q T . (2.15)
where, we denote by H α the operator of L 1 (Q T ) defined for all (x, t) ∈ Q T by

H α [µ](x, t) = t 0 H α [µ(., s)](x, t -s)ds = t 0 R N H α (t -s, x, y)µ(s, y)dyds. (2.16)
Notice that, by Duhamel's formula, there holds

u(t, •) L 1 (R N ) ≤ µ L 1 (Q T ) + ν L 1 (R N ) , ∀t ∈ (0, T ), (2.17)
and

u L 1 (Q T ) ≤ T ( µ L 1 (Q T ) + ν L 1 (R N ) ). (2.18)
The advantage of this notion of solution is that Duhamel's formula has a meaning as soon as µ and ν are integrable in their respective domains of definition. As for any continuous semigroup of bounded linear operators, a strong solution is an integral solution.

The following proposition is the Kato's type estimate which is essential tool to prove the uniqueness of solutions to (1.1). For T > 0, we denote

Q T = (0, T ) × R N . Proposition 2.4 Assume µ ∈ L 1 (Q T ) and ν ∈ L 1 (R N ).
Then there exists a unique weak solution u ∈ L 1 (Q T ) to the problem (2.14) and there exists c 12 > 0 such that

Q T |u|dxdt ≤ c 12 Q T |µ|dxdt + c 12 R N |ν|dx. (2.19)
Moreover, for any ξ ∈ Y α,T , ξ ≥ 0, we have that

Q T |u|(-∂ t ξ + (-∆) α ξ)dxdt + R N |u(T, x)|ξ(T, x)dx ≤ Q T ξsign(u)µdxdt + R N ξ(0, x)|ν|dx (2.20)
and

Q T u + (-∂ t ξ + (-∆) α ξ)dxdt + R N u + (T, x)ξ(T, x)dx ≤ Q T ξsign + (u)µdxdt + R N ξ(0, x)ν + dx.
(2.21)

In order to prove Proposition 2.4, we introduce the following notations. We say that u :

Q T → R is in C σ,σ ′ t,x (Q T ) for σ, σ ′ ∈ (0, 1) if u C σ,σ ′ t,x (Q T ) := u L ∞ (Q T ) + sup Q T |u(t, x) -u(s, y)| |t -s| σ + |x -y| σ ′ < +∞ and u ∈ C 1+σ,2α+σ ′ t,x (Q T ) if u C 1+σ,2α+σ ′ t,x (Q T ) := u L ∞ (Q T ) + ∂ t u C σ,σ ′ t,x (Q T ) + (-∆) α u C σ,σ ′ t,x (Q T ) < +∞. Lemma 2.2 Let µ ∈ C 1 (Q T ) ∩ L ∞ (Q T ), ν ∈ L ∞ (R N )
and u be an integral solution of problem (2.14), then there exists σ ∈ (0, 1)

such that u ∈ C 1+σ,2α+σ t,x in (ǫ, T )× R N for any ǫ ∈ (0, T ). In particular, if D 2 ν L ∞ (R N ) + (-∆) α ν C 1-α x (R N ) < ∞, then u ∈ C 1+σ,2α+σ t,x (Q T ). Proof. Step 1. When D 2 ν L ∞ (R N ) + (-∆) α ν C 1-α x (R N ) < ∞, it follows directly by [9, (A.1)] that u ∈ C 1+σ,2α+σ t,x (Q T ).
Step 2. When ν ∈ L ∞ (R N ), we use [10, Theorem 6.1] to obtain that u ∈ C σ 2α ,σ t,x (Q T ) for some σ > 0. For any ǫ ∈ (0, T ), let η : [0, T ] → [0, 1] be a C 2 function such that η = 0 in [0, ǫ 4 ] and η = 1 in [ǫ, T ] and v = ηu in Q T . Since η does not depend on x, we obtain that v satifies,

∂ t v + (-∆) α v = ηµ + η ′ (t)u, ∀(t, x) ∈ Q T , where ηµ+η ′ (t)u ∈ C σ 2α ,σ t,x (Q T ) and v(0, •) = 0 in R N , Then we apply the argument in Step 1 to obtain that v ∈ C 1+σ,2α+σ t,x (Q T ). Therefore, u is C 1+σ,2α+σ t,x in (ǫ, T ) × R N . The proof is complete. Lemma 2.3 (i) Let µ ∈ C 1 (Q T ) ∩ L ∞ (Q T ) and ν ∈ C 1 (R N ) ∩ L ∞ (R N ), then problem (2.14) admits a unique classical solution u. (ii) Let µ ∈ C 1 (Q T ) ∩ L ∞ (Q T ) ∩ L 1 (Q T ), ν ∈ C 2 (R N ) ∩ L ∞ (R N ) ∩ L 1 (R N ) and u be the classical solution of (2.14), then u is C 1+σ,2α+σ t,x in (ǫ, T ) × R N for any ǫ ∈ (0, T ) and for any ξ ∈ Y α,T , Q T u(t, x)[-∂ t ξ(t, x) + (-∆) α ξ(t, x)]dxdt = Q T µ(t, x)ξ(t, x)dxdt + R N ξ(0, x)νdx - R N ξ(T, x)u(T, x)dx. (2.22)
Thus u is a weak solution and it belongs to Y α,T .

(iii) Let μ ∈ C 1 (Q T ) ∩ L ∞ (Q T ) and ν ∈ C 1 (R N ) ∩ L ∞ (R N ), then problem -∂ t w + (-∆) α w = μ in Q T , w(T, •) = ν in R N (2.23) admits a unique classical solution w ∈ C 1+σ,2α+σ t,x (Q T ) for some σ ∈ (0, 1). More- over, if µ ∈ C 1 (Q T ) ∩ L ∞ (Q T ) ∩ L 1 (Q T ) and ν ∈ C 2 (R N ) ∩ L ∞ (R N ) ∩ L 1 (R N ), then
ξ is a weak solution and it belongs to Y α,T .

Proof. (i) By [10, Theorem 3.3, Theorem 6.1], if µ and ν are continuous and bounded, there exists a unique viscosity solution u ∈ C(Q T ). The higher regularity is provided by [10, Theorem 6.1] which asserts that there exist σ > 0 and a positive constant c depending on N , τ ∈ (0, T ) and α such that for all (t, x) and (s, y) belonging to

Q B 1 T -τ , there holds | u(t, x) -u(s, y) | (|x -y| + |t -s| 1 2α ) σ ≤ c u L ∞ (Q B 2 T ) + sup 0≤t≤T u(t, .) L 1 (R N + µ L ∞ (Q T ) (2.24)
where

Q Ω T = (0, T ) × Ω. Thus u ∈ C σ 2α ,σ t,x (Q T ). By Lemma 2.2 the integral solution u belongs to C 1+σ ′ ,2α+σ ′ t,x
in (ǫ, T )×R N for any ǫ ∈ (0, T ) and some σ ′ ∈ (0, min{ σ 2α , σ}). Then u is a classical solution of (2.14) and thus a viscosity solution.

(ii) By the definition of (-∆) α u, u(t, •) ∈ L 1 (R N ) for all t ∈ (0, T ). As in [9, Appendix A.2] we have Duhamel formula, thus u ∈ L 1 (Q T ) and it is an integral solution.

We claim that

(-∆) α ǫ u(t, •) L ∞ (R N ) is uniformly bounded with respect to ǫ ∈ (0, ǫ 0 ). Since u(t, •) ∈ C 2α+σ x (R N ) for some σ ∈ (0, min{2 -2α, 1}), then for x ∈ R N and y ∈ B 1 (0), |u(x + y) + u(x -y) -2u(x)| ≤ u(t, •) C 2α+σ x (R N ) |y| 2α+σ . Thus, |(-∆) α ǫ u(t, •)| L ∞ (R N ) ≤ sup x∈R N R N \B 1 (0) |u(x + y) -u(x)| |y| N +2α dy + 1 2 B 1 (0)\Bǫ(0) |u(x + y) + u(x -y) -2u(x)| |y| N +2α dy ≤ 2 u L 1 (R N ) + B 1 (0) |y| σ-N dy u(t, •) C 2α+σ x (R N ) .
Next we claim that

Q T ξ(-∆) α ǫ udxdt = Q T u(-∆) α ǫ ξdxdt ∀ξ ∈ Y α,T . (2.25)
Indeed, using the fact that for any t > 0 there holds

R N R N [u(t, z) -u(t, x)]ξ(t, x) |z -x| N +2α χ ǫ (|x -z|)dzdx = R N R N [u(t, x) -u(t, z)]ξ(t, z) |z -x| N +2α χ ǫ (|x -z|)dzdx, then we have R N ξ(t, x)(-∆) α ǫ u(t, x)dx = - 1 2 R N R N (u(t, z) -u(t, x))ξ(t, x) |z -x| N +2α + (u(t, x) -u(t, z))ξ(t, z) |z -x| N +2α χ ǫ (|x -z|)dzdx = 1 2 R N R N [u(t, z) -u(t, x)][ξ(t, z) -ξ(t, x)] |z -x| N +2α χ ǫ (|x -z|)dzdx.
Similarly,

R N u(t, x)(-∆) α ǫ ξ(t, x)dx = 1 2 R N R N [u(t, z) -u(t, x)][ξ(t, z) -ξ(t, x)] |z -x| N +2α χ ǫ (|x -z|)dzdx. Then (2.25) holds. Since u is C 1+σ,2α+σ t,x in (ǫ, T ) × R N for any ǫ ∈ (0, T ) and ξ belongs to Y α,T , (-∆) α ǫ ξ(t, •) → (-∆) α ξ(t, •) and (-∆) α ǫ u(t, •) → (-∆) α u(t, •) as ǫ → 0 in R N and (-∆) α ǫ ξ(t, •), (-∆) α ǫ u(t, •) ∈ L ∞ (R N ) and ξ(t, •), u(t, •) ∈ L 1 (R N ), then it follows by the Dominated Convergence Theorem that lim ǫ→0 + R N ξ(t, x)(-∆) α ǫ u(t, x)dx = R N ξ(t, x)(-∆) α u(t, x)dx and lim ǫ→0 + R N (-∆) α ǫ ξ(t, x)u(t, x)dx = R N (-∆) α ξ(t, x)u(t, x)dx.
Combining this with (2.25), and letting ǫ → 0 + , we have that

R N ξ(t, x)(-∆) α u(t, x)dx = R N (-∆) α ξ(t, x)u(t, x)dx,
integrating over [0, T ] and by (2.14), we conclude that (2.22) holds.

(iii) End of the proof. Let u be the weak solution of problem (2.14) obtained from (ii) with μ(T -t, .) = µ(t, .) and

w(t, x) = u(T -t, x) (t, x) ∈ [0, T ] × R N .
Then w is a solution of (2.23) and for some σ ∈ (0, 1),

w is C 1+σ,2α+σ t,x (Q T ). On the contrary, if w is a solution of (2.23), then u(t, x) = w(T -t, x) for (t, x) ∈ [0, T ] × R N
is a solution of (2.14), then the uniqueness holds since the solution of (2.14) is unique.

Since u ∈ C 1+σ,2α+σ t,x (Q T ), then (-∆) α u(t, •) ∈ C σ x and then (-∆) α ǫ u(t, •) is bounded, which implies u ∈ Y α,T . Proof of Proposition 2.4. Uniqueness. Let v ∈ L 1 (Q T ) be a weak solution of ∂ t v + (-∆) α v = 0 in Q T , v(0, •) = 0 in R N . (2.26)
We claim that v = 0 a.e. in Q T .

In fact, let ω be a Borel subset of Q T and η ω,n be the solution of

-∂ t u + (-∆) α u = ζ n in Q T , u(T, •) = 0 in R N , (2.27)
where

ζ n : QT → [0, 1] is a function C 1 c (Q T ) such that ζ n → χ ω in L ∞ ( QT ) as n → ∞.
Then η ω,n ∈ Y α,T by Lemma 2.3, and

Q T vζ n dxdt = 0.
Passing to the limit when n → ∞, we derive

ω vdxdt = 0.
This implies v = 0 a.e. in Q T .

Existence and estimate (2.20). For δ > 0, we define an even convex function φ δ by

φ δ (t) = |t| -δ 2 if |t| ≥ δ, t 2 2δ if |t| < δ/2.
(2.28)

Then for any t, s ∈ R, |φ ′ δ (t)| ≤ 1, φ δ (t) → |t| and φ ′ δ (t) → sign(t) when δ → 0 + . Moreover, φ δ (s) -φ δ (t) ≥ φ ′ δ (t)(s -t). (2.29) Let {µ n }, {ν n } be two sequences of functions in C 2 0 (Q T ), C 2 0 (R N ), respectively, such that lim n→∞ Q T |µ n -µ|dxdt = 0, lim n→∞ R N |ν n -ν|dx = 0.
We denote by u n the corresponding solution to (2.14) where µ, ν are replaced by µ n , ν n , respectively. By Lemma 2.2 and Lemma 2.

3(ii), u n ∈ C 1+σ,2α+σ t,x (Q T ) ∩ L 1 (Q T )
and then we use Lemma 2.3 in [START_REF] Chen | Semilinear fractional elliptic equations with gradient nonlinearity involving measures[END_REF] and Lemma 2.3 (ii) to obtain that for any δ > 0 and ξ ∈ Y α,T , ξ ≥ 0,

Q T φ δ (u n )[-∂ t ξ + (-∆) α ξ]dxdt + R N ξ(T, x)φ δ (u n (T, x))dx = Q T ξ[∂ t φ δ (u n ) + (-∆) α φ δ (u n )]dxdt + R N ξ(0, x)φ δ (ν n )dx ≤ Q T ξφ ′ δ (u n )[∂ t u n + (-∆) α u n ]dxdt + R N ξ(0, x)φ δ (ν n )dx = Q T ξφ ′ δ (u n )µ n dxdt + R N ξ(0, x)φ δ (ν n )dx.
Letting δ → 0 + , we obtain

Q T |u n |[-∂ t ξ + (-∆) α ξ]dxdt + R N ξ(T, x)|u n (T, x)|dx ≤ Q T ξsign(u n )µ n dxdt + R N ξ(0, x)|ν n |dx.
(2.30)

Let η k be the solution of

-∂ t u + (-∆) α u = ς k in Q T , u(T, •) = 0 in R N , (2.31)
where

ς k : Q T → [0, 1] is a C 2 0 function such that ς k = 1 in (0, T ) × B k (0). From the proof of Lemma 2.3, ηk (t, x) := η k (T -t, x) satisfies with ςk (t, x) = ς k (T -t, x) ∂ t u + (-∆) α u = ςk in Q T , u(0, •) = 0 in R N . By Lemma 2.2, ηk ∈ C 1+σ,2α+σ t,x
(Q T ) with some σ ∈ (0, 1) and

0 ≤ ηk (t, x) ≤ c 8 T t R N (s -t) -N 2α 1 + |(s -t) -1 2α (y -x)| N +2α dyds ≤ c 8 T t R N dz 1 + |z| N +2α ds = c 13 (T -t).
Taking ξ = η k in (2.30), we derive that

Q T |u n |χ (0,T )×B k (0) dxdt ≤ c 13 T Q T |µ n |dxdt + c 13 T R N |ν n |dx.
Then, letting k → ∞, we have

Q T |u n |dxdt ≤ c 13 T Q T |µ n |dxdt + c 13 T R N |ν n |dx.
(2.32)

Similarly, Remark 2.2 Other classes of uniqueness of solutions of the fractional heat equations exist. In [START_REF] Barrios | A Widders Type Theorem for the Heat Equation with Nonlocal Diffusion[END_REF] it is proved that any positive strong solution u ∈ C([0, T ) × R N ) can be represented by the convolution integral defined by

Q T |u n -u m |dx ≤ c 13 T Q T |µ n -µ m |dxdt + c 13 T R N |ν n -ν m |dx. ( 2
u(t, x) = R N P t (y)u(0, y)dy where P t (x) = 1 t N 2α e it -1 2α x.ξ-|ξ| 2α .
However the fact that u ∈ L 1 (Q T ) is a part of the definition of strong solution therein. Furthermore, the notion of weak solution used in this paper differs from ours.

3 Proof of Theorem 1.1

If h(t, .) is monotone nondecreasing, for any λ > 0, I + λh(t, .) is an homeorphism of R and the inverse function J λ (t, .) = (I + λh(t, .)) -1 is a contraction. We define the Yosida approximation by

h λ (t, .) = I -J λ (t, .) λ . (3.1)
The function h λ (t, .) is monotone nondecreasing, vanishes at 0 as h does it and it is

1 λ -Lipschitz continuous. Furthermore rh λ (t, r) ↑ rh(t, r) as λ → 0, ∀r ∈ R, (3.2
∂ t u + (-∆) α u + h λ • u = 0 in Q ∞ , u(0, •) = φ in R N . (3.3)
Moreover,

H α [φ] -H α [h λ • H α [φ + ])] ≤ u φ ≤ H α [φ] -H α [h λ • (-H α [φ -])] in Q T , (3.4)
where φ ± = max{0, ±φ} and

u φ (t, .) -u ψ (t, .) L 1 ≤ φ -ψ L 1 , ∀1 ≤ p ≤ ∞. (3.5) (i) u φ ≥ 0 if φ ≥ 0 in Ω;
(ii) the mapping φ → u φ is increasing.

Proof. Existence is a consequence of the Cauchy-Lipschitz-Picard theorem (see [START_REF] Th | An introduction to semilinear evolution equations[END_REF]Chap 4]): we write (3.3) under the integral form u

= T [u] = H α [φ] -H α [h λ • u], i.e. T [u](t, .) = H α [φ](t, .) - t 0 H α [h λ • u](t -s, .)ds. (3.6)
The space C([0, ∞); L 1 (R N )) endowed with the norm

w C-L 1 = sup e -kt w(t, .) L 1 : t ≥ 0 , (k > λ -1
), is a Banach space. Since u → h λ (t, u) is 1 λ -Lipschitz continuous, the mapping T is 1 λk -Lipschitz continuous in X p . Thus it admits a unique fixed point u φ which is an integral solution of (3.3).

u φ (t, .) = H α [φ](t, .) - t 0 H α [h λ • u φ ](t -s, .)ds.
(3.7)

The semigroup {H α [.](t, .)} t≥0 is analytic in L 1 (R N ) since it is generated by the fractional power of a closed operator. It follows from the classical regularity theory for analytic semigroups as exposed in [START_REF] Grisvard | Équations différentielles abstraites[END_REF]Sec 6] that u φ is a strong solution of (3.3). Since it is continuous, it is also a weak solution in the sense that

Q T (u φ [-∂ t ξ + (-∆) α ξ] + ξh λ • u φ ) dxdt = R N ξ(0, x)φ(x)dx - R N ξ(T, x)u φ (T, x)dx ∀ξ ∈ Y α,T . (3.8) If φ 1 , φ 2 ∈ L 1 (R N
) and u φ j are the corresponding solutions of (3.3), it follows from the positivity of H α that

(u φ 2 -u φ 1 ) + ≤ (H α [h λ • u φ 2 -h λ • u φ 1 ]) + ≤ 1 λ H α [(u φ 2 -u φ 1 ) + ].
Therefore, (u φ 2 (t, .) -u φ 1 (t, .))

+ L p ≤ 1 λ t 0 (u φ 2 (t -s) -u φ 1 (t -s)) + L p ds,
and by Gronwall inequality

(u φ 2 (t) -u φ 1 (t)) + L p ≤ e t λ (φ 2 -φ 1 ) + L p .
This implies (i) and (ii). As a consequence,

-H α [φ -] ≤ -u φ -≤ u φ ≤ u φ + ≤ H α [φ + ]
and thus

h λ • (-H α [φ -]) ≤ h λ • (-u φ -) ≤ h λ • u φ ≤ h λ • u φ + ≤ h λ • H α [φ + ].
Jointly with (3.7) it yields (3.4).

Notation. In the sequel, if η ∈ L 1 (Q τ ) and τ ≥ T , we denote by ξ η,τ the solution of

-∂ t ξ η + (-∆) α ξ η = η in Q τ , ξ η (τ, .) = 0. (3.9) If η ≥ 0, then ξ η,τ ≥ 0; if η ∈ C ∞ 0 (R N +1 ), then η ∈ Y α,τ ; if η n = η( . n ), where n ∈ N * and η ∈ C ∞ 0 (R N +1
) is nonnegative, 0 ≤ η ≤ 1, with value 1 on B 1 and 0 on B c 2 , then ξ ηn,τ ↑ τ -t as n → ∞.

In the next lemma we prove that we can replace h λ by h. Lemma 3.2 Assume that h satisfies (H)-(i) and φ ∈ L 1 (R N ). Then there exists a unique solution

u φ ∈ C([0, ∞); L 1 (R N ) of ∂ t u + (-∆) α u + h • u = 0 in Q ∞ , u(0, •) = φ in R N .
(3.10)

Moreover inequality (3.5) and statements (i) and (ii) in Lemma 3.1 hold.

Proof. We denote by u λ,φ the solution of (3.3).

Step 1-A priori estimate. Let φ ≥ 0. If we take ξ = ξ ηn,τ in (3.8) and let n → ∞, we derive

Q T (u λ,φ + (τ -t)h λ • u λ,φ ) dxdt + (τ -T ) R N u λ,φ (T, x)dx = τ R N φ(x)dx. (3.11) For 0 < λ < λ ′ we set w = u λ,φ -u λ ′ ,φ . It follows from (2.

21) and inequality

h λ ′ • u λ,φ ≤ h λ • u λ,φ , that for any nonnegative ξ in Y α,T , Q T w + [-∂ t ξ + (-∆) α ξ] + ξ h λ • u λ,φ -h λ • u λ ′ ,φ sign + (w) dxdt ≤ Q T w + h λ ′ • u λ ′ ,φ -h λ • u λ ′ ,φ dxdt - R N ξ(T, x)w + (T, x)dx, Since h λ (t, .
) is nondecreasing, we derive

Q T w + [-∂ t ξ + (-∆) α ξ]dxdt ≤ 0 ∀ξ ∈ Y α,T , ξ ≥ 0. If η ∈ C ∞ 0 (R N +1 ) is nonnegative, then ξ η ∈ Y α,T , ξ η ≥ 0 and Q T w + ηdxdt = 0.
This implies u λ,φ ≤ u λ ′ ,φ .

Step 2-Truncation. We replace φ by φ n = inf{φ, n} for n ∈ N * and denote by u λ,φn the corresponding solution of (3.3). By Step 1, the sequence {u λ,φn } λ>0 is decreasing and it converges to some nonnegative u φn when λ ↓ 0. Therefore h λ •u λ,φn → h•u φn a.e. in Q T . It follows from (3.11) and Fatou's lemma that

Q T (u φn + (τ -t)h • u φn ) dxdt + (τ -T ) R N u φn (., T )dx = τ R N φ n (x)dx. (3.12) Since 0 ≤ u λ,φn ≤ n, then 0 ≤ h λ • u λ,φn ≤ h • u λ,φn ≤ h(n) by (3.5). If E ⊂ Q T is a Borel set, E h λ • u λ,φn dxdt ≤ h(n)|E|. By Vitali convergence theorem h λ • u λ,φn → h • u φn in L 1 (Q T )
. Therefore, we can let λ → 0 in identity (3.8) and conclude that u φn is a weak solution of (3.10) with initial data φ n .

Step 3-Existence with φ bounded.

If φ = φ + -φ -∈ L 1 (R N ), set φ +,n = inf{φ + , n}
and φ -,n = inf{φ -, n}. We denote by u λ,φ +,n , u φ +,n , u λ,-φ -,n and u -φ -,n the corresponding solutions of (3.3) and (3.10). Then

u λ,-φ -,n ≤ u λ,φ +,n -φ -,n ≤ u λ,φ +,n which implies h λ • u λ,-φ -,n ≤ h λ • u λ,φ +,n -φ -,n ≤ h λ • u λ,φ +,n . (3.13) Estimate (3.11) is valid under the form Q T u λ,φ +,n + (τ -t)h λ • u λ,φ +,n dxdt + (τ -T ) R N u λ,φ +,n (., T )dx = τ R N φ +,n (x)dx. (3.14) and Q T u λ,-φ -,n + (τ -t)h λ • u λ,-φ -,n dxdt + (τ -T ) R N u λ,-φ -,n (., T )dx = -τ R N φ -,n (x)dx. (3.15) Since h λ • u λ,φ +,n and h λ • u λ,-φ -,n are bounded in L 1 (Q T ) independently of λ and n, h λ • u λ,φ +,n -φ -,n inherits the same property. Since u λ,φ +,n -φ -,n = H α [φ +,n -φ -,n ] -H α [h λ • u λ,φ +,n -φ -,n ],
it follows from [20, Sec 6] that u λ,φ +,n -φ -,n remains bounded in the interpolation space

Y 1 := L 1 ([0, T ]; D(A 1 )(R N )) ∩ W s,1 ([0, T ]; L 1 (R N )
), for any s ∈ (0, 1), where D(A 1 ) is defined in (2.4). Although a bounded subset K of Y 1 is not a relatively compact subset of L 1 (Q T ), for any ball B ⊂ R N , the set of restrictions to B of functions belonging to K is relatively compact in L 1 ((0, T ) × B). Thus, there exists a subsequence {λ k } such that {u λ k ,φ +,n -φ -,n } converges a.e. to some function

U n . Furthermore {h λ k • u λ k ,φ +,n -φ -,n } converges a.e. to h • U n . Since the sequences {u λ k ,-φ -,n } λ k , {u λ k ,φ +,n } λ k , {h λ k •u λ k ,-φ -,n } λ k and {h λ k •u λ k ,φ +,n } λ k are convergent in L 1 (Q T )
they are uniformly integrable. Because of (3.13) the same property is shared by the two sequences

{u λ k ,φ +,n -φ -,n } λ k and {h λ k • u λ k ,φ +,n -φ -,n } λ k . Letting λ k to 0 in the identity u λ k ,φ +,n -φ -,n (t, .) = H α [φ +,n -φ -,n ](t, .) - t 0 H α [h λ k • u λ k ,φ +,n -φ -,n ](t -s, .)ds (3.16) yields U n (t, .) = H α [φ +,n -φ -,n ](t, .) - t 0 H α [h • U n ](t -s, .)ds.
(3.17)

This implies that U n is an integral solution, thus a weak solution of (3.10) with initial data φ +,n -φ -,n = sgn(φ) inf{n, |φ|} and then U n = u φn .

Step 4-Existence with φ ∈ L 1 (R N ). By Kato's inequality (2.20), we obtain that

Q T (|u φ k -u φm |(-∂ t ξ + (-∆) α ξ) + ξ|h • u φ k -h • u φm |) dxdt + R N |u φ k (T, x) -u φm (T, x)|ξ(T, x)dx ≤ R N ξ(0, x)|φ k -φ m |dx,
for m, k ∈ N * and ξ ∈ Y α,T , ξ > 0. Taking ξ = ξ ηn,τ as in (3.9) and letting n → ∞ yields

Q T (|u φ k -u φm | + (τ -t)|h • u φ k -h • u φm |) dxdt + (τ -T ) R N |u φ k (T, .) -u φm (T, .)|dx ≤ τ R N |φ k -φ m |dx. (3.18) Since {φ m } is a Cauchy sequence in L 1 (R N ), {u φm } and {h • u φm } are also Cauchy sequences in C(0, T ; L 1 (R N )) and L 1 (Q T ) respectively. Set U = lim m→∞ u φm , then it satisfies Q T (U [-∂ t ξ + (-∆) α ξ] + ξh • U ) dxdt = R N ξ(0, x)φ(x)dx - R N ξ(T, x)U (T, x)dx ∀ξ ∈ Y α,T , (3.19)
and it is also an integral solution of (3.10). Thus

u φ ∈ C([0, ∞); L 1 (R N )).
Finally, we end the proof of uniqueness which is a consequence of the inequality below

Q T (|U -U ′ | + (τ -t)|h • U -h • U ′ |) dxdt + (τ -T ) R N |U (T, .) -U ′ (T, .)|dx ≤ τ R N |φ -φ ′ |dx, (3.20)
valid for two solutions U and U ′ of problem (3.10) with respective initial data φ and φ ′ , the proof of which is the same as the one of (3.18). Notice also that statement (i) and (ii) as well as inequality (3.5) follows by the above approximations.

Remark 3.1 By the same method it can be proved that for any p ∈ (1, ∞) and φ ∈ (3.10). Furthermore (3.5) holds.

L p (R N ) (resp. φ ∈ C 0 (R N )) there exists a unique solution u φ ∈ C([0, ∞); L p (R N )) (resp. u φ ∈ C([0, ∞); C 0 (R N ))) solution of
Proof of Theorem 1.1. Existence for ν ≥ 0. We consider a sequence of nonnegative functions

{ν n } n ⊂ C 2 0 (R N ) such that ν n → ν as n → ∞ in the weak sense of bounded measures, i.e. lim n→∞ R N ζν n dx = R N ζdν ∀ζ ∈ C(R N ) ∩ L ∞ (R N ). (3.21)
It follows from the Banach-Steinhaus theorem that ν n M b (R N ) is bounded independently of n and we assume that

ν n M b (R N ) ≤ 2 ν M b (R N )
. By Lemma 3.1, we denote by u νn the corresponding solution of (3.10) with initial data ν n . Then u n is nonnegative and satisfies that

0 ≤ u νn = H α [ν n ] -H α [h • u νn ] ≤ H α [ν n ] in Q T . (3.22)
Jointly with (2.7) it implies

u νn M p * β (Q T ,t β dxdt) ≤ c 5 ν M b (R N ) . (3.23)
We have also the following estimates from (2.9) and (3.12)

u νn (t, x) ≤ H α [ν n ](t, x) ≤ 2c 8 t -N 2α ν M b (R N ) , ∀(t, x) ∈ Q T (3.24)
and

Q T (u νn + (τ -t)h • u νn ) dxdt + (τ -T ) R N u νn (., T )dx = τ R N ν n (x)dx ≤ 2τ ν M b (R N ) . (3.25)
As in the proof of Lemma 3.2-Step 3, using the regularizing properties of the semigroup H α [.](t) (see [START_REF] Grisvard | Équations différentielles abstraites[END_REF]Sec 6]) we infer that there exists a subsequence {u νn k } which converges a.e. in Q T to some function U and {h • u νn k } converges a.e. to h • U .

For κ > 0, we denote

S κ = {(t, x) ∈ Q T : |u n k (t, x)| > κ} and ω(κ) = Sκ t β dxdt. Then for any Borel set E ⊂ Q T E h • u νn k dxdt ≤ E∩{uν n k ≤κ} h • u νn k dxdt + E∩Sκ h • u νn k dxdt ≤ g(κ) E t β dxdt + {Sκ t β g(u νn k )dxdt ≤ g(κ) E t β dxdt - ∞ κ g(s)dω(s),
where

∞ κ g(s)dω(s) = lim M →∞ M κ g(s)dω(s).
By (2.1) and (3.23), ω(s) ≤ c 14 s -p * β , thus

- M κ g(s)dω(s) = -g(s)ω(s) s=M s=κ + M κ ω(s)dg(s) ≤ g(κ)ω(κ) -g(M )ω(M ) + c 14 M κ s -p * β dg(s) ≤ g(κ)ω(κ) -g(M )ω(M ) + c 14 M -p * β g(M ) -κ -p * β g(κ) + c 14 p * β + 1 M κ s -1-p * β g(s)ds.
Since lim M →∞ M -p * β g(M ) = 0 by (1.12) and [15, Lemma 4.1] and ω(s) ≤ c 14 s -p * β , we derive g(κ)ω(κ) ≤ c 14 κ -p * β g(κ) and then

- ∞ κ g(s)dω(s) ≤ c 14 p * β + 1 ∞ κ s -1-p * β g(s)ds.
The above quantity on the right-hand side tends to 0 when κ → ∞. The conclusion follows: for any ǫ > 0 there exists κ > 0 such that

c 14 p * β + 1 ∞ κ s -1-p * β g(s)ds ≤ ǫ 2
and there exists δ > 0 such that

E t β dxdt ≤ δ =⇒ g(κ) E t β dxdt ≤ ǫ 2 .

This means that {h

n k • u νn k } is uniformly integrable in L 1 (Q T ) and by Vitali con- vergence theorem h n k • u νn k → h • U in L 1 (Q T ) . Letting n k → ∞ in the identity u νn k (t, .) = H α [ν n k ](t, .) - t 0 H α [h • u νn k (s, .)](t -s, .
)ds for some t > 0 such that u νn k (t, .) → U (t, .) a.e. in R N yields

U (t, .) = H α [ν](t, .) - t 0 H α [h • U (s, .)](t -s, .)ds.
This is valid for almost all t > 0 and implies that U ∈ C([0, T ]; L 1 (R N )), up to a modification on a set of t > 0 with zero measure. Moreover

Q T u νn k (-∂ t ξ + (-∆) α ξ) + ξh • u νn k dxdt = R N ξ(0, x)ν n k dx - R N u νn k (T, x)ξ(T, x)dx.
where ξ ∈ Y α,T is arbitrary. Thus, using the continuity of t → U (t, .) in L 1 (R N ), we derive

Q T (U (-∂ t ξ + (-∆) α ξ) + ξh • U ) dxdt = R N ξ(0, x)dν(x) - R N U (T, x)ξ(T, x)dx.
From this we infer that U is a weak solution of (1.1).

Existence for general ν. For ν ∈ M b (R N ), a sequence {ν n } in C 2 0 (R N ) converge to ν in the weak sense of bounded measures. Because of the monotonicity of h(t, •),

-H α [|ν n |] ≤ u -|νn| ≤ u νn ≤ u |νn| ≤ H α [|ν n |].
Then by above analysis, the sequence {h • u -|νn| )} and {h • u |νn| )} are relatively compact in L 1 (Q B T ) for any T > 0 and ball B and (3.23) holds for {u νn }. Therefore {u νn } is relatively locally compact in L 1 (Q B T ) and there exist some subsequence

{u νn k } and U ∈ L 1 (Q T ) such that u νn k → U =⇒ h • u νn k → h • U as k → ∞ a.e. in Q T .
As in the previous case it implies that U is a weak solution of (1.1) and also an integral solution.

Uniqueness. Let u 1 , u 2 be two weak solutions of (1.1) with the same initial ν and w = u 1 -u 2 . Then (2.20), for ξ ∈ Y α,T , ξ ≥ 0, we have that

∂ t w + (-∆) α w = h • u 2 -h • u 1 in Q T . Since h • u 2 -h • u 1 ∈ L 1 (Q T ), then by
Q T |w|[-∂ t ξ + (-∆) α ξ]dxdt + R N |w(T, x)|ξ(T, x)dxdt + Q T (h • u 2 -h • u 1 )sign(w)ξdxdt ≤ 0.
This implies w = 0 by monotonicity.

Statements (i) and (ii) and inequality (1.14) follows from the fact that the same relation holds for u νn by Lemma 3.2.

Stability is proved by the same approach that existence. If {ν n } converges to ν in the weak sense of measures, then ν n M b is bounded independently of n. Since the distribution function of h • u νn depends only on the supremum of ν n M b , this set of functions is uniformly integrable in Q T . This, combined with local compactness of the set {u νn } in L 1 (Q T ), implies the convergence of a subsequence (u νn k , h • u νn k ) to (u ν , h • u ν ) where u ν is the solution of (1.1). Because of uniqueness, all converging subsequences have the same limit, which imply the convergence of the whole sequence and stability.

Dirac mass as initial data

In this section, we study the properties of solutions to (1.1) when h(t, r) = t β r p with β > -1 and 0 < p < p * β and the initial data is ν = kδ 0 with k > 0.

Proposition 4.1 Assume 0 < p < p * β and that u k is the solution of (1.15), then there exists c 15 > 0 such that

lim t→0 + t N 2α u k (t, 0) = c 15 k. (4.1)
Proof. By (1.14) it follows that

u k (t, 0) ≤ kH α [δ 0 ](t, 0) = kΓ α (t, 0), t > 0. (4.2)
We claim that there exists c 16 > 0 independent of k such that

u k (t, 0) ≥ kΓ α (t, 0) -c 16 k p t -N 2α p+1+β , t ∈ (0, 1/2). (4.3)
Indeed, from (1.14), it is infered that

u k (t, 0) ≥ kΓ α (t, 0) -k p W (t, 0), t ∈ (0, 1/2),
where

W (t, x) = t 0 H α [s β (H p α [δ 0 ]](t -s, x)ds, (t, x) ∈ Q ∞ .
For t ∈ (0, 1/4), there exists c 17 , c 18 > 0 such that

W (t, 0) ≤ c 17 t 0 R N (t -s) -N 2α s β 1 + ((t -s) -1 2α |y|) N +2α s -N 2α 1 + (s -1 2α |y|) N +2α p dyds ≤ c 17 t 0 R N s β-N 2α p dzds 1 + ( t-s s ) 1 2α |z| (N +2α)p (1 + |z| N +2α ) ≤ c 17 t β+1-Np 2α 1 0 R N dτ dZ 1 + 1-τ τ (N+2α)p 2α |Z| (N +2α)p (1 + |Z| N +2α ) ≤ c 18 t β+1-Np 2α .
Combining (1.19) and

-N 2α p + 1 + β > -N 2α , we obtain that lim t→0 + t N 2α W (t, 0) = 0.
Therefore, (

In what follows we consider the limit of the solution {u k } of (1.15) as k → ∞ for p ∈ (0, 1]. Proposition 4.2 Assume 0 < p ≤ 1 and that u k is the solution of (1.15), then (1.14), for p ∈ (0, 1) and (t, x) ∈ (0, ∞) × R N , we have that

lim k→∞ u k = ∞ in Q ∞ , locally uniformly in Q ∞ . Proof. We observe that H α [δ 0 ] and H α [t β (H α [δ 0 ]) p ] are positive in (0, ∞) × R N . By
u k ≥ kH α [δ 0 ] -k p W =⇒ lim k→∞ u k = ∞. For p = 1, it is obvious that u k = ku 1 and u 1 > 0 in (0, ∞) × R N , then lim k→∞ u k = ∞ in Q ∞ .
The proof is complete. Now we deal with the range p ∈ (1, p * β ).

Lemma 4.1 Assume 1 < p < p * β and that u k is the solution of (1.15). Then for any k > 0, 0

≤ u k ≤ U p in Q ∞ , (4.4)
where U p is given by (1.21).

Proof. Let {f n,k } be a sequence of nonnegative functions in C 1 c (R N ) which converges to kδ 0 in the weak sense of measures as n → ∞. We denote by u n,k the corresponding solution of (1.17) with initial data by f n,k . We claim that

u n,k ≤ U p in Q ∞ , (4.5)
where, we recall it, U p is the maximal solution of the ODE y ′ + t β y p = 0 on R + . Indeed this implies (4.4).

Step 1. We claim that

lim |x|→∞ u n,k (t, x) = 0, ∀t > 0. (4.6)
From [START_REF] Chen | Heat kernel estimates for the Dirichlet fractional Laplacian[END_REF][START_REF] Chen | Global heat kernel estimates for fractional laplacians in unbounded open sets[END_REF], there exists c 8 > 0 such that for any x, y ∈ R N and t ∈ (0, ∞),

0 < Γ α (t, x -y) ≤ c 8 t -N 2α 1 + (|x -y|t -1 2α ) N +2α
.

Then for |x| > 1, 0 ≤ H α [f n,k ](t, x) ≤ c 8 t -N 2α R N f n,k (y) 1 + (|x -y|t -1 2α ) N +2α dy = c 8 R N f n,k (x -zt 1 2α ) 1 + |z| N +2α dz = c 8 R N \B R f n,k (x -zt 1 2α ) 1 + |z| N +2α dz + B R f n,k (x -zt 1 2α ) 1 + |z| N +2α dz , where R = 1 2 |x|t -1 2α and B R = {z ∈ R N : |z| < R}. It is obvious that |x -zt 1 2α | ≥ |x| -|z|t 1 2α ≥ |x|/2 for all z ∈ B R . Then B R f n,k (x -zt 1 2α ) 1 + |z| N +2α dz ≤ sup |y|≥ |x| 2 f n,k (y) B R 1 1 + |z| N +2α dz ≤ sup |y|≥ |x| 2 f n,k (y) R N 1 1 + |z| N +2α dz = c 16 sup |y|≥ |x| 2 f n,k (y) and R N \B R f n,k (x -zt 1 2α ) 1 + |z| N +2α dz ≤ R N \B R f n,k L ∞ (R N ) 1 + |z| N +2α dz ≤ c 18 R -2α = c 18 t |x| 2α ,
for some c 18 > 0 independent of x, t and R. Since f n,k ∈ C 1 0 (R N ), we have that lim

|x|→∞ sup |y|≥ |x| 2
f n,k (y) = 0 and then for any t > 0, 0

≤ u n,k (t, x) ≤ H α [f n,k ](t, x) → 0 as |x| → ∞.
Step 2. We claim that (4.5) holds. By contradiction, if (4.5) is not verified, there exists (t 0 , x 0 ) ∈ (0, ∞) × R N such that

(U p -u n,k )(t 0 , x 0 ) = min (t,x)∈(0,∞)×R N (U p -u n,k )(t, x) < 0, since U p (t) > 0 = lim |x|→∞ u n,k (t, x) for any t ∈ (0, ∞), U p (0) = ∞ > f n,k (x) = u n,k (0, x) for x ∈ R N and lim t→∞ U p (t) = lim t→∞ u n,k (t, x) = 0 for x ∈ R N . Then ∂ t (U p -u n,k )(t 0 , x 0 ) = 0. Moreover, (U p -u n,k )(t 0 , x 0 ) = min{U p (t 0 ) -u n,k (t 0 , x) : x ∈ R N } = U p (t 0 ) -max{u n,k (t 0 , x) : x ∈ R N } and u n,k (t 0 , x 0 ) = max{u n,k (t 0 , x) : x ∈ R N } =⇒ (-∆) α u n,k (t 0 , x 0 ) ≥ 0. Then 0 = ∂ t (U p -u n,k )(t 0 , x 0 ) -(-∆) α u n,k (t 0 , x 0 ) + t β 0 U p p (t 0 ) -t β 0 u p n,k (t 0 , x 0 ) < 0,
which is impossible. Thus (4.5) holds. Proposition 4.3 (i) Assume 0 < p < p * β and that u k is the solution of (1.15). Then u k is a classical solution of (1.17).

(ii) Assume 1 < p < p * β and that u ∞ is defined by (1.16). Then u ∞ is a classical solution of (1.17).

Proof. (i) Since

u k ≤ kH α [δ 0 ], it is infered that u k is bounded in (ǫ, ∞) × R N for ǫ > 0.
Let {g n,k } be a sequence of nonnegative functions in C 1 0 (R N ) which converges to kδ 0 as n → ∞ and u n,k the corresponding solution of (1.17) with initial data g n,k .

Then H α [g n,k ] → kH α [δ 0 ] as n → ∞ uniformly in [ǫ, ∞) × R N for any ǫ > 0 and by the Comparison Principle, there exists c 19 > 1 such that

0 ≤ u n,k (t, x) ≤ kH α [g n,k ] ≤ c 19 kH α [δ 0 ] in [ǫ, ∞) × R N ,
and there exists σ ∈ (0, 1) such that {u n,k } are uniformly bounded with respect to n in C σ 2α ,σ t,x ((ǫ, ∞)×R N ) with ǫ > 0. Therefore, by the Arzela-Ascoli theorem, u n,k converges to

u k in C σ ′ 2α ,σ ′ t,x
((ǫ, ∞)×R N ) with σ ′ ∈ (0, σ) and then u k is a viscosity solution of (1.17) in (ǫ, ∞) × R N . By estimate (A.1) in [START_REF] Caffarelli | Regularity of solutions to the parabolic fractional obstacle problem[END_REF],

u k is in C 1+σ ′ ,2α+σ ′ t,x ((ǫ, ∞) × R N ) and u k is a classical solution of (1.17) in (ǫ, ∞) × R N .
(ii) The proof is the same as part (i), just replacing u k ≤ kH α [δ 0 ] by u ∞ ≤ U p .

5 Self-similar and very singular solutions By Theorem 1.1 and (4.4), we see that {u k } is an increasing sequence of nonnegative functions bounded from above by U p . Then for p ∈ (1, p * β ), there exists u ∞ = lim k→∞ u k , which is a classical solution of (1.17) by Proposition 4.3 (ii) and satisfies

u ∞ ≤ U p in Q ∞ .
(5.1) Proposition 5.1 Assume 1 < p < p * β , then u ∞ is a self-similar solution of (1.17).

Proof. For λ > 0, we set

T λ [u](t, x) = λ 2α(1+β) p-1 u(λ 2α t, λx), (t, x) ∈ Q ∞ .
It is straightforward to verify that T λ [u k ] is the solution of

∂ t u + (-∆) α u + t β u p = 0 in Q ∞ , u(0, .) = λ 2α(1+β) p-1 -N kδ 0 in R N . (5.2) Because of uniqueness, T λ [u k ] = u kλ 2α (1+β)
p-1 -N . Letting k → ∞ and using the continuity of u → T λ [u], we have that

lim k→∞ T λ [u k ] = T λ [u ∞ ] = u ∞ , which implies that u ∞ is a self-similar solution (1.17). Let us denote U ∞ (z) = u ∞ (1, z), z ∈ R N , then U ∞ is a classical solution of (1.20). It is clear that the constant ( 1+β p-1 ) 1
p-1 is a constant positive solution of the self-similar equation (1.20). We observe that N < 2α(1+β)

p-1 < N + 2α when 1 + 2α(1+β) N +2α < p < 1 + 2α(1+β)
N . We prove below this fundamental result that u ∞ is the minimal self similar solution.

Proposition 5.2 Assume that 1 < p < 1 + 2α(1+β) N and ũ is a positive self-similar solution of (1.23). Then u ∞ ≤ ũ.

Proof. For any r > 0, we have that

Br(0) ũ(t, x)dx = t -1+β p-1 Br(0) ũ(1, t -1 2α x)dx = t -1+β p-1 + N 2α B t -1 2α r (0) ũ(1, z)dz ≥ t -1+β p-1 + N 2α B 1 (0) ũ(1, z)dz → +∞ as t → 0 + ,
where last inequality holds for t ∈ (0, r 2α ]. Let {ǫ n } be a sequence positive decreasing numbers converging to 0 as n → ∞. For ǫ n and k > 0, there exists t n,k > 0 such that

Bǫ n (0) ũ(t n,k , x)dx = k.
We observe that for any fixed k, t n,k → 0 as n → ∞ since lim n→∞ ǫ n = 0. Let η 0 : R N → [0, 1] be a C 2 function such that supp η 0 ⊂ B2 (0), η 0 = 1 in B 1 (0) and

η n (x) = η 0 (ǫ -1 n x) for x ∈ R N . Choosing {f n,k } be a sequence of C 2 functions such that 0 ≤ f n,k (x) ≤ η n (x)ũ(t n,k , x), ∀x ∈ R N and f n,k → kδ 0 as n → ∞.
Let u n,k be the solution of (1.1) with initial data f n,k , then

u n,k (t, x) ≤ u(t n,k + t, x), ∀(t, x) ∈ Q ∞
and by uniqueness of u k , lim n→∞ u n,k = u k , where u k is the solution of (1.1) with initial data kδ 0 . Then for any k, we have

u k ≤ ũ in Q ∞ , which implies that u ∞ ≤ ũ in Q ∞ .
5.1 The case 1 + 2α(1+β) N +2α < p < 1 + 2α(1+β)

N

We define the function w λ by

w λ (t, x) = λt -1+β p-1 w(t -1 2α |x|), (t, x) ∈ Q ∞ , (5.3)
where w(s) = ln(e+s 2 ) 1+s N+2α .

Lemma 5.1 Assume 1 + 2α(1+β) N +2α < p < 1 + 2α(1+β) N , then there exists Λ 0 > 0 such that for λ ≥ Λ 0 , ∂ t w λ (t, x) + (-∆) α w λ (t, x) + t β w p λ (t, x) ≥ 0, ∀(t, x) ∈ Q ∞ . (5.4)
Proof. By direct computation, we have

∂ t w λ (t, x) = - λ(1 + β) p -1 t -1+β p-1 -1 w(t -1 2α |x|) - λ 2α t -1+β p-1 -1 2α -1 |x|w ′ (t -1 2α |x|) and (-∆) α w λ (t, x) = λt -1+β p-1 -1 (-∆) α w(t -1 2α |x|), which implies ∂ t w λ (t, x) + (-∆) α w λ (t, x) + t β w p λ (t, x) = λt -1+β p-1 -1 (-∆) α w(s) - 1 2α w ′ (s)s - 1 + β p -1 w(s) + λ p-1 w p (s) , (5.5)
where s = |z| with z = t -1 2α x. Next, for s > 0, we have

- 1 2α w ′ (s)s - 1 + β p -1 w(s) = N + 2α 2α s N +2α 1 + s N +2α - 1 + β p -1 - s 2 (e + s 2 ) -1 α ln(e + s 2 ) w(s). Since N +2α 2α > 1+β p-1 , lim s→∞ s N+2α
1+s N+2α = 1 and lim s→∞ 1 ln(e+s 2 ) = 0, there exists R 0 > 0 and σ 0 > 0 such that

- 1 2α w ′ (s)s - 1 + β p -1 w(s) ≥ σ 0 w(s), ∀s ≥ R 0 . (5.6)
For |z| > 2, and using the definition of the fractional Laplacian, we have

-(-∆) α w(|z|) = 1 2 R N ln(e + |z + ỹ| 2 ) 1 + |z + ỹ| N +2α + ln(e + |z -ỹ| 2 ) 1 + |z -ỹ| N +2α - 2 ln(e + |z| 2 ) 1 + |z| N +2α dỹ |ỹ| N +2α = w(|z|) 2|z| 2α R N I z (y) |y| N +2α dy, (5.7)
where We claim that there exists c 24 > 0 such that

I z (y) = 1 + |z| N +2α 1 + |z| N +2α |e z + y| N
I z (y) |y| N +2α dy ≤ ω N 1 2 0 1 + |z| N +2α 1 + (|z|r) N +2α ln(e + |z| 2 r 2 ) ln(e + |z| 2 ) r N -1 dr + c 22 ≤ ω N w(|z|)|z| N ∞ 0 t N -1 ln(e + t 2 ) 1 + t N +2α dt + c 22
B 1 2 (0) I z (y) |y| N +2α dy ≤ c 24 .
(5.9) Indeed, since the function

I z is C 2 in B 1 2 (
0), I z (0) = 0 and I z (y) = I z (-y), then ∇I z (0) = 0 and there exists c 34 > 0 such that

|D 2 I z (y)| ≤ c 25 ∀y ∈ B 1 2 (0).
Then we have

I z (y) ≤ c 25 |y| 2 ∀y ∈ B 1 2 (0), which implies B 1 2 (0) I z (y) |y| N +2α dy ≤ c 25 B 1 2 (0) |y| 2 |y| N +2α dy ≤ c 24 .
We claim that there exists c 26 > 0 such that

A I z (y) |y| N +2α dy ≤ c 26 , (5.10)
where

A = R N \ (B 1 2 (0) ∪ B 1 2 (e z ) ∪ B 1 2
(-e z )). In fact, for y ∈ A, we observe that there exists c 27 > 0 such that I z (y) ≤ c 27 and

A I z (y) |y| N +2α dy ≤ R N \B 1 2 (0) c 27 |y| N +2α ≤ c 28 ,
for some c 28 > 0. Therefore, by (5.5)-(5.10), there exists c 29 > 0 such that which, together with (5.5), implies that (5.4) holds.

(-∆) α w(|z|) ≥ - c 29 1 + |z| N +2α , |z| ≥ 2.
Next we prove that u ∞ is not a trivial flat solution when 1 + 2α(1+β) N +2α < p < p * β .

Lemma 5.2 Assume 1 + 2α(1+β) N +2α < p < 1 + 2α(1+β)

N

, that w Λ 0 is given in (5.3) and u ∞ is given in (1.16). Then u ∞ (t, x) ≤ w Λ 0 (t, x) ∀(t, x) ∈ Q ∞ .

(5.13)

Moreover, lim -N > 0. Let u n,k ñ be the solution of (1.17) with initial data f n,k ñ . By Lemma 5.1, w Λ 0 (• + t n , •) is a super-solution of (1.17) , ∀(t, x) ∈ (0, 1) × R N .

(5.16)

Proof. We divide the proof into two steps.

Step 1. Let σ 0 = 1 + β -N 2α (p -1) > 0, η(t) = 2 -t σ 0 for t > 0 and denote

v ǫ (t, x) = ǫη(t)Γ α (t, x),
where Γ α is the fundamental solution of (1.17). In this step we prove that there exists ǫ 0 > 0 such that

u k 0 ≥ v ǫ 0 in (0, 1) × R N , (5.17)
where k 0 = 2ǫ 0 and u k 0 is the solution of (1.17) with initial data k 0 δ 0 . Indeed, ∂ t v ǫ (t, x) = ǫη ′ (t)Γ α (t, x) + ǫη(t)∂ t Γ α (t, x) and (-∆) α v ǫ (t, x) = ǫη(t)(-∆) α Γ α (t, x).

Let Γ 1 (t -1 2α x) = Γ α (1, t -1 2α x), then there exists ǫ 0 > 0 such that for any ǫ ≤ ǫ 0 and (t, x) ∈ (0, 1) × R N , we have that

∂ t v ǫ (t, x) + (-∆) α v ǫ (t, x) + t β v p ǫ (t, x) = ǫη ′ (t)t -N 2α Γ 1 (t -1 2α x) + ǫ p η p (t)t -N 2α p+β Γ p 1 (t -1 2α x) ≤ -ǫσ 0 t -N 2α -1+σ 0 Γ 1 (t -1 2α x) + 2 p ǫ p t -N 2α p+β Γ p 1 (t -1 2α x) ≤ 0,
the last inequality holds since -N 2α -1 + σ 0 = -N 2α p + β and Γ 1 is bounded. In particular, there holds ∂ t v ǫ 0 (t, x) + (-∆) α v ǫ 0 (t, x) + t β v p ǫ 0 (t, x) ≤ 0, ∀(t, x) ∈ (0, 1) × R N .

(5. (2 + |x 0 |) N +2α , ∀x ∈ B 1 (x 0 ), where t n = n -2α . Then there exists N 0 > 0 such that for any n ≥ N 0 , f n,k (x) ≤ v ǫ 0 (t n , x), ∀x ∈ B 1 (x 0 ).

Since n θN f 0 (n θ (x -x 0 )) → c 41 δ x 0 , as n → ∞ in weak sense of measures, for some c 41 > 0.

Let w n,k be the solution of (1.17) with initial data f n,k , then

w n,k (0, x) = f n,k (x) ≤ v ǫ 0 (t n , x) ≤ u ∞ (t n , x), ∀x ∈ R N .
Therefore, by the Comparison Principle

w n,k (t, x) ≤ u ∞ (t + t n , x), ∀(t, x) ∈ Q ∞ .
We observe that lim k→∞ [ lim n→∞ w n,k (t, x)] = u ∞ (t, x -x 0 ), ∀(t, x) ∈ Q ∞ .

Theorem 1 . 4

 14 (i) Assume p * * β < p < p * β . Then problem (1.20) admits a minimal positive solution v ∞ satisfying lim |η|→∞ |η| 2α(1+β) p-1 v ∞ (η) = 0.

  ii) Assume 1 < p < p * * β . Then problem (1.20) admits no positive solution satisfying (1.25). The question of uniqueness of the very singular solution in the case p * * β < p < p * β remains an open problem.

≤ c 23 w 1 2(

 231 (|z|)|z| N , where c 22 , c 23 > 0 and the last inequality holds since w(|z|)|z| N → 0 as |z| → ∞. Thus, I z (y) |y| N +2α dy = B -ez) I z (y) |y| N +2α dy ≤ c 23 w(|z|)|z| N .

(5. 11 )

 11 By(5.6) and(5.11), there existsR 1 ≥ R 0 + 2 such that for |z| > R 1 , (-∆) α w(|z|) -1 2α w ′ (|z|)|z| -1 + β p -1 w(|z|) ≥ σ 0 w(|z|) -c 29 1 + |z| N +2α = w(|z|) σ 0 -c 29 ln(e + |z| 2 ) ≥ 0. When |z| ≤ R 1 , it is clear that there exists c 30 > 0 such that (-∆) α w(|z|) -1 2α w ′ (|z|)|z| -1 + β p -1 w(|z|) ≥ -c 30 .Then there exists Λ 0 > 0 such that for λ ≥ Λ 0 ,(-∆) α w(|z|) -1 2α w ′ (|z|)|z| -1 + β p -1 w(|z|) + λ p-1 w p (|z|) ≥ 0, ∀z ∈ R N ,(5.12)

t→0u∞ 1 . 1 -

 11 (t, •) = 0 uniformly on B c ǫ , ∀ǫ > 0.(5.14)Proof. Let us denotef 0 (r) = k 0 ln(e + r 2 ) 1 + r N +2α , ∀ r ≥ 0 and f n,k (x) = kn N f 0 (n|x|), ∀x ∈ R N ,wherek 0 = ω N ∞ 0 ln(e + r 2 ) 1 + r N +2α r N -1 dr -Then for any η ∈ C c (R N ), we have that lim n→∞ R N f n,k ηdx = k lim n→∞ R N f 0 (|x|)η x n dx = kη(0).Let t n = n -2α and thenw Λ 0 (t n , x) = Λ 0 t N n N f 0 (n|x|) = f n,k ñ (x),where ñ ≤ n andk ñ = Λ 0 ñ 2α(1+β) p-1 -N . We see that k ñ = Λ 0 ñ 2α(1+β) p-1 -N → ∞ as ñ → ∞, since 2α(1+β) p-1

18 ) 5 . 1 5 . 2 For 1 1 -

 18515211 Corollary Assume 1 + 2α(1+β) N +2α < p < 1 + 2α(1+β) N . Then either ũ > u ∞ in Q ∞ (5.19) or ũ ≡ u ∞ in Q ∞ .(5.20) The case1 < p < 1 + 2α(1+β) N +2α < p < 1 + 2α(1+β) N +2α , it follows from Lemma 5.3 that lim t→0 + u ∞ (t, x) = ∞, ∀x ∈ R N . (5.21) Proof of Theorem 1.3 (i). Let f 0 ∈ C c (R N ) be a nonnegative function such that suppf 0 ⊂ B 1 (0) and max x∈B 1 (0) f 0 = 1. Denote f n,k (x) = kn θN f 0 (n θ (x -x 0 )),wherek ≤ n τ with τ = 1 2 ( 2α(1+β) p-N -2α) > 0, θ = τ N and x 0 ∈ R N . Since f n,k (x) ≤ n τ for x ∈ B 1 (x 0 ), f n (x) = 0 for x ∈ B c 1 (x 0 ) and v ǫ 0 (t n , x) ≥ c 39 n 2α(1+β)p-1 -N -2α

  with initial data w Λ 0 (t n , •), that is, for (t, x) ∈ Q ∞ , ∂ t w λ (t + t n , x) + (-∆) α w λ (t + t n , x) + (t + t n ) β w p λ (t + t n , x) ≥ 0.By the Comparison Principle,u n,k ñ (t, x) ≤ w Λ 0 (t + t n , x), ∀(t, x) ∈ Q ∞ ,for any ñ ≤ n. Letting n → ∞ we inferu k ñ (t, x) ≤ w Λ 0 (t, x), ∀(t, x) ∈ Q ∞ ,(5.15)where u k ñ is the solution of (1.17) with k ñδ 0 initial data. Thus (5.13) is obtained by letting ñ → ∞. Finally(5.14) follows by the fact that Lemma 5.3 Assume 1 < p < p * β , then there exists c 31 > 0 such that u ∞ (t, x) ≥ c 31 t -1+β

	lim t→0 +	w Λ 0 (t, x) = 0,	∀x ∈ R N \ {0},
	which completes the proof.
		p-1
	1 + |t -1 2α x| N +2α

Acknowledge: H. Chen is supported by National Natural Science Foundation of China, No:11401270 and the Project-sponsored by SRF for ROCS, SEM. L. Véron is supported by the MathAmsud collaboration program 13MATH-02 QUESP.

Let f n (x) = v ǫ 0 (t n , x) with t n = n -2α . Since lim t→0 + η(t) = 2, then we have that f n → 2ǫ 0 δ 0 as n → ∞ in the weak sense of measures. There exists N 0 > 0 such that t n ∈ (0, 1 8 ) for n ≥ N 0 . Let w n be the solution of (1.17) with initial data f n , then it infers that

Because u k 0 is uniquely defined, there holds

which imply (5.17).

Step 2. We claim that (5.16) holds. Since

p-1 -N , we observe that for any λ > 0,

which implies (5.16) and completes the proof.

Then u ∞ (t, x -x 0 ) = u ∞ (t, x) for all (t, x) ∈ Q ∞ , which implies that u ∞ is independent of x. Combining (5.1) and (5.16), implies that

The proof is complete.

In the case of p = 1 + 2α(1+β) N +2α , it derive from Lemma 5.3 that

Proof of Theorem 1.3 (ii). We note that u ∞ is a self-similar solution of (1.17). Moreover, we derive (1.24) by (5.16), which ends the proof.

The self-similar equation

In this section we prove Theorem 1.4.

Proof of Theorem 1.4 (i). We set v ∞ (η) = t 1+β p-1 u ∞ (1, η). Then relations (1.25) and (1.26) hold from Lemmas 5.2 and 5.3. Assume ṽ is another positive solution of (1.20). Then (t, x) → t -1+β p-1 ṽ(t -1 2α x) is a positive self-similar solution of (1.23). By Proposition 5.2 it is larger than u ∞ . Thus v ∞ ≤ ṽ. Assume now that there exists η 0 ∈ R N such that v ∞ (η 0 ) = ṽ(η 0 ). and set w = ṽ -v ∞ . Then

Since ∇w(η 0 ) we reach a contradiction.

Proof of Theorem 1.4 (ii). It is a consequence of the equality

Open problem. We conjecture that in the case 1 + 2α(1+β) N +2α < p < 1 + 2α(1+β) N , v ∞ is the unique positive solution of the self-similar equation satisfying (1.25). One step could be to prove that any positive solution ṽ satisfying (1.25) satisfies, for some K > 1, ṽ ≤ Kv ∞ in R N .

(5.23)

We also conjecture that v ∞ satisfies the following asymptotic behavior v ∞ (η) = c N,p,α,β |η| -N -2α as |η| → ∞.

(5.24)

Thus if any positive solution ṽ inherits the same property, the conclusion (and the uniqueness) follows.


Alexandre Mouradian
email: alexandre.mouradian@u-psud.fr

Study of Probabilistic Worst Case Inter-Beacon Delays Under Realistic Vehicular Mobility Conditions

Road safety applications are one of the main incentives to deploy vehicular networks. These applications rely on periodic message exchange among vehicles (known as beaconing). The beacon messages contain information about the environment which is used to perceive dangerous situations and alert the drivers. The inter-beacon delay is the time between two consecutive beacons received from a car. It is an essential parameter because, if this delay exceeds the application requirement, the application cannot accurately predict dangerous situations and alert the drivers on time. The worst case inter-beacon delay has thus to be bounded according to the application requirements. Unfortunately, a tight and strict bound is in fact very difficult to obtain for a real network because of the randomness of the collisions among beacons coming from: the unpredictable mobility patterns, random interferences, randomness of the MAC layer backoff, etc. In this paper, we propose to provide a probabilistic worst-case of the inter-beacon delay under realistic mobility using Extreme Value Theory (EVT). EVT provides statistical tools which allow to make predictions on extreme deviations from the average of a parameter. These statistical predictions can be made based on data gathered from simulation or experimentation. We first introduce the EVT technique. Then we discuss its application to the study of inter-beacon delays. Finally, we apply EVT on the results of extensive vehicular network simulation using a realistic mobility trace: the Cologne trace.

Introduction

The road safety applications are seen as an essential motivation for the deployment of Vehicular Ad hoc NETworks (VANETs) [START_REF] Hartenstein | VANET: vehicular applications and internetworking technologies[END_REF]. The main goal of these applications is to avoid car crashes and thus reduce the number of road traffic deaths and injuries. We can cite as examples of safety applications [START_REF] Consortium | Vehicle safety communications project: task 3 final report: identify intelligent vehicle safety applications enabled by dsrc[END_REF]:

-traffic signal violation: alert neighbor cars when a user does not stop at a red traffic light; -electronic brake: alert neighbor vehicles when a car performs an emergency braking;

The final publication is available at http://link.springer.com/chapter/10.1007/978-3-319-19662-6_27

-on-coming traffic warning: notify the driver of on-coming traffic during overtaking maneuvers.

The main building block for safety applications is the exchange of periodical one-hop broadcast messages among neighbor vehicles [START_REF] Hartenstein | VANET: vehicular applications and internetworking technologies[END_REF]. These messages are called beacons or heartbeat messages and contain, at least, information on the car positions, directions and speeds. In the rest of this paper, as in [START_REF] Stanica | Broadcast communication in vehicular adhoc network safety applications[END_REF] [START_REF] Noori | A novel study on beaconing for vanet-based vehicle to vehicle communication: Probability of beacon delivery in realistic large-scale urban area using 802[END_REF] we refer to these messages as beacons. Each safety application uses these beacons to build a representation of its environment, analyze the current situation and predict dangerous situations. The period, latency and scope (or range) of the beacons depend on the requirements of the application. For example, according to [START_REF] Hartenstein | VANET: vehicular applications and internetworking technologies[END_REF] the electronic brake application needs a beacon frequency of 10Hz, a maximum latency of 100ms and a range of 200m to be effective. In this paper, we focus on the period of these messages: the inter-beacon delay. We are thus interested in the capacity of the MAC layer to broadcast these beacons on a periodical basis which complies with the application requirements. We consider the MAC from the IEEE 802.11p standard, since it is regarded as the standard of choice for VANETs [START_REF] Hartenstein | VANET: vehicular applications and internetworking technologies[END_REF].

In order to be effective, critical road safety applications need strong timing guaranties on the delivery of beacons. It is thus of paramount importance that the MAC layer is able to provide worst case inter-beacon delays which are bounded. But more importantly, the bound has to be known and has to correspond to the target application. Nevertheless, most of the performance evaluation techniques of the literature such as stochastic models, simulation and experimentation focus on parameter averages [START_REF] Jain | The art of computer systems performance analysis[END_REF] and give very few insights on extreme cases. This is the case, in particular, concerning many performance studies of beaconing in vehicular networks [START_REF] Hartenstein | VANET: vehicular applications and internetworking technologies[END_REF] [19] [START_REF] Noori | A novel study on beaconing for vanet-based vehicle to vehicle communication: Probability of beacon delivery in realistic large-scale urban area using 802[END_REF]. On the other hand, a strict bound on the inter-beacon delay is difficult, if not impossible, to obtain because of the random nature of the beacon collisions which comes from the mobility patterns, interferences, pathloss, randomness of the MAC layer backoff, etc.

In this paper, we propose to estimate a probabilistic bound on the interbeacon delay using statistical tools. The probabilistic worst case delay can be viewed as the probability that the maximum delay is less than a given value. This quantity is very useful for the system designer which must ensure that the system can handle the timing requirements of the application with a high probability. An advantage of the statistical approach is that it allows to evaluate directly the real studied system (or a highly detailed simulation model) instead of working on an abstract theoretical model.

The statistical tools we use come from Extreme Value Theory (EVT) which is presented in sections 3. The method can be summed up in the three following steps: (1) produce/Gather data (by simulation or experimentation); (2) extract extreme values from the data; (3) fit extracted values to a probability distribution predicted by the theory.

The main contribution of this paper is to use EVT to characterize the distribution of high inter-beacon delays when using the IEEE 802.11p standard under realistic mobility conditions. We first discuss EVT applicability to study inter-beacon delays and then use it on simulation results. The simulations are performed in ns2 [1] using a highly realistic mobility trace developed by Uppor and Fiore [START_REF] Uppoor | Large-scale urban vehicular mobility for networking research[END_REF]. Up to our knowledge, this study represents the first application of EVT to the study of extreme delays in VANETs and is the first characterization of extreme inter-beacon delays under realistic mobility conditions using the IEEE 802.11p standard.

The remainder of the paper is organized as follows. In section 2 the related work about safety beaconing performance and EVT is presented and commented. In section 3, we introduce EVT main theorems and discuss the application of EVT to the study of large inter-beacon delays. In section 4, we present the simulation setup and the produced data sets. In section 5 we describe the results of the application of EVT to inter-beacon delay simulation data and provide discussion about the method and the obtained probabilistic worst-case interbeacon delays. Section 6 gives the conclusion remarks and lists future works.

Related work

In this section we present the related work regarding the study of beaconing in VANETs with 802.11p, and on the application of EVT in general and more specifically in computer science.

Many studies concerning the performance of beaconing using 802.11p exist in the literature [START_REF] Hartenstein | VANET: vehicular applications and internetworking technologies[END_REF] [START_REF] Stanica | Broadcast communication in vehicular adhoc network safety applications[END_REF] [11] [START_REF] Autolitano | An insight into Decentralized Congestion Control techniques for VANETs from ETSI TS 102 687 V1[END_REF] [17] [START_REF] Nguyen | A performance analysis of csma based broadcast protocol in vanets[END_REF]. In [START_REF] Hartenstein | VANET: vehicular applications and internetworking technologies[END_REF], the authors study the performance of beaconing in VANETs through simulation. They use the ns2 simulator. In their scenarios, the vehicles are positioned on parallel lines (straight highway scenario) and are not moving. The authors provide the probability of a successful beacon reception for different beaconing frequencies. The results show that the probability of reception decreases rapidly after few hundred meters between the emitter and receiver. In [START_REF] Stanica | Broadcast communication in vehicular adhoc network safety applications[END_REF], Stanica et al. propose an analytical model of the probability of collision and the reception probability of beacons. They highlight the low reliability of beaconing with 802.11p as with their model, the reception probability is never greater than 0.7. They confirm their findings through simulation. The authors also show that the performance depends on the 802.11 contention window and derive its optimal value. In [START_REF] Nguyen | A performance analysis of csma based broadcast protocol in vanets[END_REF], the repartition of the nodes of the VANET is modeled as a Poisson Point Process. The authors show that above a critical density of nodes, the CSMA scheme of 802.11p behaves like an ALOHA protocol and its performance is thus highly deteriorated. The authors study the reception probability for simple highway scenarios under various traffic conditions and transmission power hypotheses. Again the reliability of beaconing is very low. The authors of [START_REF] Noori | A novel study on beaconing for vanet-based vehicle to vehicle communication: Probability of beacon delivery in realistic large-scale urban area using 802[END_REF] simulate beaconing using ns2 and the realistic Cologne mobility trace [START_REF] Uppoor | Large-scale urban vehicular mobility for networking research[END_REF]. They study the probability of beacon delivery in different areas of the city. They confirm the bad reliability of 802.11p in a realistic mobility scenario. Despite the fact that many studies evaluate the reliability of beaconing with 802.11p, none of the presented study provide insight on the worst case inter-beacon delay under realistic mobility conditions (we can note that [11] is concerned about the inter-beacon delay distribution, but not the worst case). Yet, this parameter is of paramount importance to evaluate the achievable timeliness of safety applications.

In the literature, EVT has been extensively used in various contexts: extreme rainfalls [START_REF] Shukla | On the proficient use of gev distribution: a case study of subtropical monsoon region in india[END_REF], forest fires [START_REF] Alvarado | Modeling large forest fires as extreme events[END_REF], wind speed [START_REF] Beirlant | Statistics of extremes: theory and applications[END_REF], financial crashes [START_REF] Embrechts | Extreme value theory as a risk management tool[END_REF] studies. EVT has also been used in the context of computer networks for the estimation of traffic peaks or bursts [START_REF] Liu | Application of extreme value theory to the analysis of wireless network traffic[END_REF] [20] [START_REF] Dahab | Application of extreme value theory to bursts prediction[END_REF]. In [START_REF] Uchida | Traffic data analysis based on extreme value theory and its applications[END_REF], the author proposes to study the traffic on an Ethernet network in order to predict traffic peaks. The author shows that the gathered extreme deviations fit very well to a Generalized Pareto distribution as predicted by the theory. In another study [START_REF] Liu | Application of extreme value theory to the analysis of wireless network traffic[END_REF], the authors apply EVT to the study of traffic throughput in wireless networks and show that the generalized EVT distribution is a better match for large deviation prediction than exponential, gamma or log-normal distributions. In [START_REF] Dahab | Application of extreme value theory to bursts prediction[END_REF] the authors fit Ethernet traffic throughput data to a Weibull distribution (also predicted by EVT). Nevertheless, these applications of EVT focus on network traffic throughput. Here we are more interested in the possibility to use such technique to study large delays in computer networks.

In order to find applications of EVT to the study of large delays in computer science, we have to look to the field of worst case execution times. Works such as [START_REF] Edgar | Statistical analysis of wcet for scheduling[END_REF] [14] make use of EVT to derive probabilistic worst case execution times. In our work, we explore the case of inter-beacon delays instead of task executions. We comment, in section 3 and 5, the issues which may arise when applying EVT to this specific case.

The novelty of our work is twofold: we are interested in characterizing large inter-beacon delays behavior instead of their probability of reception in realistic mobility conditions, and we use the EVT method which up to our knowledge have never been apply to the study of delays in networks.

Extreme Value Theory applied to delays in networks

A brief introduction to EVT

Extreme Value Theory has been developed during the 20th century and is now a well established tool to study extreme deviations from the average of a measured phenomenon [START_REF] Beirlant | Statistics of extremes: theory and applications[END_REF]. EVT is built around two main theorems: the Fisher-Tippett-Gnedenko theorem and the Pickands-Balkema-de Haan theorem. As we will detail below, the former is interested in the maximum value of a sequence of variables, whereas the latter focuses on the values of a sequence which are above a given threshold.

The Fisher-Tippett-Gnedenko theorem states that given {X 1 , ..., X n } a sequence of independent and identically distributed (i.i.d.) variables, the distribution of M n = max{X 1 , ..., X n } the variable representing the maximum value of the sequence converges (for large n) toward one of these three distribution families characterized by their CDF (Cumulative Distribution Function):

-Fréchet: G(x) = e -( x-m s ) -α , for x > m (1)
if the distribution has a heavy tail; -Gumbel:

G(x) = e -e -x-m s (2)
if the distribution has an exponential tail;

-Weibull G(x) = e -(-( x-m s )) α , for x < m (3)
if the distribution has a finite maximum; with m, s and α the distribution parameters and α > 0 in all cases.

The second theorem of EVT is the Pickands-Balkema-de Haan theorem. It states that given {X 1 , ..., X n } a sequence of i.i.d. variables, their conditional distribution F u (y) = P (Xu < y|X > u) converges toward a generalized Pareto distribution for large u:

� G(y) = 1 -(1 + (y-m)γ s ) -1 γ , if γ � = 0 G(y) = 1 -e -y-m s , if γ = 0 (4)
To each theorem corresponds a method which can be applied to characterize the distributions of extreme variation of a phenomenon. For the first theorem the method is the Block Maxima (BM) method in which the sequence of measured data is divided into blocks and the maximum of each block is computed. The maxima are then fitted to one of the three previously mentioned distributions. The second theorem of EVT corresponds to the Peak Over Threshold (POT) method. In this method, a threshold value is chosen and the data points which are above the threshold are collected and fitted to a generalized Pareto distribution.

Application to the study of inter-beacon delays

According to Fisher-Tippett-Gnedenko and the Pickands-Balkema-de Haan theorems, to apply EVT, the sequence of variables must be independent and identically distributed. The question then is: can it be the case for inter-beacon delays in VANETs ?

Let's first consider two nodes A and B. Can we assume that their interbeacon delays are identically distributed ? We would tend to answer yes if the network environment and measure conditions are the same for both A and B: same channel conditions, same node density, fair access to the medium, etc. Because in this case the packet collisions and packet loss inducing inter-beacon delays would be identically distributed. This leads us to believe that to apply EVT on inter-beacon delays we have to be careful not to have great disparities in the studied network like very dense and very sparse areas in the same network. This statement will be verified in section 5.

Concerning the independence hypothesis, we can argue that it will not always be true in the case of inter-beacon delays. If we consider the successive interbeacon delays between two nodes with bad channel conditions, they will tend to be correlated (if the channel conditions are stable between two beacons).

Nevertheless, in the literature, we find many EVT application cases on data sequences which seem correlated in time. For instance, in [START_REF] Beirlant | Statistics of extremes: theory and applications[END_REF] it is mentioned that it is possible to fit very accurately maximum wind speeds to an extreme value distribution. Nevertheless, wind speed measures may appear to be temporally correlated as studied in [START_REF] Koçak | Practical ways of evaluating wind speed persistence[END_REF]. This shows that EVT may accurately model the large values of a phenomenon even if the measures are partially correlated.

It is not clear from the literature, what are the cases where EVT can or cannot be applied. In section 5, we will consider the i.i.d hypothesis true, try to apply EVT and evaluate how the data fits the model (the fit can be assessed by a statistical test as explained in section 3.4). We will then discuss in which cases EVT is meaningful.

Due to the lack of space, in this paper, we consider only the application of the Fisher-Tippett-Gnedenko theorem. The second theorem of EVT is left as a future work.

Gathering and arranging data for EVT

The first element needed in order to apply EVT is a set of data which is a realization of the sequence of random variables {X 1 , ..., X n } mentioned in the previous section. In our case it will be measures of inter-beacon delays obtained from simulation. In this paper, we consider technique associated with the Fisher-Tippett-Gnedenko theorem to process the data. This technique is known as Block Maxima (BM). The principle of the BM technique is to divide the data sequence into blocks and to take the maximum of each block. In the literature, a block is often defined as a time interval [START_REF] Beirlant | Statistics of extremes: theory and applications[END_REF]. According to the Fisher-Tippett-Gnedenko theorem the sequence of block maxima must converge to an extreme value distribution. The observed sequence of block maxima can thus be fitted to one of the Gumbel, Weibull or Fréchet distributions. The type of distribution the data will converge to is difficult to predict from the raw data. In section 5 we thus try to fit the data to the three considered distribution and discuss which is the most accurate model for the worst-case inter-beacon delay.

Fitting technique and statistical test

In the previous subsection, we have described how to retrieve extreme value data from the original data set. We then have to fit this extreme value data to one of the extreme value distributions predicted by EVT and assess that the fitted distribution is an acceptable representation of the data thanks to a goodness-of-fit statistical test. In this paper we use the Maximum Likelihood Estimation (MLE) technique to estimate the parameters of the distribution, and the Pearson's chi-squared test to assess the goodness-of-fit.

The MLE technique is based on the likelihood function, which is defined as follows:

L(x, θ) = n � i=1 f (x i , θ) (5)
with x a vector of observed values, f the pdf we want to fit to the values and θ the vector of parameters for the pdf.

The MLE method consists in finding a vector of parameters θ such that L(x, θ) is maximized. In practice, for the problem to be tractable, it is actually the logarithm of the likelihood which is maximized (the logarithm function preserves the optimum). In this paper, in order to solve this optimization problem we use the Nelder-Mead method [START_REF] Nelder | A simplex method for function minimization[END_REF].

Once the optimal distribution parameters are obtained, we have to verify that the fitted distribution is actually a convincing representation for the data. For that purpose, we use two tools: the Pearson chi-squared test and quantilequantile plots [START_REF] Jain | The art of computer systems performance analysis[END_REF]. The former is a statistical test which assesses if there is a statistical difference between an observed data frequency distribution and a theoretical distribution. We can note that this test statistic offers a quality indicator for the fit which can be used in the case several distributions pass the chi-squared test as in [START_REF] Shukla | On the proficient use of gev distribution: a case study of subtropical monsoon region in india[END_REF].

The quantile-quantile plot [START_REF] Jain | The art of computer systems performance analysis[END_REF] (or Q-Q plot) is a tool which allows to graphically compare two distributions. In our case, it consists in plotting the quantiles of the collected data against the quantiles of the fitted distribution. If the two distributions match, we should obtain the relation x = y. As described in [START_REF] Beirlant | Statistics of extremes: theory and applications[END_REF] and [START_REF] Jain | The art of computer systems performance analysis[END_REF], the plot consists in a set of points (x, y) where x ∈ x 1 , ..., x n with x 1 ...x n the ordered set of data points (in increasing order) and y = F -1 ( i n+1 ) with n the number of data points, i = 1, ..., n and F -1 the inverse CDF of the fitted distribution (the inverse CDF corresponds to the quantile function).

Simulation

In this section we present the simulation parameters and the obtained data sets on which we apply EVT.

Simulation setup

We use the discrete-event simulator ns2 [START_REF]The Network Simulator[END_REF] to perform the simulations, the main parameters are described in Table 1. The simulation setup is actually quite classic, we thus focus on the description of the realistic mobility trace and how we use it in the simulator.

The mobility trace we use in the simulations is a realistic micro-mobility trace of the city of Cologne generated by Uppoor and Fiore [START_REF] Uppoor | Large-scale urban vehicular mobility for networking research[END_REF]. The trace covers a 400 km 2 area and a period of 24 hours and contains about 700000 vehicle travels. In this paper we use the part of the trace available online which covers the 6am-8am period. In Figure 1, we plot the instantaneous positions of the nodes at 8am. Each point corresponds to a vehicle position in the coordinate system provided by the trace (the coordinates are expressed in meters). The trace from 6am to 8am contains more than 300000 different vehicles, it is thus not possible to simulate the whole scenario in ns2. As we are interested in local communications (onehop broadcasts), we decide to divide the network and restrict the simulations to In each simulation, we monitor the inter-beacon delay as well as the emitterreceiver distance. A simulation typically provides around 2.5 millions measurements of inter-beacon delays (it varies depending on the number of cars present during the simulation).

Obtained data sets

For the BM technique, we need to divide the data into blocks and retrieve the maximum of each block. We choose to use three different ways of producing the data which is then fitted to EVT distributions:

1. The first one consists in taking the maximum of each block of 200 seconds (corresponding to one run of the simulator) for the whole trace duration and all the highlighted squares in Figure 1. 2. In the second one we take only one square and one portion of time and re-run the simulation for that particular square and block several times. 3. The third case is the same as the second but we change the order of the beacon start of the nodes at each run (in the second case, the order of beacon start dates is generated randomly once and the same order is repeated in every simulation).

These different setups allow us to understand how disparities in the simulation data affects the applicability of the BM method. Indeed, in the first case, the data collected comes from various situations in terms of car traffic amount, network density, etc. Whereas in the two last cases the blocks are more similar to one another (in the second case there are more correlations between the runs because the beacon start dates are the same).

For setup 1, we run simulations for 13 successive blocks of 200 seconds for each of the 12 considered squares. We take the maximum inter-beacon delay for each run and thus obtain 156 values. For setup 2 and 3 we run 300 simulation of the considered block, so we obtain 300 maximum values. In the results presented in the following sections, we first consider the receivers in a 500 meters range from the sender, and we then observe the impact of the range on the maximum delay distribution.

Results and discussion

In this section we present the results of the application of the EVT method presented in section 3 to the data sets presented in section 4.2.

Figure 2a is a histogram representation of the maximum delays for the first data set. First, we have to note that the measured maximum inter-beacon delays are very large compared to the inter-beacon emission period (0.1 seconds). This is due to two main reasons: the mobility and the broadcast scheme used. When a node broadcasts its beacon using 802.11p, it cannot detect collisions. Indeed, collisions happen at the level of the receivers and the sender does not know which of its neighbors will actually receive the packet and does not wait for acknowledgments in the case of broadcasting. Moreover, broadcast messages are subject to the hidden terminal problem since the RTS/CTS messages are not used for broadcast. These problems have been highlighted several times in the literature [START_REF] Hartenstein | VANET: vehicular applications and internetworking technologies[END_REF] [19] [START_REF] Nguyen | A performance analysis of csma based broadcast protocol in vanets[END_REF]. In order to better comprehend the observed long delays, let's consider the following scenario: first, a node A periodically receives a beacon Fig. 2: Results for all the squares and blocks and 500m range from one of its neighbors B, then another node C moves in the neighborhood of A and the beacons from C collide with those of B (in the case of the hidden terminal problem the beacons may constantly collide), then C moves out of range of A again. In this scenario, when C at last get out of the range of A, it puts an end to the collisions with beacons from B. When A receives the first beacon from B after C went out, the inter-beacon delay is approximately equal to the duration of the presence of C in the range of A. In the cologne trace case, this delay can be of the order of tens of seconds as we observe in the simulations results.

Let's now focus on the fitting of the data of the first data set to the different extreme value distribution families. Figures 2b, 2c and2d respectively represent the QQ-plots of the fitted Fréchet, Gumbel and Weibull distributions (fitting has been realized using the MLE method described in section 3). First we observe that none of the distribution fits well to the data. Indeed, none of the graphs show a x = y curve. Nevertheless, we remark that in the Fréchet case the curve is piecewise linear which seems to indicate that there are linear relations between the data quantiles and the fitted Fréchet distribution quantiles. In the case of the fitted Weibull distribution, we observe that for x and y lesser than 60, the points are approximately on the x = y curve. The fact that the QQ-plots are piecewise, seems to indicate that the actual maximum inter-delay distribution is multi-modal. In fact, from the simulation data, we observe that the different modes correspond to different areas (squares) of the network and different time periods. We conclude that the EVT hypothesis which states that the set of inter-beacon delays are identically distributed does not hold for this data set and thus it is not possible to apply EVT. As a matter of fact, the results of the chi-squared tests for all three EVT distributions for this data set are negative (the tests are performed with a p = 0.05 significance level).

Figures 3a and3b respectively depict the frequency plot of the inter-beacon delays with the expected frequencies from the fitted Fréchet distribution and the QQ-plot of the data against the fitted distribution for the second data set. As stated in section 4.2, this set consists of one square and one portion of 200 seconds run 300 times. The chosen square is defined as x ∈ [11000, 12000] and y ∈ [11000, 12000] (cf. Figure 1) and the considered time block is from 1200 to 1400 seconds of the original trace. In this case, only the fitted Fréchet distribution successfully passes the Pearson chi-squared test. The QQ-plot (Figure 3b) shows that the fit is good for the lowest values and of a lower quality for the highest values. We can also still discern at least two modes in the frequency distribution in Figure 3a (also visible in Figure 3b, because the points lie over the x = y curve for x < 15 and then under until approximately x = 18). 4b present the results for the third simulation setup (the third data set): it is the same as the previous (one square for one 200 second interval run multiple times) but the nodes are starting their beacon emission at different dates in each simulation. The starting dates are in fact uniformly distributed in the first second of the simulation. For this data set, all three fitted EVT distributions pass the chi-squared test. Figure 4a depicts the frequency plot of the inter-beacon delays with the expected frequencies from the three EVT distribution. In Figure 4b, we present only the QQ-plot for the fitted Fréchet distribution because, even if all the distributions pass the chi-squared test, the Fréchet distribution is the best fit (the one with the lowest test statistic). In this QQ-plot, we observe, as in the last, that the fit is good for the lower values and less good for higher values. Nevertheless, for this data set, we do not observe multimodal tendency which seems to indicate that the identically distributed hypothesis holds. We can add that for larger blocks (more than 200 seconds) the fit is even better. Unfortunately we cannot present the results here due to the lack of space. The discussion on the choice of the size of the block for the BM method will thus be presented in future works. We can also note that the provided probabilistic bound (corresponding to the fitted Fréchet distribution) is more pessimistic in third data set case than in the second as can be seen by comparing the distributions of Figures 3a and4a. In the last case, larger inter-beacon delays are more probable.

These results show that the Fréchet distribution is an adequate model for the distribution of large inter-beacon delays in VANETs with realistic mobility. Nevertheless, the distribution models accurately model the data only if the network conditions are coherent in every locations of the network where measures are taken (as shown by the failure of EVT for data set 1). It means for example, that for different network densities different fitted Fréchet distributions apply.

Fig. 5: CDF for different beacon ranges

All the results we have presented so far are for a 500 meters range: we compute the inter-beacon delays for receivers within 500 meters of the sender. Nevertheless, for most of critical safety applications, the range of beacon emission can be much lower [START_REF] Consortium | Vehicle safety communications project: task 3 final report: identify intelligent vehicle safety applications enabled by dsrc[END_REF]. In Figure 5, we plot the CDFs of fitted Fréchet distribution for different ranges. The CDF of the maximum delay expresses the probability that the maximum delay is under a given value. It corresponds to the probabilistic worst case delay. Thanks to this probabilistic worst case delay, the system designer can state that, for example, "the probability that the maximum interbeacon delay in a 100 meters range from the sender is less than 5 seconds is close to one". Whether this probabilistic delay bound is sufficient or not depends on the considered application. In Figure 5, we can notice that the CDF for higher ranges is lower. This means that the probability that the maximum delay is over a given value is higher for higher ranges. This can be explained by two phenomena: the higher probability of packet loss at longer distances and the hidden terminal problem as explained above in this section.

Conclusion and future works

In this paper, we study large inter-beacon delays in VANETs under realistic mobility with IEEE 802.11p. Understanding such delays is useful to assert if safety application requirements will be met. We show that large inter-beacon delays are Fréchet distributed. This result can be used in order to evaluate the performance of vehicular safety applications. We also show that the EVT method used to reach that conclusion is applicable for the study of delays in large scale wireless networks such as VANETs. Finally we confirm, in realistic mobility conditions, the results [START_REF] Stanica | Broadcast communication in vehicular adhoc network safety applications[END_REF] [START_REF] Nguyen | A performance analysis of csma based broadcast protocol in vanets[END_REF] which predict the bad performance of IEEE 802.11 broadcast. In future works we plan to: apply the second theorem of EVT to the study of delays in VANETs, evaluate the impact of the choice of the block size (in the BM method) on the quality of the obtained distribution (the goodness of fit), and compare the distribution obtained from various data sets (mobility traces), to evaluate how general the EVT results are.

Fig. 1 :

 1 Fig. 1: Snapshot of the Cologne data set

  (a) Histogram of the maximum inter-beacon delays (b) QQ-plot: data against Fréchet distribution (c) QQ-plot: data against Gumbel distribution (d) QQ-plot: data against Weibull distribution

Fig. 3 :

 3 Fig. 3: Results for one square during 200 seconds

Fig. 4 :

 4 Fig. 4: Results for one square during 200 seconds with random beacon start at each run

Table 1 :

 1 Simulation parameters.

	Parameter	Value
	Bitrate	6 Mbps
	Transmission power 10 dBm
	Simulation area	1000x1000 m
	Beacon size	400 bytes
	MAC and Phy	802.11p
	Propagation model Nakagami m=1
	Beacon frequency 10Hz