Quoc-Hung Nguyen
email: quoc-hung.nguyen@epfl.ch

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

Wiener criteria for existence of large solutions of nonlinear parabolic equations with absorption in a non-cylindrical domain

Keywords: Bessel capacities, Hausdorff capacities, parabolic boundary, Riesz potential, maximal solutions. 2010 Mathematics Subject Classification. 35K58, 28A12, 46E35

We obtain a necessary condition and a sufficient condition, both expressed in terms of Wiener type tests involving the parabolic W 2,1 q ′ -capacity, where q ′ = q q-1 and q > 1, for the existence of large solutions to equation ∂tu -∆u + u q = 0 in a non-cylindrical domain. We provide also a sufficient condition for the existence of such solutions to equation ∂tu -∆u + e u -1 = 0. Besides, we apply our results to equation: ∂tu -∆u + a|∇u| p + bu q = 0 for a, b > 0, 1 < p < 2 and q > 1.

Introduction

The aim of this paper is to study the problem of existence of large solutions to some nonlinear parabolic equations with superlinear absorption in an arbitrary bounded open set O ⊂ R N +1 , N ≥ 2. These are functions u ∈ C 2,1 (O), solutions of

∂ t u -∆u + |u| q-1 u = 0 in O, lim δ→0 inf O∩Q δ (x,t) u = ∞ for all (x, t) ∈ ∂ p O, (1.1)
with q > 1 and

∂ t u -∆u + sign(u)(e |u| -1) = 0 in O, lim δ→0 inf O∩Q δ (x,t) u = ∞ for all (x, t) ∈ ∂ p O, (1.2)
in which expressions ∂ p O denotes the parabolic boundary of O, i.e. the set all points X = (x, t) ∈ ∂O such that the intersection of the cylinder Q δ (x, t) := B δ (x) × (tδ 2 , t) with O c is not empty for any δ > 0. By the maximal principle for parabolic equations we can assume that all solutions of (1.1) and (1.2) are positive. Henceforth we consider only positive solutions of the preceding equations.

In [START_REF] Véron | A note on maximal solutions of nonlinear parabolic equations with absorption[END_REF], we studied the existence and the uniqueness of solution of semilinear heat equations in a cylindrical domain,

∂ t u -∆u + f (u) = 0 in Ω × (0, ∞), u = ∞ in ∂ p (Ω × (0, ∞)) , (1.3)
where Ω is a bounded open set in R N and f a continuous nondecreasing real-valued function such that f (0) ≥ 0 and f (a) > 0 for some a > 0. In order to obtain the existence of a maximal solution of ∂ t u -∆u + f (u) = 0 in Ω × (0, ∞) there is need to introduce the following assumptions (i) ˆ∞ a ˆs 0 f (τ )dτ

-1 2 ds < ∞, (ii)
ˆ∞ a (f (s)) -1 ds < ∞.

(1.4) Condition (i), due to Keller and Osserman, is a necessary and sufficient for the existence of a maximal solution to

-∆u + f (u) = 0 in Ω. (1.5)
Condition (ii) is a necessary and sufficient for the existence of a maximal solution of the differential equation

ϕ ′ + f (ϕ) = 0 in (0, ∞), (1.6)
and this solution tends to ∞ at 0. In [START_REF] Véron | A note on maximal solutions of nonlinear parabolic equations with absorption[END_REF], it is shown that if for any m ∈ R there exists L = L(m) > 0 such that for any x, y ≥ m ⇒ f (x + y) ≥ f (x) + f (y) -L, and if (1.5) has a large solution, then (1.3) admits a solution.

It is not alway true that the maximal solution to (1.5) is a large solution. However, if f satisfies ˆ∞ 1 s -2(N -1)/(N -2) f (s)ds < ∞ if N ≥ 3, or inf a ≥ 0 : ˆ∞ 0 f (s)e -as ds < ∞ < ∞ if N = 2, then (1.5) has a large solution for any bounded domain Ω, see [START_REF] Marcus | Nonlinear second order elliptic equations involving measures[END_REF]. When f (u) = u q , q > 1 and N ≥ 3, the first above condition is satisfied if and only if q < q c := N N -2 , this is called the sub-critical case. When q ≥ q c , a necessary and sufficient condition for the existence of a large solution to

-∆u + u q = 0 in Ω (1.7)
is expressed in term of a Wiener-type test,

ˆ1 0 Cap 2,q ′ (Ω c ∩ B r (x)) r N -2
dr r = ∞ for all x ∈ ∂Ω.

(1.8)

In the case q = 2 it is obtained by Dhersin and Le Gall [START_REF] Dhersin | Wieners test for super-Brownian motion and the Brownian snake[END_REF], see also [START_REF] Gall | Branching processes, random trees and superprocesses[END_REF][START_REF] Gall | Spatial branching processes, random snakes and partial differential equations[END_REF], using probabilistic methods involving the Brownian snake; this method can be extended for 1 < q ≤ 2 by using ideas from [START_REF] Dynkin | Fine topology and fine trace on the boundary associated with a class of semilinear differential equations[END_REF][START_REF] Dynkin | Superdiffusions and removable singularities for quasilinear partial differential equations[END_REF]. In the general case the result is proved by Labutin, by purely analytic methods [START_REF] Labutin | Wiener regularity for large solutions of nonlinear equations[END_REF]. Note that q ′ = q q-1 and Cap 2,q ′ is the capacity associated to the Sobolev space W 2,q ′ (R N ).

In [START_REF] Nguyen | Wiener criteria for existence of large solutions of quasilinear elliptic equations with absorption[END_REF] we obtain sufficient conditions for the existence of a large solution to -∆u + e u -1 = 0 in Ω, (1.9) expressed in terms of the Hausdorff H N -2

1

-capacity in R N , and more precisely

ˆ1 0 H N -2 1 (Ω c ∩ B r (x)) r N -2
dr r = ∞ for all x ∈ ∂Ω.

(1.10)

We refer to [START_REF] Marcus | Capacitary estimates of solutions of semilinear parabolic equations[END_REF] for investigation of the initial trace theory of (1.3).

In [START_REF] Evans | Wiener criterion for the heat equation[END_REF], Evans and Gariepy establish a Wiener criterion for the regularity of a boundary point (in the sense of potential theory) for the heat operator L = ∂ t -∆ in an arbitrary bounded set of R N +1 . We denote by M(R N +1 ) the set of Radon measures in R N +1 and, for any compact set K ⊂ R N +1 , by M K (R N +1 ) the subset of M(R N +1 ) of measures with support in K. Their positive cones are respectively denoted by M + (R N +1 ) and M + K (R N +1 ). The capacity used in this criterion is the thermal capacity defined by

Cap H (K) = sup{µ(K) : µ ∈ M + K (R N +1 ), H * µ ≤ 1}, for any K ⊂ R N +1 compact, where H is the heat kernel in R N +1 . It coincides with the parabolic Bessel G 1 -capacity Cap G1,2 , Cap G1,2 (K) = sup ˆRN+1 |f | 2 dxdt : f ∈ L 2 + (R N +1 ), G 1 * f ≥ χ K ,
here G 1 is the parabolic Bessel kernel of first order, see [START_REF] Nguyen | Potential estimates and quasilinear equations with measure data[END_REF]Remark 4.12]. Garofalo and Lanconelli [START_REF] Garofalo | Wiener's criterion for parabolic equations with variable coefficients and its consequences[END_REF] extend this result to the parabolic operator

L = ∂ t -div(A(x, t)∇), where A(x, t) = (a i,j (x, t)), i, j = 1, 2, ..., N is a real, symmetric, matrix-valued function on R N +1 with C ∞ entries satisfying C -1 |ξ| 2 ≤ N i,j=1 a i,j (x, t)ξ i ξ j ≤ C|ξ| 2 ∀(x, t) ∈ R N +1 , ∀ξ ∈ R N ,
for some constant C > 0.

Much less is known concerning the equation

∂ t u -∆u + f (u) = 0 (1.11) in a bounded open set O ⊂ of R N +1
, where f is a continuous function in R. Gariepy and Ziemer [START_REF] Gariepy | Thermal capacity and boundary regularity[END_REF][START_REF] Ziemer | Behavior at the boundary of solution of quasilinear parabolic equations[END_REF] prove that if there exist (x 0 , t 0 ) ∈ ∂ p O, l ∈ R and a weak solution

u ∈ W 1,2 (O) ∩ L ∞ (O) of (1.11) such that η(-l -ε + u) + , η(l -ε -u) + ∈ W 1,2 0 (O) for any ε > 0 and η ∈ C ∞ c (B r (x 0 ) × (-r 2 + t 0 , r 2 + t 0 )
) for some r > 0, and if there holds The main purpose of this article is to extend Labutin's result [START_REF] Labutin | Wiener regularity for large solutions of nonlinear equations[END_REF] to the semilinear parabolic equation (1.1). Namely, we give a necessary and a sufficient condition for the existence of solutions to problem (1.1) in a bounded non-cylindrical domain O ⊂ R N +1 , expressed in terms of a Wiener test based upon the parabolic W 2,1 q ′ -capacity in R N +1 . We also give a sufficient condition for solving problem (1.2) expressed in terms of a Wiener test based upon the parabolic Hausdorff PH N ρ -capacity. These capacities are defined as follows:

ˆ1 0 Cap H O c ∩ B ρ (x 0 ) × (t 0 -9 4 αρ 2 , t 0 -5 4 αρ 2 ) ρ N dρ ρ = ∞ for some α > 0, then lim (x,t)→(x0,t0) u(x, t) = l.
if K ⊂ R N +1 is a compact set, we set Cap 2,1,q ′ (K) = inf{||ϕ|| q ′ W 2,1 q ′ (R N +1 ) : ϕ ∈ S(R N +1 ), ϕ ≥ 1 in a neighborhood of K}, where ||ϕ|| W 2,1 q ′ (R N +1 ) = ||ϕ|| L q ′ (R N +1 ) + || ∂ϕ ∂t || L q ′ (R N +1 ) + ||∇ϕ|| L q ′ (R N +1 ) + i,j || ∂ 2 ϕ ∂x i ∂x j || L q ′ (R N +1 ) ,
and for a Suslin set

E ⊂ R N +1 , Cap 2,1,q ′ (E) = sup{Cap 2,1,q ′ (D) : D ⊂ E, D compact}.
This capacity has been used in order to obtain estimates expressed with the help of potential that are most helpful for studying quasilinear parabolic equations (see e.g. [START_REF] Baras | Problèmes paraboliques semi-linéaires avec données mesures[END_REF][START_REF] Baras | Critère d'existence des solutions positives pour des équations semilinéaires non monotones[END_REF][START_REF] Nguyen | Potential estimates and quasilinear equations with measure data[END_REF]). Thanks to a result due to Richard and Bagby [START_REF] Bagby | Lebesgue spaces of parabolic potentials[END_REF], the capacities Cap 2,1,p and Cap G2,p are equivalent in the sense that, for any Suslin set K ⊂ R N +1 , there holds

C -1 Cap 2,1,q ′ (K) ≤ Cap G2,q ′ (K) ≤ CCap 2,1,q ′ (K),
for some C = C(N, q), where Cap G2,q ′ is the parabolic Bessel G 2 -capacity, see [START_REF] Nguyen | Potential estimates and quasilinear equations with measure data[END_REF].

For a set E ⊂ R N +1 , we define PH N ρ (E) by

PH N ρ (E) = inf    j r N j : E ⊂ B rj (x j ) × (t j -r 2 j , t j + r 2 j ), r j ≤ ρ    .
It is easy to see that, for 0 < σ ≤ ρ and E ⊂ R N +1 , there holds

PH N ρ (E) ≤ PH N σ (E) ≤ C(N ) ρ σ 2 PH N ρ (E). (1.12)
With these notations, we can state the two main results of this paper.

Theorem 1.1 Let N ≥ 2 and q ≥ q * := N +2 N . Then (i) The equation

∂ t u -∆u + u q = 0 in O (1.13)
admits a large solution if there holds

ˆ1 0 Cap 2,1,q ′ (O c ∩ (B ρ 30 (x) × (t -30ρ 2 , t -ρ 2 ))) ρ N dρ ρ = ∞, (1.14)
for any (x, t) ∈ ∂ p O and q > q * or q = q * when N ≥ 3.

(ii) If equation (1.13) admits a large solution, then

ˆ1 0 Cap 2,1,q ′ (O c ∩ Q ρ (x, t)) ρ N dρ ρ = ∞, (1.15
)

for any (x, t) ∈ ∂ p O, where Q ρ (x, t) = B ρ (x) × (t -ρ 2 , t).
It is an open problem to prove that the maximal solution is unique whenever it exists as it holds in the elliptic case for equation (1.7), see Remark p. 25.

Theorem 1.2 Let N ≥ 2. The equation ∂ t u -∆u + e u -1 = 0 in O (1.16)
admits a large solution if there holds

ˆ1 0 PH N 1 (O c ∩ (B ρ 30 (x) × (t -30ρ 2 , t -ρ 2 ))) ρ N dρ ρ = ∞, (1.17)
for any (x, t) ∈ ∂ p O.

From properties of the W 2,1 q ′ -capacity, relation (1.14) is satisfied if the following relations hold in which | | denotes the Lebesgue measure in R N +1 ,

ˆ1 0 |O c ∩ (B ρ 30 (x) × (t -30ρ 2 , t -ρ 2 )| 1-2q ′ N +2 ρ N dρ ρ = ∞ when q > q * , and
ˆ1 0 log + O c ∩ (B ρ 30 (x) × (t -30ρ 2 , t -ρ 2 ) -1 -N 2 ρ N dρ ρ = ∞ when q = q * .
Similarly, it follows from properties of the PH N 1 -capacity that identity (1.17) is verified if As a consequence of Theorem 1.1 we derive a sufficient condition for the existence of a large solution of a class of viscous parabolic Hamilton-Jacobi equations.

ˆ1 0 |O c ∩ (B ρ 30 (x) × (t -30ρ 2 , t -ρ 2 )| N N +2 ρ N dρ ρ = ∞. When O = {(x, t) ∈ R N +1 : |x| 2 + |t| 2 λ < 1}
Theorem 1.3 Let q 1 > 1. If there exists a large solution v ∈ C 2,1 (O) of ∂ t v -∆v + v q1 = 0 in O,
then, for any a, b > 0, 1 < q < q 1 and 1 < p < 2q1 q1+1 , the problem

∂ t u -∆u + a|∇u| p + bu q = 0 in O, u = ∞ on ∂ p O, (1.18
)

admits a solution u ∈ C 2,1 (O) which satisfies u(x, t) ≥ C min a -1 p-1 R -2-p p-1 + 2 α(q 1 -1) , b -1 q-1 R -2 q-1 + 2 α(q 1 -1) (v(x, t)) 1 α ,
for all (x, t) ∈ O where R > 0 is such that O ⊂ QR (x 0 , t 0 ), C = C(N, p, q, q 1 ) > 0 and α = max

2(p-1) (q1-1)(2-p) , q-1 q1-1 ∈ (0, 1).

Preliminaries

Throughout the paper, we denote

Q ρ (x, t) = B ρ (x) × (t -ρ 2 , t],
and Qρ (x, t) = B ρ (x) × (tρ 2 , t + ρ 2 ), for (x, t) ∈ R N +1 and ρ > 0, and r k = 4 -k for all k ∈ Z. We also denote A ( )B if A ≤ (≥)CB for some C depending on some structural constants, A ≍ B if A B A.

Definition 2.1 Let R ∈ (0, ∞] and µ ∈ M + (R N +1 ). We define R-truncated Riesz parabolic potential I R 2 of µ by I R 2 [µ](x, t) = ˆR 0 µ( Qρ (x, t)) ρ N dρ ρ for all (x, t) ∈ R N +1 ,
and the R-truncated fractional maximal parabolic potential M R 2 of µ by

M R 2 [µ](x, t) = sup 0<ρ<R µ( Qρ (x, t)) ρ N for all (x, t) ∈ R N +1 .
We also set

I ∞ 2 = I 2 and M ∞ 2 = M 2 .
We recall two results in [START_REF] Nguyen | Potential estimates and quasilinear equations with measure data[END_REF].

Theorem 2.2 Let q > 1, R > 0 and K be a compact set in R N +1 . There exists µ :

= µ K ∈ M + (R N +1 ) with compact support in K such that µ(K) ≍ Cap 2,1,q ′ (K) ≍ ˆRN+1 I 2R 2 [µ] q dxdt
where the constants of equivalence depend on N, q and R. The measure µ K is called the capacitary measure of K.

Theorem 2.3 For any R > 0, there exist positive constants C 1 , C 2 such that for any

µ ∈ M + (R N +1 ) such that ||M R 2 [µ]|| L ∞ (R N +1 ) ≤ 1, there holds 1 |Q| ˆQ exp(C 1 I R 2 [χ Q µ])dxdt := Q exp(C 1 I R 2 [χ Q µ])dxdt ≤ C 2 ,
for all Q = Qr (y, s) ⊂ R N +1 , r > 0 , where χ Q is the indicator function of Q.

Frostman's Lemma in [START_REF] Tureson | Nonlinear Potential Theory and weighted Sobolev Spaces[END_REF]Th. 3.4.27] is at the core of the dual definition of Hausdorff capacities with doubling weight. It is easy to see that it is valid for the parabolic Hausdorff PH N ρ -capacity version. As a consequence we have Theorem 2.4 There holds

sup µ(K) : µ ∈ M + (R N +1 ), supp(µ) ⊂ K, ||M ρ 2 [µ]|| L ∞ (R N +1 ) ≤ 1 ≍ PH N ρ (K),
for any compact set K ⊂ R N +1 and ρ > 0, where equivalent constant depends on N .

For our purpose, we need the some results about the behavior of the capacity with respect to dilations.

Proposition 2.5 Let K ⊂ Q100 (0, 0) be a compact set and

1 < p < N +2 2 . Then Cap 2,1,p (K) |K| 1-2p N +2 and Cap 2,1, N +2 2 (K) log | Q200 (0, 0)| |K| -N 2 .
(2.1)

Furthermore Cap 2,1,p (K ρ ) ≍ ρ N +2-2p Cap 2,1,p (K), (2.2) 1 Cap 2,1, N +2 2 (K ρ ) ≍ 1 Cap 2,1, N +2 2 (K) + (log(2/ρ)) N/2 , (2.3)
for any 0 < ρ < 1, where K ρ = {(ρx, ρ 2 t) : (x, t) ∈ K}.

Proposition 2.6 Let K ⊂ Q1 (0, 0) be a compact set and

1 < p ≤ N +2 2 . Then, there exists a function ϕ ∈ C ∞ c ( Q3/2 (0, 0)) with 0 ≤ ϕ ≤ 1 and ϕ| D = 1 for some open set D ⊃ K such that ˆRN+1 |D 2 ϕ| p + |∇ϕ| p + |ϕ| p + |∂ t ϕ| p dxdt Cap 2,1,p (K). (2.4)
We will give proofs of the above two propositions in the Appendix. Let {e t∆ } t≥0 be the semigroup of contractions in L p (1 ≤ p < ∞) generated by ∆. It is wellknown that the solution u of the problem

∂ t u -∆u = µ in QR (0, 0), u = 0 on ∂ p QR (0, 0), (2.5)
with µ ∈ C ∞ ( QR (0, 0)), can be expressed by Duhamel's formula u(x, t) = ˆt 0 e (t-s)∆ µ (x, s)ds for all (x, t) ∈ QR (0, 0).

We denote by H the Gaussian kernel in R N +1 :

H(x, t) = 1 (4πt) N 2 e -|x| 2 4t χ t>0 .
We have

|u(x, t)| ≤ (H * µ)(x, t) for all (x, t) ∈ QR (0, 0).
In [21, Proof of Proposition 4.8] we show that

|(H * µ)|(x, t) ≤ C 1 (N )I 2R 2 [|µ|](x, t) for all (x, t) ∈ QR (0, 0).
Here µ is extended by 0 in ( QR (0, 0)) c . Thus,

| ˆt 0 e (t-s)∆ µ (x, s)ds| ≤ C 1 (N )I 2R 2 [|µ|](x, t) for all (x, t) ∈ QR (0, 0). (2.6)
Moreover, we also prove in [START_REF] Nguyen | Potential estimates and quasilinear equations with measure data[END_REF], that if µ ≥ 0 then for (x, t) ∈ QR (0, 0) and

B ρ (x) ⊂ B R (0), ˆt 0 e (t-s)∆ µ (x, s)ds ≥ C 2 (N ) ∞ k=0 µ(Q ρ k 8 (x, t -35 128 ρ 2 k )) ρ N k , (2.7)
with

ρ k = 4 -k ρ.
It is easy to see that estimates (2.6) and (2.7) also holds for any bounded Radon measure µ in QR (0, 0). The following result is proved in [START_REF] Baras | Problèmes paraboliques semi-linéaires avec données mesures[END_REF] and [START_REF] Nguyen | Phuoc Parabolic equations with exponential nonlinearity and measure data[END_REF], and also in [START_REF] Nguyen | Potential estimates and quasilinear equations with measure data[END_REF] in a more general framework.

Theorem 2.7 Let q > 1, R > 0 and µ be a bounded Radon measure in QR (0, 0). (i) If µ is absolutely continuous with respect to Cap 2,1,q ′ in QR (0, 0), then there exists a unique weak solution u to equation

∂ t u -∆u + |u| q-1 u = µ in QR (0, 0), u = 0 on ∂ p QR (0, 0). (ii) If exp C 1 (N )I 2R 2 [|µ|] ∈ L 1 ( QR (0, 0))
, then there exists a unique weak solution v to equation

∂ t v -∆v + sign(v)(e |v| -1) = µ in QR (0, 0), v = 0 on ∂ p QR (0, 0),
where the constant C 1 (N ) is the one of inequality (2.6).

From estimates (2.6) and (2.7) and using comparison principle we get the estimates from below of the solutions u and v obtained in Theorem 2.7.

Proposition 2.8 If µ is nonnegative, then the functions u and v of the previous theorem are nonnegative too and satisfy

u(x, t) ≥ C 2 (N ) ∞ k=0 µ(Q ρ k 8 (x, t -35 128 ρ 2 k )) ρ N k -C 1 (N ) q+1 I 2R 2 I 2R 2 [µ] q (x, t), (2.8
)

and v(x, t) ≥ C 2 (N ) ∞ k=0 µ(Q ρ k 8 (x, t -35 128 ρ 2 k )) ρ N k -C 1 (N )I 2R 2 exp C 1 (N )I 2R 2 [µ] -1 (x, t), (2.9)
for any (x, t) ∈ QR (0, 0) and B ρ (x) ⊂ B R (0) and ρ k = 4 -k ρ.

Maximal solutions

In this section we assume that O is an arbitrary non-cylindrical and bounded open set in R N +1 and q > 1. We will prove the existence of a maximal solution of

∂ t u -∆u + u q = 0 (3.1)
in O. We also get an analogous result when u q is replaced by e u -1.

It is easy to see that if u satisfies (3.1) in Qr (0, 0

) ( Q r (0, 0) ) then u a (x, t) = a -2/(q-1) u(ax, a 2 t) satisfies (3.1) in Qr/a (0, 0) (Q r/a (0, 0)) for any a > 0. If X = (x, t) ∈ O, the parabolic dis- tance from X to the parabolic boundary ∂ p O of O is defined by d(X, ∂ p O) = inf (y,s)∈∂pO s≤t max{|x -y|, (t -s) 1 2 }.
It is easy to see that there exists C = C(N, q) > 0 such that the function V defined by

V (x, t) = C (ρ 2 + t) -1 q-1 + ρ 2 -|x| 2 ρ -2 q-1 in B ρ (0) × (-ρ 2 , 0), satisfies ∂ t V -∆V + V q ≥ 0 in B ρ (0) × (-ρ 2 , 0). (3.2) Proposition 3.1 There exists a maximal solution u ∈ C 2,1 (O) of (3.1) and it satisfies u(x, t) ≤ C(d((x, t), ∂ p O)) -2 q-1 for all (x, t) ∈ O, (3.3)
for some C = C(N, q).

Proof. Let D k , k ∈ Z be the collection of all the dyadic parabolic cubes (abridged p-cubes) of the form

{(x 1 , ..., x N , t) : m j 2 -k ≤ x j ≤ (m j + 1)2 -k , j = 1, ..., N, m N +1 4 -k ≤ t ≤ (m N +1 + 1)4 -k }
where m j ∈ Z. The following properties hold, a. for each integer k, D k is a partition of R N +1 and all p-cubes in D k have the same sidelengths.

b. if the interiors of two p-cubes Q in D k1 and P in D k2 , denoted

• Q, • P , have nonempty intersection then either Q is contained in P or Q contains P . c. Each Q in D k is union of 2 N +2 p-cubes in D k+1 with disjoint interiors. Let k 0 ∈ N be such that Q ⊂ O for some Q ∈ D k0 . Set O k = Q∈D k Q⊂O Q, ∀k ≥ k 0 , we have O k ⊂ O k+1 and O = k≥k0 O k = k≥k0 •
Ok. More precisely, there exist real numbers

a 1 , a 2 , ...., a n(k) and open sets Ω 1 , Ω 2 , .., Ω n(k) in R N such that a i < a i + 4 -k ≤ a i+1 < a i+1 + 4 k for i = 1, ..., n(k) -1,
and

• Ok = n(k)-1 i=1 Ω i × (a i , a i + 4 -k ] Ω n(k) × (a n(k) , a n(k) + 4 -k ) .
For k ≥ k 0 , we claim that there exists a solution

u k ∈ C 2,1 ( • Ok) to problem ∂ t u k -∆u k + u q k = 0 in • Ok, u k (x, t) → ∞ as d((x, t), ∂ p • Ok) → 0. (3.4)
Indeed, by [START_REF] Dibenedetto | Degenerate parabolic equations[END_REF][START_REF] Lieberman | Second Order Parabolic Differential Equations[END_REF] for m > 0, one can find nonnegative solutions

v i ∈ C 2,1 (Ω i × (a i , a i + 4 -k ]) ∩ C(Ω i × [a i , a i + 4 -k ]) for i = 1, .., n(k) to equations ∂ t v 1 -∆v 1 + v q 1 = 0 in Ω 1 × (a 1 , a 1 + 4 -k ), v 1 (x, t) = m on ∂Ω 1 × (a 1 , a 1 + 4 -k ), v 1 (x, a 1 ) = m in Ω 1 ,
and

∂ t v i -∆v i + v q i = 0 in Ω i × (a i , a i + 4 -k ), v i (x, t) = m on ∂Ω i × (a i , a i + 4 -k ), v i (x, a i ) = m i in Ω i ,
where

m i = m in Ω i if a i > a i-1 + 4 -k , mχ Ωi\Ωi-1 (x) + v i-1 (x, a i-1 + 4 -k )χ Ωi-1 (x) otherwise . Clearly, u k,m = v i in Ω i × (a i , a i + 4 -k ] for i = 1, ..., n(k) is a solution in C 2,1 ( • Ok) ∩ C(O k ) to equation ∂ t u k,m -∆u k,m + u q k,m = 0 in • Ok, u k,m = m on ∂ p • Ok. Moreover, for (x, t) ∈ • Ok, we see that B d 2 (x) × (t -d 2 4 , t) ⊂ • Ok where d = d((x, t), ∂ p • Ok). From (3.2), we verify that U (y, s) := V (y -x, s -t) = C (ρ 2 + s -t) -1 q-1 + ρ 2 -|x -y| 2 ρ -2 q-1 (x) × (t - d 2 4 , t). (3.5)
Applying the comparison principle we get

u k,m (y, s) ≤ U (y, s) in B d 2 (x) × (t - d 2 4 , t],
which implies

u k,m (x, t) ≤ C d((x, t), ∂ p • Ok) -2 q-1 for all (x, t) ∈ • Ok. (3.6)
From this, we obtain also uniform local bounds for {u k,m } m . By standard regularity theory see, [START_REF] Dibenedetto | Degenerate parabolic equations[END_REF][START_REF] Lieberman | Second Order Parabolic Differential Equations[END_REF], {u k,m } m is uniformly locally bounded in C 2,1 . Hence, up to a subsequence,

u k,m → u k C 1,0 loc ( • Ok) as m → ∞. We derive that u k is a weak solution of (3.4) in • Ok, which satisfies u k (x, t) → ∞ as d((x, t), ∂ p • Ok) → 0 and u k (x, t) ≤ C d((x, t), ∂ p • Ok) -2 q-1 for all (x, t) ∈ • Ok. Let m > 0 and k ≥ k 0 . Since u k+1,m ≤ m in O k and O k ⊂ O k+1
, it follows by the comparison principle applied to u k+1,m and u k,m in the following n(k) sub-domains of

• Ok: Ω 1 × (a 1 , a 1 + 4 -k ), Ω 2 × (a 2 , a 2 + 4 -k ),..., Ω n(k) × (a n(k) , a n(k) + 4 -k ), that u k+1,m ≤ u k,m in • Ok, and thus u k+1 ≤ u k in • Ok by letting m → ∞. In particular, {u k } k is uniformly locally bounded in L ∞ loc .
We use the same compactness property as above to infer that u k → u as k → ∞. Then u is a solution of (3.1) and it satisfies (3.3). By construction u is the maximal solution.

Remark 3.2 Let R ≥ 2r ≥ 2, K be a compact subset in Qr (0, 0). As in the proof of Proposition 3.1, we can show that there exists a maximal solution of

∂ t u -∆u + u q = 0 in QR (0, 0)\K, u = 0 on ∂ p QR (0, 0), (3.7
)

which satisfies u(x, t) ≤ C(d((x, t), ∂ p ( QR (0, 0)\K)) -2 q-1 ∀ (x, t) ∈ QR (0, 0)\K, (3.8)
for some C = C(N, q). Furthermore, assume K 1 , K 2 , , , , K m are compact subsets in Qr (0, 0) and

K = K 1 ∪ ... ∪ K m . Let u, u 1 , ..., u m be the maximal solutions of (3.7) in QR (0, 0)\K, QR (0, 0)\K 1 , QR (0, 0)\K 2 , , , , QR (0, 0)\K m , respectively, then u ≤ m j=1 u j in QR (0, 0)\K. (3.9)
Remark 3.3 If the equation (3.1) admits a large solution for some q > 1, then for any 1 < q 1 < q, the equation

∂ t u -∆u + u q1 = 0 in O (3.10)
admits also a large solution. Indeed, assume that u is a large solution of (3.1) and v the maximal solution of (3.10).

Take R > 0 such that O ⊂ B R (0) × (-R 2 , R 2 ), then the function V defined by V (x, t) = (q -1) -1 q-1 (2R 2 + t) -1 q-1 , satisfies (3.1). It follows for all (x, t) ∈ O u(x, t) ≥ inf (y,s)∈O V (x, t) ≥ (q -1) -1 q-1 R -2 q-1 =: a 0 .
Then ũ = a q-q 1 q 1 -1 0 u is a subsolution of (3.10). Therefore v ≥ a q-q 1 q 1 -1 0 u in O, thus v is a large solution.

Remark 3.4 (Sub-critical case) Assume that 1 < q < q * . It is easy to check that the function

U (x, t) = C t 1 q-1 e -|x| 2 4t χ t>0 (3.11) is a subsolution of (3.1) in R N +1 \{(0, 0)}, where C = 2 q-1 -N 2 1 q-1 .
Therefore, the maximal solution u of

(3.1) in O verifies u(x, t) ≥ C 1 (t -s) 1 q-1 e -|x-y| 2 4(t-s) χ t>s , (3.12)
for all (x, t) ∈ O and (y, s) ∈ O c . If for any (x, t) ∈ ∂ p O there exist ε ∈ (0, 1) and a decreasing sequence {δ n } ⊂ (0, 1) converging to 0 as n → ∞ such that B δn (x) × (-δ 2 n + t, -εδ 2 n + t) ∩ O c = ∅ for any n ∈ N, then u is a large solution. For proving this, we need to show that

lim ρ→0 inf O∩(Bρ(x)×(-ρ 2 +t,ρ 2 +t)) u = ∞. Let 0 < ρ < ε 2 δ 1 and n ∈ N such that ε 2 δ n+1 ≤ ρ < ε 2 δ n . Since B δn (x) × (-δ 2 n + t, -εδ 2 n + t) ∩ O c = ∅, there is (x n , t n ) ∈ O c such that |x n -x| < δ n and -δ 2 n + t < t n < -εδ 2 n + t. So if (y, s) ∈ O ∩ (B ρ (x) × (-ρ 2 + t, ρ 2 + t)) then |y -x n | < ( √ ε + 1)δ n and ε 2 δ 2 n < s -t n < (ε + 1)δ 2 n .
Hence, thanks to (3.12) we have for any (y, s)

∈ O ∩ (B ρ (x) × (-ρ 2 + t, ρ 2 + t)) u(y, s) ≥ C 1 (s -t n ) 1 q-1 e -|y-xn | 2 4(s-tn ) ≥ C(ε + 1) -1 q-1 e -( √ ε+1) 2 2ε δ -2 q-1 n , which implies inf O∩(Bρ(x)×(-ρ 2 +t,ρ 2 +t)) u ≥ C(ε + 1) -1 q-1 e -( √ ε+1) 2 2ε δ -2 q-1 n → ∞ as ρ → 0. Remark 3.5 Note that if u ∈ C 2,1 (O) is a solution of (3.1) for some q > 1 then, for a, b > 0 and 1 < p ≤ 2, the function v = b -1 q-1 u is a super-solution of ∂ t v -∆v + a|∇v| p + bv q = 0 in O. (3.13)
Thus, we can apply the argument of the previous proof, with equation (3.1) replaced by (3.13), and deduce that there exists a maximal solution

v ∈ C 2,1 (O) of (3.13) satisfying v(x, t) ≤ Cb -1 q-1 (d((x, t), ∂ p O)) -2 q-1
for all (x, t) ∈ O.

Furthermore, if 1 < q < q * , q = 2p p+1 , a, b > 0 then the function U expressed by (3.11) in Remark 3.4 is a subsolution of (3.13) in R N +1 \{(0, 0)}, provided the explicit constant C given therein is replaced by some C = C(N, p, q, a, b). Therefore, we conclude that every maximal solution of v ∈ C 2,1 (O) of (3.13) satisfy v(x, t) ≥ C 1 (ts)

1 q-1 e -|x-
converging to 0 as n → ∞ such that B δn (x) × (-δ 2 n + t, -εδ 2 n + t) ∩ O c = ∅ for any n ∈ N, then v is a large solution.
Next, we consider the following equation

∂ t u -∆u + e u -1 = 0. (3.15)
It is easy to see that the two functions

V 1 (t) = -log t + ρ 2 1 + ρ 2 and V 2 (x) = C -2 log ρ 2 -|x| 2 ρ satisfy V ′ 1 + e V1 -1 ≥ 0 in (-ρ 2 , 0],
and

-∆V 2 + e V2 -1 ≥ 0 in B ρ (0), for some C = C(N ). Using e a + e b ≤ e a+b -1 for a, b ≥ 0, we obtain that V 1 + V 2 is a supersolution of equation (3.15) in B ρ (0)×(-ρ 2 , 0]
. By the same argument as in Proposition 3.1 and the estimate of the above supersolution, we infer the following: Proposition 3.6 There exists a maximal solution u ∈ C 2,1 (O) of

∂ t u -∆u + e u -1 = 0 in O, (3.16)
and it satisfies

u(x, t) ≤ C -log (d((x, t), ∂ p O)) 3 4 + (d((x, t), ∂ p O)) 2 for all (x, t) ∈ O, (3.17)
for some C = C(N ).

The next three propositions will be useful to prove Theorem 1.1-(ii).

Proposition 3.7 Let K ⊂ Q1 (0, 0) be a compact set and q > 1, R ≥ 100. Let u be a solution of (3.7) in QR (0, 0)\K and ϕ as in Proposition 2.6 with p = q ′ . Set ξ = (1ϕ) 2q ′ . Then,

ˆQR(0,0) u (|∆ξ| + |∇ξ| + |∂ t ξ|) dxdt Cap 2,1,q ′ (K), (3.18) u(x, t) Cap 2,1,q ′ (K) + R -2 q-1
for any (x, t) ∈ QR/5 (0, 0)\ Q2 (0, 0), (3.19) and

ˆQ2(0,0) uξdxdt Cap 2,1,q ′ (K) + R -2 q-1 , (3.20)
where the constants in above inequalities depend only on N and q.

Proof.

Step 1. We claim that ˆQR(0,0) u q ξdxdt Cap 2,1,q ′ (K).

(3.21)

Actually, using integration by parts and the Green formula, one has ˆQR(0,0)

u q ξdxdt = - ˆQR(0,0) ∂ t uξdxdt + ˆQR(0,0) ξ∆udxdt = ˆQR(0,0) u∂ t ξdxdt + ˆQR(0,0) u∆ξdxdt + ˆR2 -R 2 ˆ∂BR(0) ξ ∂u ∂ν -u ∂ξ ∂ν dSdt,
where ν is the outer normal unit vector on ∂B R (0). Clearly, ∂u ∂ν ≤ 0 and ∂ξ ∂ν = 0 on ∂B R (0).

Thus,

ˆQR(0,0) u q ξdxdt ≤ ˆQR(0,0) u|∂ t ξ|dxdt + ˆQR(0,0) u|∆ξ|dxdt ≤ 2q ′ ˆQR(0,0) u(1 -ϕ) 2q ′ -1 |∂ t ϕ|dxdt + 2q ′ (2q ′ -1) ˆQR(0,0) u(1 -ϕ) 2q ′ -2 |∇ϕ| 2 dxdt + 2q ′ ˆQR(0,0) u(1 -ϕ) 2q ′ -1 |∆ϕ|dxdt ≤ 2q ′ ˆQR(0,0) uξ 1/q |∂ t ϕ|dxdt + 2q ′ (2q ′ -1) ˆQR(0,0) uξ 1/q |∇ϕ| 2 dxdt + 2q ′ ˆQR(0,0) uξ 1/q |∆ϕ|dxdt. (3.22)
In the last inequality, we have used the fact that (1φ) 2q ′ -1 ≤ (1φ) 2q ′ -2 = ξ 1/q . Hence, by Hölder's inequality, ˆQR(0,0) u q ξdxdt ˆQR(0,0) |∂ t ϕ| q ′ dxdt + ˆQR(0,0)

|∇ϕ| 2q ′ dxdt + ˆQR(0,0) |∆ϕ| q ′ dxdt.
By the Gagliardo-Nirenberg inequality,

ˆQR(0,0) |∇ϕ| 2q ′ dxdt ||ϕ|| q ′ L ∞ ( QR(0,0)) ˆQR(0,0) |D 2 ϕ| q ′ dxdt ˆQR(0,0) |D 2 ϕ| q ′ dxdt.
Hence, we find ˆQR(0,0) u q ξdxdt ˆQR(0,0) (|∂ t ϕ| q ′ + |D 2 ϕ| q ′ )dxdt, and derive (3.21) from (2.4). In view of (3.22), we also obtain

ˆQR(0,0) u(|∆ξ| + |∂ t ξ|)dxdt Cap 2,1,q ′ (K),
and

ˆQR(0,0) u|∇ξ|dxdt Cap 2,1,q ′ (K), since ˆQR(0,0) u|∇ξ|dxdt = 2q ′ ˆQR(0,0) uξ (2q ′ -1)/2q ′ |∇ϕ|dxdt ≤ 2q ′ ˆQR(0,0) uξ 1/q |∇ϕ|dxdt ˆQR(0,0) u q ξdxdt + ˆQR(0,0) |∇ϕ| q ′ dxdt.
It yields (3.18).

Step 2. Relation (3.19) holds. Let η be a cut off function on QR/4 (0, 0) with respect to QR/3 (0, 0) such that |∂ t η| + |D 2 η| R -2 and |∇η| R -1 . We have

∂ t (ηξu) -∆(ηξu) = F ∈ C c ( QR/3 (0, 0)).
Hence, we can write

(ηξu)(x, t) = ˆRN ˆt -∞ 1 (4π(t -s)) N 2 e -|x-y| 2 4(t-s) F (y, s)dsdy ∀(x, t) ∈ R N +1 .
Now, we fix (x, t) ∈ QR/5 (0, 0)\ Q2 (0, 0). Since supp{|∇η|} ∩ supp{|∇ξ|} = ∅ and

F = ηξ (∂ t u -∆u) -2 (η∇ξ + ξ∇η) ∇u + (ξ∂ t η + η∂ t ξ -2∇η∇ξ -∆ηξ -η∆ξ) u ≤ -2 (η∇ξ + ξ∇η) ∇u + (ξ∂ t η + η∂ t ξ -ξ∆η -η∆ξ) u, there holds u(x, t) = (ηξu)(x, t) ≤ -2 ˆRN ˆt -∞ 1 (4π(t -s))
N 2

e -|x-y| 2 4(t-s) (η∇ξ + ξ∇η) ∇udsdy

+ ˆRN ˆt -∞ 1 (4π(t -s))
N 2

e -|x-y| 2 4(t-s) (η∂ t ξ -η∆ξ) udsdy

+ ˆRN ˆt -∞ 1 (4π(t -s))
N 2

e -|x-y| 2 4(t-s) (∂ t ηξ -ξ∆η) udsdy.

= I 1 + I 2 + I 3 .
By integration by parts,

I 1 = 2(4π) -N/2 ˆt -∞ ˆRN (x -y) 2(t -s) (N +2)/2 e -|x-y| 2 4(t-s) (η∇ξ + ξ∇η) udyds + 2(4π) -N/2 ˆt -∞ ˆRN 1 (t -s) N/2 e -|x-y| 2 4(t-s) (ξ∆η + η∆ξ) u dyds. Note that 1 (t -s) N/2 e -|x-y| 2 4(t-s) max{|x -y|, |t -s| 1/2 } -N , (x -y) 2(t -s) (N +2)/2 e -|x-y| 2 4(t-s) max{|x -y|, |t -s| 1/2 } -N -1 , and max{|x -y|, |t -s| 1/2 } 1 ∀(y, s) ∈ supp{|D α ξ|} ∪ supp{|∂ t ξ|}, max{|x -y|, |t -s| 1/2 } R ∀(y, s) ∈ supp{|D α η|} ∪ supp{|∂ t η|} ∀|α| ≥ 1.
We deduce Combining this inequality with (3.18) and (3.8), we obtain (3.19).

I 1 ˆRN+1 max{|x -y|, |t -s| 1/2 } -N -1 (η|∇ξ| + ξ|∇η|)u dyds + ˆRN+1 max{|x -y|, |t -s| 1/2 } -N (ξ|∆η| + η|∆ξ|) u dyds ˆRN+1 (|∇ξ| + |∆ξ|)u dyds + ˆQ R/3 (0,0)\ QR/4 (0,0) (R -N -1 |∇η| + R -N |∆η|)u dyds ˆRN+1 (|∇ξ| + |∆ξ|)u dyds + sup QR/3 (0,0)\ QR/4 (0,0) u, I 2 ˆRN+1 max{|x -y|, |t -s| 1/2 } -N (|∂ t ξ| + |∆ξ|)u dyds ˆRN+1 (|∂ t ξ| + |∆ξ|)u dyds, and
I 3 ˆRN+1 max{|x -y|, |t -s| 1/2 } -N (|∂ t η| + |∆η|)u dyds ˆQ R/3 (0,0)\ QR/4 (0,0) R -N (|∂ t η| + |∆η|)u dyds
Step 3. End of the proof. Let θ be a cut off function on Q3 (0, 0) with respect to Q4 (0, 0). As above, we have for any (x, t) ∈ R N +1

(θξu)(x, t)

ˆRN+1 (max{|x -y|, |t -s| 1/2 }) -N -1 (θ|∇ξ| + ξ|∇θ|)u dyds + ˆRN+1 (max{|x -y|, |t -s| 1/2 }) -N (θ|∆ξ| + ξ|∆θ|)u dyds + ˆRN+1 (max{|x -y|, |t -s| 1/2 }) -N (θ|∂ t ξ| + θ|∆ξ|)u dyds + ˆRN+1 (max{|x -y|, |t -s| 1/2 }) -N (ξ|∂ t θ| + ξ|∆θ|)u dyds.
Hence, by Fubini theorem,

ˆQ2(0,0) ηudxdt = ˆQ2(0,0) θηudxdt A ˆRN+1 (θ|∇ξ| + ξ|∇θ| + θ|∆ξ| + ξ|∆θ| + θ|∂ t ξ| + ξ|∂ t θ|) u dyds ˆRN+1 (|∂ t ξ| + |∇ξ| + |∆ξ|)u dyds + sup Q4(0,0)\ Q3(0,0) u, we derive u(x 0 , t 0 ) ≤ jε j=1 u j (x 0 , t 0 ) ≤ jε j=1 L(N ) k=1 u k,j (x 0 , t 0 ) jε j=0 Cap 2,1,q ′ (K j ) ρ 2 q-1 j + j ε R -2 q-1 ,
which yields (3.24). If q > q * , then by (2.2) in Proposition 2.5, we have

Cap 2,1,q ′ (K j ) ρ -N -2+2q ′ j+3 Cap 2,1,q ′ (K ∩ Qρj-2 (0, 0)),
which implies (3.23).

Proposition 3.9 Let K, u, ξ be as in Proposition 3.7. For any compact set K 0 in Q1 (0, 0)

with positive measure |K 0 |, there exists ε = ε(N, q, |K 0 |) > 0 such that Cap 2,1,q ′ (K) ≤ ε ⇒ inf K0 u ˆQ2(0,0) uξdxdt,
where the constant in the inequality depends on K 0 . In particular,

Cap 2,1,q ′ (K) ≤ ε ⇒ inf K0 u Cap 2,1,q ′ (K) + R -2 q-1 . (3.25)
Proof. It is enough to prove that there exists ε > 0 such that

Cap 2,1,q ′ (K) ≤ ε ⇒ |K 1 | ≥ 1/2|K 0 |, (3.26)
where K 1 = {(x, t) ∈ K 0 : ξ(x, t) ≥ 1/2}. By (2.1) in Proposition 2.5, we have the following estimates

|K 0 \K 1 | 1-2q ′ N +2
Cap 2,1,q ′ (K 0 \K 1 ), if q > q * , and

log | Q200 (0, 0)| |K 0 \K 1 | -N 2 Cap 2,1,q ′ (K 0 \K 1 ), if q = q * . On the other hand, Cap 2,1,q ′ (K 0 \K 1 ) = Cap 2,1,q ′ ({K 0 : ϕ > 1 -(1/2) 1/(2q ′ ) }) ≤ (1 -(1/2) 1/(2q ′ ) ) -q ′ ˆRN+1 |D 2 ϕ| q ′ + |∇ϕ| q ′ + |ϕ| q ′ + |∂ t ϕ| q ′ dxdt Cap 2,1,q ′ (K),
where ϕ is in Proposition 3.7. Henceforth, one can find

ε = ε(N, q, |K 0 |) > 0 such that Cap 2,1,q ′ (K) ≤ ε ⇒ |K 0 \K 1 | ≤ 1/2 |K 0 |.
This implies (3.26).

Large solutions

In the first part of this section, we prove theorem 1.1-(ii), then we prove theorems 1.1-(i) and 1.2. At end weapply our result to a parabolic viscous Hamilton-Jacobi equation.

Proof of Theorem 1.1-(ii)

Let R 0 ≥ 4 such that O ⊂⊂ QR0 (0, 0). Assume that the equation (1.13) has a large solution u. We claim that (1.15) holds with (x, t) ∈ ∂ p O, and without loss of generality, we can assume (x, t) = (0, 0). Set K = Q2R0 (0, 0)\O and define

T j = {x : ρ j+1 ≤ max{|x|, |t| 1/2 } ≤ ρ j , t ≤ 0}, Tj = {x : ρ j+3 ≤ max{|x|, |t| 1/2 } ≤ ρ j-2 , t ≤ 0}.
Here ρ j = 2 -j . For j ≥ 3, let u 1 , u 2 , u 3 , u 4 be the maximal solutions of (3.7) when K is replaced by K ∩Q ρj+3 (0, 0), K ∩ Tj , K ∩ Q 1 (0, 0) \Q ρj-2 (0, 0) and K\Q 1 (0, 0) respectively and R ≥ 100R 0 . From (3.9) in Remark 3.2, we can assert that

u ≤ u 1 + u 2 + u 3 + u 4 in O ∩ {(x, t) ∈ R N +1 : t ≤ 0}. Thus, inf Tj u ≤ ||u 1 || L ∞ (Tj ) + ||u 3 || L ∞ (Tj ) + ||u 4 || L ∞ (Tj ) + inf Tj u 2 . (4.1)
Case 1: q > q * . By (3.8) in Remark 3.2,

||u 4 || L ∞ (Tj ) 1. (4.2)
By (3.23) in Proposition 3.8,

||u 3 || L ∞ (Tj ) j-4 i=-2 Cap 2,1,q ′ (K ∩ Q ρi (0, 0)) ρ N i + jR -2 q-1 . (4.3) Since (x, t) → u 1 (x, t) = ρ 2/(q-1) j+3 u 1 (ρ j+3 x, ρ 2 j+3 t)
is the maximal solution of (3.7) when (K, QR (0, 0)) is replaced by ({(y/ρ j+3 , s/ρ 2 j+3 ) : (y, s) ∈ K ∩ Q ρj+3 (0, 0)}, QR/ρj+3 (0, 0)), we derive

||u 1 || L ∞ (T-3) Cap 2,1,q ′ (K ∩ Q ρj+2 (0, 0)) ρ N +2-2q ′ j + (R/ρ j+3 ) -2 q-1 ,
thanks to (3.19) in Proposition 3.7 and (2.2) in Proposition 2.5, from which follows

||u 1 || L ∞ (Tj ) Cap 2,1,q ′ (K ∩ Q ρj+2 (0, 0)) ρ N j + R -2 q-1 . (4.4) Since (x, t) → u 2 (x, t) = ρ 2/(q-1) j-2 u 2 (ρ j-2 x, ρ 2 j-2 t)
is the maximal solution of (3.7) when the couple (K, QR (0, 0)) is replaced by ({(y/ρ j-2 , s/ρ 2 j-2 ) : (y, s) ∈ K ∩ Tj }, QR/ρj-2 (0, 0)), Proposition 3.9 and relation (2.2) in Proposition 2.5 yield Cap 2,1,q ′ (K ∩ Tj )

ρ N +2-2q ′ j-2 ≤ ε ⇒ inf T2 u 2 Cap 2,1,q ′ (K ∩ Tj ) ρ N +2-2q ′ j-2 + (R/ρ j-2 ) -2 q-1 , which implies Cap 2,1,q ′ (K ∩ Q ρj-3 (0, 0)) ρ N +2-2q ′ j-2 ≤ ε ⇒ inf Tj u 2 Cap 2,1,q ′ (K ∩ Q ρj-3 (0, 0)) ρ N j-2 + R -2 q-1 , (4.5)
for some ε = ε(N, q) > 0. First, we assume that there exists J ∈ N, J ≥ 10 such that Cap 2,1,q ′ (K ∩ Q ρj-3 (0, 0))

ρ N +2-2q ′ j-2 ≤ ε ∀ j ≥ J.
Then, from (4.1) and (4.2), (4.3), (4.4), (4.5), we have inf

Tj u j+2 i=-2 Cap 2,1,q ′ (K ∩ Q ρi (0, 0)) ρ N i + jR -2 q-1 + 1,
for any j ≥ J, and letting R → ∞,

inf Tj u j+2 i=-2 Cap 2,1,q ′ (K ∩ Q ρi (0, 0)) ρ N i + 1. Since inf Tj u → ∞ as j → ∞, we get ∞ i=0 Cap 2,1,q ′ (K ∩ Q ρi (0, 0)) ρ N i = ∞,
which implies that (1.15) holds with (x, t) = (0, 0). Alternatively, assume that for infinitely many j

Cap 2,1,q ′ (K ∩ Q ρj-3 (0, 0)) ρ N +2-2q ′ j-2 > ε, then, Cap 2,1,q ′ (K ∩ Q ρj-3 (0, 0)) ρ N j-2 > ρ 2-2q ′ j-2 ε → ∞ when j → ∞.
We also derive that (1.15) holds with (x, t) = (0, 0). This proves the case q > q * . Case 2: q = q * . Similarly to Case 1, we have: for j ≥ 6

||u 4 || L ∞ (Tj ) 1, (4.6
)

||u 3 || L ∞ (Tj ) j-2 i=0 Cap 2,1,q ′ (K j ) ρ N i + jR -2 q-1 , (4.7
)

||u 1 || L ∞ (Tj ) Cap 2,1,q ′ (K j ) ρ N j + R -2 q-1 , (4.8)
Cap 2,1,q ′ (K j-5 ) ≤ ε ⇒ inf Tj u 2 Cap 2,1,q ′ (K j-5 ) ρ N j + R -2 q-1 , (4.9)
where

K j = {(x/ρ j+3 , t/ρ 2 j+3 ) : (x, t) ∈ K ∩ Q ρj-3 (0, 0)} and ε = ε(N ) > 0. From (2.2) in Proposition 2.5, we have 1 Cap 2,1,q ′ (K ∩ Q ρj-3 (0, 0)) ≤ c Cap 2,1,q ′ (K j ) + cj N/2
for any j ≥ 4 where c = c(N ). If there are infinitely many j ≥ 4 such that

Cap 2,1,q ′ (K ∩ Q ρj-3 (0, 0)) > 1 2cj N/2 , then (1.15) holds with (x, t) = (0, 0) since Cap 2,1,q ′ (K ∩ Q ρj-3 (0, 0)) ρ N j-3 > 2 j-3 2cj N/2 → ∞ when j → ∞.
Now, we assume that there exists J ≥ 6 such that Cap

2,1,q ′ (K ∩ Q ρj-3 (0, 0)) ≤ 1 2cj N/2 ∀ j ≥ J. Then, Cap 2,1,q ′ (K j ) ≤ 2cCap 2,1,q ′ (K ∩ Q ρj-3 (0, 0)) ∀ j ≥ J. This leads to Cap 2,1,q ′ (K j ) ≤ 2cCap 2,1,q ′ (K ∩ Q ρj-3 (0, 0)) ≤ ε ∀ j ≥ J ′ + J,
for some J ′ = J ′ (N ). Hence, from (4.6)-(4.9) we have, for any j ≥ J ′ + J + 3,

||u 4 || L ∞ (Tj ) 1, ||u 3 || L ∞ (Tj ) j-2 i=J ′ +J+1 Cap 2,1,q ′ (K ∩ Q ρi-3 (0, 0)) ρ N i + C(J ′ + J) + jR -2 q-1 , ||u 1 || L ∞ (Tj ) Cap 2,1,q ′ (K ∩ Q ρj-3 (0, 0)) ρ N j + R -2 q-1 , inf Tj u 2 Cap 2,1,q ′ (K ∩ Q ρj-8 (0, 0)) ρ N j + R -2 q-1 ,
where

C(J ′ + J) = J ′ +J i=0 Cap 2,1,q ′ (Kj ) ρ N i . Consequently we derive inf Tj u j i=0 Cap 2,1,q ′ (K ∩ Q ρi (0, 0)) ρ N i + C(J ′ + J) + 1 + jR -2 q-1 ∀ j ≥ J ′ + J + 3 from (4.1). Letting R → ∞ and j → ∞ we obtain ∞ i=0 Cap 2,1,q ′ (K ∩ Q ρi (0, 0)) ρ N i = ∞,
i.e. (1.15) holds with (x, t) = (0, 0). This completes the proof of Theorem 1.1-(ii).

Proof of Theorem 1.1-(i) and Theorem 1.2

Fix (x 0 , t 0 ) ∈ ∂ p O. We can assume that (x 0 , t 0 ) = 0. Let δ ∈ (0, 1/100). For (y 0 , s 0 ) ∈

(B δ (0) × (-δ 2 , δ 2 )) ∩ O, we set M k = O c ∩ B r k+2 (y 0 ) × [s 0 -(73 + 1 2 )r 2 k+2 , s 0 -(70 + 1 2 )r 2 k+2 ] ,
and

S k = {(x, t) : r k+1 ≤ max{|x -y 0 |, |t -s 0 | 1 2 } < r k } for k = 1, 2, ...,
where r k = 4 -k . Note that M k = ∅ for k large enough and M k ⊂ S k for all k. Let R 0 ≥ 4 such that O ⊂⊂ QR0 (0, 0). By Theorems 2.2 and 2.4 and estimate (1.12) there exist two sequences {µ k } k and {ν k } k of nonnegative Radon measures such that

supp(µ k ) ⊂ M k , supp(ν k ) ⊂ M k , (4.10)
µ k (M k ) ≍ Cap 2,1,q ′ (M k ) ≍ ˆRN+1 I 2R0 2 [µ k ] q dxdt (4.11)
and

ν k (M k ) ≍ PH N 1 (M k ), ||M 2R0 1 [ν k ]|| L ∞ (R N +1 ) ≤ 1 for k = 1, 2, ..., (4.12)
where the constants of equivalence depend on N, q, R 0 .

Take ε > 0 such that exp

C 1 εI 2R0 2 [ ∞ k=1 ν k ] ∈ L 1 ( QR0 (0, 0)), in which expression the constant C 1 = C 1 (N )
is the one of inequality (2.6). By Theorem 2.7 and Proposition 2.8, there exist two nonnegative solutions U 1 , U 2 of problems

∂ t U 1 -∆U 1 + U q 1 = ε ∞ k=1 µ k in QR0 (0, 0), U 1 = 0 on ∂ p QR0 (0, 0), and
∂ t U 2 -∆U 2 + e U2 -1 = ε ∞ k=1 ν k in QR0 (0, 0), U 2 = 0 on ∂ p QR0 (0, 0),
respectively which satisfy

U 1 (y 0 , z 0 ) ∞ i=0 ∞ k=1 ε µ k (B r i 8 (y 0 ) × (s 0 -37 128 r 2 i , s 0 -35 128 r 2 i )) r N i -I 2R0 2 I 2R0 2 [ε ∞ k=1 µ k ] q (y 0 , s 0 ) =: A, (4.13)
and

U 2 (y 0 , z 0 ) ∞ i=0 ∞ k=1 ε ν k (B r i 8 (y 0 ) × (s 0 -37 128 r 2 i , s 0 -35 128 r 2 i )) r N i -I 2R0 2 exp C 1 I 2R0 2 [ε ∞ k=1 ν k ] -1 (y 0 , s 0 ) =: B, (4.14)
and

U 1 , U 2 ∈ C 2,1 (O).
Let u 1 , u 2 be the maximal solutions of equations (3.1) and (3.16) respectively. We have u 1 (y 0 , s 0 ) ≥ U 1 (y 0 , s 0 ) and u 2 (y 0 , s 0 ) ≥ U 2 (y 0 , s 0 ). Now, we claim that

A ∞ k=1 Cap 2,1,q ′ (M k ) r N k , (4.15)
and

B -c 1 (R 0 ) + ∞ k=1 PH N 1 (M k ) r N k . (4.16)
Proof of assertion (4.15). From (4.11) we have

A ε ∞ k=1 Cap 2,1q ′ (M k ) r N k -ε q A 0 , (4.17)
with

A 0 = I 2R0 2 I 2R0 2 [ ∞ k=1 µ k ] q (y 0 , s 0 ). Take i 0 ∈ Z such that r i0+1 < max{2R 0 , 1} ≤ r i0 . Then A 0 ∞ i=i0 r -N i ˆQr i (y0,s0) I 2R0 2 [ ∞ k=1 µ k ] q dxdt = ∞ i=i0 ∞ j=i r -N i ˆSj I 2R0 2 [ ∞ k=1 µ k ] q dxdt = ∞ j=k0 j i=i0 r -N i ˆSj I 2R0 2 [ ∞ k=1 µ k ] q dxdt ∞ j=i0 r -N j ˆSj I 2R0 2 [ ∞ k=1 µ k ] q dxdt.
Here we have used the fact that

j i=i0 r -N i ≤ 4 3 r -N j
for all j. If we set µ k ≡ 0 for all i 0 -1 ≤ k ≤ 0, the previous inequality becomes

A 0 ∞ j=i0 r -N j ˆSj   I 2R0 2 [µ j + j-1 k=i0-1 µ k + ∞ k=j+1 µ k ]   q dxdt ∞ j=i0 r -N j ˆSj I 2R0 2 [µ j ] q dxdt + ∞ j=i0 r 2 j j-1 k=i0-1 ||I 2R0 2 [µ k ]|| L ∞ (Sj ) q + ∞ j=i0 r 2 j   ∞ k=j+1 ||I 2R0 2 [µ k ]|| L ∞ (Sj )   q = A 1 + A 2 + A 3 . (4.18)
Using (4.11) we obtain

A 1 ≤ ∞ k=1 Cap 2,1,q ′ (M k ) r N k . (4.19)
Next, using (4.10) we have for any (x, t) ∈ S j

I 2R0 2 [µ k ](x, t) = ˆ2R0 rj+1 µ k ( Qρ (x, t)) ρ N dρ ρ µ k (R N +1 ) r N j (4.20)
if k ≥ j + 1, and

I 2R0 2 [µ k ](x, t) = ˆ2R0 r k+1 µ k ( Qρ (x, t)) ρ N dρ ρ µ k (R N +1 ) r N k (4.21) if k ≤ j -1. Thus, A 2 ∞ j=i0 r 2 j j-1 k=i0-1 µ k (R N +1 ) r N k q , and
A 3 ∞ j=i0 r 2-N q j   ∞ k=j+1 µ k (R N +1 )   q .
Noticing that (a + b) qa q ≤ q(a + b) q-1 b for any a, b ≥ 0, we get

(1 -4 -2 ) ∞ j=i0 r 2 j j-1 k=i0-1 µ k (R N +1 ) r N k q = ∞ j=i0 r 2 j j-1 k=i0-1 µ k (R N +1 ) r N k q - ∞ j=i0+1 r 2 j j-2 k=i0-1 µ k (R N +1 ) r N k q ≤ ∞ j=i0 qr 2 j j-1 k=i0-1 µ k (R N +1 ) r N k q-1 µ j-1 (R N +1 ) r N j-1
.

Similarly, we also have

(1 -4 2-N q ) ∞ j=i0 r 2-N q j   ∞ k=j+1 µ k (R N +1 )   q ≤ ∞ j=i0 qr 2-N q j   ∞ k=j+1 µ k (R N +1 )   q-1 µ j+1 (R N +1 ).
Therefore,

A 2 + A 3 ∞ j=i0 r 2 j j-1 k=i0-1 µ k (R N +1 ) r N k q-1 µ j-1 (R N +1 ) r N j-1 + ∞ j=i0 r 2-N q j   ∞ k=j+1 µ k (R N +1 )   q-1 µ j+1 (R N +1 ).
Since µ k (R N +1 ) r N +2-2q ′ k if q > q * and µ k (R N +1 ) min{k -1 q-1 , 1} if q = q * for any k, we infer that

r 2 j j-1 k=i0-1 µ k (R N +1 ) r N k q-1
1, and

r 2-N q j   ∞ k=j+1 µ k (R N +1 )   q-1 r -N j+1
for any j.

In the case q = q * we assume N ≥ 3 in order to ensure that

∞ j=1 µ k (R N +1 ) ∞ k=1 k -1 q-1 < ∞.
This leads to

A 2 + A 3 ∞ k=1 µ k (R N +1 ) r N k .
Combining this with (4.19) and (4.18), we deduce

A 0 ∞ k=1 Cap 2,1,q ′ (M k ) r N k .
From (4.12) we have µ k (R N +1 ) r N k for all k, therefore

B 2 + B 3 ∞ j=i0 r 2 j exp (c 3 ε(j -i 0 )) + ∞ j=i0 r 2 j exp (c 3 ε) ∞ j=i0 exp (c 3 ε(j -i 0 ) -4 log(2)j) + r 2 i0
≤ c 4 (N, q, R 0 ) for ε small enough.

Combining this with (4.24) and (4.23) we obtain (4.22).

This implies straightforwardly exp

C 1 εI 2R0 2 [ ∞ k=1 ν k ] ∈ L 1 ( QR0 (0, 0)). We conclude that for any (y 0 , s 0 ) ∈ (B δ (0) × (-δ 2 , δ 2 )) ∩ O, u 1 (y 0 , s 0 ) ∞ k=1 Cap 2,1,q ′ (M k (y 0 , s 0 )) r N k , and
u 2 (y 0 , s 0 ) -c 1 (R 0 ) + ∞ k=1 PH N 1 (M k (y 0 , s 0 )) r N k , where r k = 4 -k and M k (y 0 , s 0 ) = O c ∩ B r k+2 (y 0 ) × [s 0 -(73 + 1 2 )r 2 k+2 , s 0 -(70 + 1 2 )r 2 k+2 ] . If we take r k δ +4 ≤ δ < r k δ +3 , we have for 1 ≤ k ≤ k δ M k (y 0 , s 0 ) ⊃ O c ∩ B r k+2 -δ (0) × δ 2 -(73 + 1 2 )r 2 k+2 , -δ 2 -(70 + 1 2 )r 2 k+2 ⊃ O c ∩ B r k+3 (0) × -73r 2 k+2 , -71r 2 k+2 = O c ∩ B r k+3 (0) × -1168r 2 k+3 , -1136r 2 k+3 . Finally, inf (y0,s0)∈(B δ (0)×(-δ 2 ,δ 2 ))∩O u 1 (y 0 , s 0 ) -1 + ˆ1 r k δ +3 Cap 2,1,q ′ (O c ∩ (B ρ (0) × (-17bρ 2 , -bρ 2 ))) ρ N dρ ρ with b = 1136 -1 + ˆ1 30r k δ +3 Cap 2,1,q ′ (O c ∩ (B ρ 30 (0) × (-30ρ 2 , -ρ 2 ))) ρ N dρ ρ → ∞ as δ → 0, and inf (y0,s0)∈(B δ (0)×(-δ 2 ,δ 2 ))∩O u 2 (y 0 , s 0 ) -1 + ˆ1 30r k δ +3 PH N 1 (O c ∩ (B ρ 30 (0) × (-30ρ 2 , -ρ 2 ))) ρ N dρ ρ → ∞ as δ → 0.
This completes the proof of Theorem 1.1-(i) and Theorem 1.2.

Remark 4.1 (Uniqueness) In [START_REF] Marcus | Maximal solutions for -∆u + u q = 0 in open and finely open sets[END_REF], Marcus and Véron prove that condition (1.8) is not only a necessary and sufficient condition for the existence of a large solution to (1.7), but it implies the uniqueness of a such a large solution when it is fulfilled. The main step for this proof is to show that there exists a constant c = c(Ω, q > 0) such that any couple of large solutions (u, û) satisfies u(x) ≤ cû(x) ∀x ∈ Ω. (4.25)

The above estimate which is the key stone for proving uniqueness cannot be obtained in the case of the parabolic equation (1.13) since the necessary condition and the sufficient condition in Theorem 1.1 do not complement completely.

The viscous Hamilton-Jacobi parabolic equations

In this section we apply our previous result to the question of existence of a large solution of the following type of parabolic viscous Hamilton-Jacobi equation

∂ t u -∆u + a|∇u| p + bu q = 0 in O, u = ∞ on ∂ p O, (4.26)
where a > 0, b > 0 and 1 < p ≤ 2, q ≥ 1. First, we show that such a large solution to (4.26) does not exist when q = 1. Equivalently, there is no function u ∈ C 2,1 (O) satisfying

∂ t u -∆u + a|∇u| p ≥ -bu in O, u = ∞ on ∂ p O. (4.27) for a > 0, b > 0 and p > 1. Indeed, assuming that such a function u ∈ C 2,1 (O) exists, we define U (x, t) = u(x, t)e bt - ε 2 |x| 2 ,
for ε > 0 and denote by (x 0 , t 0 ) ∈ O\∂ p O the point where U achieves it minimum in O, i.e. U (x 0 , t 0 ) = inf{U (x, t) : (x, t) ∈ O}. Clearly, we have ∂ t U (x 0 , t 0 ) ≤ 0, ∆U (x 0 , t 0 ) ≥ 0 and ∇U (x 0 , t 0 ) = 0.

Thus, ∂ t u(x 0 , t 0 ) ≤ -bu(x 0 , t 0 ), -∆u(x 0 , t 0 ) ≤ -εN e -bt0 and a|∇u(x 0 , t 0 )| p = aε p |x 0 | p e -pbt0 , from which follows

∂ t u(x 0 , t 0 ) -∆u(x 0 , t 0 ) + a|∇u(x 0 , t 0 )| p ≤ -bu(x 0 , t 0 ) + εe -bt0 -N + aε p-1 |x 0 | p e -(p-1)bt0
< -bu(x 0 , t 0 ) for ε small enough, which is a contradiction.

Proof of Theorem 1.3. By Remark 3.3, we have

inf{v(x, t); (x, t) ∈ O} ≥ (q 1 -1) -1 q 1 -1 R -2 q 1 -1 . Take V = λv 1 α ∈ C 2,1 (O) for λ > 0. Thus v = λ -α V α , inf{V (x, t); (x, t) ∈ O} > 0} ≥ λ(q 1 -1) - 1 α(q 1 -1) R - 2 α(q 1 -1) , and
∂ t v -∆v + v q1 = αλ -α V α-1 ∂ t V -αλ -α V α-1 ∆V + α(1 -α)λ -α V α-1 |∇V | 2 V + λ -αq1 V αq1 .
This leads to

∂ t V -∆V + (1 -α) |∇V | 2 V + α -1 λ -α(q1-1) V αq1-α+1 = 0 in O.
Using Hölder's inequality we obtain

(1 -α) |∇V | 2 V + (2α) -1 λ -α(q1-1) V αq1-α+1 ≥ c 1 |∇V | p λ -α(q 1 -1)(2-p) 2 V α(q 1 -1)(2-p) 2 -(p-1) ≥ c 2 |∇V | p λ -(p-1) R -2+p+ 2(p-1)
α(q 1 -1) , and (2α) -1 λ -α(q1-1) V αq1-α+1 ≥ c 3 λ -(q-1) R -2+ 2(q-1)

α(q 1 -1) V q .

If we choose λ = min{c

1 p-1 2 , c 1 q-1 3 } min a -1 p-1 R -2-p p-1 + 2 α(q 1 -1) , b -1 q-1 R -2 q-1 + 2 α(q 1 -1)
, then c 2 λ -(p-1) R -2+p+ 2(p-1) α(q 1 -1) ≥ a, c 3 λ -(q-1) R -2+ 2(q-1) α(q 1 -1) ≥ b, from what follows

∂ t V -∆V + a|∇V | p + bV q ≤ 0 in O.
By Remark 3.5, there exists a maximal solution u ∈ C 2,1 (O) of ∂ t u -∆u + a|∇u| p + bu q = 0 in O.

Therefore, u ≥ V = λv 1 α and u is a large solution of (4.26). This completes the proof of Theorem 1.3.

Appendix

Proof of Proposition 2.5.

Step 1. We claim that the following relation holds:

ˆRN+1

I 1 2 [µ](x, t) (N +2)/N dxdt ≍ ˆRN+1 ˆ1 0 (µ( Qr (x, t))) 2/N dr r dµ(x, t).

(5.1)

In fact, we have for ρ j = 2 -j , j ∈ Z, ∞ j=1 ˆRN+1

(µ( Qρj (x, t))) 2/N dµ(x, t) ˆRN+1 ˆ1 0 (µ( Qr (x, t))) 2/N dr r dµ(x, t)

∞ j=0 ˆRN+1
(µ( Qρj (x, t))) 2/N dµ(x, t).

Note that for any j ∈ Z ρ -N -2 j ˆRN+1 (µ( Qρj+1 (x, t))) (N +2)/N dxdt ˆRN+1 (µ( Qρj (x, t))) 2/N dµ(x, t) ρ -N -2 j ˆRN+1 (µ( Qρj-1 (x, t))) (N +2)/N dxdt. Step 2. End of the proof. The first inequality in (2.1) is proved in [START_REF] Nguyen | Potential estimates and quasilinear equations with measure data[END_REF]. We now prove the second inequality. By Theorem 2.4 there is µ ∈ M + (R N +1 ), supp(µ) ⊂ K such that

||M 2 2 [µ]|| L ∞ (R N +1 ) ≤ 1 and µ(K) ≍ PH N 2 (K) |K| N/(N +2) . (5.2)
Thanks to (5.1), we have for δ = min{1, (µ(K)) (5.4)

As in [12,

  t) ≤ I 1 + I 2 + I 3 ˆRN+1 (|∂ t ξ| + |∇ξ| + |∆ξ|)u dyds + sup QR/3 (0,0)\ QR/4 (0,0)u.

( 2 (L

 2 N +2)/N (µ(K)) (N +2)/N 1 + log + (µ(K)) -1 (µ(K)) (N +2)/N log | Q200 (0, 0)| |K| . Set μ = log | Q200(0,0)| |K| -N/(N +2) µ/µ(K), then ||I 1 2 [μ]|| L (N +2)/N (R N +1 ) 1. It is well known that Cap 2,1, N +2 K) ≍ sup{(ω(K)) (N +2)/2 : ω ∈ M + (K), ||I 1 2 [ω]|| L (N +2)/N (R N +1 ) 1} (5.3)see[START_REF] Nguyen | Potential estimates and quasilinear equations with measure data[END_REF] Section 4]. This gives the second inequality in (2.1).It is easy to prove (2.2) from its definition. Moreover, (5(N +2)/N (R N +1 ) : ω ∈ M + (K), ω(K) = 1}.We deduce from (5.1) that1 Cap 2,1, N +2 2 (K) 2/N ≍ inf ˆRN+1 ˆ1 0(ω( Qr (x, t))) 2/N dr r dµ(x, t) : ω ∈ M + (K), ω(K) = 1 .

  for some λ > 0, we see that ∂O = ∂ p O.

	Therefore (1.15) holds for any (x, t) ∈ ∂ p O, and (1.14)-(1.17) hold for any (x, t) ∈ ∂ p O\{(0, √ λ)}. However, (1.14) and (1.17) are also valid at (x, t) = (0, √ λ) if λ > 1800 2 , but not valid if
	λ < 1800 2 .

  y| 2 4(t-s) χ t>s (3.14) for all (x, t) ∈ O and (y, s) ∈ ∂ p O. Arguing as in Remark 3.4, if for any (x, t) ∈ ∂ p O there exist ε ∈ (0, 1) and a decreasing sequence {δ n } ⊂ (0, 1)

  proof of Lemma 2.2], it is easy to derive (2.3) from(5.4).Proof of Proposition 2.6. Thanks to the Poincaré inequality, it is enough to show that there exists ϕ ∈ C ∞ c ( Q3/2 (0, 0)) such that 0 ≤ ϕ ≤ 1, with ϕ = 1 in an open neighborhood of K andˆRN+1(|D 2 ϕ| p + |∂ t ϕ| p )dxdt Cap 2,1,p (K).(5.5)By definition, one can find 0≤ φ ∈ S(R N +1 ), φ ≥ 1 in a neighborhood of K such that ˆRN+1 (|D 2 φ| p + |∇φ| p + |φ| p + |∂ t φ| p )dxdt ≤ 2Cap 2,1,p (K).Let η be a cut off function on Q1 (0, 0) with respect to Q3/2 (0, 0) andH ∈ C ∞ (R) such that 0 ≤ H(t) ≤ t + , |t||H ′′ (t)| 1 for all t ∈ R, H(t) = 0 for t ≤ 1/4 and H(t) = 1 for t ≥ 3/4. ϕ| p + |∂ t ϕ| p )dxdt ˆRN+1 (|D 2 φ| p + |∇φ| p + |φ| p + |∂ t φ| p )dxdt,(5.6)where ϕ = ηH(φ). Indeed, we have|D 2 ϕ| |D 2 η|H(φ) + |∇η||H ′ (φ)||∇φ| + η|H ′′ (φ)||∇φ| 2 + η|H ′ (φ)||D2 φ|, and |∂ t ϕ| |∂ t η|H(φ) + η|H ′ (φ)||φ t |, H(φ) ≤ φ, φ|H ′′ (φ)| 1. ˆRN+1 (|D 2 ϕ| p + |∂ t ϕ| p )dxdt ˆRN+1 (|D 2 φ| p + |∇φ| p + |φ| p + |∂ t φ| p )dxdt

	We claim that
	ˆRN+1
	(|D 2 Thus,
		+	ˆRN+1	|∇φ| 2p φ p dxdt.
	This implies (5.6) since, according to [1], one has
	ˆRN	|∇φ(t)| 2p (φ(t))

p dx ˆRN |D 2 φ(t)| p dx ∀t ∈ R.

with ρ = d/2, satisfies ∂ t U -∆U + U q ≥ 0 in B d where A = sup (y,s)∈ Q4(0,0) ˆQ2(0,0) ((max{|x -y|, |t -s| 1/2 }) -N + (max{|x -y|, |t -s| 1/2 }) -N -1 )dxdt.

Therefore we obtain (3.20) from (3.18) and (3.19).

Proposition 3.8 Let 0 < ε < 1, K ⊂ {(x, t) : ε < max{|x|, |t| 1/2 } < 1} be a compact set and u the maximal solution of (3.7) in QR (0, 0)\K with R ≥ 100. Then

and sup Qε/4 (0,0)

where

Proof. For j ∈ N , we define S j = {x :

where (x k,j , t k,j ) ∈ S j . For k = 1, ..., L(N ), let u j , u k,j be the maximal solutions of (3.7) where K is replaced by K ∩ S j and K ∩ Qρj+3 (x k,j , t k,j ), respectively. Clearly the function ũk,j defined by ũk,j (x, t) = ρ

is the maximal solution of (3.7) provided (K k,j , QR/ρj+3 (-x k,j /ρ j+3 , -t k,j /ρ 2 j+3 )) with

is replacing (K, QR (0, 0)). Let u k,j be the maximal solution of (3.7) with (K, QR (0, 0)) replaced by (K k,j , Q2R/ρj+3 (0, 0)). Since QR/ρj+3 (-x k,j /ρ j+3 , -t k,j /ρ 2 j+3 ) ⊂ Q2R/ρj+3 (0, 0), then, by the comparison principle as in the proof of Proposition 3.1, we get ũk,j ≤ u k,j in QR/ρj+3 (-x k,j /ρ j+3 , -t k,j /ρ 2 j+3 )\K k,j , and thus

Therefore, using (3.9) in Remark 3.2 and the fact that

Consequently, we obtain (4.15) from (4.17), for ε small enough.

Proof of assertion (4.16). From (4.12) we get

where

We show that

In fact, as above we have

Consequently,


Yanbo Shou
email: yshou@femto-st.fr

Hervé Guyennet
email: hguyennet@femto-st.fr

A Fault Tolerant Parallel Computing Scheme of Scalar Multiplication for Wireless Sensor Networks

Keywords: Wireless sensor networks, Elliptic curves, Scalar multiplication, Parallel computing, Fault tolerance

In event-driven sensor networks, when a critical event occurs, sensors should transmit messages back to base station in a secure and reliable manner. We choose Elliptic Curve Cryptography to secure the network since it offers faster computation and good security with shorter keys. In order to minimize the running time, we propose to split and distribute the computation of scalar multiplications by involving neighboring nodes in this operation. In order to improve the reliability, we have also proposed a fault tolerance mechanism. It uses half of the available cluster members as backup nodes which take over the work of faulty nodes in case of system failure. Parallel computing does consume more resources, but the results of simulation show that the computation can be significantly accelerated. This method is designed specially for applications where running time is the most important factor.

Introduction

The advances of micro-electro-mechanical technology in recent years have enabled the fast development of smart sensor node. A sensor node is small electronic device which is usually composed of processing, sensing, radio communication and power supply units [START_REF] Akyildiz | Wireless sensor networks: a survey[END_REF][START_REF] Yick | Wireless sensor network survey[END_REF]. It is programmable and is able to collect environmental data and communicate with other sensor nodes by using wireless technologies. Such sensor nodes have limited resources and are often batteryoperated, they are not designed to handle complicated tasks and that's why a sensor node rarely works alone. Sensor nodes are supposed be deployed massively and be programmed to form automatically a fully functional network, called wireless sensor network.

Ideally, a sensor network consists of a large number of low-cost and lowpower sensor nodes, which are interconnected with each other and operate in unattended manner. This kind of networks are often deployed in inaccessible and hostile zones where human interventions are not always possible. Today we can find a wide range of applications using wireless sensor technologies, such as environment monitoring, industrial sensing, medical care and military surveillance [START_REF] Werner-Allen | Deploying a wireless sensor network on an active volcano[END_REF][START_REF] Kim | Wireless sensor networks for structural health monitoring[END_REF][START_REF] Baker | Wireless sensor networks for home health care[END_REF][START_REF] Malan | Codeblue: An ad hoc sensor network infrastructure for emergency medical care[END_REF][START_REF] Gosnell | Gamma-ray identification of nuclear weapon materials[END_REF].

However sensor nodes are quite fragile and are vulnerable to various attacks due to the lack of resources and unreliability of wireless communication. [START_REF] Walters | Wireless sensor network security: A survey. Security in distributed, grid, mobile[END_REF] gives a detailed presentation of almost all possible attacks in wireless sensor networks. An efficient way to protect sensor networks is to use cryptographic techniques [START_REF] Walters | Wireless sensor network security: A survey. Security in distributed, grid, mobile[END_REF][START_REF] Zhou | Securing wireless sensor networks: a survey[END_REF]. On one hand, symmetric cryptographic algorithms are usually computational less expensive and easier to implement in hardware and software, but as we use the same key for data encryption and decryption, the key management in sensor networks becomes a challenging issue. On the other hand, asymmetric algorithms need more complicated computation, but as we use two different keys respectively for encryption and decryption, a compromised node can not provide clue to the private keys of other nodes.

In this paper we choose Elliptic Curve Cryptography (ECC) to secure the communications in sensor networks. It has become recently one of the most famous asymmetric cryptographic mechanism because of its shorter key length requirement and faster computation compared with other asymmetric mechanisms, such as RSA [START_REF] Rivest | A method for obtaining digital signatures and public-key cryptosystems[END_REF]. ECC has successfully attracted the interest of researchers, especially in the domain of embedded system where most of the devices have strict resource restriction.

The performance of ECC can be significantly improved by using mathematical tools, such as NAF, windowed method, projective coordinates [START_REF] Gordon | A survey of fast exponentiation methods[END_REF]. Besides traditional algorithmic optimization, parallel computing is an other choice for accelerating the computation on elliptic curves. The computation task is split into smaller independent ones which are then distributed to neighboring sensor nodes and carried out simultaneously. However, sensor nodes communicate with each other through wireless connection, which is considered as unreliable and sensitive to radio interference. In addition, sensor nodes might suffer external attacks during computation. In this paper we present our new fault tolerant parallel computing scheme for elliptic curve cryptography in wireless sensor networks, which is actually the following of our research published in [START_REF] Shou | Parallel scalar multiplication on elliptic curves in wireless sensor networks[END_REF].

The rest of the paper is organized as follows. Section 2 presents basic concepts of ECC, and section 3 describes our parallel computing scheme. In section 4 and 5 we present possible fault tolerance issues and the related work. In section 6, we propose a fault tolerance mechanism designed specially for our parallel computing scheme, then it is tested by a simulator and the results are illustrated in 7. Section 8 concludes the paper.

Basic Concepts of Elliptic Curve

Elliptic curve cryptography (ECC) is an approach to public-key cryptography which is proposed independently by Koblitz [START_REF] Koblitz | Elliptic curve cryptosystems[END_REF] and Miller [START_REF] Miller | Use of elliptic curves in cryptography[END_REF] in the 80's. It's suitable for creating lightweight and efficient cryptosystem, especially for embedded systems which have limited resources. Experiments prove that a 160-bit elliptic curve key can provide the same security robustness as a 1024-bit RSA key [START_REF] Gura | Comparing elliptic curve cryptography and rsa on 8-bit cpus[END_REF].

In cryptography we work with the curves which are defined over finite field F q where q = p m . p is a prime number, called the characteristic of F. F is finite prime field when m = 1 and p = 2, 3, which is also the configuration that we use in this paper. Such curves can be represented by the simplified Weierstrass equation (see formula 1).

y 2 = x 3 + ax + b (1)
whose discriminant ∆ = -16(4a 3 + 27b 2 ) and ∆ = 0. All points on an elliptic curve, including the point at infinity, constitute an abelian group whose law is point addition, denoted by +, which combines 2 points on the curve to form a third point which is on the same curve. Suppose that P 1 (x 1 , y 1 ) and P 2 (x 2 , y 2 ) are 2 points on an elliptic curve, then P 3 (x 3 , y 3 ) = P 1 (x 1 , y 1 ) + P 2 (x 2 , y 2 ) can be calculated by using the formulas 2 and 3.

x 3 = s 2 -x 1 -x 2 y 3 = s(x 1 -x 3 ) -y 1 (2)
where

s = (y 2 -y 1 )/(x 2 -x 1 ) if P 1 = P 2 (3x 2 1 + a)/2y 1 if P 1 = P 2 (3)
Based on point addition, we may also perform point multiplication, also called scalar multiplication. For example Q = kP , where Q and P are 2 points on an elliptic curve, and k is a big integer. The most basic method to calculate kP is using Double-and-Add algorithm [START_REF] Hankerson | Guide to elliptic curve cryptography[END_REF].

The security of ECC relies on the difficulty of solving the discrete logarithm problem. For Q = kP , given points Q and P , it's extremely difficult to compute the value of k if it's big enough. However it's not always easy to transform the data we want to protect into a point on elliptic curve. Thus in most of the cases, we only use ECC to establish a shared key between both parties, then we can use a symmetric-key cryptographic algorithm to encrypt our data.

A example of key exchange protocol is Elliptic Curve Diffie-Hellman [START_REF] Diffie | New directions in cryptography[END_REF]. Suppose that Alice and Bob share the same elliptic curve E whose generator point is G. To establish a shared key, Alice computes and sends her public key Q A = s A G to Bob where s A is her private key, meanwhile Bob sends Q B = s B G to Alice where s B is his private key. Then the shared key

K AB = s A Q B = s B Q A = s A s B G.
We can notice that the scalar multiplication is the most essential operation on elliptic curve, and it's also the most computationally intensive operation we need to perform. It is obvious that this kind of computation is too complicated for sensor nodes, performance optimization is then absolutely necessary.

We suppose that in a cluster based sensor network, cluster members send periodically environmental data to their cluster head which is responsible for data processing. Whenever a critical event is detected, the cluster head has to warn the base station as fast as possible in a secure and reliable manner.

Our parallel scalar multiplication method is based on the idea of [START_REF] Lim | More flexible exponentiation with precomputation[END_REF] which doesn't need shared memory and offers an efficient scalar decomposition. For example we need to compute Q = kG where Q and G are 2 points on an elliptic curve, and k is a large integer of length l which can be represented in binary

k = l-1 i=0 k i 2 i . The cluster head splits k into n blocks B i of b = l n bits
according to the number of available cluster members.

B i = (i+1)b-1 j=ib k j 2 j
As all sensor nodes share the same elliptic curve which is configured and preloaded before deployment, then it is possible to precompute points G i = 2 ib G where G is the generator point. Thus the computation of kP can be rewritten in the following manner.

Q = kG = l-1 i=0 k i 2 i = Q 0 + Q 1 + . . . + Q n-1 = B 0 G + B 1 2 b G + . . . + B n-1 2 b(n-1) G (4)
We can notice in equation 4 that each Q i can be considered as an independent task and can be treated separately. The entire process of parallel computing is graphically illustrated in figure 1.

1. Cluster members send data to cluster head periodically. 2. Cluster head detects a critical event. It broadcasts a call for parallel computing and then waits for responses. 3. Available cluster members send positive responses back to cluster head and become Slave nodes of parallel computing. 4. Cluster head decomposes the computation according to the number of available slaves. Then it sends the tasks to slaves, and it become the Master node of parallel computing. 5. Slaves complete the task and send the results back to their master. 6. Master nodes combines received results to obtain the final result, by which the master can encrypt its message and send it to the node of next hop.

As we can see that the parallel scalar multiplication on elliptic curve is a complicated operation, and every step must be carried out carefully. Moreover, the process needs reliable wireless communication which is not always possible in wireless sensor networks.

Cluster Member

Cluster Head

Fault Tolerance Issues

Three terms have been introduced in [START_REF] Mishra | Fault tolerance in wireless sensor networks[END_REF] in order to explain the fault tolerance issues in wireless sensor networks.

A fault is a defect causing an error, which represents an incorrect status of system. Such erroneous status may lead the entire network to a failure in which the system deviates from its specification and loses its intended functionality.

As we know, sensor nodes are prone to suffer various faults and attacks[1] [START_REF] Yick | Wireless sensor network survey[END_REF]. There are many different types of fault in wireless sensor networks, it is extremely hard to design a fault tolerance technique which solves all kinds of problem. For our parallel computing scheme, we have identified three main possible errors during computation.

-Missing result : The master node needs to receive every single result from its slaves to form the final result. Missing results will make the master unable to complete the parallel computing. The missing result may due to various faults such as battery depletion, radio interference and physical attack. -Incorrect result : The results sent back to master node might be wrong because of low battery level, radio interference and other internal errors of sensor nodes. If the master node compute the final result based on incorrect slave results, the entire encryption process will fail. -Faulty cluster head : The cluster head plays the role of master node during parallel computing, if the cluster head becomes faulty in the course of task distribution and results reception, the parallel computing process has to be aborted.

As we can notice that either of these errors may cause the parallel computing scheme to fail. Before elaborating our fault tolerance mechanism, we will see in the following section a few fault tolerance techniques proposed in literature.

Fault tolerance techniques mainly consist of 2 actions, fault detection which aims to detect malfunction in sensor networks, and fault recovery which recovers the system from incorrect status [START_REF] Mishra | Fault tolerance in wireless sensor networks[END_REF]. The main technique is to replace the faulty component by a new one, which is logical, since sensor network is constituted of large number of low-cost sensors. When a sensor node becomes faulty, it is worth nothing that we try to locate and replace it manually. The best way is to replace it by an other node which is already deployed in the zone.

A anomaly detection method based on analysis of sensed data is presented in [START_REF] Rajasegarar | Distributed anomaly detection in wireless sensor networks[END_REF]. We suppose sensors S = {s 1 , s 2 , . . . , s n } are connected in a hierarchical topology. At every time interval ∆ k , each sensor measures a feature vector x i k , and very vector contains attributes x i k = {v i kj : j = 1 . . . d}. Two approaches are presented, in centralized way, every sensor sends its measured vectors X i = {x i k : k = 1 . . . m} to its immediate parent, which merges the received data with its own vectors and then send them to its parent. All vectors are sent back step by step to the gateway node, on which a clustering algorithm [START_REF] Eskin | A geometric framework for unsupervised anomaly detection[END_REF] is executed. Vectors belonging to a cluster are kept, the others are discarded. The inconvenience of this approach is the big data transmission load which may reduce the lifetime of the network. In distributed way, the clustering algorithm is applied at every step, only necessary statistics are sent to the next step for following analysis. The volume of data to transmit can be considerably reduced.

An other method to identify faulty node is to launch a vote. In [START_REF] Chen | Distributed fault detection of wireless sensor networks[END_REF] sensors are deployed randomly in a zone, and we suppose that every sensor has at least 3 neighbors. Sensors are considered as neighboring sensors if they're in the radio range of each other, and each sensor broadcasts periodically to the neighbors its measured data which is then stored in their memory.

Sensors deployed in the same area are supposed to have similar measured values. A sensor S i is interested in history data when more then half of its neighbors have significantly different values. It can use ∆d ∆t l ij which represents the difference between measured value of sensor S i and S j from time t l+1 to t l . The status of the sensor T i is likely faulty (LF) if the measurements change over time significantly.

The status of sensor S i is good(GD) if ∀S j ∈ N (S i ) and

T j = LG (likely good), (1 -c ij ) -c ij ≥ |N (S i )|/2 ,
where N (S i ) represents all neighbors of S i and c ij is a test result generated by S i and its neighbors. c ij = 0 if S i and S j have the same status, otherwise c ij = 1.

In [START_REF] Gupta | Fault-tolerant clustering of wireless sensor networks[END_REF], a fault tolerant clustering mechanism is proposed for sensor networks. Cluster heads evaluate their status and diffuse status update messages to other cluster heads through inter-cluster communication. Faulty cluster head can be identified by vote of all cluster heads. Once the identification is done, the network pass to the next step, system recovery.

If a cluster head becomes faulty, all its cluster members are then divided into smaller groups and merged with neighboring clusters. A sensor S j belongs to cluster head

G i if S i ∈ RSet Gi ⇔ (R Gi > d Sj →Gi ) ∧ (R Sj.max > d Sj →Gi ) where
RSet Gi is the range set of G i , d Sj →Gi is the distance between G i and S j , R Gi and R Sj.max are respectively the ranges of G i and S j . Once S j is assigned to G i , a final set F Set Gi is constructed based on minimum communication cost between sensors and cluster head.

A cluster head can still construct a backup set, denoted BSet, containing sensors which belong to its RSet, but not its F Set. S j ∈ BSet Gi ⇔ (S j ∈ RSet Gi ) ∧ (S j / ∈ F Set Gi ). If the cluster head of S j becomes faulty, it can be recovered if it's present in an other cluster head's BSet.

In a landslide monitoring application presented in [START_REF] Raj | Fault tolerant clustering approaches in wireless sensor network for landslide area monitoring[END_REF], sensors are formed in clusters which contain a cluster head (CH) and a node leader (NL). The NL aggregates the data collected from other members and sends it to CH, which will then forward the data to base station via multi-hop transmission.

For fault tolerance purpose, the intersection zone of 2 overlapping clusters is considered as a sub cluster, which contains only a NL. When a CH is failed, its NL will send the latest aggregated data to the NL of the sub cluster, then the data will be forwarded to the CH of the other cluster. If NL is failed, its CH will take over its work, and a NL election algorithm will be executed at the same time.

We can notice that in sensor networks, one sensor node is barely able to detect system malfunction or recover the network from faulty status alone. The idea of sensor networks is to make sensor nodes cooperate together to achieve a common goal. For fault tolerance, sensor nodes should also work collectively for fault detection and system recovery. We can also see that the basic method of system recovery is to prepare a backup node which takes over the work of the faulty node.

Countermeasures Against Sensor Node Failure

According to [START_REF] Koushanfar | Fault tolerance in wireless sensor networks[END_REF], fault tolerance techniques can be designed for and applied to (from lowest to highest) hardware, system software, middleware and application layers. Generally speaking, techniques designed for lower layer, such as hardware layer, are more generic, which means they can usually be applied to different applications. However higher layer techniques are more application-specific, a fault tolerance method is designed specially for a particular application, and it can hardly be used in other applications.

Certainly, fault tolerance methods for application layer are not as portable as the other ones designed for lower layers, but they can be carefully tailored to meet the specific needs of applications. For our parallel computing scheme, we have decided to design and deploy our fault tolerance technique on application layer.

Our fault tolerance method is also based on the idea of backup nodes. Suppose that there are n available members in a cluster, and the cluster head CH needs to perform a parallel scalar multiplication. Instead of involve all available cluster members in parallel computing, we use only n 2 members for parallel computing, the rest of cluster members are used as backup nodes. If the probability of a member is faulty is p, which also means the probability that a parallel scalar multiplication fails. The use of backup node can reduce the probability of system failure to p/2.

The selection of slaves nodes and of backup nodes is driven by algorithm 1 where L a is the list of all available cluster members, L s and L b are respectively 2 lists of selected slave nodes and backup nodes.

Algorithm 1: Algorithm of backup selection

Data: La Result: Ls, L b Sort(La); forall the ϕ ∈ La do ω ← getBackup(ϕ); La ← La -ϕ; La ← La -ω; Ls ← Ls + ϕ; L b ← L b + ω; end
Before node selection, we sort L a by the number of its available neighbors in ascending order, since we want the ϕ having the least neighbors to choose its backup node first. getBackup() is a function which returns the most suitable backup node for ϕ. Every ϕ is evaluated by multiple parameters, such as RSSI, remaining energy, distance to ϕ. We always choose a backup node having the most reliable connection.

As we have explained in our last paper [START_REF] Shou | Parallel scalar multiplication on elliptic curves in wireless sensor networks[END_REF], to avoid large radio communication overhead, the number of slave nodes participating in parallel computing should be limited at less than 5. Thus the possibility for performing fault tolerant parallel scalar multiplication depends on the number of available cluster members, denoted by n :

n ∈ [0, 1]: Unable to perform fault tolerant parallel computing.

n ∈]1, 8[: Parallel computing using n 2 slaves and n 2 backup nodes.

n ∈ [8, ∞]: Parallel computing using 4 slaves and 4 backup nodes.

Result detection

To deal with the missing result error, we ask the backup node to prepare an other copy of result for the slave that it's monitoring. We can see in figure 2 that tasks are sent not only to slave nodes, but also to backup nodes. Both slave node and backup node have the same task. After computation, if a backup node detects that the slave node doesn't send result back to master node, it will send its local copy instead of the slave. Suppose that all cluster members have the same distance to their cluster head. The energy cost of wireless transmission is a function of message size m and distance d, E Tx = e(m, d), and the energy cost of reception is a constant E Rx . We may notice that the parallel scalar multiplication with result detection doesn't consume more energy than the original parallel computing scheme. However, the fault tolerant version takes a little more time, since backup nodes have to wait a small period before sending its result.

Result verification

For incorrect result error, the slave node does send result back to its master, but an incorrect one. Just like the case of missing result, both slave node and backup node have the same task.

Backup node is supposed to verify the result sent by the slave node. When slave node sends its result, as the backup node is in its radio range, it can also receive the result. Backup node compares the result with its local one. If the results are different, the backup node will send immediately a warning message to their master node so that the result is discarded, and the task in question will be repeated locally by the master node.

When no fault occurs, the system consumes almost the same energy and time than the original version. But when some slave nodes become faulty, the energy and time cost is proportional to the number of faulty nodes.

For the third case, when cluster head is faulty, the system has to execute a cluster head election algorithm such as the one described in [START_REF] Heinzelman | Energy-efficient communication protocol for wireless microsensor networks[END_REF].

Simulation and Performance Evaluation

To assess the performance of our method, we have created a simulator in Java which estimates the running time and the energy cost of our fault tolerant parallel computing scheme.

The simulator creates a random sensor cluster in an ideal free space without obstacle, and asks the cluster members to perform parallel scalar multiplications. We suppose that the cluster uses a TDMA based MAC protocol which allows only one sensor to send data at a time. Moreover, all nodes share the same elliptic curve which is defined over a finite prime field and uses N IST 192 recommended parameters [START_REF] Hankerson | Guide to elliptic curve cryptography[END_REF] We also suppose that the sensor nodes communicate with each other using Zigbee protocol whose throughput is studied in [START_REF] Burchfield | Maximizing throughput in zigbee wireless networks through analysis, simulations and implementations[END_REF]. As in sensor networks, most of the energy is consumed during wireless communication, only the energy cost of radio communication is taken into account, and the formula for energy cost estimation is presented in [START_REF] Heinzelman | Energy-efficient communication protocol for wireless microsensor networks[END_REF] and [START_REF] Heinzelman | An applicationspecific protocol architecture for wireless microsensor networks[END_REF]. In addition, the running time of scalar multiplication is estimated based on the results published in our last paper [START_REF] Shou | Parallel scalar multiplication on elliptic curves in wireless sensor networks[END_REF].

The simulator mainly focus on the execution of our parallel computing protocol. Our goal is not to retrieve the precise values of the running time and the energy cost, but to illustrate the impact of our fault tolerance method on the parallel computing scheme. Tests are run is three different cases:

-There are just enough nodes for parallel computing without fault tolerance, but not enough to have a backup node for each slave participating in computation. -There are enough nodes for parallel computing with the proposed fault tolerance mechanism activated.

-There are enough nodes for fault tolerant parallel computing, but faults occur during the computation.

In our last paper, we have concluded that for efficiency reasons, the maximum number of slaves participating in computation should be limited at 4. We define that n is the number of available cluster members, and the tables 1 and 2 contain the simulation results of the first case where n ≤ 4. The results are also graphically illustrated in figures 4. As the node usage is doubled when our fault tolerance method is applied, so on the one hand, only n/2 nodes can participate in parallel computing and it takes more time to finish the tasks, on the other hand backup nodes don't need to communicate with the cluster head when no fault occurs, the cluster can consumes less energy. In the second case where n ≤ 8 which means the cluster head always has enough slaves to perform parallel scalar multiplication no matter if our fault tolerance method is used. As we can see in table 3 and in figure 5, when our fault tolerance method is used, the cluster needs slightly more time to complete the computation, since more nodes are involved in parallel computing, and backup nodes also need to receive tasks from cluster head during task distribution. The total energy cost increases gradually with increase in number of slaves. The difference of energy cost between the tests with and without fault tolerance is mainly due to the randomness of node deployment. For the 3rd test, we suppose that we may use up to 4 slaves to perform parallel computing, and each slave has a backup node. Some slaves don't function properly during the computation, in this case, backup nodes are supposed to detect the fault and try to recover the computation.

Nb of cluster members

When a backup node detects an erroneous result, it has to send a warning to the cluster head, and the latter will redo the faulty slave's task locally. Thus compared with results without faulty node, we can see in table 4 and figure 6 that the cluster needs more time and energy to finish the computation. The total energy cost increases slightly when more slaves become faulty, since more backup nodes need to warn the cluster head. However as the size of warning message is relatively smaller, the energy cost doesn't increase very fast, its value remains between 0.44mJ and 0.45mJ.

A backup node is also supposed to send result back to cluster head when the slave that it's monitoring doesn't respond at all. However we can notice in table 5 that the running time doesn't increase when more slaves become faulty, since all backup nodes monitor the behavior of slaves and react simultaneously. The irregularity of energy cost in figure 7 is due to the random deployment of sensor nodes. In our power model, the energy consumption is proportional to the distance of wireless communication.

Conclusion

In this paper we have proposed parallel computing scheme to accelerate the scalar multiplication on elliptic curves, we have also designed a fault tolerance mechanism which can significantly improve the reliability of the system. We have tested our method by using a simulator, and the results show a considerable acceleration of computation. Even in case of anomaly, such as sensor failure and incorrect result, the system is still able to detect the error and recover the system from errors. In addition, when no fault occurs, this fault tolerance mechanism doesn't have serious negative impact in term of running time and energy cost. The only drawback of the parallel computing scheme is the energy consumption since nodes have to communicate with each other for task distribution and result retrieval. Thus it shouldn't be used as the default computation scheme in wireless sensor networks. it can only be used in cases where running time is the most critical factor, like in disaster monitoring and military applications.

Fig. 1 .

 1 Fig. 1. Parallel Scalar Multiplication Process

Fig. 2 .

 2 Fig. 2. Parallel scalar multiplication with result detection

Fig. 3 .

 3 Fig. 3. Parallel scalar multiplication with result verification

Fig. 4 .

 4 Fig. 4. Running time (ms) and energy cost (mJ) (n ≤ 4)

Fig. 5 .

 5 Fig. 5. Running time (ms) and energy consumption (mJ) (n ≤ 8)

Fig. 7 .

 7 Fig. 7. Performance of the fault tolerance method when missing result occurs

Table 1 .

 1 Performance of the parallel computing scheme (n ≤ 4)

		Running time (ms) Energy cost (mJ)
	0	2308.77	0.06639066
	1	1158.88	0.17869498
	2	777.75	0.27243610
	3	588.81	0.34194090
	4	476.75	0.41024986
	Nb of cluster members Running time (ms) Energy cost (mJ)
	0	2308.70	0.06639066
	1	2308.77	0.06639066
	2	1159.01	0.16637882
	3	1159.01	0.16637882
	4	778.01	0.24765018

Table 2 .

 2 Performance of the fault tolerant parallel computing scheme (n ≤ 4)

Table 3 .

 3 Performance of the fault tolerant parallel computing scheme (n ≤ 8)

	Nb of slaves Running time (ms) Energy cost (mJ)
	0	2308.70	0.06639066
	1	1159.01	0.16637882
	2	778.01	0.24765018
	3	589.20	0.34410666
	4	477.27	0.43921370

Table 4 .

 4 Nb of faulty slaves Running time (ms) Energy cost (mJ) Performance of the fault tolerance method when erroneous result occurs Fig.6. Performance of the fault tolerance method when erroneous result occurs

	0	477.27	0.43921370
	1	938.86	0.44161946
	2	1400.44	0.44564062
	3	1862.03	0.44965998
	4	2323.61	0.45207022
	0	477.27	0.43921370
	1	517.27	0.43859930
	2	517.27	0.47736890
	3	517.27	0.49638458
	4	517.27	0.47632058

Nb of faulty slaves Running time (ms) Energy cost (mJ)

Table 5 .

 5 Performance of the fault tolerance method when missing result occurs