Moshe Marcus

Laurent Veron

Capacitary estimates of solutions of semilinear parabolic equations

Keywords: 2000 Mathematics Subject Classification. 35K05, 35K55, 31C15, 31B10, 31C40 Heat equation, singularities, Borel measures, Besov spaces, real interpolation, Bessel capacities, quasiadditivity, capacitary measures, Wiener type test, initial trace

We prove that any positive solution of ∂ t u -∆u + u q = 0 (q > 1) in R N × (0, ∞) with initial trace (F, 0), where F is a closed subset of R N can be represented, up to two universal multiplicative constants, by a series involving the Bessel capacity C 2/q,q ′ . As a consequence we prove that there exists a unique positive solution of the equation with such an initial trace. We also characterize the blow-up set of u(x, t) when t ↓ 0 , by using the "density" of F expressed in terms of the C 2/q,q ′ -Bessel capacity.

Introduction

Let T ∈ (0, ∞] and Q T = R N × (0, T ] (N ≥ 1). If q > 1 and u ∈ C 2 (Q T ) is nonnegative and verifies

∂ t u -∆u + u q = 0 in Q T , (1.1)
it has been proven by Marcus and Véron [START_REF] Véron | The initial trace of positive solutions of semilinear parabolic equations[END_REF] that there exists a unique outer-regular positive Borel measure ν in R N such that lim

t→0 u(., t) = ν, (1.2)
in the sense of Borel measures; the set of such measures is denoted by B reg + (R N ). To each of its element ν is associated a unique couple (S ν , µ ν ) (we write ν ≈ (S ν , µ ν )) where S ν , the singular part of ν, is a closed subset of R N and µ ν , the regular part is a nonnegative Radon measure on R ν = R N \ S ν . In this setting, relation (1.2) has the following meaning :

(i) lim t→0 Rν u(., t)ζdx = Rν ζdµ ν , ∀ζ ∈ C 0 (R ν ), (ii) lim t→0 O u(., t)dx = ∞, ∀O ⊂ R N open, O ∩ S ν = ∅.
(1.

3)

The measure ν is by definition the initial trace of u and denoted by T r R N (u). It is wellknown that equation (1.1) admits a critical exponent

1 < q < q c = 1 + N 2 .
This is due to the fact, proven by Brezis and Friedman [START_REF] Brezis | Nonlinear parabolic equations involving measures as initial conditions[END_REF], that if q ≥ q c , isolated singularities of solutions of (1.1) in R N \ {0} are removable. Conversely, if 1 < q < q c , it is proven by the same authors that for any k > 0, equation (1.1) admits a unique solution u kδ 0 with initial data kδ 0 . This existence and uniqueness results extends in a simple way if the initial data kδ 0 is replaced by any Radon measure µ in R N (see [START_REF] Brezis | Semilinear equations in R N without condition at infinity[END_REF]). Furthermore, if k → ∞, u kδ 0 increases and converges to a positive, radial and self-similar solution u ∞ of (1.1). Writing it under the form u ∞ (x, t) = t -1

q-1 f (|x| / √ t), f is a positive solution of ∆f + 1 2 y.Df + 1 q-1 ff q = 0 in R N lim |y|→∞ |y| 2 q-1 f (y) = 0.

(1.4)

The existence, uniqueness and the expression of the asymptotics of f has been studied thoroughly by Brezis, Peletier and Terman in [START_REF] Brezis | A very singular solution of the heat equation with absorption[END_REF]. Later on, Marcus and Véron proved in [START_REF] Véron | The initial trace of positive solutions of semilinear parabolic equations[END_REF] that in the same range of exponents, for any ν ∈ B reg + (R N ), the Cauchy problem

∂ t u -∆u + u q = 0 in Q ∞ , T r R N (u) = ν, (1.5)
admits a unique positive solution. This result means that the initial trace establishes a one to one correspondence between the set of positive solutions of (1.1) and B reg + (R N ). A key step for proving the uniqueness is the following inequalities

t -1 q-1 f (|x -a| / √ t) ≤ u(x, t) ≤ ((q -1)t) -1 q-1 ∀(x, t) ∈ Q ∞ , (1.6)
valid for any a ∈ S ν . As a consequence of Brezis and Friedman's result, if q ≥ q c , i.e. in the supercritical range, Problem (1.5) may admit no solution at all. If ν ∈ B reg + (R N ), ν ≈ (S ν , µ ν ), the necessary and sufficient conditions for the existence of a maximal solution u = u ν to Problem (1.5) are obtained in [START_REF] Véron | The initial trace of positive solutions of semilinear parabolic equations[END_REF] and expressed in terms of the the Bessel capacity C 2/q,q ′ , (with q ′ = q/(q -1)). Furthermore, uniqueness does not hold in general as it was pointed out by Le Gall [START_REF] Legall | A probabilistic approach to the trace at the boundary for solutions of a semilinear parabolic partial differential equation[END_REF]. In the particular case where S ν = ∅ and ν is simply the Radon measure µ ν , the necessary and sufficient condition for solvability is that µ ν does not charge Borel subsets with C 2/q,q ′ -capacity zero. This result was already proven by Baras and Pierre [START_REF] Baras | Problèmes paraboliques semi-linéaires avec données mesures[END_REF] in the particular case of bounded measures and extended by Marcus and Véron [START_REF] Véron | The initial trace of positive solutions of semilinear parabolic equations[END_REF] to the general case. We denote by M q + (R N ) the positive cone of the space M q (R N ) of Radon measures which do not charge Borel subsets with zero C 2/q,q ′ -capacity. Notice that W -2/q,q (R N ) ∩ M b + (R N ) is a subset of M q + (R N ) where M b + (R N ) is the cone of positive bounded Radon mesures in R N . For such measures, uniqueness always holds and we denote u µν = u µν .

In view of the already known results concerning the parabolic equation, it is useful to recall the main advanced results previously obtained for the stationary equation -∆u + u q = 0 in Ω, (1.7) in a smooth bounded domain Ω of R N . This equation has been intensively studied since 1993, both by probabilists (Le Gall, Dynkin, Kuznetsov) and by analysts (Marcus, Véron). The existence of a boundary trace for positive solutions, in the class of outer-regular positive Borel measures on ∂Ω, is proven by Le Gall [START_REF] Legall | The Brownian snake and solutions of ∆u = u 2 in a domain[END_REF], [START_REF] Legall | A probabilistic approach to the trace at the boundary for solutions of a semilinear parabolic partial differential equation[END_REF] in the case q = N = 2, by probabilistic methods, and by Marcus and Véron in [START_REF] Marcus | The boundary trace of positive solutions of semilinear elliptic equations: the subcritical case[END_REF], [START_REF] Marcus | The boundary trace of positive solutions of semilinear elliptic equations: the supercritical case[END_REF] in the general case q > 1, N > 1. The existence of a critical exponent q e = (N + 1)/(N -1) is due to Gmira and Véron [START_REF] Gmira | Boundary singularities of solutions of some semilinear elliptic equation[END_REF] who shew that, if q ≥ q e boundary isolated singularities of solutions of (1.7) are removable, which is not the case if 1 < q < q e . In this subcritical case Le Gall and Marcus and Véron proved that the boundary trace establishes a one to one correspondence between positive solutions of (1.7) in Ω and outer regular positive Borel measures on ∂Ω. This fundamental result does not hold in the supercritical case q ≥ q e . In [START_REF] Dynkin | Solutions of Lu = u α dominated by harmonic functions[END_REF] Dynkin and Kuznetsov introduced the notion of σ-moderate solution which means that u is a positive solution of (1.7) such that there exists an increasing sequence of positive Radon measures on ∂Ω {µ n } belonging to W -2/q,q ′ (∂Ω) such that the corresponding solutions v = v µn of -∆v

+ v q = 0 in Ω v = µ n in ∂Ω (1.8)
converges to u locally uniformly in Ω. This class of solutions plays a fundamental role since Dynkin and Kuznetsov proved that a σ-moderate solution of (1.7) is uniquely determined by its fine trace, a new notion of trace introduced in order to avoid the non-uniqueness phenomena.

Later on, it is proved by Mselati (if q = 2) [START_REF] Mselati | Classification and probabilistic representation of the positive solutions of a semilinear elliptic equation[END_REF], then by Dynkin (if q e ≤ q ≤ 2) [START_REF] Dynkin | Superdiffusions and positive solutions of nonlinear partial differential equations[END_REF] and finally by Marcus with no restriction on q [START_REF] Marcus | Complete classification of the positive solutions of -∆u + u q = 0[END_REF], that all the positive solutions of (1.7) are σmoderate. One of the key-stones element in their proof (partially probabilistic) is the fact that the maximal solution u K of (1.7) with a boundary trace vanishing outside a compact subset K ⊂ ∂Ω is indeed σ-moderate. This deep result was obtained by a combination of probabilistic and analytic methods by Mselati [START_REF] Mselati | Classification and probabilistic representation of the positive solutions of a semilinear elliptic equation[END_REF] in the case q = 2 and by purely analytic tools by Marcus and Véron [START_REF] Marcus | Capacitary estimates of solutions of a class of nonlinear elliptic equations[END_REF], [START_REF] Marcus | Capacitary estimates of positive solutions of semilinear elliptic equations with absorption[END_REF] in the case q ≥ q e . Defining u K as the largest σ-moderate solution of (1.7) with a boundary trace concentrated on K, the crucial step in Marcus-Véron's proof (non probabilistic) is the bilateral estimate satisfied by u K and u K C -1 ρ(x)W K (x) ≤ u K (x) ≤ u K (x) ≤ Cρ(x)W K (x).

(1.9)

In this expression C = C(Ω, q), ρ(x) = dist (x, ∂Ω) and W F (x) is the elliptic capacitary potential of K defined by

W K (x) = ∞ -∞ 2 -m(q+1)
q-1 C 2/q,q ′ (2 m K m (x)), (1.10) where K m (x) = K ∩ {z : 2 -m-1 ≤ |z -x| ≤ 2 -m }, the Bessel capacity being relative to R N -1 . Note that, using a technique introduced in [START_REF] Marcus | The boundary trace of positive solutions of semilinear elliptic equations: the supercritical case[END_REF], inequality

u K ≤ C 2 u K implies u K = u K .
The aim of this article is to initiate the fine study of the complete initial trace problem for positive solutions of (1.1) in the supercritical case q ≥ q c and to give in particular the parabolic counterparts of the results of [START_REF] Mselati | Classification and probabilistic representation of the positive solutions of a semilinear elliptic equation[END_REF], [START_REF] Marcus | Capacitary estimates of solutions of a class of nonlinear elliptic equations[END_REF] and [START_REF] Marcus | Capacitary estimates of positive solutions of semilinear elliptic equations with absorption[END_REF]. Extending Dynkin's ideas to the parabolic case, we introduce the following notion

Definition 1.1 A positive solution u of (1.1) is called σ-moderate if their exists an increasing sequence {µ n } ⊂ W -2/q,q (R N ) ∩ M b + (R N ) such that the corresponding solution u := u µn of ∂ t u -∆u + u q = 0 in Q ∞ u(x, 0) = µ n in R N , (1.11)
converges to u locally uniformly in Q ∞ .
If F is a closed subset of R N , we denote by u F the maximal solution of (1.1) with an initial trace vanishing on F c , and by u F the maximal σ-moderate solution of (1.1) with an initial trace vanishing on F c . Thus u F is defined by

u F = sup{u µ : µ ∈ W -2/q,q (R N ) ∩ M b + (R N ), µ(F c ) = 0}, (1.12)
(and clearly W -2/q,q (R N ) ∩ M b + (R N ) can be replaced by M q + (R N )). One of the main goal of this article is to prove that u F is σ-moderate and more precisely, Theorem 1.2 For any q > 1 and any closed subset

F of R N , u F = u F .
We define below a set function which will play a fundamental role in the sequel.

Definition 1.3 Let F be a closed subset of R N . The Bessel parabolic capacitary potential W F of F is defined by W F (x, t) = 1 t N 2 ∞ n=0 d N -2 q-1 n+1 e -n 4 C 2/q,q ′ F n d n+1 ∀(x, t) ∈ Q ∞ , (1.13)
where C 2/q,q ′ is the N -dimensional Bessel capacity,

d n = √ nt and F n = {y ∈ F : d n ≤ |x -y| ≤ d n+1 }.
In our study, it is useful to introduce a variant of W F with the help of the Besov capacity: if Ω ⊂ R N is a bounded domain, we set

φ B 2/q,q ′ = Ω×Ω |φ(x) -φ(y)| q ′ |x -y| N + 2 q-1 dxdy 1/q ′ , (1.14)
if 1 < 2/q < 1, and φ B 1,2 = ∇φ L 2 if 2/q = 1 (i.e. N = 2 and q = 2). The Besov capacity of a compact set K ⊂ Ω relative to Ω is expressed by

R Ω 2/q,q ′ = inf φ q ′ B 2/q,q ′ : φ ∈ C ∞ 0 (Ω), 0 ≤ φ ≤ 1, η = 1 on K . (1.15)
The Besov-parabolic capacitary potential WF of F is defined by

WF (x, t) = t -N 2 ∞ n=0 d N -2 q-1 n+1 e -n 4 R Γn 2/q,q ′ F n d n+1 ∀(x, t) ∈ Q ∞ , (1.16)
where

Γ n = B d n+1 \ B dn .
The Besov-parabolic capacitary potential is equivariant with respect to the same scaling transformation which let (1.1) invariant in the sense that, for any ℓ > 0,

ℓ 1 q-1 WF ( √ ℓx, ℓt) = WF/ √ ℓ (x, t) ∀(x, t) ∈ Q ∞ . (1.17)
and we prove that there exists c = c(N, q) > 0 such that

c -1 WF (x, t) ≤ W F (x, t) ≤ c WF (x, t) ∀(x, t) ∈ Q ∞ . (1.18)
One of the tool for proving Theorem 1.2 is the following bilateral estimate which is only meaningful in the supercritical case, otherwhile it reduces to (1.6); Theorem 1.4 For any q ≥ q c there exist two positive constants C 1 ≥ C 2 > 0, depending only on N and q such that for any closed subset F of R N , there holds

C 2 W F (x, t) ≤ u F (x, t) ≤ u F (x, t) ≤ C 1 W F (x, t) ∀(x, t) ∈ Q ∞ . (1.19)
Actually our result is more general since the upper estimate in (1. [START_REF] Kuznetsov | σ-moderate solutions of Lu = u α and fine trace on the boundary[END_REF]) is valid for any positive solution of

∂ t u -∆u + u q ≤ 0 in Q T (1.20) satisfying lim t→0 u(x, t) = 0 locally uniformly in F c . (1.21)
Extension to positive solutions of

∂ t u -∆u + f (u) = 0 in Q T (1.22)
where f is continuous from R + to R + and satisfies

c 2 r q ≤ f (r) ≤ c 1 r q ∀r ≥ 0 (1.23)
for some 0 < c 2 ≤ c 1 is straightforward. This quasi representation, up to uniformly upper and lower bounded functions, is also interesting in the sense that it indicates precisely how to characterize the blow-up points of u F = u F := u F . Introducing an integral expression comparable to W F , we show in particular the following results

lim τ →0 τ 2 q-1 -N C 2/q,q ′ (F ∩ B τ (x)) = γ ∈ [0, ∞) =⇒ lim t→0 t 1 q-1 u F (x, t) = Cγ (1.24)
for some C γ = C(N, q, γ) > 0, and lim sup

τ →0 τ 2 q-1 C 2/q,q ′ F τ ∩ B 1 (x) < ∞ =⇒ lim sup t→0 u F (x, t) < ∞. (1.25)
Our paper is organized as follows. In Section1 we recall some properties of the Besov spaces with fractional derivatives B s,p and their links with heat equation. In Section 2 we obtain estimates from above on u F . In Section 3 we give estimates from below on u F . In Section 4 we prove the main theorems and expose various consequences. In Appendix we derive a series of sharp integral inequalities.

Aknowledgements

Q Ω T := Ω × (0, T ), ∂ ℓ Q Ω T = ∂Ω × (0, T ), Q T := Q R N T , Q ∞ := Q R N ∞ . Let H Ω [.] (resp. H[.]
) denote the heat potential in Ω with zero lateral boundary data (resp. the heat potential in R N ) with corresponding kernel

(x, y, t) → H Ω (x, y, t) (resp.(x, y, t) → H(x, y, t) = (4πt) -N 2 e -|x-y| 2 4t
).

We denote by q c := 1 + N 2 , the Brezis-Friedman critical exponent.

Theorem 2.1 Let q ≥ q c . Then there exists a positive constant

C 1 = C 1 (N, q) such that for any closed subset F of R N and any u ∈ C 2 (Q ∞ ) ∩ C(Q ∞ \ F ) satisfying ∂ t u -∆u + u q = 0 in Q ∞ lim t→0 u(x, t) = 0 locally uniformly in F c , (2.1
)

there holds u(x, t) ≤ C 1 W F (x, t) ∀(x, t) ∈ Q ∞ , (2.2)
where W F is the (2/q, q ′ )-parabolic capacitary potential of F defined by (1.13).

First we consider the case where F = K is compact and

K ⊂ B r ⊂ B r , (2.3)
and then we extend to the general case by a covering argument.

Capacities and Besov spaces

L p regularity

Throughout this paper C will denote a generic positive constant, depending only on N , q and sometimes T , the value of which may vary from one occurrence to another. We also use sometimes the notation A ≈ B for meaning that there exists a constant C > 0 independent of the data such that

C -1 A ≤ B ≤ CA.
We recall some classical results dealing with L p capacities as they are developed in [START_REF] Baras | Problèmes paraboliques semi-linéaires avec données mesures[END_REF]: if 1 < p < ∞ we denote

W 2,1 p (R N +1 ) := {φ ∈ L p (R N +1 ) : ∂ t φ, ∇φ, D 2 φ ∈ L p (R N +1 )}, (2.4)
with the associated norm

φ W 2,1 p = φ L p + ∇φ L p + ∂ t φ L p + D 2 φ L p . (2.5)
We define a corresponding capacity on compact sets, that we extend it classicaly on capacitable sets.

C 2,1,p (E) = inf{ φ W 2,1 p : φ ∈ C ∞ 0 (R N +1 ) : φ ≥ 1 in a neighborhood of E}, (2.6)
We extend the heat kernel H in R N +1 = {(x, t) ∈ R N × R} by assigning the value 0 for t < 0. Then, for any η ∈ C 0 (R N ),

H[η](x, t) = 0 if t < 0 H * (η ⊗ δ 0 )(x, t) if t > 0, (2.7)
where δ 0 has to be understood as the Dirac measure on R at t = 0. For any subset

E ∈ R N +1 C H,p (E) = inf{ f L p : f ∈ L p (R N +1 ), H * f ≥ 1 on E}. (2.8)
The following result is proved in [5, Prop 2.1].

Proposition 2.2 For any T > 0, there exists c = c(T, p, N ) such that

c -1 C H,p (E) ≤ C 2,1,p (E) ≤ cC H,p (E) ∀E ⊂ R N ×] -T, T [, E Borel.
(2.9)

We recall the Gagliardo Nirenberg inequality valid for any φ ∈ C ∞ 0 (R d )

∇φ 2p L 2p ≤ c d,p φ p L ∞ D 2 φ p L p .
(2.10) Furthermore, the trace at t = 0 of functions in W

c -1 c 2-2 p ,p (E) ≤ C 2,1,p (E × {0}) ≤ cc 2-2 p ,p (E) ∀E ⊂ R N , E Borel. (2.
12)

The c 2-2 p ,p -capacity is equivalent to the Bessel capacity C 2-2 p ,p defined by

C 2-2 p ,p (E) = inf{ f L p : f ∈ L p (R N ), G 2-2 p * f ≥ 1 on E} (2.13)
where

G 2-2 p = F[(1+|ξ| 2 ) 1 p -1 ]
denotes the Bessel kernel associated to the operator (-∆+I) 1-1 p .

The Aronszajn-Slobodeckij integral

If Ω is a domain in R N and 0 < s < 1, we denote by . Ḃs,p (Ω) the Aronszajn-Slobodeckij norm defined on C ∞ 0 (Ω) by

η Ḃs,p = Ω×Ω |η(x) -η(y)| p |x -y| N +sp dxdy 1/p ∀η ∈ C ∞ 0 (Ω).
(2.14)

In the case 1 < s < 2, all the results which are presented still holds by replacing the function by its gradient. We also consider the case s = 1, but in our range of exponents the corresponding exponent for p is 2, in which case the space under consideration is just H 1 0 (Ω). Since the imbedding of W 1,p (Ω) is compact, it follows the imbedding of B s,p (Ω) into L p (Ω) is compact too. Therefore the following Poincaré type inequality holds [39, p. 134]. Actually, the proof, obtained by contradiction, is given with W 1,p (Ω) instead of B s,p (Ω), but it depends only on the compactness of the imbedding.

Proposition 2.4

Let Ω be a bounded domain and, p ∈ (1, ∞) and 0 < s ≤ 1 such that sp ≤ N . Then there exists λ = λ(Ω, N, p) > 0 such that

Ω×Ω |η(x) -η(y)| p |x -y| N +sp dxdy ≥ λ Ω |η(x)| p dx ∀η ∈ C ∞ 0 (Ω). ( 2

.15)

Remark. If sp > N , the same proof re holds for all η ∈ C ∞ 0 (Ω) (see the proof of [9, Th 8.2])

Ω×Ω |η(x) -η(y)| p |x -y| N +sp dxdy 1/p ≥ C |η(z) -η(z ′ )| |z -z ′ | α ∀(z, z ′ ) ∈ Ω × Ω, z = z ′ , (2.16
) with α = s -N/p and C = C(s, N, p). This estimate implies

Ω×Ω |η(x) -η(y)| p |x -y| N +sp dxdy 1/p ≥ Cd -α u L ∞ , (2.17)
where d is the width of Ω, i.e. the smallest of δ > 0 such that there exists an isometry R such that R(Ω) ⊂ D δ := {x = (x 1 , x ′ ) : 0 < x 1 < δ}.

The related unpublished result due to L. Tartar [40] will be useful in the sequel. We reproduce its proof for the sake of completeness.

Proposition 2.5 Assume b > a and Ω ⊂ Γ a,b := {x = (x 1 , x ′ ) : a < x 1 < b} is a domain. If sp ≤ N there exists C = C(s, p, N, b/a) > 0 such that that Ω×Ω |η(x) -η(y)| p |x -y| N +sp dxdy ≥ λ (b -a) sp Ω |η(x)| p dx ∀η ∈ C ∞ 0 (Ω). (2.18)
Proof. Using the notation of [START_REF] Lions | Espaces d'interpolation[END_REF],

W s,p (R N ) is the interpolation space [W 1,p (R N ), L p (R N )] s,p and subset of L p (R N -1 ; [W 1,p (R), L p R] s,p ) = L p (R N -1 ; W s,p (R))
, with continuous imbedding. Thus there exist C > 0 such that

η p L p + R N-1 R×R |η(x 1 , x ′ ) -η(y 1 , x ′ )| p |x 1 -y 1 | 1+sp dx 1 dy 1 dx ′ ≤ C η p L p + R N ×R N |η(x) -η(y)| p |x -y| N +sp dxdy (2.19)
for all η ∈ C ∞ 0 (R N ). This inequality is valid if η is replaced by η τ where η τ (x) = η(τ x) and τ > 0. This gives

η p L p + τ sp-N R N-1 R×R |η(x 1 , x ′ ) -η(y 1 , x ′ )| p |x 1 -y 1 | 1+sp dx 1 dy 1 dx ′ ≤ C η p L p + τ sp-N R N ×R N |η(x) -η(y)| p |x -y| N +sp dxdy .
Letting τ → 0, we obtained

R N-1 R×R |η(x 1 ) -η(y 1 )| p |x 1 -y 1 | 1+sp dx 1 dy 1 dx ′ ≤ C R N ×R N |η(x) -η(y)| p |x -y| N +sp dxdy ∀η ∈ C ∞ 0 (R N ).
(2.20) Using Proposition 2.4 with N = 1 we get

1 0 1 0 |η(x 1 , x ′ ) -η(y 1 , x ′ )| p |x 1 -y 1 | 1+sp |dx 1 dy 1 ≥ λ 1 0 |η(x 1 , x ′ )| p dx 1 ∀η ∈ C ∞ 0 (0, 1) × R N -1 for all x ′ ∈ R N -1 . Using a standard change of scale, it transforms into b a b a |η(x 1 , x ′ ) -η(y 1 , x ′ )| p |x 1 -y 1 | 1+sp |dx 1 dy 1 ≥ λ(b -a) sp b a |η(x 1 , x ′ )| p dx 1 ∀η ∈ C ∞ 0 (a, b) × R N -1
Integrating over R N -1 and using (2.20), we derive (2.18). It is well known (see e.g. [START_REF] Berens | Semigroups of operators and approximations[END_REF]) that the Besov space B s,p (Ω) can be defined directly as the space of η ∈ L p (Ω) functions such that η Ḃs,p < ∞ or or such that η Bs,p < ∞. It coincides with the the interpolation space W 2,p (Ω), L p (Ω) s/2,p (see [START_REF] Lions | Espaces d'interpolation[END_REF]). Furthermore, there exists

C = C(s, p, N ) > 0 such that C -1 η L p + η Ḃs,p ≤ η L p + η Bs,p ≤ C η L p + η Ḃs,p ∀η ∈ B s,p (Ω).
(2.24)

Lemma 2.8 Assume 0 < s < 1 and 1 < p < ∞ or s = 1 and p = 2. Then there exists a positive constant C, depending only on s, p, N , such that for any domain Ω, there holds

C -1 η Ḃs,p ≤ η Bs,p ≤ C η Ḃs,p ∀η ∈ C ∞ 0 (Ω). (2.25) Proof. Let η ∈ C ∞ 0 (R N
) and τ > 0. Set η τ (x) = η(τ x), then (2.25) applied to η τ yields to

C -1 η L p + τ s η Ḃs,p ≤ η L p + τ s η Bs,p ≤ C η L p + τ s η Ḃs,p .
Since it holds for any arbitrary large τ and η ∈ C ∞ 0 (R N ), (2.25) follows. We denote by

T Ω (K) the set of functions η ∈ C ∞ 0 (Ω) such that 0 ≤ η ≤ 1 and η = 1 on K. If Ω is a bounded subset of R N , we define the Besov capacity of a compact set K ⊂ Ω ⊂ R N by R Ω s,p (K) = inf{ η p Ḃs,p : η ∈ T Ω (K)}, (2.26)
and the Bessel capacity relative to Ω by

C Ω s,p (K) = inf{ η p B s,p : η ∈ T Ω (K)}. (2.27)
We extend classicaly this capacity to any capacitable set K ⊂ Ω. This capacity has the following scaling property.

Lemma 2.9 For any τ > 0 and any capacitable set K ⊂ Ω, there holds

R Ω s,p (K) = τ N -sp R τ -1 Ω s,p (τ -1 K).
(2.28)

Furthermore, if Ω ⊂ B b \ B a , there exists c = c(b -a, b/a, N, s, p) > 0 such that c -1 C Ω s,p (K) ≤ R Ω s,p (K) ≤ cC Ω s,p (K).
(2.29)

Finally, if K ⊂ Ω ′ ⊂ Ω ′ ⊂ Ω, there exists c = c(N, s, p, dist (Ω ′ , Ω c )) such that C s,p (K) ≤ C Ω s,p (K) ≤ cC s,p (K). (2.30)
Proof. The scaling property (2.28) is clear by change of variable. Estimate (2.29) is a consequence of Definition 2.6 and Proposition 2.5. For the last statement, the left-hand side is obvious. For the right-hand side, consider a smooth nonnegative cut-off function ζ which is 1 on Ω ′ , has value between 0 and 1 and has compact support in Ω. If η ∈ T R N (K), ζη ∈ T Ω (K) and

ζη p B s,p = ζη p L p (Ω) + ζη p Ḃs,p ≤ η p L p (Ω) + η p Ḃs,p + ζ p Ḃs,∞ η p L p ≤ c η p B s,p , where ζ Ḃs,∞ = sup x =y |ζ(x) -ζ(y)| |x -y| s and c ≈ 1 + (dist (Ω ′ , Ω c )) -s . The proof follows.
In the sequel we assume that q ≥ q c and we take p = q ′ and s = 2/q. If K ⊂ Ω, Ω is bounded and η ∈ T Ω (K), we set

R[η] = |∇H[η]| 2 + |∂ t H[η]| .
(2.31)

Lemma 2.10 There exists C = C(N, q) > 0 such that for every η ∈ T Ω (K)

η q ′ B2/q,q ′ ≤ Q∞ (R[η]) q ′ dx dt := R[η] q ′ L q ′ ≤ C η q ′ B2/q,q ′ (2.32)
Proof. Using (2.23) and Lemma 2.8, it follows from Corollary 2.7 that

η q ′ B2/q,q ′ ≈ Q∞ |∂ t H[η]| q ′ dxdt.
Using the Gagliardo-Nirenberg inequality in R N , an elementary elliptic estimate and the fact that 0 ≤ H[η] ≤ 1, we see that

R N |∇(H[η](., t))| 2q ′ dx ≤ C D 2 H[η](., t) q ′ L q ′ H[η](., t) q ′ L ∞ ≤ C ∆H[η](., t) q ′ L q ′ , (2.33) for all t > 0. Since ∂ t H[η] = ∆H[η], it implies (2.32).
The dual space B -2/q,q (Ω) of B 2/q,q ′ (Ω) is naturally endowed with the norm

µ B -2/q,q = sup µ(η) : η ∈ B 2/q,q ′ (Ω), η B 2/q,q ′ ≤ 1 .
The following result is may be already known, but we have not found it in the literature. If µ is a bounded measure in R N , we denote by H[µ] the solution of heat equation in Q ∞ with initial data µ. Lemma 2.11 Assume q ≥ q c . For any T > 0, there exist a constant c > 0 such that, for any bounded measure µ belonging to B -2/q,q (R N ), there holds

c -1 µ B -2/q,q (R N ) ≤ H[µ] L q (Q T ) ≤ c µ B -2/q,q (R N ) .
(2.34)

Furthermore, if q > q c there holds

c -1 µ B -2/q,q (R N ) ≤ H[µ] L q (Q∞) ≤ c µ B -2/q,q (R N ) + c µ M(R N ) . (2.35)
.

Proof. If µ ∈ B -2/q,q (R N ), there exists a unique ω ∈ B 2-2/q,q (R N ) such that µ = (I -∆)ω, and µ B -2/q,q ≈ ω B 2-2/q,q . Applying standard interpolation methods to the analytic semi-group e -t(I-∆) = e -t e t∆ (see e.g. [START_REF] Berens | Semigroups of operators and approximations[END_REF], [START_REF] Triebel | Interpolation theory, function spaces, Differential operators[END_REF]) we obtain,

Q∞ t 1/q (I -∆)H[ω]
q dx e -qt dt t

1/q = Q∞ t 1/q H[µ]
q dx e -qt dt t

1/q ≈ ω B 2-2/q,q
≈ µ B -2/q,q .

(2.36)

Clearly

e -qT Q T t 1/q H[µ] q dx dt t ≤ Q∞ t 1/q H[µ]
q dx e -qt dt t , and

Q∞ t 1/q H[µ] q dx e -qt dt t = ∞ n=0 Q T +n+1 \Q T +n t 1/q H[µ] q dx e -qt dt t = ∞ n=0 Q T |H[µ](s + n)| q e -q(s+n) ds ≤ ∞ n=0 e -qn Q T t 1/q H[µ] q dt t .
This implies (2.34). Furthermore,

|H[µ](., t)| q L q ≤ ct -N (q-1)/2 µ q M , thus H[µ] ∈ L q (Q ∞ ) if q > q c (but this does not hold if q = q c ). If q > q c (equivalently N (q -1)/2 > 1), Q∞ t 1/q H[µ] q dx dt t = ∞ n=0 Q T +n+1 \Q T +n t 1/q H[µ] q dx dt t = Q T t 1/q H[µ] q dx dt t + Q T ∞ n=1 |H[µ](s + n)| q dxds ≤ Q T t 1/q H[µ] q dx dt t + C ∞ n=1 n -N (q-1)/2 µ q M .
Thus we obtain (2.35).

Global L q -estimates

Let ρ > 0, we assume (2.3) holds. With the previous notations, T r,r+ρ (K) denotes the set of functions η ∈ C ∞ 0 (B r+ρ ), such that 0 ≤ η ≤ 1 and value 1 on K. If η ∈ T r,ρ (K), we set

η * = 1 -η and ζ = H[η * ] 2q ′ .
Lemma 2.12 Assume u is a positive solution of (2.1) in Q ∞ . There exists C = C(N, q) > 0 such that for every T > 0 and every compact set K ⊂ B r ,

Q T u q ζdx dt + R N (uζ)(x, T )dx ≤ C R[η] q ′ L q ′ ∀η ∈ T r,ρ (K). (2.37)
Proof. We recall that there always holds 0 ≤ u(x, t) ≤ 1 t(q -1)

1 q-1 ∀(x, t) ∈ Q ∞ , (2.38)
and

0 ≤ u(x, t) ≤ C t + (|x| -r) 2 1 q-1 ∀(x, t) ∈ Q ∞ \ B r × R, (2.39)
by the Brezis-Friedman estimate [START_REF] Brezis | Nonlinear parabolic equations involving measures as initial conditions[END_REF]. Since

η * vanishes in an open neighborhood N 1 , for any open subset N 2 such that K ⊂ N 2 ⊂ N 2 ⊂ N 1 there exist c 2 = c N 2 > 0 and C 2 = C N 2 > 0 such that H[η * ](x, t) ≤ C 2 e -c 2 t , ∀(x, t) ∈ Q N 2 T . Therefore lim t→0 R N (uζ)(x, t)dx = 0.
Thus ζ is an admissible test function and one has

Q T u q ζdx dt + R N (uζ)(x, T )dx = Q T u(∂ t ζ + ∆ζ)dx dt. (2.40)
Notice that the two terms on the left-hand side are nonnegative. Put H η * = H[η * ], then

∂ t ζ + ∆ζ = 2q ′ H 2q ′ -1 η * (∂ t H η * + ∆H η * ) + 2q ′ (2q ′ -1)H 2q ′ -2 η * |∇H η * | 2 , = 2q ′ H 2q ′ -1 η * (∂ t H η + ∆H η ) + 2q ′ (2q ′ -1)H 2q ′ -2 η |∇H η | 2 , because H η * = 1 -H η , hence u(∂ t ζ + ∆ζ) = uH 2q ′ /q η * 2q ′ (2q ′ -1)H 2q ′ -2-2q ′ /q η * |∇H η | 2 -2q ′ H 2q ′ -1-2q ′ /q η * (∆H η + ∂ t H η ) .
Finally, since 2q ′ -2 -2q ′ /q = 0 and 0 ≤ H η * ≤ 1, there holds

Q T u(∂ t ζ + ∆ζ)dx dt ≤ C(q) Q T u q ζdx dt 1/q Q T R q ′ (η)dx dt 1/q ′ , where R(η) = |∇H η | 2 + |∆H η + ∂ t H η | .
Using Lemma 2.10 one obtains (2.37).

Proposition 2.13 Under the assumptions of Lemma 2.12, let r > 0, ρ > 0, T ≥ (r + ρ) 2

E r+ρ := {(x, t) : |x| 2 + t ≤ (r + ρ) 2 } and Q r+ρ,T = Q T \ E r+ρ . There exists C = C(N, q, T ) > 0 such that Q r+ρ,T u q dx dt + R N u(x, T )dx ≤ C R[η] q ′ L q ′ ∀η ∈ T r,ρ (K). (2.41)
Proof. In view of Lemma 2.12 we only have to show that there exists a positive constant c(N, q) such that, for η as above and

T ≥ (r + ρ) 2 , ζ = Hη * 2q ′ > c(N, q).
Since, by assumption

K ⊂ B r , η * ≡ 1 outside B r+ρ and 0 ≤ η * ≤ 1, H[η * ](x, t) ≥ H[1 -χ B r+ρ ](x, t) = 1 4πt N 2 |y|≥r+ρ e -|x-y| 2 4t dy, = 1 - 1 4πt N 2 |y|≤r+ρ e -|x-y| 2 4t
dy. We claim that max 1 4πτ

For (x, t) ∈ Q r+ρ,T , put x = (r + ρ)ξ, y = (r + ρ)υ and t = (r + ρ) 2 τ . Then (ξ, τ ) ∈ Q 1, T (
N 2 |υ|≤1 e -|ξ-υ| 2 4τ dυ : (ξ, τ ) ∈ Q 1, T (r+ρ) 2 = ℓ, (2.42)
for some ℓ = ℓ(N, T (r+ρ) 2 ) ∈ (0, 1], and ℓ is actually independent of T (r+ρ) 2 if this quantity is larger than 1. We recall that 1 4πτ

N 2 |υ|≤1 e -|ξ-υ| 2 4τ dυ < 1 ∀τ > 0. (2.43) If the maximum is achieved for some ( ξ, τ ) ∈ Q 1, T (r+ρ) 2
, it is smaller than 1 and

H[η * ](x, t) ≥ H[1 -χ B r+ρ ](x, t) ≥ 1 -ℓ > 0, ∀(x, t) ∈ Q r+ρ,T . (2.44)
Let us assume that the maximum is achieved following a sequence {(ξ n , τ n )} with τ n → 0 and

|ξ n | → α ≥ 1. Then 1 4πτ n N 2 |υ|≤1 e -|ξn-υ| 2 4τn dυ = 1 4πτ n N 2 B 1 (ξn) e -|υ| 2 4τn dυ ≤ 1 2 .
To verify this, note that

B 1 (ξ n ) ∩ B 1 (-ξ n ) = ∅, so that B 1 (ξn) e -|υ| 2 4τn dυ + B 1 (-ξn) e -|υ| 2 4τn dυ < R N e -|υ|
C = (1 -ℓ) -1 , then Q r,T u q dx dt + R N u(., T )dx ≤ C R[η] q ′ L q ′ , (2.45)
and (2.41) follows.

Pointwise estimates

In this subsection u is a positive solution of (2.1) in Q ∞ and the assumptions of Lemma 2.12 hold. We first derive a rough pointwise estimate.

Lemma 2.14 There exists a constant C = C(N, q) > 0 such that, for any η ∈ T r,ρ (K),

u(x, (r + 2ρ) 2 ) ≤ C R[η] q ′ L q ′ (ρ(r + ρ)) N 2 , ∀x ∈ R N . (2.46)
Proof. We recall that

T s R N u q dx dt + R N u(x, T )dx = R N u(x, s)dx ∀T > s > 0, (2.47) and R N u(., s)dx ≤ C R[η] q ′ L q ′ ∀T > s ≥ (r + ρ) 2 , (2.48)
by Proposition 2.13. Using the fact that

u(x, τ + s) ≤ H[u(., s)](x, τ ) ≤ 1 4πτ N 2 R N u(., s)dx, (2.46) follows from (2.48) with s = (r + ρ) 2 and τ = (r + 2ρ) 2 -(r + ρ) 2 ≈ ρ(r + ρ).
The above estimate does not take into account the fact that u(x, 0) = 0 if |x| ≥ r. It is mainly interesting if |x| ≤ r. In order to derive a sharper estimate which takes this fact into account, we need some lateral boundary estimates.

Lemma 2.15 Let γ ≥ r + 2ρ and c > 0 and either N = 1 or 2 and 0 ≤ t ≤ cγ 2 for some c > 0, or N ≥ 3 and t > 0. Then, for any η ∈ T r,ρ (K), there holds

t 0 ∂Bγ udSdτ ≤ C 5 γ R[η] q ′ L q ′ . (2.49)
where C > 0 depends on N , q and c if N = 1, 2 or depends only on N and q if N ≥ 3.

Proof. First we assume

N = 1 or 2. Put G γ := B c γ × (-∞, 0) and ∂ ℓ G γ = ∂B γ × (-∞, 0). We set h γ (x) = 1 - γ |x| ,
and let ψ γ be the solution of

∂ τ ψ γ + ∆ψ γ = 0 in G γ , ψ γ = 0 on ∂ ℓ G γ , ψ γ (., 0) = h γ in B c γ .
(2.50)

Thus the function ψ(x, τ ) = ψ γ (γx, γ 2 τ ) satisfies ∂ t ψ + ∆ ψ = 0 in G 1 ψ = 0 on ∂ ℓ G 1 ψ(., 0) = h in B c 1 ,
(2.51) and h(x) = 1 -|x| -1 . By the maximum principle 0 ≤ ψ ≤ 1, and by Hopf Lemma

- ∂ ψ ∂n ∂B 1 ×[-c,0] ≥ θ > 0, (2.52)
where θ = θ(N, c). Then 0 ≤ ψ γ ≤ 1 and

- ∂ψ γ ∂n ∂Bγ×[-γ 2 ,0] ≥ θ/γ. (2.53) Multiplying (1.1) by ψ γ (x, τ -t) = ψ * γ (x, τ ) and integrating on B c γ × (0, t) yields to t 0 B c γ u q ψ * r dxdτ + B c γ (uh γ )(x,
c γ × (0, t) ⊂ E c γ ), first by taking t = T = γ 2 ≥ (r + 2ρ) 2
, and then for any t ≤ γ 2 . If N ≥ 3, we proceed as above except that we take

h γ (x) = 1 - γ |x| N -2
.

Then ψ γ (x, t) = h γ (x) and θ = N -2 is independent of the length of the time interval. This leads to the conclusion.

Lemma 2.16 I-Let M, a > 0 and η ∈ L ∞ (R N ) such that 0 ≤ η(x) ≤ M e -a|x| 2 a.e. in R N . (2.55)
Then, for any t > 0,

0 ≤ H[η](x, t) ≤ M (4at + 1) N 2 e -a|x| 2 4at+1 ∀x ∈ R N .
(2.56)

II-Let M, a, b > 0 and η ∈ L ∞ (R N ) such that 0 ≤ η(x) ≤ M e -a(|x|-b) 2 + a.e. in R N . (2.57)
Then, for any t > 0,

0 ≤ H[η](x, t) ≤ M e - a(|x|-b) 2 + 4at+1 (4at + 1) N 2 ∀x ∈ R N , ∀t > 0.
(2.58)

Proof. For the first statement, put a = 1 4 s. Then

0 ≤ η(x) ≤ M (4πs) N 2 1 (4πs) N 2 e -|x| 2 4s = C(4πs) N 2 H[δ 0 ](x, s).
By the order property of the heat kernel,

0 ≤ H[η](x, t) ≤ M (4πs) N 2 H[δ 0 ](x, t + s) = M s t + s N 2 e -|x| 2
4(t+s) , and (2.56) follows by replacing s by 1 4 a. For the second statement, let ã < a and R = max{e Applying the statement I, we derive

0 ≤ H[η](x, t) ≤ Ce aãb 2 a-ã (4ãt + 1) N 2 e -ã|x| 2 4ãt+1 ∀x ∈ R N , ∀t > 0.
(2.59)

Since for any x ∈ R N and t > 0,

(4ãt + 1) -N 2 e -ã|x| 2 4ãt+1 ≤ e -aãb 2 a-ã (4at + 1) -N 2 e -a(|x|-b) 2 4at+1 ,
(2.58) follows from (2.59).

Lemma 2.17 There exists a constant C = C(N, q) > 0 such that, for any η ∈ T r,ρ (K), there holds

u(x, (r + 2ρ) 2 ) ≤ C max r + ρ (|x| -r -2ρ) N +1 , |x| -r -2ρ (r + ρ) N +1 e - (|x|-(r+2ρ)) 2 4(r+2ρ) 2 R[η] q ′ L q ′ , (2.60) for any x ∈ R N \ B r+3ρ .
Proof. It is classical that the Dirichlet heat kernel H B c 1 in the complement of B 1 satisfies, for some C = C(N ) > 0,

H B c 1 (x ′ , y ′ , t ′ , s ′ ) ≤ C 7 (t ′ -s ′ ) -(N +2)/2 (|x ′ | -1)e - |x ′ -y ′ | 2 4(t ′ -s ′ ) , (2.61)
for t ′ > s ′ . By performing the change of variable

x ′ → (r + 2ρ)x ′ , t ′ → (r + 2ρ) 2 t ′ , for any x ∈ R N \ B r+2ρ and 0 ≤ t ≤ T , one obtains u(x, t) ≤ C(|x| -r -2ρ) t 0 ∂B r+2ρ e -|x-y| 2 4(t-s) (t -s) 1+ N 2
u(y, s)dσ(y)ds.

(2.62)

The right-hand side term in (2.62) is smaller than

max C(|x| -r -2ρ) (t -s) 1+ N 2 e -(|x|-r-2ρ) 2 4(t-s) : s ∈ (0, t) t 0 ∂B r+2ρ
u(y, s)dσ(y)ds.

We fix t = (r + 2ρ) 2 and |x| ≥ r + 3ρ. Since max

   e -(|x|-r-2ρ) 2 4s s 1+ N 2 : s ∈ 0, (r + 2ρ) 2    = (|x| -r -2ρ) -2-N max e -1 4σ σ 1+ N 2 : 0 < σ < r + 2ρ |x| -r -2ρ 2 , a direct computation gives max e -1 4 σ σ 1+ N 2 : 0 < σ < r + 2ρ |x| -r -2ρ 2 =      (2N + 4) 1+ N 2 e -(N +2)/2 if r + 3ρ ≤ |x| ≤ (r + 2ρ)(1 + √ 4 + 2N ), |x| -r -2ρ r + 2ρ 2+N e - |x|-r-2ρ 2r+4ρ 2 if |x| ≥ (r + 2ρ)(1 + √ 4 + 2N ).
Thus there exists a constant C(N ) > 0 such that max

   e -(|x|-r-2ρ) 2 4s s 1+ N 2 : s ∈ 0, (r + 2ρ) 2    ≤ C(N )ρ -2-N e - |x|-(r+2ρ) 2r+4ρ 2
.

(2.63)

Combining this estimate with (2.49) with γ = r + 2ρ and (2.62), one derives (2.60).

Lemma 2.18 There exists a constant C = C(N, q) > 0 such that

0 ≤ u(x, (r + 2ρ) 2 ) ≤ C max (r + ρ) 3 ρ(|x| -r -2ρ) N +1 , 1 (r + ρ) N -1 ρ e -|x|-r-3ρ 2r+4ρ 2 R[η] q ′ L q ′ , (2.64) for every x ∈ R N \ B r+3ρ .
Proof. This is a direct consequence of the inequality

(|x| -r -2ρ)e - |x|-r-2ρ 2r+4ρ 2 ≤ C(r + ρ) 2 ρ e - |x|-r-3ρ 2r+4ρ 2 , ∀x ∈ B c r+2ρ , (2.65)
and Lemma 2.17.

Lemma 2.19 There exists a constant C = C(N, q) > 0 such that, for any η ∈ T r,ρ (K), the following estimate holds

u(x, t) ≤ C Me - (|x|-r-3ρ) 2 + 4t t N 2 R[η] q ′ L q ′ , ∀x ∈ R N , ∀t ≥ (r + 2ρ) 2 , (2.66)
where

M = M (x, r, ρ) =          1 + r ρ N 2 if |x| < r + 3ρ (r+ρ) N+3 ρ(|x|-r-2ρ) N+2 if r + 3ρ ≤ |x| ≤ c * N (r + 2ρ) 1 + r ρ if |x| ≥ c * N (r + 2ρ) (2.67) with c * N = 1 + √ 4 + 2N .
Proof. It follows by the maximum principle

u(x, t) ≤ H[u(., (r + 2ρ) 2 )](x, t -(r + 2ρ) 2 ).
for t ≥ (r + 2ρ) 2 and x ∈ R N . By Lemma 2.14 and Lemma 2.18

u(x, (r + 2ρ) 2 ) ≤ C 10 M e -(|x|-r-3ρ) 2 4(r+2ρ) 2 R[η] q ′ L q ′ , where M =          ((r + ρ)ρ) -N 2 if |x| < r + 3ρ (r+ρ) 3 ρ (|x| -r -2ρ)) N +2 if r + 3ρ ≤ |x| ≤ c * N (r + 2ρ) 1 (r+ρ) N-1 ρ if |x| ≥ c * N (r + 2ρ)
Applying Lemma 2.16 with a = (2r + 4ρ) -2 , b = r + 3ρ and t replaced by t -(r + 2ρ)

2 implies u(x, t) ≤ C (r + 2ρ) N M t N 2 e -(|x|-r-3ρ) 2 4t R[η] q ′ L q ′ , (2.68)
for all x ∈ B c r+3ρ and t ≥ (r + 2ρ) 2 , which is (2.66). The next estimate gives a precise upper bound for u when t is not bounded from below.

Lemma 2.20 Assume that 0 < t ≤ (r + 2ρ) 2 , then there exists a constant C = C(N, q) > 0 such that the following estimate holds

u(x, t) ≤ C(r + ρ) max 1 (|x| -r -2ρ) N +1 , 1 ρt N 2 e -(|x|-r-3ρ) 2 4t R[η] q ′ L q ′ , (2.69
)

for any (x, t) ∈ R N \ B r+3ρ × (0, (r + 2ρ) 2 ].
Proof. Thanks to (2.49) the following estimate is a straightforward variant of (2.60) for any |x| ≥ r + 2ρ,

u(x, t) ≤ C 8 (|x| -r -2ρ)(r + 2ρ) max    e -(|x|-r-2ρ) 2 4s s 1+ N 2 : 0 < s ≤ t    R[η] q ′ L q ′ . (2.70) Clearly max    e -(|x|-r-2ρ) 2 4s s 1+ N 2 : 0 < s ≤ t    =        (2N + 4) 1+ N 2 (|x| -r -2ρ) -N -2 e -N+2 2 if 0 < |x| ≤ r + 2ρ + 2t(N + 2) e -(|x|-r-2ρ) 2 4t t 1+ N 2 if |x| > r + 2ρ + 2t(N + 2). By elementary analysis, if x ∈ B c r+3ρ , (|x| -r -2ρ)e -(|x|-r-2ρ) 2 4t ≤ e -(|x|-r-3ρ) 2 4t      ρe -ρ 2 4t if 2t < ρ 2 2t ρ e -1+ ρ 2 4t if ρ 2 ≤ 2t ≤ 2(r + 2ρ) 2 .
However, since

ρ t e -ρ 2 4t ≤ 4 ρ , we derive (|x| -r -2ρ)e -(|x|-r-2ρ) 2 4t ≤ Ct ρ e -(|x|-r-3ρ) 2 4t
, and (2.69) follows.

Remark. In the subcritical case 1 < q < q c , it is easy to show by using Lemma 2.20, that any positive solution u of (2.1), such that u(x, 0) = 0 for x = 0, satisfies

u(x, t) ≤ Ct -1 q-1 min 1, |x| √ t 2 q-1 -N e -|x| 2 4t ∀(x, t) ∈ Q ∞ . (2.71)
This upper estimate corresponds to the one obtained in [START_REF] Brezis | A very singular solution of the heat equation with absorption[END_REF]. If F = B r the upper estimate is less esthetic. However, it is proved in [START_REF] Véron | The initial trace of positive solutions of semilinear parabolic equations[END_REF] by a barrier method that, if the initial trace of positive solution u of (2.1), vanishes outside F, and if 1 < q < 3, there holds

u(x, t) ≤ t -1 q-1 f 1 ((|x| -r)/ √ t) ∀(x, t) ∈ Q ∞ , |x| ≥ r, (2.72)
where f = f 1 is the unique positive (and radial) solution of

   f ′′ + y 2 f ′ + 1 q -1 f -f q = 0 in (0, ∞) f ′ (0) = 0 , lim y→∞ |y| 2 q-1 f (y) = 0.
(2.73)

Notice that the existence of f 1 follows from [START_REF] Brezis | A very singular solution of the heat equation with absorption[END_REF] since q belongs to the subcritical range on exponents in dimension one. Furthermore f 1 has the following asymptotic expansion f 1 (y) = Cy (3-q)/(q-1) e -y 2 /4t (1 + •(1))) as y → ∞. (2.75)

The upper Wiener test

If K ⊂ R N and i = 2, ∞, δ i [(x, t), K] = inf{δ i [(x, t), (y, 0)] : y ∈ K} =    max dist (x, K), |t| if i = ∞, dist 2 (x, K) + |t| if i = 2.
For β > 0 and i = 2, ∞, we denote by B i β (m) the parabolic ball of center m = (x, t) and radius β in the parabolic distance δ i .

Let K be any compact subset of R N and u K the maximal solution of (1.1) which blows up on K. The function u K is constructed in [START_REF] Véron | The initial trace of positive solutions of semilinear parabolic equations[END_REF] as being the decreasing limit of the u Kǫ (ǫ > 0) when ǫ → 0, where

K ǫ = {x ∈ R N : dist (x, K) ≤ ǫ}
and u Kǫ = lim k→∞ u k,Kǫ = u K , where u k is the solution of the classical problem,

       ∂ t u k -∆u k + u q k = 0 in Q T , u k = 0 on ∂ ℓ Q T , u k (., 0) = kχ Kǫ in R N .
(2.76)

If (x, t) = m ∈ R N × (0, T ], we set d K = dist (x, K), D K = max{|x -y| : y ∈ K} and λ = d 2 K + t = δ 2 [m, K].
We define a slicing of K, by setting

d n = d n (K, t) := √ nt (n ∈ N), d ± n = √ nt ± √ t √ n +
(the positive part is only needed when n = 0) and

T * n = B d + n+1 (x) \ B d - n (x) , T n = B d n+1 (x) \ B dn (x), ∀n ∈ N, thus T * 0 = B 2 √ t (x), T 0 = B √ t (x)
, and

K n (x, t) = K ∩ T n (x, t) for n ∈ N and Q n (x, t) = K ∩ B d n+1 (x, t).
When there is no ambiguity, we will skip the (x, t) variable in the above sets. The main result of this section is the following discrete upper Wiener-type estimate.

Theorem 2.22 Assume q ≥ q c . Then there exists C = C(N, q, T ) > 0 such that

u K (x, t) ≤ C t N 2 at n=0 d N -2 q-1 n+1 e -n 4 C 2/q,q ′ K n d n+1 ∀(x, t) ∈ Q T , (2.77)
where a t is the largest integer j such that K j = ∅.

With no loss of generality, we can assume that x = 0. Furthermore, in considering the scaling transformation u ℓ (y, t) = ℓ 1 q-1 u( √ ℓy, ℓt), with ℓ > 0, we can assume t = 1. Thus the new compact singular set of the initial trace becomes K/ √ ℓ, that we still denote K. We also set

a K = a K,1 For n ∈ N * set δ n = d n+1 -d n , then 1 2 √ n+1 ≤ δ n ≤ 1 2 √ n . By convention δ 0 = 1. It is possible to exhibit a collection Θ n of points a n,j with center on the sphere Σ n = {y ∈ R N : |y| = (d n+1 + d n )/2}, such that T n ⊂ a n,j ∈Θn B δn (a n,j ), |a n,j -a n,k | ≥ δ n and #Θ n ≤ Cn N -1 ,
for some constant C = C(N ). If K n,j = K n ∩ B δn (a n,j ), there holds

K = 0≤n≤a K a n,j ∈Θn K n,j .
The first intermediate step is based on the quasi-additivity property of capacities developed in [START_REF] Aikawa | Quasiadditivity and measure property of capacity and the tangential boundary behavior of harmonic functions[END_REF]. Lemma 2.23 Let q ≥ q c . There exists a constant C = C(N, q) such that

a n,j ∈Θn R B 2δn (a n,j ) 2/q,q ′ (K n,j ) ≤ Cd N -2 q-1 n+1 C 2/q,q ′ K n d n+1 ∀n ∈ N * . (2.78) Proof. The following result is proved in [2, Th 3]: if the spheres B ρ θ j (b j ), θ = 1 -2/N (q -1), are disjoint in R N and G is an analytic subset of B ρ j (b j )
where the ρ j are positive and smaller than some ρ * > 0, there holds

C 2/q,q ′ (G) ≤ j C 2/q,q ′ (G ∩ B ρ j (b j )) ≤ AC 2/q,q ′ (G), (2.79)
for some A depending on N , q and ρ * . This property is called quasi-additivity. We define for n ∈ N * , Tn = d n+1 T n , Kn = d n+1 K n and Qn = d n+1 Q n .

Since K n,j ⊂ B δn (a n,j ), it follows that Kn,j := d n+1 K n,j ⊂ B d n+1 δn (ã n,j ).

Note that by Lemma 2.9

R B 2δn (a n,j ) 2/q,q ′ (K n,j ) = d 2 q-1 -N n+1 R B 2δnd n+1 (d n+1 a n,j ) 2/q,q ′ ( Kn,j ) ≈ d 2 q-1 -N n+1 C B 2δnd n+1 (d n+1 a n,j ) 2/q,q ′ ( Kn,j ) ≈ d 2 q-1 -N n+1 C 2/q,q ′ ( Kn,j ) (2.80)
where Kn,j = d n+1 K n,j . For a fixed n > 0 and each repartition Λ of points ãn,j = d n+1 a n,j such that the balls B 2 θ (ã n,j ) are disjoint, the quasi-additivity property holds: if we set

K n,Λ = a n,j ∈Λ K n,j , Kn,Λ = d n+1 K n,Λ = a n,j ∈Λ
Kn,j and Kn = d n+1 K n , then a n,j ∈Λ C 2/q,q ′ ( Kn,j ) ≈ C 2/q,q ′ ( Kn,Λ ).

(2.81)

The maximal cardinal of any such repartition Λ is of the order of Cn N -1 for some positive constant C = C(N ), therefore, the number of repartitions needed for a full covering of the set Tn is of finite order depending upon the dimension. Because Kn is the union of the Kn,Λ , a n,j ∈Θn C 2/q,q ′ ( Kn,j ) = Λ a n,j ∈Λ C 2/q,q ′ ( Kn,j ) ≈ C 2/q,q ′ ( Kn ).

(2.82) By Lemma 2.9,

C 2/q,q ′ ( Kn ) ≤ C B 2d n+1 2/q,q ′ ( Kn ) ≈ d N -1 q-1 n+1 C B 2 2/q,q ′ K n d n+1 ≈ d N -1 q-1 n+1 C 2/q,q ′ K n d n+1 ,
we obtain (2.78) by combining this last inequality with (2.80) and (2.82).

Proof of Theorem 2.22.

Step 1. We first notice that

u K ≤ 0≤n≤a K a n,j ∈Θn u K n,j . (2.83)
Actually, since K = n a n,j K n,j , for any 0 < ǫ ′ < ǫ, there holds K ǫ ′ ⊂ n a n,j K n,j ǫ . Because a finite sum of positive solutions of (1.1) is a super solution,

u K ǫ ′ ≤ 0≤n≤a K a n,j ∈Θn u K n,j ǫ .
(2.84)

Letting successively ǫ ′ and ǫ go to 0 implies (2.83).

Step 2. Let n ∈ N. Since K n,j ⊂ B δn (a n,j ) and |xa n,j | = (d n + d n+1 )/2, we can apply the previous lemmas with r = δ n and ρ = r. For n ≥ n N , there holds t = 1 ≥ (r + 2ρ) 2 = 9/(n + 1) and |xa n,j | = (

√ n + 1 - √ n)/2 ≥ (2 + C N )(3/ √ n + 1) (notice that n N ≥ 8). Thus u K n,j (0, 1) ≤ Ce ( √ n-3/ √ n+1) 2 /4 R B 2δn (a n,j ) 2/q,q ′ (K n,j ) ≤ Ce 3/2 e -n 4 R B 2δn (a n,j ) 2/q,q ′ (K n,j ). ( 2

.85)

Using Lemma 2.23 we obtain, with

d n = d n (1) = √ n + 1 a K n=n N a n,j ∈Θn u K n,j (0, 1) ≤ C a K n=n N d N -2 q-1 n+1 e -n 4 C 2/q,q ′ K n d n+1 .
(2.86)

Finally, we apply Lemma 2.14 if 1 ≤ n < n N and get

n N -1 1 a n,j ∈Θn u K n,j (0, 1) ≤ C n N -1 1 C 2/q,q ′ K n d n+1 ≤ C ′ n N -1 1 d N -2 q-1 n+1 e -n 4 C 2/q,q ′ K n d n+1 .
(2.87)

For n = 0, we proceed similarly, in splitting K 1 in a finite number of K 1,i , depending only on the dimension, such that diam K 1,i < 1/3. Combining (2.86) and (2.87), we derive

u K (0, 1) ≤ C a K n=0 d N -2 q-1 n+1 e -n 4 C 2/q,q ′ K n d n+1 .
(2.88)

In order to derive the same result for any t > 0, we notice that

u K (y, t) = t -1 q-1 u K/ √ t (y/ √ t, 1).
Going back to the definition of

d n = d n (K, t) = √ nt = d n (K √ t, 1
), we derive from (2.88) and the fact that a

K,t = a K √ t,1 u K (0, t) ≤ Ct -N 2 a K n=0 d N -2 q-1 n+1 e -n 4 C 2/q,q ′ K n d n+1 , (2.89)
with d n = d n (t) = t(n + 1) . This is (2.77) with x = 0, and a space translation leads to the final result.

Proof of Theorem 2.1. Let m > 0 and F m = F ∩ B m . We denote by U B c m the maximal solution of (1.1) in Q ∞ the initial trace of which vanishes on B m . Such a solution is actually the unique solution of (2.1) which satisfies lim

U B c m ℓ (y, t) = ℓ 1 q-1 U B c m ( √ ℓy, ℓt) = U B c m/ √ ℓ (y, t). Furthermore lim m→∞ U B c m (y, t) = lim m→∞ m -2 q-1 U B c 1 (y/m, t/m 2 ) = 0 uniformly on any compact subset of Q ∞ . Since u Fm + U B c
m is a super-solution, it is larger that u F and therefore u Fm ↑ u F . Because W Fm (x, t) ≤ W F (x, t) and u Fm ≤ C 1 W Fm (x, t), the result follows.

Remark. It is clear that Theorem 2.1 still holds if u is a positive subsolution of (1.1) satisfying the initial trace condition (1.21). Theorem 2.1 admits the following integral expression. Theorem 2.24 Assume q ≥ q c . Then there exists a positive constant C * 1 = C * (N, q, T ) such that, for any closed subset F of R N , there holds

u F (x, t) ≤ C * 1 t 1+ N 2 √ t(at+2) √ t e -s 2 4t s N -2 q-1 C 2/q,q ′ 1 s F ∩ B 1 (x) s ds, (2.90)
where a t = min{n :

F ⊂ B √ n+1)t (x)}.
Proof. We first use

C 2/q,q ′ F n d n+1 ≤ C 2/q,q ′ F d n+1 ∩ B 1 ,
and we denote

Φ(s) = C 2/q,q ′ F s ∩ B 1 ∀s > 0.
(2.91)

Step 1. The following inequality holds

c 1 Φ(αs) ≤ Φ(s) ≤ c 2 Φ(βs) ∀s > 0, ∀1/2 ≤ α ≤ 1 ≤ β ≤ 2, (2.92)
for some positive constants c 1 , c 2 depending on N and q. See [START_REF] Adams | Function spaces and potential theory[END_REF] and [START_REF] Marcus | Capacitary estimates of positive solutions of semilinear elliptic equations with absorption[END_REF].

If β ∈ [1, 2], Φ(βs) = C 2/q,q ′ 1 β F s ∩ B β ≈ C 2/q,q ′ F s ∩ B β ≥ c 1 Φ(s). If α ∈ [1/2, 1], Φ(αs) = C 2/q,q ′ 1 α F s ∩ B α ≈ C 2/q,q ′ F s ∩ B α ≤ c 2 Φ(s).
Step 2. By (2.92)

C 2/q,q ′ F d n+1 ∩ B 1 ≤ c 2 C 2/q,q ′ F s ∩ B 1 ∀ s ∈ [d n+1 , d n+2 ],
and n ≤ a t . Then

c 2 d n+2 d n+1 s N -2 q-1 e -s 2 /4t C 2/q,q ′ F s ∩ B 1 s ds ≥ C 2/q,q ′ F d n+1 ∩ B 1 d n+2 d n+1 s N -2
q-1 e -s 2 /4t s ds.

Using the fact that N -2 q-1 ≥ 0, we get,

d n+2 d n+1 s N -2 q-1 e -s 2 4t s ds ≥ e -n+2 4 d N -2 q-1 +1 n+1 (d n+2 -d n+1 ) (2.93) ≥ t 4e 2 d N -2 q-1 n+1 e -n 4 . (2.94) Thus u F (x, t) ≤ C t 1+ N 2 √ t(at+2) √ t s N -2 q-1 e -s 2 4t C 2/q,q ′ 1 s F ∩ B 1 s ds, (2.95)
which ends the proof.

Estimate from below of the solution of the heat equation

The purely spatial slicing used is the trace on R N × {0} of an extended slicing in Q T which is constructed as follows: if K is a compact subset of R N , m = (x, t), we define d K , λ, d n and a t as in Section 2.3. Let α ∈ (0, 1) to be fixed later on, we define T n for n ∈ Z by

T n =    B 2 √ t(n+1) (m) \ B 2 √ tn (m) if n ≥ 1, B 2 α -n √ t (m) \ B 2 α 1-n √ t (m) if n ≤ 0,
and put

T * n = T n ∩ {s : 0 ≤ s ≤ t}, for n ∈ Z. We recall that for n ∈ N * , Q n = K ∩ B 2 √ t(n+1) (m) = K ∩ B dn (x)
and

K n = K ∩ T n+1 = K ∩ B d n+1 (x) \ B dn (x) . Let ν n ∈ M b + (R N ) ∩ W -2/q,q (R N ) be the q-capacitary measure of the set K n /d n+1 . See [1, Sec. 2.2]. Such a measure has support in K n /d n+1 and ν n (K n /d n+1 ) = C 2/q,q ′ (K n /d n+1 ) and ν n W -2/q,q ′ (R N ) = C 2/q,q ′ (K n /d n+1 ) 1/q . (3.9)
We define µ n as follows

µ n (A) = d N -2 q-1 n+1 ν n (A/d n+1 ) ∀A ⊂ K n , A Borel , (3.10)
and set

µ t,K = at n=0 µ n , and
H µ t,K = at n=0 H µn . (3.11)
Proposition 3.2 Let q ≥ q c , then there holds

H µ t,K (x, t) ≥ 1 (4πt) N 2 at n=0 e -n+1 4 d N -2 q-1 n+1 C 2/q,q ′ K n d n+1 , (3.12) in R N × (0, T ).
Proof. Since

Estimate from above of the nonlinear term

We write (3.4) under the form

u µ (x, t) ≥ n∈Z H µn (x, t) - t 0 R N H(x, y, t -s)   n∈A K H µn (y, s)   q dyds = I 1 -I 2 . (3.14) since µ n = 0 if n / ∈ A K = N ∩ [1, a t ],
and

I 2 = 1 (4π) N 2 t 0 R N (t -s) -N 2 e - |x-y| 2 4(t-s)   n∈A K H µn (y, s)   q dyds = 1 (4π) N 2 (J ℓ + J ′ ℓ ), (3.15)
for some ℓ ∈ N * to be fixed later on, where

J ℓ = p∈Z T * p (t -s) -N 2 e -|x-y| 2 4(t-s)   n<p+ℓ H µn (y, s)   q dyds, and
J ′ ℓ = p∈Z T * p (t -s) -N 2 e -|x-y| 2 4(t-s)   n≥p+ℓ H µn (y, s)   q dyds.
The next estimate will be used several times in the sequel.

Lemma 3.3 Let 0 < a < b and t > 0, then,

max σ -N 2 e -ρ 2 4σ : 0 ≤ σ ≤ t, at ≤ ρ 2 + σ ≤ bt = e 1 4          t -N 2 e -a 4 if a 2N > 1, 2N at N 2 e -N 2 if a 2N ≤ 1. Proof. Set J (ρ, σ) = σ -N 2 e -ρ 2 4σ and K a,b,t = (ρ, σ) ∈ [0, ∞) × (0, t] : at ≤ ρ 2 + σ ≤ bt .
We first notice that, for fixed σ, the maximum of J (., σ) is achieved for ρ minimal. If σ ∈ [at, bt] the minimal value of ρ is 0, while if σ ∈ (0, at), the minimum of ρ is √ ats.

-Assume first a ≥ 1, then

J ( √ at -σ, σ) = e 1 4 σ -N 4 e -at 4σ . Thus if 1 ≤ a/2N , the minimal value of J ( √ at -σ, σ) is e 1-2N 4 2N at N 2 , while if a/2N < 1 ≤ a, the minimum is e 1 4 t -N 2 e -a 4 .

29

-Assume now a ≤ 1. Then max{J (ρ, σ) : (ρ, σ) ∈ K a,b,t } = max max σ∈(at,t] J (0, σ), max σ∈(0,at]

J ( √ at -σ, σ) = max (at) -N 2 , e 1-2N 4
2N at

N 2 = e 1-2N 4 2N at N 2 .
Combining these two estimates, we derive the result.

Remark. The following variant of Lemma 3.3 will be useful in the sequel: For any θ ≥ 1/2N there holds

max{J (ρ, σ) : (ρ, σ) ∈ K(a, b, t)} ≤ e 1 4 2N θ t N 2 e -a 4 if θa ≥ 1. (3.16)
Lemma 3.4 There exists a positive constant C = C(N, ℓ, q) such that

J ℓ ≤ Ct -N 2 at n=1 d N -2 q-1 n+1 e -(1+(n-ℓ) + )/4 C 2/q,q ′ K n d n+1 . (3.17)
Proof. The set of the p's for the summation in J ℓ is reduced to Z ∩ [-ℓ + 2, ∞), thus we write

J ℓ = J 1,ℓ + J 2,ℓ
where

J 1,ℓ = 0 p=2-ℓ T * p (t -s) -N 2 e -|x-y| 2 4(t-s)   n<p+ℓ H µn (y, s)   q and J 2,ℓ = ∞ p=1 T * p (t -s) -N 2 e -|x-y| 2 4(t-s)   n<p+ℓ H µn (y, s)   q . If p = 2 -ℓ, . . . , 0, (y, s) ∈ T * p =⇒ tα 2-2p ≤ |x -y| 2 + t -s ≤ tα -2p , and, if p ≥ 1 (y, s) ∈ T * p =⇒ pt ≤ |x -y| 2 + t -s ≤ (p + 1)t. By Lemma 3.3 and (3.16), there exists C = C(N, ℓ, α) > 0 such that max (t -s) -N 2 e -|x-y| 2 4(t-s) : (y, s) ∈ T * p ≤ Ct -N 2 e -α 2-2p /4 , (3.18)
if p = 2ℓ, . . . , 0, and

max (t -s) -N 2 e -|x-y| 2 4(t-s) : (y, s) ∈ T * p ≤ Ct -N 2 e -p/4 , (3.19) if p ≥ 1. When p = 2 -ℓ, . . . , 0 p+ℓ-1 1 H µn (y, s) q ≤ C p+ℓ-1 1 H q µn (y, s), (3.20)
for some C = C(ℓ, q) > 0, thus

J 1,ℓ ≤ Ct -N 2 0 p=2-ℓ e -α 2-2p 4 p+ℓ-1 n=1 H µn q L q (Qt) ≤ Ct -N 2 ℓ-1 n=1 H µn q L q (Qt) 0 p=n-ℓ+1 e -α 2-2p 4 (3.21) ≤ Ct -N 2 e -α 2ℓ-2 4 ℓ-1 n=1 H µn q L q (Qt) .
If the set of p's is not upper bounded, we introduce some parameter δ > 0 to be made precise later on. Then

p+ℓ-1 1 H µn (y, s) q ≤ p+ℓ-1 1 e δq ′ n 4 q/q ′ p+ℓ-1 1 e -δqn 4 H q µn (y, s), (3.22)
with q ′ = q/(q -1). If, by convention µ n = 0 whenever n > a t , we obtain, for some C > 0 which depends also on δ,

J 2,ℓ ≤ Ct -N 2 ∞ p=1 e δ(p+ℓ-1)q-p 4 p+ℓ-1 n=1 e -δqn 4 H µn q L q (Qt) ≤ Ct -N 2 ∞ n=1 H µn q L q (Qt) e -δqn 4 ∞ p=(n-ℓ+1)∨1
e δ(p+ℓ-1)q-p 4

(3.23)

≤ Ct -N 2 ∞ n=1 e -1+(n-ℓ) + 4 H µn q L q (Qt) .
Notice that we choose δ such that δℓq < 1. Combining (3.21) and (3.23), we derive (3.17) from Lemma 2.11, (3.9) and (3.10).

The set of indices p for which the µ n terms are not zero in

J ′ ℓ is Z ∩ (-∞, a t -ℓ].
We write

J ′ ℓ = J ′ 1,ℓ + J ′ 2,ℓ ,
where

J ′ 1,ℓ = 0 p=-∞ T * p (t -s) -N 2 e - |x-y| 2 4(t-s)   ∞ n=1∨p+ℓ H µn (y, s)   q dyds, and
J ′ 2,ℓ = at-ℓ p=1 T * p (t -s) -N 2 e - |x-y| 2 4(t-s)   ∞ n=p+ℓ H µn (y, s)   q dyds.
Lemma 3.5 There exists a constant C = C(N, q, ℓ) > 0 such that

J ′ 1,ℓ ≤ Ct 1-Nq 2 at n=0 e -(1+β 0 )(n-h) + 4 d N q-2q ′ n+1 C q 2/q,q ′ K n d n+1 , (3.24)
where β 0 = (q -1)/4 and h = 2q(q + 1)/(q -1) 2 .

Proof. Since (y, s) ∈ T * p , and (z, 0)

∈ K n =⇒ |y -z| ≥ ( √ n -α -p ) √ t, (3.25)
there holds

H µn (y, s) ≤ (4πs) -N 2 e -( √ n-α -p ) 2 t 4s µ n (K n ) ≤ Ct -N 2 e -( √ n-α -p ) 2 4 µ n (K n ),
by Lemma 3.3. Let {ǫ n } be a sequence of positive numbers such that

A ǫ = ∞ n=1 ǫ q ′ n < ∞, then J ′ 1,ℓ ≤ CA q/q ′ ǫ t -Nq 2 0 p=-∞ T * p (t -s) -N 2 e - |x-y| 2 4(t-s) ∞ n=1∨(p+ℓ) ǫ -q n e -q ( √ n-α -p ) 2 4 µ q n (K n )ds dy ≤ CA q/q ′ ǫ t -Nq 2 ∞ n=1 ǫ -q n µ q n (K n ) p=0∧(n-ℓ) -∞ e -q( √ n-α -p ) 2 4 T * p (t -s) -N 2 e - |x-y| 2
4(t-s) ds dy

≤ CA q/q ′ ǫ t -Nq 2 ∞ n=1 ǫ -q n µ q n (K n )e -q( √ n-1) 2 4 {∪ p≤0 T * p } (t -s) -N 2 e
-|x-y| 2 4(t-s) ds dy

≤ CA q/q ′ ǫ t 1-Nq 2 ∞ n=1 ǫ -q n µ q n (K n )e -q( √ n-1) 2 4 . (3.26)
Set h = 2q(q + 1)/(q -1)

2 and Q = (1 + q)/2, then q( √ n -1) 2 ≥ Q(n -h) + for any n ≥ 1. If we choose ǫ n = e -(q-1)(n-h) + 16q
, there holds ǫ -q n e -q( √ n-1)

2 4 ≤ e -(q+3)(n-h) + 16 . Finally J ′ 1,ℓ ≤ Ct 1-Nq 2 ∞ n=1 e -(1+β 0 )(n-h) + 4 µ q n (K n ),
with β 0 = (q -1)/4, which yields to (3.24) by the choice of the µ n .

In order to make easier the obtention of the estimate of the term J ′ 2,ℓ , we first give the proof in dimension 1. Lemma 3.6 Assume N = 1 and ℓ is an integer larger than 1. There exists a positive constant C = C(q, ℓ) > 0 such that

J ′ 2,ℓ ≤ Ct -1/2 at n=ℓ e -n 4 d q-3 q-1 n+1 C 2/q,q ′ K n d n+1 . (3.27) Proof. If (y, s) ∈ T * p and z ∈ K n (p ≥ 1, n ≥ p = ℓ) , there holds |x -y| ≥ √ t √ p and |y -z| ≥ √ t( √ n - √ p + 1). Therefore J ′ 2,ℓ ≤ C √ t at-ℓ p=1 1 √ p t 0 e -pt 4(t-s)   at n=p+ℓ s -1/2 e -( √ n-√ p+1 ) 2 t 4s µ n (K n )   q .
If ǫ ∈ (0, q) is some positive parameter which will be made more precise later on, there holds

  at n=p+ℓ s -1/2 e -( √ n-√ p+1 ) 2 t 4s µ n (K n )   q ≤   at n=p+ℓ e -ǫq ′ ( √ n-√ p+1 ) 2 t 4s   q/q ′ at n=p+ℓ s -q 2 e -(q-ǫ) ( √ n-√ p+1 ) 2 t 4s µ q n (K n ),
by Hölder's inequality. By comparison between series and integrals and using Gauss integral

at n=p+ℓ e -ǫq ′ ( √ n-√ p+1) 2 t 4s ≤ ∞ p+ℓ e -ǫq ′ ( √ x-√ p+1) 2 t 4s dx = 2 ∞ √ p+ℓ- √ p+1 e -ǫq ′ x 2 t 4s (x + √ p + 1)dx ≤ 4s ǫq ′ t e -ǫq ′ ( √ p+ℓ-√ p+1) 2 t 4s + 2 √ p + 1 ∞ √ p+ℓ- √ p+1 e -ǫq ′ x 2 t 4s dx ≤ C (p + 1)s t e -ǫq ′ ( √ p+ℓ-√ p+1) 2 t 2s ≤ C (p + 1)s t .
If we set q ǫ = qǫ, then

J ′ 2,ℓ ≤ Cǫ -q ′ /q t 1-q 2 ∞ n=ℓ+1 µ q n (K n ) n-ℓ p=1 p q-2 2 t 0 (t -s) -1/2 s -1/2 e -pt 4(t-s) e -qǫ ( √ n-√ p+1 ) 2 t 4s ds.
where C = C(ǫ, q) > 0. Since

t 0 (t -s) -1/2 s -1/2 e -pt 4(t-s) e -qǫ ( √ n-√ p+1 ) 2 t 4s ds = 1 0 (1 -s) -1/2 s -1/2 e -p 4(1-s) e -qǫ ( √ n-√ p+1 ) 2 4s ds, we can apply Lemma A.1 with a = 1/2, b = 1/2, A = √ p and B = √ q ǫ ( √ n - √ p + 1). In this range of indices B ≥ √ q ǫ ( √ p + ℓ - √ p + 1) ≥ √ qǫ(ℓ-1) √ p
, thus κ = √ q ǫ (ℓ -1) and

A A + B B A + B ≤ p 1 4 n -1/2 ( √ n - √ p) 1/2 . Therefore t 0 (t -s) -1/2 s -q 2 e -pt 4(t-s) e -q ( √ n-√ p+1) 2 t 4s ds ≤ Cp 1 4 ( √ n - √ p) 1/2 √ n e -( √ p+ √ qǫ( √ n-√ p+1)) 2 4 , (3.28)
which implies

J ′ 2,ℓ ≤ Ct 1-q 2 at n=ℓ+1 µ q n (K n ) √ n n-ℓ p=1 p 2q-3 4 ( √ n - √ p) 1/2 e -( √ p+ √ qǫ( √ n-√ p+1)) 2 4 , (3.29)
where C depends of ǫ, q and ℓ. By Lemma A.2

J ′ 2,ℓ ≤ Ct 1-q 2 at n=ℓ+1 n q-3 2 e -n 4 µ q n (K n ) (3.30) Because µ n (K n ) = d q-3 q-1 n+1 C 2/q,q ′ K n d n+1 (remember N = 1) and diam Kn d n+1 ≤ n -1 , there holds µ q n (K n ) ≤ C √ t √ n q-3 µ n (K n ) = C √ t √ n q-3 d q-3 q-1 n+1 C 2/q,q ′ (K n /d n+1 ) (3.31)
and inequality (3.27) follows.

Next we give the general proof. For this task we will use again the quasi-additivity with separated partitions. Lemma 3.7 Assume N ≥ 2 and ℓ is an integer larger than 1. There exists a positive constant

C 1 = C 1 (q, N, ℓ) > 0 such that J ′ 2,ℓ ≤ C 1 t -N 2 at n=ℓ e -n 4 d N -2 q-1 n+1 C 2/q,q ′ K n d n+1 . (3.32)
Proof. As in the proof of Theorem 2.22, we know that there exists a finite number J, depending only on the dimension N , of separated sub-partitions {#Θ h t,n } J h=1 of the rescaled sets Tn = n+1 t T n by the N -dim balls B 2 (ã n,j ) where ãn,j =

n+1 t a n,j , |a n,j | = d n+1 + d n 2 and |a n,j -a n,k | ≥ 4t n+1 . Furthermore #Θ h t,n ≤ Cn N -1 . We denote K n,j = K n ∩ B t n+1 (a n,j ).
We write µ n = J h=1 µ h n , and accordingly

J ′ 2,ℓ = J h=1 J ′ 2,ℓ h , where µ h n = j∈Θ h t,n
µ n,j , and µ n,j are the capacitary measures of K n,j relative to B n,j = B 6t/5 √ n (a n , j), which means

ν n,j (K n,j ) = C B n,j
2/q,q ′ (K n,j ) and ν n,j W -2/q,q ′ (B n,j ) = C

B n,j 2/q,q ′ (K n,j )

1/q . ( 3.33)
Thus

J ′ 2,ℓ = at-ℓ p=1 T * p (t -s) -N 2 e -|x-y| 2 4(t-s)    ∞ n=p+ℓ J h=1 j∈Θ h t,n H µ n,j (y, s)    q dyds.
We denote

J ′ 2,ℓ h = at-ℓ p=1 T * p (t -s) -N 2 e -|x-y| 2 4(t-s)    ∞ n=p+ℓ j∈Θ h t,n H µ n,j (y, s)    q dyds,
and clearly

J ′ 2,ℓ ≤ C J h=1 J ′ 2,ℓ h , (3.34)
where C depends only on N and q. For integers n and p such that n ≥ ℓ + 1, we set

λ n,j,y = inf{|y -z| : z ∈ B √ t/ √ n+1 (a n,j )} = |y -a n,j | - √ t √ n + 1 . Therefore at n=p+ℓ Kn e -|y-z| 2 4s dµ h n (z) = at n=p+ℓ j∈Θ h t,n K n,j e -|y-z| 2 4s dµ n,j (z) ≤    at n=p+ℓ j∈Θ h t,n e -ǫq ′ λ 2 n,j,y 4s    1/q ′    at n=p+ℓ j∈Θ h t,n e -qλ 2 n,j,y 1-ǫ 4s µ q n,j (K n,j )    1/q
where ǫ > 0 will be made precise later on.

Step 1 We claim that

at n=p+ℓ j∈Θt,n e -ǫq ′ λ 2 n,j,y 4s ≤ C ps t (3.35)
where C depends on ǫ, q and N . If y is fixed in T p , we denote by z y the point of T n which solves |y -

z y | = dist (y, T n ). Thus √ t( √ n -p + 1) ≤ |y -z y | ≤ t( √ n - √ p).
Let Y = y t(p + 1)/ |y|. On the axis

-→ 0Y we set e = Y / |Y |, consider the points b k = (k √ t/ √ n)e
where -n ≤ k ≤ n and denote by G n,k the spherical shell obtained by intersecting the spherical shell T n with the domain H n,k which is the set of points in R N limited by the hyperplanes orthogonal to -→ 0Y going through ((k + 1)

√ t/ √ n)e and ((k -1) √ t/ √ n)e.
The number of points

a n,j ∈ G n,k is smaller than C(n + 1 -|k|) N -2
, where C depends only on N , and we denote by Λ n,k the set of j ∈ Θ t,n such that a n,j ∈ G n,k . Furthermore, if a n,j ∈ G n,k elementary geometric considerations (Pythagora's theorem) imply that λ 2 n,j,y is greater than t(n+p+1-2k

√ p + 1/ √ n). Therefore at n=p+ℓ j∈Θt,n e -ǫq ′ λ 2 n,j,y 4s ≤ C at n=p+ℓ n k=-n (n + 1 -|k|) N -2 e - ǫq ′ (n+p+1-2k √ p+1/)t 4s √ n . ( 3
≤ C at n=p+ℓ 1 + s √ n t √ p e -ǫq ′ ( √ n-√ p+1 ) 2 4s ≤ C ∞ p+1 e -ǫq ′ ( √ x-√ p+1 ) 2 t 4s dx + Cs t √ p ∞ p+1 √ xe -ǫq ′ ( √ x-√ p+1 ) 2 t 4s dx. (3.38) Next ∞ p+1 e -ǫq ′ ( √ x-√ p+1 ) 2 t 4s dx = 2 ∞ √ p+1 e -ǫq ′ (y-√ p+1 ) 2 t 4s ydy = 2 ∞ 0 e -ǫq ′ y 2 t 4s ydy + 2 √ p + 1 ∞ 0 e -ǫq ′ y 2 t 4s dy = 2s t ∞ 0 e -ǫq ′ z 2 4 zdz + 2 (p + 1)s t ∞ 0 e -ǫq ′ z 2 4 dz, (3.39)
and

∞ p+1 √ xe -ǫq ′ ( √ x-√ p+1 ) 2 t 4s dx = 2 ∞ √ p+1 e -ǫq ′ (y-√ p+1 ) 2 t 4s y 2 dy = 2 ∞ 0 e -ǫq ′
(n + 1 -|k|) d e ǫq ′ (k √ p+1)t 2s √ n ≤ 2 n k=0 (n + 1 -k) d e ǫq ′ (k √ p+1)t 2s √ n . (3.42)
We set

α = ǫq ′ t √ p + 1 2s
√ n and

I d = n k=0 (n + 1 -k) d e kα .
Since e kα = e (k+1)αe kα e α -1 ,

we use Abel's transform to obtain

I d = 1 e α -1 e (n+1)α -(n + 1) d + n k=1 (n + 2 -k) d -(n + 1 -k) d e kα ≤ 1 e α -1 (1 -d)e (n+1)α -(n + 1) d + de α n k=1 (n + 1 -k) d-1 e kα .
Therefore the following induction holds

I d ≤ de α e α -1 I d-1 . (3.43)
In (3.37), we have already used the fact that

de α e α -1 ≤ C 1 + s √ n t √ p ,
and

I d ≤ C 1 + s √ n t √ p d+1 I 0 .
Thus (3.38) is replaced by

at n=p+ℓ j∈Θt,n e - ǫq ′ λ 2 n,j,y 4s ≤ C at n=p+ℓ 1 + s √ n t √ p d+1 e -ǫq ′ ( √ n-√ p+1 ) 2 t 4s ≤ C ∞ p+1 e -ǫq ′ ( √ x-√ p+1 ) 2 t 4s dx + Cs t √ p d+1 ∞ p+1
x (d+1)/2 e -ǫq ′ ( √ x-√ p+1 ) 2 t 4s dx.

(3.44)

The first integral on the right-hand side has already been estimated in (3.39), for the second integral, there holds Step 2.

∞ p+1 x (d+1)/2 e -ǫq ′ ( √ x-√ p+1 ) 2 t 4s dx = ∞ 0 (y + √ p + 1 ) d+2 e -ǫq ′ y 2 t 4s dx ≤ C ∞ 0 y d+2 e -ǫq ′ y 2 t 4s dy + Cp 1+ d 2 ∞ 0 e -ǫq ′ y 2 t 4s dy ≤ C s t 2+ d 2 ∞ 0 z (d+1)/2 e -ǫq ′ z 2 4 dz + C s t 3/2 p 1+ d 2 ∞ 0 e -ǫq ′ z 2 4 dz.
Since T * p ⊂ Γ p × [0, t] where Γ p = B d p+1 (x) \ B d p-1 (x), (y, s) ∈ T * p implies that |x -y| 2 ≥ (p -1)t, thus J ′ 2,ℓ h satisfies J ′ 2,ℓ h ≤ Ct 1-q 2 ∞ p=1 p q-1 2 t 0 Γp (t -s) -N 2 s -(q(N -1)+1)/2 e -|x-y| 2 4(t-s) × at n=p+ℓ j∈Θ h t,n e - qλ 2 n,j,y (1-ǫ) 4s
µ q n,j (K n,j )dsdy

≤ Ct 1-q 2 at n=ℓ+1 j∈Θ h t,n µ q n,j (K n,j ) × n-ℓ p=1 p q-1 2 t 0 Γp
(ts) -N 2 s -(q(N -1)+1)/2 e -|x-y| 2 /4(t-s) e - and the constant C depends on N, q and ǫ. Next we set q ǫ = (1ǫ)q. Writting

|y -a n,j | 2 = |x -y| 2 + |x -a n,j | 2 -2 y -x, a n,j -x ≥ pt + |x -a n,j | 2 -2 y -x, a n,j -x , we get Γp e - qǫ | y-a n,j | 2 4s dy = e - qǫ | x-a n,j | 2 4s √ t(p+1)
√ tp e -qǫr 2 4s

|x-y|=r e 2qǫ y-x,a n,j -x /4s dS r (y)dr.

For estimating the value of the spherical integral, we can assume that a n,j -x = (0, . . . , 0, |a n,j -x|), y = (y 1 , . . . , y N ) and, using spherical coordinates with center at x, that the unit sphere has the representation (ts) -N 2 s -(q(N -1)+1)/2 e -|x-y| 2 4(t-s) e -qǫ λ 2 n,j,y 4s dy ds

S N -1 = {(sin φ.σ, cos φ) ∈ R N -1 × R : σ ∈ S N -2 , φ ∈ [0, π]}.
≤ C √ tp N-3 4 n N-1 4 t 0 (t -s) -N 2 s -(q-1)(N-1)+1 2 e -pt 4(t-s) e -qǫ ( √ tn- √ t(p+1) ) 2 4s ds ≤ C t 1-q(N-1) 2 p N-3 4 n N-1 4 1 0 (1 -s) -N 2 s -(q-1)(N-1)+1 2 e -p 4(1-s) e -qǫ ( √ n-√ p+1 ) 2 4s
.

(3.49)

We apply Lemma A.1, with

A = √ p, B = √ q ǫ ( √ n - √ p + 1), b = (q-1)(N -1)+1 2
, a = N 2 and κ = √ q ǫ (ℓ -1)/8 as in the case N = 1, and noticing that, for these specific values,

A 1-a B 1-b (A + B) a+b-2 = p 2-N 4 ( √ q ǫ ( √ n - √ p + 1)) 1-(q-1)(N-1) 2 × ( √ p + √ q ǫ ( √ n - √ p + 1)) (q-1)(N-1)+N-3 2 ≤ C n p N 4 -1/2 √ n - √ p √ n 1-(q-1)(N-1) 2
, where C depends on N , q and κ. Therefore

t 0 Γp (t -s) -N 2 s -N 2 e
-|x-y| 2 4(t-s) e -qǫ|y-z| 2 /4s dy ds

≤ C t (1-q(N -1))/2 p N-3 4 n N-1 4 n p N 4 -1/2 √ n - √ p √ n 1-(q-1)(N-1) 2 e -( √ p+ √ qǫ( √ n-√ p+1)) 2 4 ≤ Ct 1-q(N-1) 2 p -1 4 n (q-1)(N-1)-2 4 ( √ n - √ p) 1-(q-1)(N-1) 2 e -( √ p+ √ qǫ( √ n-√ p+1)) 2 4 .
(3.50)

We derive from (3.46), (3.50),

J ′ 2,ℓ h ≤ Ct 1-Nq 2 × at n=ℓ+1 j∈Θ h t,n n (q-1)(N-1)-2 4 µ q n,j (K n,j ) n-ℓ p=1 p 2q-3 4 ( √ n - √ p) 1-(q-1)(N-1) 2 e -( √ p+ √ qǫ( √ n-√ p+1 )) 2 4 .
(3.51) By Lemma A.2 with α = 2q -3 4 , β = 1-(q-1)(N -1)

2

, δ = 1 4 and γ = q ǫ , we obtain

n-ℓ p=1 p 2q-3 4 ( √ n - √ p) 1-(q-1)(N-1) 2 e -( √ p+ √ qǫ( √ n-√ p+1 )) 2 4 ≤ Cn N(q-1)+q-3 4 e -n 4 , (3.52) thus J ′ 2,ℓ h ≤ Ct 1-Nq 2 at n=ℓ+1 n N(q-1) 2 -1 e -n 4 j∈Θ h t,n µ q n,j (K n,j ). (3.53) Because µ n,j (K n,j ) = C B n,j
2/q,q ′ (K n,j ), we use the rescaling procedure as in the proof of Lemma 2.23, except that the scale factor is (n + 1)t instead of √ n + 1 so that the sets Tn , Kn , Qn and Kn remains unchanged Using again the quasi-additivity and the fact that

J ′ 2,ℓ = J h=1 J ′ 2,ℓ h , we deduce J 2,ℓ ≤ C ′ t -N 2 at n=ℓ+1 d N -2 q-1 n+1 e -n 4 C 2/q,q ′ K n d n+1 , (3.54)
which implies (3.32).

The proof of Theorem 3.1 follows from the previous estimates on J 1 and J 2 . Furthermore the following integral expression holds Theorem 3.8 Assume q ≥ q c . Then there exists a positive constants C * 2 , depending on N ,q and T , such that for any closed set F , there holds

u F (x, t) ≥ C * 2 t 1+ N 2 √ tat 0 e -s 2 4t s N -2 q-1 C 2/q,q ′ F s ∩ B 1 (x) s ds, (3.55)
where a t is the smallest integer j such that F ⊂ B √ jt (x).

Proof. We distinguish according q = q c , or q > q c , and for simplicity we denote B r = B r (x) for the various values of r.

Case

1: q = q c ⇐⇒ N -2 q-1 = 0. Because F n = F ∩ (B d n+1 \ B dn ) there holds C 2/q,q ′ F n d n+1 ≥ C 2/q,q ′ F d n+1 ∩ B 1 -C 2/q,q ′ F ∩ B dn d n+1 , Furthermore, since d n+1 ≥ d n , C 2/q,q ′ F ∩ B dn d n+1 = C 2/q,q ′ d n d n+1 F ∩ B dn d n ≤ C 2/q,q ′ F d n ∩ B 1 , thus C 2/q,q ′ F n d n+1 ≥ C 2/q,q ′ F d n+1 ∩ B 1 -C 2/q,q ′ F d n ∩ B 1 , 40 it follows at n=1 e -n 4 C 2/q,q ′ F n d n+1 ≥ at n=1 e -n 4 C 2/q,q ′ F d n+1 ∩ B 1 - at n=1 e -n 4 C 2/q,q ′ F d n ∩ B 1 ≥ at n=1 e -n 4 C 2/q,q ′ F d n+1 ∩ B 1 -e -1 4 a t -1 n=0 e -n 4 C 2/q,q ′ F d n+1 ∩ B 1 ≥ (1 -e -1 4 ) a t -1 n=1 e -n 4 C 2/q,q ′ F d n+1 ∩ B 1 -e -1 4 C 2/q,q ′ F √ t ∩ B 1 .
Since, by (2.92),

C 2/q,q ′ F s ′ ∩ B 1 ≥ C 2/q,q ′ F d n+1 ∩ B 1 ≥ C 2/q,q ′ F s ∩ B 1 , for any s ′ ∈ [d n+1 , d n+2 ] and s ∈ [d n , d n+1 ], there holds te -n 4 C 2/q,q ′ F d n+1 ∩ B 1 ≥ C 2/q,q ′ F d n+1 ∩ B 1 d n+1 dn e -s 2 /4t s ds ≥ d n+1 dn e -s 2 /4t C 2/q,q ′ F s ∩ B 1 s ds.
This implies

W F (x, t) ≥ (1 -e -1 4 )t -(1+ N 2 )
√ tat 0 e -s 2 /4t C 2/q,q ′ F s ∩ B 1 s ds.

Case 2: q > q c ⇐⇒ N -2 q-1 > 0. In that case it follows from Lemma 2.9 that

C 2/q,q ′ F n d n+1 ≈ d 2 q-1 -N n+1 C 2/q,q ′ (F n ) . Thus W F (x, t) ≈ t -1-N 2 at n=0 e -n 4 C 2/q,q ′ (F n ) . Since C 2/q,q ′ (F n ) ≥ C 2/q,q ′ F ∩ B d n+1 -C 2/q,q ′ (F ∩ B dn ) ,
and again

t -N 2 at n=0 e -n 4 C 2/q,q ′ (F n ) ≥ (1 -e -1 4 )t -N 2 a t -1 n=0 e -n 4 C 2/q,q ′ F ∩ B d n+1 ≥ (1 -e -1 4 )t -(1+ N 2 ) √ tat 0 e -s 2 4t C 2/q,q ′ (F ∩ B s ) s ds. Because C 2/q,q ′ (F ∩ B s ) ≈ s N -2 q-1 C 2/q,q ′ s -1 F ∩ B 1 , (3.55) follows.

Applications

The first result of this section is the following Theorem 4.1 Assume N ≥ 1 and q > 1. Then u K = u K .

Proof. If 1 < q < q c , the result is already proved in [START_REF] Véron | The initial trace of positive solutions of semilinear parabolic equations[END_REF]. The proof in the super-critical case is an adaptation that we recall, for the sake of completeness. By Theorem 2.24 and Theorem 3.8 there exists a positive constant C, depending on N , q and T such that

u F (x, t) ≤ Cu F (x, t) ∀(x, t) ∈ Q T . By convexity ũ = u F - 1 2C (u F -u F ) is a super-solution, which is smaller than u F if we assume that u F = u F . If we set θ := 1/2 + 1/(2C), then u θ = θu F is a subsolution. Therefore there exists a solution u 1 of (1.1) in Q ∞ such that u θ ≤ u 1 ≤ ũ < u F . If µ ∈ M q + (R N ) satisfies µ(F c ) = 0,
then u θµ is the smallest solution of (1.1) which is above the subsolution θu µ . Thus u θµ ≤ u 1 < u F and finally u F ≤ u 1 < u F , a contradiction.

If we combine Theorem 2.24 and Theorem 3.8 we derive the following integral approximation of the parabolic capacitary potential Proposition 4.2 Assume q ≥ q c . Then there exist two positive constants C † 1 , C † 2 , depending only on N , q and T such that

C † 2 t -(1+ N 2 ) √ tat 0 s N -2 q-1 e -s 2 4t C 2/q,q ′ F s ∩ B 1 (x) s ds ≤ W F (x, t) ≤ C † 1 t -(1+ N 2 ) √ t(at+2) √ t s N -2 q-1 e -s 2 4t C 2/q,q ′ F s ∩ B 1 (x) s ds (4.56) for any (x, t) ∈ Q T . Definition 4.3 If F is a closed subset of R N , we define the (2/q, q ′ )-integral parabolic capacitary potential W F by W F (x, t) = t -1-N 2 D F (x) 0 s N -2 q-1 e -s 2 /4t C 2/q,q ′ F s ∩ B 1 (x) s ds ∀(x, t) ∈ Q ∞ , (4.57)
where D F (x) = max{|x -y| : y ∈ F }.

An easy computation shows that 0

≤ W F (x, t) -t -(1+ N 2 ) √ tat 0 s N -2 q-1 e -s 2 4t C 2/q,q ′ F s ∩ B 1 (x) s ds ≤ C t (q-3)/2(q-1) D F (x) e -D 2 F (x)/4t , (4.58)
and

0 ≤ t -(1+ N 2 ) √ t(at+2) 0 s N -2 q-1 e -s 2 4t C 2/q,q ′ F s ∩ B 1 (x) s ds -W F (x, t) ≤ C t (q-3)/2(q-1) D F (x) e -D 2 F (x) 4t , (4.59)
for some C = C(N, q) > 0. Furthermore

W F (x, t) = t -1 q-1 D F (x)/ √ t 0 s N -2 q-1 e -s 2 4 C 2/q,q ′ F s √ t ∩ B 1 (x) s ds. ( 4

.60)

The following result gives a sufficient condition in order that u F does not have a strong blow-up at a point x.

Proposition 4.4 Assume q ≥ q c and F is a closed subset of R N . If there exists γ ∈ [0, ∞) such that lim τ →0 C 2/q,q ′ F τ ∩ B 1 (x) = γ, (4.61) then lim t→0 t 1 q-1 u F (x, t) = Cγ, (4.62)
for some C = C(N, q) > 0.

Proof. Clearly, condition (4.61) implies lim t→0

C 2/q,q ′ F √ ts ∩ B 1 (x) = γ
for any s > 0. Then (4.62) follows by Lebesgue's theorem. Notice also that the set of γ is bounded from above by a constant depending on N and q.

In the next result we give a condition in order that the solution remains bounded at a point x. The proof is similar to the previous one. Proposition 4.5 Assume q ≥ q c and F is a closed subset of R N . If

lim sup τ →0 τ -2 q-1 C 2/q,q ′ F τ ∩ B 1 (x) < ∞, (4.63)
then u F (x, t) remains bounded when t → 0.

Remark. If we assume that f is a convex function on R + satisfying c 2 r q ≤ f (r) ≤ c 1 r q ∀r ≥ 0 (4.64) for some 0 < c 2 ≤ c 1 we can construct in the same way as for (1.1) the solutions u F and u F for equation Because a > 1, (A.12) turns into

∂ t u -∆u + f (u) = 0 in Q T . ( 4
Ψ(v) ≤ CA 2-2b (v + (A + B) 2 ) b-1 (v + A 2 + AB) a-1 (v + B 2 + AB) 1-b v(v + κ) ≤ Ce -(A+B) 2 /4
y(p 0 ) √ n = 1 - 1 √ n √ p 0 + 1 - √ p 0 √ γ , 1 - y(n -ℓ + 1) √ n = 1 - √ n -ℓ + 2 √ n + √ n -ℓ + 1 √ nγ = 1 √ γ 1 + √ γ (ℓ -2) -ℓ + 1 2n + √ γ (ℓ -2) 2 -(ℓ -1) 2 8n 2
+ O(n -3 ). Remark. Although it does not has any importance for our use, it must be noticed that I N can be expressed either with hyperbolic functions if N is odd, or with Bessel functions if N is even.

  The authors are grateful to the European RTN Contract N • HPRN-CT-2002-00274 for the support provided in the realization of this work. The authors are grateful to Luc Tartar for providing them the proof of the sharp Poincaré inequality Proposition 2.5 and related references.

2

  Estimates from above Some notations. Let Ω be a domain in R N with a compact C 2 boundary and T > 0. Set B r (a) the open ball of radius r > 0 and center a (and B r (0) := B r ) and

2 :

 2 -a(r-b) 2 + +ãr r ≥ 0}. A direct computation gives R = e aãb 2 a-ã , and (2.58) implies 0 ≤ η(x) ≤ M e aãb 2 a-ã e -ã|x| 2 .

Definition 2 . 21

 221 We define on R N × R the two parabolic distances δ 2 and δ ∞ by δ 2 [(x, t), (y, s)] := |x -y| 2 + |t -s|, (2.74) and δ ∞ [(x, t), (y, s)] := max{|x -y|, |t -s|}.

t→0u

  (x, t) = ∞ uniformly on B cm ′ , for any m ′ > m: this can be easily proved by noticing that

  n =⇒ |x -y| ≤ d n+1 ,(3.12) follows because of (3.10) and(3.11).

  [START_REF] Tartar | Sur un lemme d'équivalence utilisé en analyse Numérique[END_REF],(3.44) and (3.45), we derive(3.35).

2 π 0 e

 20 With this representation, dS r = r N -1 sin N -2 φ dφ dσ and yx, a n,jx = |a n,j -x| |y -x| cos φ. Therefore |x-y|=r e 2qǫ y-x,a n,j -x 4s dS r (y) = r N -1 S N -2qǫ | a n,j -x | r cos φ 4s sin N -2 φ dφ. By Lemma A.3 |x-y|=r e 2qǫ y-x,a n,j -x 4s dS r (y) ≤ C r N -1 e 2qǫ r | a n,j -x | |a n,j -x| ≥ √ tn,

  .65) The bilateral estimate estimate (1.19) is still valid (up to change of the C i ). Since only convexity of f is used in the proof of Theorem 4.1, there still holds u F = u F . Similar extensions of Proposition 4.4 and Proposition 4.5 are also clear.

1 0Ψ

 1 A 2-2b (A + B) 2b-2 × v a-b + (A 2 + AB) a-1 v 1-b + (B 2 + AB) 1-b v a-1 + A a-1 B 1-b (A + B) a-b v(v + κ) .(A.20) Because AB ≥ κ, there exists a positive constant C, depending on κ, such that∞ 0 v a-b + (A 2 + AB) a-1 v 1-b + (B 2 + AB) 1-b v a-1 v(v + κ) e -v dv ≤ CA a-1 B 1-b (A + B) a-b ∞ 0 e -v dv v(v + κ) . (A.21)Combining (A.20) and (A.21) yields tox (x)dx ≤ Ce -(A+B) 2 /4 A 1-a B 1-b (A + B) a+b-2 . (A.22)This, again, implies that (A.1) holds.

Case 3 : 2 . 1 =( 1 -

 3211 max{a, b} < 1. Inequalities (A.7)-(A.11) hold, but (A.12) has to be replaced by Φ(v) ≤ CB2-2b v + (A + B)2 v + A(A + B)a-1 v + B 2 + AB b-1 v(v + κ) ≤ CB 1-b (A + B) 2a+b-3 v 1-a + A 2 + AB a e -v dv v(v + κ) ≤ C A 2 + AB 1-a ∞ 0 e -v dv v(v + κ) ,it follows that (A.13) holds. Finally (A.14) holds by exchanging (A, a) and (B, b).Lemma A.2 . Let α, β, γ, δ be real numbers and ℓ an integer. We assume γ > 1, δ > 0 and ℓ ≥ Then there exists a positive constant C such that, for any integer n > ℓ p) β e -δ( √ p+ √ γ( √ n-√ p+1)) 2 ≤ Cn α-β/2 e -δn . (A.24) Proof. The function x → ( √ x+ √ γ( √ n-√ x + 1)) 2 is decreasing on [(γ -1) -1 , ∞). Furthermore there exists C > 0 depending on ℓ, α and β such thatp α ( √ n -√ p ) β ≤ Cx α ( √ n -√ x + 1 ) β for x ∈ [p, p + 1]If we denote by p 0 the smallest integer larger than (γ -1) -1 , we deriveS = n-ℓ p=1 p α ( √ n -√ p ) β e -( √ p+ √ γ( √ p ) β e -δ( √ p+ √ γ( √ p ) β e -δ( √ p+ √ γ( x ≤ nℓ). Clearly p 0 -1 p=1 p α ( √ n -√ p ) β e -δ( √ p+ √ γ( √ n-√ p+1)) 2 ≤ C 0 n α ( √ n -√ nℓ ) β e -δn (A.25)for some C 0 independent of n. We set y = y(x) = √ x 0 ,and their exists ǫ = ǫ(δ, γ) > 0 such that √ 2 √ x ≥ y(x) ≥ ǫ √ x and y ′ (x) ≥ ǫ/ √ x. Furthermore √ x = √ γ y + γy 2 + 1γ γ -1 , √ n -√ x =√ n(γ -1) -√ γy -√ γ γy 2 + 1γ γn(γ -1) + γ -2y √ γnγy 2 √ n(γ -1) -z) 2α+1 (z + 1 + 1/n -1 -1/ √ γ) β e -γδnz 2 dz.

(A. 27 )( 1 -( 1 -( 1 -( 1 -

 271111 Let θ fixed such that 1 -y(nℓ + 1) √ n < θ < 1 -y(p 0 ) √ n for any n > p 0 . Then1-y(p 0 )/ √ n θ z) 2α+1 (z + 1 + 1/n -1 -1/ √ γ) β e -γδnz 2 dz ≤ C θ 1-y(p 0 )/ √ n θ z) 2α+1 e -γδnz 2 dz ≤ C θ e -γδnθ 2 1-y(p 0 )/ √ n θ z) 2α+1 dz ≤ C e -γδnθ 2 max{1, n -α-1/2 }.Because γθ 2 > 1 we derive1-y(p 0 )/ √ n θ (1z) 2α+1 (z + 1 + 1/n -1 -1/ √ γ) β e -γδnz 2 dz ≤ Cn -β e -δn , (A.28)for some constant C > 0. On the other handθ 1-y(n+1-ℓ)/ √ n z) 2α+1 (z + 1 + 1/n -1 -1/ √ γ) β e -γδnz 2 dz 1 + 1/n -1 -1/ √ γ) β e -γδnz 2 dz. The minimum of z → (z + 1 + 1/n -1 -1/ √ γ) β is achieved at 1y(n + 1ℓ) with value √ γ(ℓ + 1) + 1ℓ 2n √ γ + O(n -2 ),and the maximum of the exponential term is achieved at the same point with value e -nδ+((ℓ-2) √ γ+1-ℓ)/2 (1 + •(1)) = C γ e -nδ (1 + •(1)).

0 e 0 e 0 e m cos θ cos θ dθ and I ′′ 2 (m) = π 0 e 0 e m cos θ sin 2 θ dθ = I 2 (m) - 1 m π 0 e 3 I 3 -3 m π 0 ee 0 + N -3 m 2 π 0 e 5 (

 000200203300205 We denote z γ,n = 1 + 1/ √ γ -1 + 1/n and I β = θ 1-y(n+1-ℓ)/ √ n (zz γ,n ) β e -γδnz 2 dz.Since 1y(n + 1ℓ) ≥ 1/ √ 2γ for n large enough, z γ,n) β ze -γδnz 2 dz z γ,n ) β e -γδnz 2 θ z γ,n ) β-1 ze -γδnz 2 dz But 1y(n + 1ℓ)/ √ nz γ,n = (ℓ -1)(1 -1/ √ γ)/2n, therefore I β ≤ C 1 n -β-1 e -δn + βC ′ 1 n -1 I β-1 . (A.29)If β ≤ 0 , we deriveI β ≤ C 1 n -β-1 e -δn ,which inequality, combined with (A.26) and (A.28), yields to (A.24). If β > 0, we iterate and getI β ≤ C 1 n -β-1 e -δn + C ′ 1 n -1 (C 1 n -β e -δn + (β -1)C ′ 1 n -1 I β-2 ) If β -1 ≤ 0 we derive I β ≤ C 1 n -β-1 e -δn + C 1 C ′ 1 n -1-β e -δn = C 2 n -β-1 e -δn ,which again yields to (A.24). If β -1 > 0, we continue up we find a positive integer k such that βk ≤ 0, which again yields toI β ≤ C k n -β-1 e -δnand to (A.24).The next estimate is fundamental in deriving the N -dimensional estimate.Lemma A.3 For any integer N ≥ 2 there exists a constant c N > 0 such thatπ m cos θ sin N -2 θ dθ ≤ c N e m (1 + m) (N -1)/2 ∀m > 0. (A.30) Proof. Put I N (m) = π m cos θ sin N -2 θ dθ. Then I ′ 2 (m) = π m cos θ cos 2 θ dθ = I 2 (m) -π m cos θ cos θ dθ = I 2 (m) -1 m I ′ 2 (m).50 Thus I 2 satisfies a Bessel equation of order 0. Since I 2 (0) = π and I ′ 2 (0) = 0, π -1 I 2 is the modified Bessel function of index 0 (usually denoted by I 0 ) the asymptotic behaviour of which is well known, thus (A.30) holds. If N = (m) = π 0 e m cos θ sin θ dθ = -e m cos θ m cos θ ) sin N -3 θ dθ = N m cos θ cos θ sin N -4 θ dθ. m cos θ cos θ dθ = I ′ 2 (m), and, again (A.30) holds since I ′ 0 (m) has the same behaviour as I 0 (m) at infinity. For N ≥ 5 I N (m) = 3 -N m 2 e m cos θ cos θ sin N -5 θ π m cos θ d dθ cos θ sin N -5 θ dθ. Differentiating cos θ sin N -5 θ and using (A.31), we obtain I -4 (m) -I N -2 (m)) , (A.32) for N ≥ 6. Since the estimate (A.30) for I 2 , I 3 , I 4 and I 5 has already been obtained, a straigthforward induction yields to the general result.

  Definition 2.6 Assume s ∈ (0, 1) and sp < 1 or s = 1 and p = 2. If Ω is any domain in R N ,

	the Besov space B s,p 0 (Ω) is the closure of C ∞ 0 (Ω) with respect to the norm
		η B s,p = η Ḃs,p + η L p .		(2.21)
	The following result is derived from Proposition 2.5.
	Corollary 2.7 Let b > a > 0 and Ω be an open domain of R N such that Ω ⊂ B b \ B a . Then there exists a constant C = C(s, p, N ) > 0 such that for any η ∈ C ∞ 0 (Ω)
	η Ḃs,p ≤ η B s,p ≤ C(b -a) sp η Ḃs,p .	(2.22)
	2.1.3 Heat potential and Besov space
	If η ∈ C ∞ 0 (Ω), we extend it by 0 outside Ω and set
	η Bs,p =	Q∞	t 1-s/2 ∂ t H[η]	p	dx	dt t	1/p	(2.23)

  t)dx -

	t 0 ∂Bγ	∂u ∂n	ψ * γ dSdτ = -	t 0 ∂Bγ	∂ψ * γ ∂n	udσdτ.	(2.54)
	Since ψ * γ is bounded from above by 1, estimate (2.49) follows from (2.53) and Proposition 2.13
	(notice that B

  .36) Case N = 2. Summing a geometric series and using the inequality e u e u -1 ≤ 1 + u -1 for u > 0, we obtain

			n k=-n e	ǫq ′ (k 2s √ p+1)t √ n	≤ e ≤ e	ǫq ′ t ǫq ′ t	√ 2s n(p+1) √ n(p+1) 2s	e	ǫq ′ t 2s √ p+1 √ n ǫq ′ t e √ p+1 2s √ n -1 1 + 2s ǫq ′ t √ p + 1 √ n	.	(3.37)
	Thus, by comparison between series and integrals,
	at	e -	ǫq ′ λ 2 n,j,y 4s
	n=p+ℓ j∈Θt,n

( √ n -√ x ) β ≈ √ n + 1 -√ n/ √ γy β .

Estimate from below

If µ ∈ M q + (R N ) ∩ M b (R N ), we denote by u µ = u µ,0 the solution of

The maximal σ-moderate solution of (1.1) which has an initial trace vanishing outside a closed set F is defined by

The main result of this section is the next one Theorem 3.1 Assume q ≥ q c . There exists a constant C 2 = C 2 (N, q, T ) > 0 such that, for any closed subset F ⊂ R N , there holds

We first assume that F is compact, and we denote it by K. The first observation is that if

where G is the parabolic Green potential in Q T defined by

The main idea of the proof is as follows. For any (x, t) ∈ Q T , construct a measure µ = µ(x, t) ∈ M q + (R N ) such that there holds

and

with constants C depends only on N , q, and T . Then replace µ by µ ǫ = ǫµ with ǫ = (2C)

and the proof of Theorem 3.1 with C 2 = 2 -1 C. In the following sections we describe the construction of measures µ(x, t) satisfying (3.5) and (3.6).

A Appendix

The next estimate is crucial in our study of semilinear parabolic equations.

Lemma A.1 Let a and b be two real numbers, a > 0 and κ > 0. Then there exists a constant C = C(a, b, κ) > 0 such that for any A > 0, B > κ/A there holds

Proof. We first notice that

and it is achieved for

If 0 < x < x 0 this equation admits the solution

Using the explicit value of the root x(u), we finally get

and the factorization below holds

We set u = υ + (A + B) 2 /4 and obtain

) .

We introduce the relation ≈ linking two positive quantities depending on A and B. It means that the two sided-inequalities up to multiplicative constants independent of A and B. Therefore

where C depends on a, b and κ.

The estimate of J a,b is obtained by exchanging (A, a) with (B, b) and replacing x by 1x. Mutadis mutandis, this yields directely to the same expression as in A.13 and finally

Case 2: a ≥ 1, b < 1. Estimates (A.7), (A.8), (A.9), (A.10) and (A.11) are valid. Because v → (v + B(A + B)) b-1 is decreasing, (A.12) has to be replaced by

This implies (A.13) directly. The estimate of J a,b is performed by the change of variable x → 1x. If x 1 = 1x 0 , there holds

Then