Mohamed Abderrahman Ebde

Hatem Zaag

Construction and stability of a blow up solution for a nonlinear heat equation with a gradient term

We consider a nonlinear heat equation with a gradient term. We construct a blow-up solution for this equation with a prescribed blow-up profile. For that, we translate the question in selfsimilar variables and reduce the problem to a finite dimensional one. We then solve the finite dimensional problem using index theory. The interpretation of the finite dimensional parameters allows us to derive the stability of the constructed solution with respect to initial data.

Introduction

We consider the problem

u t = ∆u + |u| p-1 u + h(u, ∇u), (1)
u(., 0) = u(0) ∈ H ≡ W 1,r (R N ) ∩ W 1,∞ (R N ),
where r is large enough, u(t) : x ∈ R N → u(x, t) ∈ R,

h : R × R N → R such that |h(u, v)| ≤ C + C|u| ᾱ + C|v| α , p > 1, 0 ≤ ᾱ < p and 0 ≤ α < 2p p + 1 . (2)
Note that we make no assumption on the sign of h.

The problem of local existence was already done by Alfonsi and Weissler [START_REF] Alfonsi | Blow up in R n for a parabolic equation with a damping nonlinear gradient term[END_REF]. They proved the uniqueness of the solution u(t) for t ∈ [0, T ), with T ≤ ∞. When T < ∞, the solution u(t) blows up in the following sense : u(t) H → +∞ when t → T.

When α = 2p p+1 , Souplet, Tayachi and Weissler [START_REF] Souplet | Exact self-similar blow-up of solutions of a semilinear parabolic equation with a nonlinear gradient term[END_REF] have managed to construct radial blow-up selfsimilar solutions for the equation [START_REF] Abramowitz | Handbook of mathematical functions with formulas, graphs, and mathematical tables[END_REF], using a shooting method to solve the differential equation. Other blowup results can be found in Chipot and Weissler [START_REF] Chipot | Some blowup results for a nonlinear parabolic equation with a gradient term[END_REF], Bebernes and Eberly [START_REF] Bebernes | Characterization of blow-up for a semilinear heat equation with a convection term[END_REF], Fila [START_REF] Fila | Remarks on blow up for a nonlinear parabolic equation with a gradient term[END_REF], Kawohl and Peletier [START_REF] Kawohl | Observations on blow up and dead cores for nonlinear parabolic equations[END_REF]. In this paper, we focus on the subcritical case in α (α < 2p p+1 ), and aim at constructing a stable blow-up solution of the equation [START_REF] Abramowitz | Handbook of mathematical functions with formulas, graphs, and mathematical tables[END_REF] and to give its behavior at blow-up. Compared to the paper [START_REF] Souplet | Exact self-similar blow-up of solutions of a semilinear parabolic equation with a nonlinear gradient term[END_REF] of Souplet, Tayachi and Weissler, our work has a double interest :

1-We construct a non selfsimilar solution, and give exactly its profile which depends on the variable

z = x (T -t)| log (T -t)| .
Note that the solution of Souplet, Tayachi and Weissler, [START_REF] Souplet | Exact self-similar blow-up of solutions of a semilinear parabolic equation with a nonlinear gradient term[END_REF] depends on another variable

y = x √ T -t .
2-We prove the stability of our solution with respect to perturbations in the initial data.

Our main result is the following :

Theorem 1 (Existence of a blow-up solution for equation [START_REF] Abramowitz | Handbook of mathematical functions with formulas, graphs, and mathematical tables[END_REF] with the description of its profile) There exists T > 0 such that equation [START_REF] Abramowitz | Handbook of mathematical functions with formulas, graphs, and mathematical tables[END_REF] has a solution u(x, t) in R N × [0, T ) such that : i) the solution u blows up in finite time T at the point a = 0, ii)

(T -t) 1 (p-1) u(• √ T -t, t) -f ( . | log (T -t)| ) W 1,∞ ≤ C | log(T -t)| , (3)
where

f (z) = p -1 + (p -1) 2 4p |z| 2 -1 p-1 . (4)
Remark : We note that the profile f is the same as in the nonlinear heat equation without the gradient term see (Bricmont and Kupiainen [4]).

Remark : Note that (i) follows from (ii). Indeed, (ii) implies that u(0, t) ∼ κ(T -t)

-1
p-1 → +∞ as t → T, with κ = f (0) = (p -1)

-1 p-1 . (5)
Remark : Note that estimate (3) holds in W 1,∞ (R), which a novelty of our paper. Indeed in the previous literature about neighboring equations ( [START_REF] Merle | Stability of the blow-up profile for equations of the type u t = ∆u+ |u| p-1 u[END_REF], [START_REF] Zaag | Blow-up results for vector-valued nonlinear heat equations with no gradient structure[END_REF], [START_REF] Masmoudi | Blow-up profile for the complex Ginzburg-Landau equation[END_REF]), the authors have only L ∞ estimates.

Remark : Note that classical methods such as energy estimates or the maximum principle break down for the equation [START_REF] Abramowitz | Handbook of mathematical functions with formulas, graphs, and mathematical tables[END_REF]. Indeed, there is no Lyapunov functional for (1), and the general form of h(u, ∇u) prevents any use of the maximum principle. To our knowledge, theorem 1 is the first existence result of a blow-up solution for equation [START_REF] Abramowitz | Handbook of mathematical functions with formulas, graphs, and mathematical tables[END_REF]. As we point out in theorem 2 below, this solution is stable with respect to perturbations in initial data.

Our approach in this paper is inspired by the method of Bricmont and Kupiainen [START_REF] Bricmont | Universality in blow-up for nonlinear heat equations[END_REF], Merle and Zaag [START_REF] Merle | Stability of the blow-up profile for equations of the type u t = ∆u+ |u| p-1 u[END_REF] for the semilinear heat equation

u t = ∆u + |u| p-1 u. (6)
In some sense, we show in this paper that the term h(u, ∇u) in (1) has a subcritical size when α < 2p p+1 and ᾱ < p. One may think then that our paper is just a simple perturbation of [START_REF] Merle | Stability of the blow-up profile for equations of the type u t = ∆u+ |u| p-1 u[END_REF]. If this is true in the statements, it is not the case for the proof, where we need some involved arguments to control the h(u, ∇u) term (see below Proposition 3.3 page 10 and Lemma 4.1 page 15). Therefore, with respect to [START_REF] Merle | Stability of the blow-up profile for equations of the type u t = ∆u+ |u| p-1 u[END_REF], all that we need to do is to control the contribution h(u, ∇u). We will then refer to [START_REF] Merle | Stability of the blow-up profile for equations of the type u t = ∆u+ |u| p-1 u[END_REF] for the other contributions. However, for the reader's convenience, we will recall the main steps of the proof from [START_REF] Merle | Stability of the blow-up profile for equations of the type u t = ∆u+ |u| p-1 u[END_REF] and give details only for the new term. We wonder whether for α = 2p p+1 , one can show the same result as Theorem 1, namely that equation ( 1) has a blow-up profile depending on the variable

z = x √ (T -t)| log(T -t)| .
We would like to mention that the method of [START_REF] Merle | Stability of the blow-up profile for equations of the type u t = ∆u+ |u| p-1 u[END_REF] has been successful for the following semilinear heat equation with no gradient structure :

u t = ∆u + (1 + iδ)|u| p-1 u (7)
where δ ∈ R is small and u : R N × [0, T ) → C see [START_REF] Zaag | Blow-up results for vector-valued nonlinear heat equations with no gradient structure[END_REF]. Unlike our equation ( 1), note that with respect to [START_REF] Fila | Remarks on blow up for a nonlinear parabolic equation with a gradient term[END_REF], equation ( 7) has the additional term iδ|u| p-1 u whose size is critical with respect to the original nonlinearity |u| p-1 u. Note also that the method of [START_REF] Merle | Stability of the blow-up profile for equations of the type u t = ∆u+ |u| p-1 u[END_REF] gives the existence of a blow-up solution for the following Ginzburg-Landau equation :

u t = (1 + iβ)∆u + (1 + iδ)|u| p-1 u -γu,
where p > 1, u : R N × [0, T ) → C and p -δ 2 -βδ(p + 1) > 0 (see Masmoudi and Zaag [START_REF] Masmoudi | Blow-up profile for the complex Ginzburg-Landau equation[END_REF]).

For simplicity in the parameters, we will give the proof in the particular case where h is equal to its bound, namely when

h(u, v) = μ|u| ᾱ + µ|v| α + µ 0 (8)
for some real numbers, μ, µ and µ 0 .

The proof is done in the selfsimilar variables' framework which we introduce :

y = x √ T -t , s = -log (T -t), w T (y, s) = (T -t) 1 p-1 u(x, t)) ( 9
)
where T is the time where we want the solution to blow up. For y ∈ R N and s ∈ [-log T, +∞), the equation of w T = w is the following

∂ s w = ∆w - 1 2 y • ∇w - 1 p -1 w + |w| p-1 w + µ|∇w| α e -βs + μ|w| ᾱe -βs + µ 0 e -p p-1 s (10)
where β = 2p -α(p + 1) 2(p -1) > 0 and β = p- ᾱ p-1 > 0. Remark : The fact that α and ᾱ are subcritical ( α < 2p p+1 and ᾱ < p) is reflected in the fact that β > 0 and β > 0, which explains the little effect of the gradient term for large times.

From this transformation we see that Theorem 1 reduces to proving the existence of a solution w(y, s) for equation [START_REF] Souplet | Exact self-similar blow-up of solutions of a semilinear parabolic equation with a nonlinear gradient term[END_REF] such that for some s 0 ∈ R, and for all s ≥ s 0 ,

w(y, s) -f ( y √ s ) W 1,∞ ≤ C √ s .
This is "reasonable" in the sense that, instead of being an exact solution of [START_REF] Souplet | Exact self-similar blow-up of solutions of a semilinear parabolic equation with a nonlinear gradient term[END_REF], the function f ( y √ s ) is an approximate solution of (10) (up to the order 1 s ). This is clear from the fact that f (z) satisfies the following

∀z ∈ R N , 0 = 1 2 z • ∇f (z) - f (z) p -1 + f (z) p .
In the w(y, s) framework, the proof relies on the understanding of the dynamics of the selfsimilar version of (1) (see equation ( 10)) around the profile (4). We proceed in 2 steps :

-In Step 1, we reduce the problem to a finite-dimensional problem : we show that it is enough to control a two-dimensional variable in order to control the solution (which is infinite-dimensional) near the profile.

-In Step 2, we proceed by contradiction to solve the finite-dimensional problem and conclude using index theory. As in [START_REF] Merle | Stability of the blow-up profile for equations of the type u t = ∆u+ |u| p-1 u[END_REF], [START_REF] Zaag | Blow-up results for vector-valued nonlinear heat equations with no gradient structure[END_REF] and [START_REF] Masmoudi | Blow-up profile for the complex Ginzburg-Landau equation[END_REF], it is possible to make the interpretation of the finite-dimensional variable in terms of the blow-up time and the blow-up point. This allows us to derive the stability of the profile (3) in Theorem 1 with respect to perturbations in the initial data. More precisely, we have the following.

Theorem 2 (Stability of the solution constructed in Theorem 1). Let us denote by û(x, t) the solution constructed in Theorem 1 and by T its blow-up time. Then, there exists a neighborhood V 0 of û(x, 0) in W 1,∞ such that for any u 0 ∈ V 0 , equation [START_REF] Abramowitz | Handbook of mathematical functions with formulas, graphs, and mathematical tables[END_REF] has a unique solution u(x, t) with initial data u 0 , and u(x, t) blows up in finite time T (u 0 ) at some blow-up point a(u 0 ). Moreover, estimate (3) is satisfied by u(x -a, t) and

T (u 0 ) → T , a(u 0 ) → 0 as u 0 → û0 in W 1,∞ (R N )
The stability result follows from the reduction to a finite dimensional case as in [START_REF] Merle | Stability of the blow-up profile for equations of the type u t = ∆u+ |u| p-1 u[END_REF] for the semilinear heat equation [START_REF] Fila | Remarks on blow up for a nonlinear parabolic equation with a gradient term[END_REF] with the same proof. Since the new term h(u, ∇u) affects the proof of the finite dimensional reduction and not the derivation of the stability from this reduction, we only prove the reduction and refer to [START_REF] Merle | Stability of the blow-up profile for equations of the type u t = ∆u+ |u| p-1 u[END_REF] for the stability.

The paper is organized as follows : -In Section 2, we recall from [START_REF] Merle | Stability of the blow-up profile for equations of the type u t = ∆u+ |u| p-1 u[END_REF] the formulation of the problem.

-In Section 3, we follow the method of [START_REF] Merle | Stability of the blow-up profile for equations of the type u t = ∆u+ |u| p-1 u[END_REF] to prove Theorem 1. Note that we do not give the proof of Theorem 2 since it follows from the proof of Theorem 1 exactly as in the case of the semilinear heat equation 6 treated in [START_REF] Merle | Stability of the blow-up profile for equations of the type u t = ∆u+ |u| p-1 u[END_REF]. We give details only when we handle the new term h(u, ∇u).

Formulation of the problem

For simplicity in the notation, we give the proof in one dimension (N =1), when h is given by [START_REF] Masmoudi | Blow-up profile for the complex Ginzburg-Landau equation[END_REF]. The proof remains the same for N ≥ 2, and for general h satisfying [START_REF] Alfonsi | Blow up in R n for a parabolic equation with a damping nonlinear gradient term[END_REF]. We would like to find u 0 initial data such that the solution u of the equation (1) blows up in finite time T and lim t→T (T -t)

1 (p-1) u(• √ T -t, t)-f ( . | log (T -t)| ) W 1,∞ = 0, where f (z) = (p -1+ (p -1) 2 4p |z| 2 ) -1 p-1 .
Using the change of variables [START_REF] Merle | Stability of the blow-up profile for equations of the type u t = ∆u+ |u| p-1 u[END_REF], this is equivalent to finding s 0 > 0 and w 0 (y) such that the solution w(y, s) of ( 10) with initial data w(y, s 0 ) satisfies

lim s→∞ w(y, s) -f y √ s W 1,∞ = 0. Introducing w = ϕ + q where ϕ = f y √ s + κ 2ps , (11)
where f and κ are defined in (4), and ( 5), the problem is then reduced to constructing a function q such that lim

s→+∞ q(., s) W 1,∞ = 0 ( 12
)
and q is a solution of the following equation for all (y, s) ∈ R × [s 0 , ∞)

∂ s q = (L + V )q + B(q) + R(y, s) + N (y, s), (13)
where

L = ∆ - 1 2 y • ∇ + 1, V (y, s) = p ϕ(y, s) p-1 - p p -1 , (14)
B(q) = |ϕ + q| p-1 (ϕ + q) -ϕ q -pϕ p-1 q, (15) R(y, s) = ∆ϕ - 1 2 y ∇ϕ - ϕ p -1 + ϕ p - ∂ϕ ∂s and N (y, s) = µ|∇ϕ + ∇q| α e -βs + μ|ϕ + q| ᾱe -βs + µ 0 e -p p-1 s (16)
Let K(s, σ) be the fundamental solution of the operator L + V . Then, for each s ≥ σ ≥ s 0 , we have :

q(s) = K(s, σ)q(σ) + s σ dτ K(s, τ )B(q(τ )) + s σ dτ K(s, τ )R(τ ) + s σ dτ K(s, τ )N (τ ). ( 17
)
In comparison with the case of equation ( 6) treated in [START_REF] Merle | Stability of the blow-up profile for equations of the type u t = ∆u+ |u| p-1 u[END_REF], all the terms in (13) were already present in [START_REF] Merle | Stability of the blow-up profile for equations of the type u t = ∆u+ |u| p-1 u[END_REF], except N (y, s) which needs to be carefully handled. Therefore, using Lemma 3.15 page 168 and Lemma 3.16 page 169 of [START_REF] Merle | Stability of the blow-up profile for equations of the type u t = ∆u+ |u| p-1 u[END_REF], we see that

|B(q)| ≤ C|q| p and R(., s) L ∞ ≤ C s (18)
for s large enough, where p = min(p, 2). Using the definition (11) of ϕ, we see that

|N (y, s)| ≤ Ce -βs ∇q α L ∞ + Ce -βs q ᾱ L ∞ + Ce -β 0 s (19)
where β 0 = min (β, β) > 0. It is then reasonable to think that the dynamics of equation ( 17) are influenced by the linear part. Hence, we first study the operator L (see page 543 in Bricmont and Kubiainen [START_REF] Bricmont | Universality in blow-up for nonlinear heat equations[END_REF] and pages 773-775 in Abramowitz and Stegun [START_REF] Abramowitz | Handbook of mathematical functions with formulas, graphs, and mathematical tables[END_REF]). The operator

L is self-adjoint in D(L) ⊂ L 2 ρ (R)
where

ρ(y) = e -|y| 2 4 √ 4π and L 2 ρ (R) = {v ∈ L 2 loc (R) such that R v(y) 2 ρ(y)dx < +∞}.
The spectrum of L is explicitly given by

spec(L) = {1 - m 2 |m ∈ N}.
It consists only in eigenvalues. All the eigenvalues are simple, and the eigenfunctions are dilation of Hermite's polynomial : the eigenvalue 1 -m 2 corresponds to the following eigenfunction :

h m (y) = [ m 2 ] n=0 m! n!(m -2n)! (-1) n y m-2n . (20)
Notice that h m satisfies :

h n h m ρdx = 2 n !δ nm . We also introduce k m = h m / h m 2 L 2 ρ (R) .
As it is mentioned in [START_REF] Merle | Stability of the blow-up profile for equations of the type u t = ∆u+ |u| p-1 u[END_REF], the potential V has two fundamental properties which will strongly influence our strategy. (a) we have V (., s) → 0 in L 2 ρ (R) when s → +∞. In practice, the effect of V in the blow-up area (|y| ≤ C √ s) is regarded as a perturbation of the effect of L.

(b) outside of the blow-up area, we have the following property : for all ǫ > 0, there exists C ǫ > 0 and s ǫ such that

sup s≥sǫ, |y| √ s ≥Cǫ V (y, s) -(- p p -1 ) ≤ ǫ, with -p p-1 < -1.
As 1 is the largest eigenvalue of the operator L, outside the blow-up area we can consider that the operator L + V is an operator with negative eigenvalues, hence, easily controlled. Considering the fact that the behavior of V is not the same inside and outside the blow-up area, we decompose q as follows :

If

χ 0 ∈ C ∞ 0 ([0, +∞)) with supp(χ 0 ) ⊂ [0, 2] and χ 0 ≡ 1 in [0, 1], we define χ(y, s) = χ 0 |y| K 0 s 1 2 (21)
with K 0 > 0 to be fixed large enough. If q = q b + q e , with q b = qχ and q e = q(1 -χ),

we remark that supp q b (s) ⊂ B(0, 2K 0 √ s), and supp q e (s) ⊂ R N \ B(0, K 0 √ s).

We write

q b (y, s) = 2 m=0 q m (s)h m (y) + q -(y, s), (23)
with q m is the projection of q b on h m and q -(y, s) = P -(q b ) where P -is the projection in the negative subspace of the L. Thus, we can decompose q in 5 components as follows :

q(y, s) = 2 m=0 q m (s)h m (y) + q -(y, s) + q e (y, s). (24)
Here and throughout the paper, we call q -the negative mode of q, q 0 the null mode of q, and the subspace spanned by {h m /m ≥ 3} will be referred to as the negative subspace.

Proof of the existence result

This section is devoted to the proof of the existence result (Theorem 1). We proceed in 4 steps, each of them making a separate subsection.

-In the first subsection, we define a shrinking set V A (s) and translate our goal of making q(s) go to 0 in L ∞ (R) in terms of belonging to V A (s). We also exhibit a two-parameter initial data family for equation (13) whose coordinates are very small (with respect to the requirements of V A (s)), except the two first q 0 and q 1 . -In the second subsection, we solve the local in time Cauchy problem for equation (13). -In the third subsection, using the spectral properties of equation (13), we reduce our goal from the control of q(s) (an infinite-dimensional variable) in V A (s) to the control of its two first components (q 0 , q 1 ) (a two-dimensional variable) in [-A s 2 , A s 2 ] 2 . -In the fourth subsection, we solve the finite-dimensional problem using index theory and conclude the proof of the theorem 1.

Definition of a shrinking set V A (s) and preparation of initial data

Let first introduce the following proposition : Proposition 3.1 (A set shrinking to zero) For all A ≥ 1 and s ≥ e, we define V A (s) as the set of all functions r in L ∞ such that :

|r m (s)| ≤ As -2 , m = 0, 1, |r 2 (s)| ≤ A 2 (log s)s -2 , ∀y ∈ R, |r -(y, s)| ≤ A(1 + |y 3 |)s -2 , ||r e (s)|| L ∞ ≤ A 2 s -1 2 ,
where r -, r e and r m are defined in (24). Then we have for all s ≥ e and r ∈ V A (s),

(i) for all y ∈ R, |r b (y, s)| ≤ CA 2 log s s 2 (1 + |y| 3 ), (ii) r(s) L ∞ ≤ CA 2 √ s ,
.

Proof : (i) For all s ≥ e and r(s) ∈ V A (s), we have

r b (y, s) = 2 m=0 r m (s)h m (y) + r -(y, s) • 1 {|y|≤2K 0 √ s} (y, s)
Using the definition (20) of h m we get : If we consider as initial data for the equation ( 13) the following affine function :

|r b (y, s)| ≤ C(1 + |y|) A s 2 + C(1 + |y| 2 )A 2 log s s 2 + C(1 + |y| 3 ) A s 2 ≤ CA 2 log s s 2 (1 + |y| 3 ) (25
q d 0 ,d 1 (y, s 0 ) = f ( y √ s 0 ) p (d 0 + d 1 y √ s 0 ) - κ 2ps 0 ( 26
)
where f is defined in ( 4) and s 0 =log T, then : (i) There exists a rectangle

D T ⊂ [- c s 0 , c s 0 ] 2 (27)
such that the mapping (d 0 , d 1 ) → (q 0 (s 0 ), q 1 (s 0 )) (where q stands q d 0 ,d 1 (s 0 )) is linear and one to one from

D T onto [-A s 2 0 , A s 2 0 ] 2 and maps ∂D T into ∂[-A s 2 0 A s 2 0 ] 2 .
Moreover, it is of degree one on the boundary. (ii) For all (d 0 , d 1 ) ∈ D T we have

|q 2 (s 0 )| ≤ log s 0 s 2 0 , |q -(y, s 0 )| ≤ c s 2 0 (1 + |y| 3 ) and q e (., s 0 ) L ∞ ≤ 1 √ s 0 , |d 0 | + |d 1 | ≤ c s 0 , (28)
∇q(•, s 0 ) L ∞ ≤ C 1 |d 0 | √ s 0 + C 2 |d 1 | √ s 0 ≤ C √ s 0 . ( 29
) (iii) For all (d 0 , d 1 ) ∈ D T , q d 0 ,d 1 ∈ V A (s 0 )
with "strict inequalities" except for (q 0 (s 0 ), q 1 (s 0 )), in the sense that

|q m (s)| ≤ As -2 , m = 0, 1, |q 2 (s)| < A 2 (log s)s -2 , ∀y ∈ R, |q -(y, s)| < A(1 + |y 3 |)s -2 , ||q e (s)|| L ∞ < A 2 s -1 2 ,
Proof : Since we have the same definition of the set V A , and the same expression (26) of initial data q(d 0 ) as in [START_REF] Merle | Stability of the blow-up profile for equations of the type u t = ∆u+ |u| p-1 u[END_REF], we refer the reader to Lemma 3.5 and the Lemma 3.9 in [START_REF] Merle | Stability of the blow-up profile for equations of the type u t = ∆u+ |u| p-1 u[END_REF] except for the bound (29) which is new and which we prove in the following (note that although ( 27), (28) are not stated explicitly in Lemma 3.5 of [START_REF] Merle | Stability of the blow-up profile for equations of the type u t = ∆u+ |u| p-1 u[END_REF], they are clearly written in its proof). In addition, for the readers convenience, we give some hint of the proof of (i).

Hint of the proof of (i)(for details, see pages 156-157 in [START_REF] Merle | Stability of the blow-up profile for equations of the type u t = ∆u+ |u| p-1 u[END_REF]) : We recall form page 157 in [START_REF] Merle | Stability of the blow-up profile for equations of the type u t = ∆u+ |u| p-1 u[END_REF] the mapping

(d 0 , d 1 ) → (q 0 (s 0 ), q 1 (s 0 )) = (d 0 a 0 (s 0 ) + b 0 (s 0 ), d 1 a 1 (s 0 ) + b 1 (s 0 )),
which reduces in our case to a linear mapping :

(d 0 , d 1 ) → (q 0 (s 0 ), q 1 (s 0 )) = (d 0 a 0 (s 0 ), d 1 a 1 (s 0 )),
with a 0 (s 0 ) = 0 and a 1 (s 0 ) = 0. Therefore, it is clear that

D T is rectangle (D T is by construction the set inverse image of [-A s 2 0 , A s 2 0 ] 2 )
and if (d 0 , d 1 ) ∈ ∂D T , then (q 0 (s 0 ),

q 1 (s 0 )) ∈ ∂[-A s 2 0 , A s 2 0
] 2 . Proof of (29) : Using(26), we have

∇q(y, s 0 ) = p d 0 √ s 0 f ′ (z)f p-1 (z) + d 1 √ s 0 f p (z) + p d 1 s 0 yf ′ (z)f p-1 (z)
, where z = y √ s 0 and f is given in (4). Since f ′ (z) = -p-1 2p zf (z) p , by (4), this gives,

∇q(y, s 0 ) = f (z) p √ s 0 (z p d 0 f (z) p-1 + d 1 + p d 1 z 2 f (z) p-1 ).
From (4) we can see clearly that f p , zf p-1 and z 2 f p-1 are in L ∞ (R) and we get from (28) :

|∇q(y, s 0 )| ≤ C 1 (p) |d 0 | √ s 0 + C 2 (p) |d 1 | √ s 0 ≤ C √ s 0
and ((29)) follows. This concludes the proof of Proposition 3.2.

Because of the presence of |∇q| α in the equation (13), we need the following parabolic regularity estimate for equation (13), with q(s 0 ) given by (26) and q(s) ∈ V A (s). More precisely, we have the following :

Proposition 3.3 (Parabolic regularity for equation ( 13)) For all A ≥ 1, there exists T1 (A) ≤ T 1 (A) such that for all T ≤ T1 (A), if q(s) is a solution of equation ( 13) on [s 0 , s 1 ] where s 1 ≥ s 0 = log T , q(s 0 ) is given by ( 26) with (d 0 , d 1 ) ∈ D T and

q(s) ∈ V A (s) for all s ∈ [s 0 , s 1 ], (30)
then, for some C 1 > 0, we have

∇q(s) L ∞ ≤ C 1 A 2 √ s for all s ∈ [s 0 , s 1 ].
Proof : We consider A ≥ 1, T ≤ T 1 (A) and q(s) a solution of equation ( 13) defined on [s 0 , s 1 ] where s 1 ≥ s 0 =log T and q(s 0 ) given by ( 26) with (d 0 , d 1 ) ∈ D T . We also assume that q(s) ∈ V A (s) for all s ∈ [s 0 , s 1 ]. In the following, we handle two cases : s ≤ s 0 + 1 and s ≥ s 0 + 1.

Case 1 : s ≤ s 0 + 1. Let s ′ 1 = min(s 0 + 1, s 1 ) and take s ∈ [s 0 , s ′ 1 ]. Then we have for any t ∈ [s 0 , s],

s 0 ≤ t ≤ s ≤ s 0 + 1 ≤ 2s 0 , hence 1 2s 0 ≤ 1 s ≤ 1 t ≤ 1 s 0 . ( 31
)
From equation (13), we write for any s ∈ [s 0 , s ′ 1 ],

q(s) = e (s-s 0 )L q(s 0 ) + s s 0 e (s-t)L F (t)dt, ( 32
)
where

F (x, t) = V (x, t)q + B(q) + R(x, t) + N (x, t) (33)
and from page 545 of [START_REF] Bricmont | Universality in blow-up for nonlinear heat equations[END_REF], for all θ > 0,

e θL (y, x) = e θ 4π(1 -e -θ ) exp [- (ye -θ/2 -x) 2 4(1 -e -θ ) ]. ( 34
)
Since we easily see from (34) that for any r ∈ W 1,∞ and θ > 0,

∇(e θL r) L ∞ ≤ Ce θ 2 ∇r L ∞ ,
we write from (32), for all s ∈ [s 0 , s

′ 1 ] ∇q(s) L ∞ ≤ ∇e (s-s 0 )L q(s 0 ) L ∞ + s s 0 ∇(e (s-t)L )F (t) L ∞ dt ≤ C ∇q(s 0 ) L ∞ + C s s 0 F (t) L ∞ 1 -e -(s-t) dt. (35)
Using ( 29) and (31), we write

∇q(s 0 ) L ∞ ≤ C √ s 0 ≤ C √ s . ( 36
)
Since V (t) L ∞ ≤ C (see ( 14)), using (30), (ii) of Proposition 3.1, ( 18) and ( 19) and (31), we write for all t ∈ [s 0 , s] and x ∈ R,

|F (x, t)| ≤ CA 2 √ t + e -βt ∇q(t) α L ∞ ≤ C(A 2 s -1 2 + e -βs ∇q(t) α L ∞ ). ( 37
)
Therefore, from ( 35), ( 36) and (37) we write with g(s

) = ∇q(s) L ∞ , g(s) ≤ CA 2 √ s + C s s 0 s -1 2 + e -βs g(t) α 1 -e -(s-t) dt ≤ CA 2 √ s + Ce -βs s s 0 g(t) α 1 -e -(s-t) dt. (38)
Using a Gronwall's argument, we see that for s 0 large enough, we have

∀s ∈ [s 0 , s ′ 1 ], g(s) ≤ 2CA 2 √ s .
Case 2 : s ≥ s 0 + 1 (note that this case does not occur when s 1 ≤ s 0 + 1). From equation (13), we write for any

s ′ ∈ [s -1, s], q(s ′ ) = e (s ′ -s+1)L q(s -1) + s ′ s-1 e (s ′ -t)L F (t)dt, ( 39
)
where F (x, t) and e θL are given in ( 33) and (34). Since we easily see from (34) that for any r ∈ L ∞ and θ > 0,

∇(e θL r) L ∞ ≤ Ce θ 2 √ 1 -e -θ r L ∞ ,
we write from (39), for all 30) and (ii) of Proposition 3.1, we use the fact that V (s -1) L ∞ ≤ C (see ( 14)) and ( 18) and ( 19) to write for all t ∈ [s -1, s ′ ] and x ∈ R,

s ′ ∈ [s -1, s] ∇q(s ′ ) L ∞ ≤ ∇e (s ′ -s+1)L q(s -1) L ∞ + s ′ s-1 ∇(e (s ′ -t)L )F (t) L ∞ dt ≤ C 1 -e -(s ′ -s+1) q(s -1) L ∞ + C s ′ s-1 F (t) L ∞ 1 -e (s ′ -t) dt. ( 40
) Since q(s ′ ) L ∞ ≤ CA 2 √ s ′ ≤ CA 2 √ s-1 ≤ CA 2 √ s from (
|F (x, t)| ≤ CA 2 √ t + e -βt ∇q(t) α L ∞ ≤ C((A 2 s ′ -1 2 + e -βs ′ ∇q(t) α L ∞ ).
Therefore, from (40), we write with g(s

′ ) = ∇q(s ′ ) L ∞ g(s ′ ) ≤ CA 2 √ s ′ 1 -e -(s ′ -s+1) + C s ′ s-1 s ′ -1 2 + e -βs ′ g(t) α 1 -e -(s ′ -t) dt ≤ CA 2 √ s ′ 1 -e -(s ′ -s+1) + Ce -βs ′ s ′ s-1 g(t) α 1 -e -(s ′ -t) dt. (41)
Using a Gronwall's argument, we see that for s large enough,

∀s ′ ∈ [s -1, s], g(s ′ ) ≤ 2CA 2 √ s ′ 1 -e -(s ′ -s+1) .
Taking s ′ = s concludes the proof of Proposition 3.3.

Local in time solution of equation (13)

In the following, we find a local in time solution for equation (13).

Proposition 3.4 (Local in time solution for equation (13) with initial data (26)) For all A ≥ 1, there exists T 2 (A) ≤ T1 (A) such that for all T ≤ T 2 , the following holds :

For all (d 0 , d 1 ) ∈ D T with initial data at s = s 0 q 0 (s 0 ) defined in (26), there exists a s max > s 0 such that equation ( 13) has a unique solution satisfying q(s) ∈ V A+1 (s) for all s ∈ [s 0 , s max ).

Proof. From the definition of q in (11) we can see that the Cauchy problem of ( 13) is equivalent to that of equation [START_REF] Souplet | Exact self-similar blow-up of solutions of a semilinear parabolic equation with a nonlinear gradient term[END_REF] which is equivalent to the Cauchy problem of equation [START_REF] Abramowitz | Handbook of mathematical functions with formulas, graphs, and mathematical tables[END_REF]. Moreover, the initial data q 0 defined in (26) gives the following initial data of (1) :

u 0,d 1 ,d 2 (x) = T -1 (p-1)    f (z)   1 + d 0 + d 1 z p -1 + (p-1) 2 4p z 2      , where z = x(| log T |T ) -1 2 .
which belongs to H ≡ W 1,r (R N ) ∩ W 1,∞ (R N ) for r which insures the local existence (see the introduction) of u in H. Now, since we have from (iii) of Proposition 3.2, q 0 ∈ V A (s 0 ) V A+1 (s 0 ), there exists s 3 such that for all s ∈ [s 0 , s 3 ), q(s) ∈ V A+1 (s). This concludes the proof of Proposition 3.4.

Reduction to a finite-dimensional problem

This step is crucial in the proof of Theorem 1. In this step, we will prove through a priori estimates that for each s ≥ s 0 , the control of q(s) in V A (s) is reduced to the control of (q 0 , q 1 )(s) in [-A s 2 , A s 2 ] 2 . In fact, this result implies that if for some s 1 ≥ s 0 , q(s 1 ) ∈ ∂V A (s 1 ) then (q 0 (s 1 ),

q 1 (s 1 )) ∈ ∂[-A s 2 , A s 2 ] 2 .
Proposition 3.5 (Control of q(s) by (q 0 (s), q 1 (s)) in V A (s)) There exists A 3 > 0 such that for each A ≥ A 3 there exists T 3 (A) ≤ T 2 (A) such that for all T ≤ T 3 (A), the following holds : If q is a solution of ( 13) with initial data at s = s 0 =log T given by ( 26) with (d 0 , d 1 ) ∈ D T , and q(s) ∈ V A (s) for all s ∈ [s 0 , s 1 ], with q(s 1 ) ∈ ∂V A (s 1 ) for some s 1 ≥ s 0 , then :

(i) (Reduction to a finite dimensional problem) (q 0 (s 1 ), q 1 (s 1 )) ∈ ∂[-A s 2 1 , A s 2 1 ] 2 . (ii) (Transverse crossing) There exist m ∈ {0, 1} and ω ∈ {-1, 1} such that ωq m (s 1 ) = A s 2 1
and ω dq ds (s 1 ) > 0.

Remark. In N dimensions, q 0 ∈ R and q 1 ∈ R N . In particular, the finite-dimensional problem is of dimension N + 1. This is why in initial data (26), one has to take d 0 ∈ R and d 1 ∈ R N .

Proof : Let us consider A ≥ 1 and T ≤ T 2 (A). We then consider q a solution of (13) with initial data at s = s 0 =log T given by ( 26) with (d 0 , d 1 ) ∈ D T , and q(s) ∈ V A (s) for all s ∈ [s 0 , s 1 ], with q(s 1 ) ∈ ∂V A (s 1 ) for some s 1 ≥ s 0 . We now claim the following : Proposition 3.6 There exists A 4 ≥ 1 such that for all A ≥ A 4 and ρ ≥ 0, there exists s 4 (A, ρ) ≥ log T 2 (A) such that the following holds for all s 0 ≥ s 4 (A, ρ) :

Assume that for all s ∈ [τ, τ + ρ] for some τ ≥ s 0 , q(s) ∈ V A (s) and ∇q(s)

L ∞ ≤ CA 2 √ s .
Then, the following holds for all s ∈ [τ, τ + ρ] : (i) (ODE satisfied by the expanding modes) For m = 0 and 1, we have

q ′ m (s) -(1 - m 2 )q m (s) ≤ C s 2 .
(ii) (Control of the null and negative modes) we have

|q 2 (s)| ≤ τ 2 s 2 |q 2 (τ )| + CA(s -τ ) s 3 , q -(s) 1 + |y| 3 L ∞ ≤ Ce -(s-τ ) 2 q -(τ ) 1 + |y| 3 L ∞ + C e -(s-τ ) 2 q e (τ ) L ∞ s 3/2 + C(s -τ ) s 2 , q e (s) L ∞ ≤ e -(s-τ ) p q e (τ ) L ∞ + e s-τ s 3/2 q -(τ ) 1 + |y| 3 L ∞ + (s -τ ) s 1/2 .
Proof : See Section 4. Using Propositions 3.6 and 3.2, one can see that Proposition 3.5 follows exactly as in the case of semilinear equation ( 6) treated in [START_REF] Merle | Stability of the blow-up profile for equations of the type u t = ∆u+ |u| p-1 u[END_REF]. The proof is easy, however, a bit technical. That is the reason why it is omitted. The interested reader can find details in pages (164-160) of [START_REF] Merle | Stability of the blow-up profile for equations of the type u t = ∆u+ |u| p-1 u[END_REF] for (i), and in page 158 of [START_REF] Merle | Stability of the blow-up profile for equations of the type u t = ∆u+ |u| p-1 u[END_REF] for (ii). This concludes the proof of Proposition 3.5.

Proof of the finite dimensional problem and proof of Theorem 1

We prove Theorem 1 using the previous subsections. We proceed in two parts.

-In Part 1, we solve the finite-dimensional problem and prove the existence of A ≥ 1, T > 0, (d 0 , d 1 ) ∈ D T such that problem (13) with initial data at s = s 0 =log T, q d 0 ,d 1 (s 0 ) given by (12) has a solution q(s) defined for all s ∈ [-log T, ∞) such that q(s) ∈ V A (s) for all s ∈ [-log T, +∞).

(

) 42
-In Part 2, we show that the solution constructed in Part 1 provides a blow-up solution of equation ( 1) which satisfies all the properties stated in Theorem 1, which concludes the proof.

Part 1 : Solution of the 2-dimensional problem This part in not new. Indeed, given Proposition 3.5, the solution of the 2-dimensional problem is exactly the same as in [START_REF] Merle | Stability of the blow-up profile for equations of the type u t = ∆u+ |u| p-1 u[END_REF] (see also [START_REF] Masmoudi | Blow-up profile for the complex Ginzburg-Landau equation[END_REF]). Nevertheless, for the sake of completeness we give a sketch of the argument here. We take A = A 3 and T = T 3 (A) = min(T 1 (A), T1 (A), T 2 (A), T 3 (A)) so that Propositions 3.2, 3.3, 3.4 and 3.5 apply. We will find the parameter (d 0 , d 1 ) in the set D T defined in Proposition 3.2. We proceed by contradiction and assume from (iii) of Proposition 3.2 that for all (d 0 , d 1 ) ∈ D T , there exists s * (d 0 , d 1 ) ≥log T such that q d 0 ,d 1 (s) ∈ V A (s) for all s ∈ [-log T, s * ] and q d 0 ,d 1 (s * ) ∈ ∂V A (s * ). From (ii) of Proposition 3.5, we see that (q 0 (s * ), q 1 (s

* )) ∈ ∂[-A s 2 * , A s 2 *
] 2 and the following function is well defined :

Φ(y, s) : D T → ∂[-1, 1] 2 (d 0 , d 1 ) → s 2 * A (q 0 , q 1 ) (d 0 ,d 1 ) (s * ).
From (ii) of Proposition 3.5, Φ is continuous. If we manage to prove that Φ is of degree 1 on the boundary, then we have a contradiction from the degree theory. Let us prove that. Using (i) and (iii) of Proposition 3.2 and the fact that q(log T ) = q d 0 ,d 1 , we see that when (d 0 , d 1 ) is on the boundary of the rectangle D T , (q 0 , q 1 )(-log

T ) ∈ ∂[-A (log T ) 2 , A (log T ) 2 ]
2 and q(log T ) ∈ V A (-log T ) with strict inequalities for the other components. Applying the transverse crossing property of (ii) in Proposition 3.5, we see that q(s) leaves V A (s) at s =log T , hence s * =log T . Using (i) of Proposition 3.2, we see that restriction of φ to the boundary is of degree 1. A contradiction then follows. Thus, there exists a value (d 0 , d 1 ) ∈ D T such that for all s ≥log T, q d 0 ,d 1 (s) ∈ V A (s). This concludes the proof of (42).

Part 2 : Proof of Theorem 1 : Since q d 0 ,d 1 satisfies (42), we clearly see, from section 2 and (ii) in Proposition 3.1 that u defined from q d 0 ,d 1 through the transformations [START_REF] Souplet | Exact self-similar blow-up of solutions of a semilinear parabolic equation with a nonlinear gradient term[END_REF] and ( 11) is well defined for all (x, t) ∈ R × [0, T ) and satisfies (3) in the L ∞ norm. Using the parabolic estimate of Proposition 3.3, we see that (3) holds in the W 1,∞ norm as well. In particular from (3), we have

u(0, t) ∼ κ(T -t) -1 p-1 ,
hence, u blows up at time T at the point a = 0. This concludes the proof of Theorem1.

4 Proof Proposition 3.6

In this section, we prove Proposition 3.6. In the following, we will find the main contribution in the projection given in the decomposition (24) of the four terms appearing in the right-hand side of equation ( 13).

Let us recall the equation ( 17

) of q q(s) = K(s, τ )q(τ ) + s τ dσK(s, σ)B(q(σ)) + s τ dσK(s, σ)R(σ) + s τ dσK(s, σ)N (σ) ( 43
)
where K is the fundamental solution of the operator L + V.

We write q = α + β + γ + δ where

α(s) = K(s, τ )q(τ ), β(s) = s τ dσK(s, σ)B(q(σ)), (44) γ(s)
We assume that q(s) is in V A (s) for each s ∈ [τ, τ + ρ]. Using (41), we derive new bounds on α, β, γ and τ.

Clearly, Proposition 3.6 follows from the following : Lemma 4.1 There exists A 5 > 0 such that for all A ≥ A 5 , and ρ > 0 there exists T 5 (A, ρ) ≤ T 2 (A), such that for all T ≤ T 5 (A, ρ), if we assume that for all s ∈ [τ, τ + ρ], q(s) satisfies (13), q(s) ∈ V A (s) and ∇q(s) L ∞ ≤ CA 2 √ s with τ ≥ s 0 =log T. Then, we have the following results for all s ∈ [τ, τ + ρ] : (i)(Linear term) Proof of Lemma 4.1 : We consider A ≥ 1, ρ > 0 and T ≤ e -ρ (so that s 0 =log T ≥ ρ). We then consider q(s) a solution of (13) satisfying q(s) ∈ V A (s) and ∇q(s)

L ∞ ≤ CA 2 √ s for all s ∈ [τ, τ + ρ] (46)
for some τ ≥ s 0 . The terms α, β and γ are already present with case of the semilinear heat equation ( 6), so we refer to Lemma 3.13 page 167 in [START_REF] Merle | Stability of the blow-up profile for equations of the type u t = ∆u+ |u| p-1 u[END_REF] for their proof, and we only focus on the new term δ(y, s). Note that since s 0 ≥ ρ, if we take τ ≥ s 0 , then τ + ρ ≤ 2τ and if τ ≤ σ ≤ s ≤ τ + ρ then

1 2τ ≤ 1 s ≤ 1 σ ≤ 1 τ . ( 47

Proposition 3 . 2 (

 32 ) which gives (i). (ii) Since we have r(y, s) = r b (y, s) + r e (y, s), we just use (25) for (|y| ≤ 2K √ s) together with the fact that r e ∞ ≤ A 2 s -1/2 . This ends the proof of Proposition 3.1. Initial data (at time s = s 0 ≡log T ) for the equation (13) will depend on two real parameters d 0 and d 1 as given in the following proposition. Decomposition of initial data on the different components) For each A ≥ 1, there exists T 1 (A) ∈ (0, 1/e) such that for all T ≤ T 1 :

  , σ)R(σ), and δ(s) = s τ dσK(s, σ)N (σ).

|α 2 2 ,

 22 (y, s)| ≤ τ 2 s 2 |q 2 (τ )| + (s -τ )CAs -3 , α -(y, s) 1 + |y| 3 L ∞ ≤ Ce -1 2 (s-τ ) q -(τ ) 1 + |y| 3 L ∞ + Ce -(s-τ ) 2 q e (τ ) L ∞ s 3/2 , α e (s) L ∞ ≤ Ce -(s-τ ) p q e (τ ) L ∞ + Ce s-τ q -(τ ) 1 + |y| 3 L ∞ s -3 2 .(ii) (Nonlinear term)|β 2 (s)| ≤ (s -τ ) s 3 , |β -(y, s)| ≤ (s -τ )(1 + |y| 3 )s -2 , β e (s) L ∞ ≤ (s -τ )s -1 (iii) (Corrective term) |γ 2 (s)| ≤ C(s -τ )s -3 , |γ -(y, s)| ≤ C(s -τ )(1 + |y| 3 )s -2 , γ e (s) L ∞ ≤ (s -τ )s -1/2 , (iv) (New perturbation term) |δ 2 (s)| ≤ C(s -τ )s -3 , |δ -(y, s)| ≤ C(s -τ )(1 + |y| 3 )s -3 , δ e (s) L ∞ ≤ C(s -τ )s -3 .

2 + 2 + e -β 0 σ ≤ 1 σ 4 e 4 ≤ C(s -τ ) e ρ s 4 ≤ (s -τ ) s 3 for s 0 2 i=0δ

 22444302 )Using (19), (46), and (ii) of Proposition 3.1, we write for τ ≤ σ ≤ s ≤ τ + ρ and x ∈ R|N (x, t)| ≤ Ce -βσ A 2α σ α Ce -βσ A 2α σα s 0 large enough (that is T small enough), where we used (19) in the last inequality. Recalling from Bricmont and Kupiainen[START_REF] Bricmont | Universality in blow-up for nonlinear heat equations[END_REF] that for all y, x ∈ R|K(s, σ, y, x)| ≤ Ce (s-σ)L (y, x)(49)where e θL is given in (34) (see page 545 in[START_REF] Bricmont | Universality in blow-up for nonlinear heat equations[END_REF]), we write from (45), (49), 48 and (34), |δ(y, s)| ≤ (s-σ)L (y, x) C s 4 dx ≤ (s -τ )e (s-τ ) C s large enough. By definitions (22) and (23) of q m , q -and q e , we write for m ≤ 2,|δ m (s)| ≤ R |χ(y, s)δ(y, s)k m (y)ρ(y)dy ≤ C R |δ(y, s)|(1 + |y| 2 )ρ(y)dy ≤ C s 3 (s -τ ) |δ -(y, s)| = |δ(y, s)i (y, s)k i (y)| ≤ (s -τ )(1 + |y| 3 ) C s 3|δ e (y, s)| = |(1 -χ(y, s))δ(y, s)| ≤ |δ(y, s)| ≤ (s -τ ) C s 3 , which concludes the proof of Lemma 4.1 and Proposition 3.6 too.

Acknowledgments : This work was supported by the Marie Curie Actions of the European Commission in the frame of the DEASE project (MEST-CT-2005-021122).

The first author wishes to thank Prof. Dr. Christian Schmeiser for his hospitality in WPI in Vienna where a part of this work was done.


Karim Kellay
email: kellay@cmi.univ-mrs.fr

Javad Mashreghi
email: javad.mashreghi@mat.ulaval.ca

On zero sets in the Dirichlet space

Keywords: 2010 Mathematics Subject Classification. Primary: 30C15, Secondary: 30D50, 30D55, 31C25 Zero sets, Dirichlet space, logarithmic capacity

We study the zeros sets of functions in the Dirichlet space. Using Carleson formula for Dirichlet integral, we obtain some new families of zero sets. We also show that any closed subset of E ⊂ T with logarithmic capacity zero is the accumulation points of the zeros of a function in the Dirichlet space. The zeros satisfy a growth restriction which depends on E.

Introduction

The Dirichlet space D consists of all analytic functions f = n≥0 a n z n , defined on the open unit disk D, such that

D(f ) := D |f ′ (z)| 2 dA(z) = ∞ n=1 n |a n | 2 < ∞,
where dA(z) = dxdy/π istands for the normalized area measure in D . A sequence (z n ) n≥1 in D is called a zero set for D provided that there is a function f ∈ D, f ≡ 0, such that f (z n ) = 0, n ≥ 1. Since D is contained in the Hardy space H 2 (D), the zero set (z n ) n≥1 must satisfy the Blaschke conditions

∞ n=1 (1 -|z n |) < ∞.
However, the complete characterization of zero sets of D is still an open question. For a short history of this topic, see [START_REF] Mashreghi | Admissible functions for the Dirichlet space[END_REF]. Very briefly, let us mention that Shapiro-Shields [START_REF] Shapiro | On the zeros of functions with finite Dirichlet integral and some related function spaces[END_REF] improved an earlier result of Carleson [START_REF] Carleson | On the zeros of functions with bounded Dirichlet integrals[END_REF] by showing that

∞ n=1 1 | log(1 -|z n |)| < ∞ (1.1)
is enough to ensure that (z n ) n≥1 is a zero set for D. On the other hand, Nagel-Rudin-Shapiro [START_REF] Nagel | Tangential boundary behavior of functions in Dirichlet-type spaces[END_REF] and Richter-Ross-Sundberg [START_REF] Richter | Zeros of functions with finite Dirichlet integrals[END_REF] showed that for any sequence (r n ) n≥1 which do not do not fulfill (1.1), one can choose (θ n ) n≥1 such that (r n e iθn ) n≥1 is a uniqueness set for D. This line of research has been continued in [START_REF] Bogdan | On the zeros of functions with finite Dirichlet integral[END_REF][START_REF] Caughran | Two results concerning the zeros of functions with finite Dirichlet integral[END_REF][START_REF] Mashreghi | Admissible functions for the Dirichlet space[END_REF][START_REF] Mashreghi | Argument of the zero sets in the Dirichlet space[END_REF][START_REF] Pau | On the zeros of functions in Dirichlet spaces[END_REF].

In this note, we use Carleson's formula for the Dirichlet integral [START_REF] Carleson | A representation formula for the Dirichlet integral[END_REF] to obtain some new families of the zero sets which are not given by the classical theorems mentioned above. We highlight two main results below. Then every Blaschke sequence

(z n ) n≥1 satisfying ∞ n=1 ω 2d(z n , E) < ∞ is a zero set for D.
The function ω(t) = t is close to the borderline for the hypothesis of Theorem 1.1. The most demanding condition in this theorem is the existence of a function f ∈ D which satisfies (1.3). However, under some mild extra conditions, such a function can be explicitly constructed. In Section 3, for Carleson sets or for sets of logarithmic capacity zero we construct a function f which fulfills (1.2) and (1.3). This is analogue to the result of that Pau-Peláez [17, Theorem 1] about the zeros of function in the Dirichlet type space

D α := f ∈ Hol(D) : D |f ′ (z)| 2 (1 -|z| 2 ) α dA(z) < ∞
for 0 < α < 1, in which more conditions are required on the zero set.

In order to state our second result, we introduce the notion of logarithmic capacity. The energy of a Borel probability measure µ on T is defined by

I(µ) = log 1 |ζ -ξ| dµ(ζ) dµ(ξ).
A simple calculation shows that

I(µ) = ∞ n=1 | µ(n)| 2 n ,
where µ(n) n∈Z is the sequence of Fourier coefficients of µ. Given a Borel subset E of T, we denote by P(E) the set of all probability measures supported on a compact subset of E. Then we define the logarithmic capacity of E by

cap(E) = 1/ inf{I(µ) : µ ∈ P(E)}.
According to a well-known result of Beurling [START_REF] Carleson | Sets of uniqueness for functions regular in the unit circle[END_REF], for each function f ∈ D, the radial limits

f * (ζ) = lim r→1-f (rζ) exist q.e. on T, that is cap {ζ ∈ T : f * (ζ) does not exist} = 0.
If E is any closed subset of T, we can take (r n ) n≥1 to be any positive sequence which satisfies the Shapiro-Shields condition (1.1), and then adjust the arguments (θ n ) n≥1 such that (r n e iθn ) n≥1 accumulates precisely at all points of E. But such a sequence converges very rapidly to the boundary. However, for a closed set of logarithmic capacity zero, using Theorem 1.1, we can construct a sequence which fulfils the above property and at the same time grows much slower to its accumulation points on T. See also Remark 3.3.

For any subset E of T, we define

E t = {ζ ∈ T : d(ζ, E) ≤ t}, t > 0.
Theorem 1.2. Let E be a closed subset of T with logarithmic capacity zero. Let ψ be any positive continuous decreasing function such that ψ(x) -→ 0 as x -→ +∞ and +∞ ψ(x) x dx < ∞, and then define

ω(t) = exp -e ψ -1 (cap(Et)) .
Let (r n ) n≥1 be any positive sequence which satisfies the growth restriction

∞ n=1 (1 -r n ) 2(1-rn) ω(t) t 2 dt < ∞. (1.4)
Then we can choose the arguments (θ n ) n≥1 such that (r n e iθn ) n≥1 is a zero set for D, and the set of its accumulation points on T is precisely equal to E.

In Section 4, we show that if (z n ) n≥1 is a zero sequence satisfying the Shapiro-Shields condition (1.1), then the angular derivative of the Blaschke product formed with (z n ) n≥1 exists-q.e. On the other direction, using Theorem 1.1, we give a zero sequence (z n ) n≥1 for D such that the angular derivative of the corresponding Blaschke product exists precisely on T\E, where E is a set with cap(E) > 0.

Proofs

For the proofs of above theorems, we need the following key lemma. Lemma 2.1. Let ω : [0, 2] -→ [0, ∞) be a continuous increasing function such that ω(0) = 0. Let E ⊂ T and suppose that there exists a function f ∈ D, f = 0, such that

|f (ζ)| 2 ≤ ω d(ζ, E) ,
a.e. on T.

Then every Blaschke sequence

(z n ) n≥1 satisfying ∞ n=1 ω 2d(z n , E) + (1 -|z n |) 2d(zn,E) ω(t) t 2 dt < ∞ is a zero set for D. Proof. Fix z ∈ D. Let Γ = {ζ ∈ T : d(ζ, E) ≥ d(z, E)},
and consider the decomposition

Γ = ∞ n=0 Γ n z , where Γ n z = ζ ∈ T : 2 n d(z, E) ≤ d(ζ, E) < 2 n+1 d(z, E) . On Γ 0 z , we have Γ 0 z |f (ζ)| 2 |ζ -z| 2 |dζ| ≤ Γ 0 z ω( d(ζ, E)) |ζ -z| 2 |dζ| ≤ ω(2d(z, E)) T |dζ| |ζ -z| 2 = 2π ω(2d(z, E)) 1 -|z| . Let ζ 0 be the closest point of E to z. Then, for each ζ ∈ Γ n z , n ≥ 1, we have |ζ -z| ≥ |ζ -ζ 0 | -|ζ 0 -z| ≥ d(ζ, E) -d(z, E) ≥ 2 n-1 d(z, E). Hence, Γ z n ⊂ {ζ ∈ T : |ζ -z| ≥ 2 n-1 d(z, E
)}, and we obtain

Γ n z |f (ζ)| 2 |ζ -z| 2 |dζ| ≤ Γ n z ω(d(ζ, E)) |ζ -z| 2 |dζ| ≤ ω(2 n+1 d(z, E)) Γ n z |dζ| |ζ -z| 2 ≤ ω(2 n+1 d(z, E)) {ζ :|ζ-z|≥2 n-1 d(z,E)} |dζ| |ζ -z| 2 ≤ ω(2 n+1 d(z, E)) ∞ 2 n-1 d(z,E) dx x 2 = 4 ω(2 n+1 d(z, E)) 2 n+1 d(z, E) . Therefore, Γ |f (ζ)| 2 |ζ -z| 2 |dζ| = ∞ n=0 Γ n z |f (ζ)| 2 |ζ -z| 2 |dζ| ≤ 2π ω(2d(z, E)) 1 -|z| + 4 ∞ n=1 ω(2 n+1 d(z, E)) 2 n+1 d(z, E) ≤ 2π ω(2d(z, E)) 1 -|z| + 4 2 2d(z,E) ω(t) t 2 dt. (2.1)
On the other hand, on

T \ Γ = {ζ ∈ T : d(ζ, E) < d(z, E)}, we have T\Γ |f (ζ)| 2 |ζ -z| 2 |dζ| ≤ T\Γ ω(d(ζ, E)) |ζ -z| 2 |dζ| ≤ ω(d(z, E)) T |dζ| |ζ -z| 2 = 2π ω (d(z, E)) 1 -|z| . (2.2)
Hence, by (2.1) and (2.2),

∞ n=1 (1 -|z n | 2 ) T |f (ζ)| 2 |ζ -z n | 2 |dζ| ≤ 4π ∞ n=1 ω(2d(z n , E)) + 4(1 -|z n | 2 ) 2 2d(zn,E) ω(t) t 2 < ∞.
Let B be the Blaschke product formed by (z n ) n≥1 . It follows from Carleson's formula [START_REF] Carleson | A representation formula for the Dirichlet integral[END_REF] that

D(Bf ) = D(f ) + 1 2π T ∞ n=1 1 -|z n | 2 |ζ -z n | 2 |f (ζ)| 2 |dζ| < ∞. (2.3)
Thus, (z n ) n≥1 is a zero set for the Dirichlet space.

Proof of Theorem 1.1.

Since (1 -|z n | 2 ) 2d(zn,E) ω(t) t 2 dt ≤ c (1 -|z n | 2 ) + c (1 -|z n | 2 ) ω(2d(z n , E)) 2d(z n , E) ≤ c (1 -|z n | 2 ) + c ω(2d(z n , E))
for some constant c > and (z n ) n≥1 is a Blaschke sequence, the proof follows immediately from Lemma 2.1.

Proof of Theorem 1.2. Since cap(E) = 0, we have cap(E t ) -→ 0 as t -→ 0. Set

η(t) = ψ -1 (cap(E t )). (2.4)
Then η(t) -→ ∞, as t -→ 0, and Define

0 cap(E t ) |dη 2 (t)| ≍ 0 ψ(η(t)) |dη 2 (t)| ≍ +∞ ψ(x) xdx < ∞.
f = exp - √ 2 2 e ϕ .
Note that by harmonicity |Im ϕ(z)| ≤ π/4 on D and

D |f ′ (z)| 2 dA(z) = 1 2 D |ϕ ′ (z)| 2 e 2Re ϕ(z) e - √ 2 cos(Im ϕ(z))e Re ϕ(z) dA(z) ≤ D |ϕ ′ (z)| 2 e 2Re ϕ(z) e -e Re ϕ(z) dA(z) ≤ sup x≥0 (x 2 e -x ) D |ϕ ′ (z)| 2 dA(z).
Hence f ∈ D. Also, we have

|f (ζ)| 2 ≤ exp -e η(d(ζ,E)) = ω(d(ζ, E)), ζ ∈ T. Write T \ E = k≥1 I k ,
where I k = (e iθ 2k , e iθ 2k+1 ) are the complementary intervals of the closed set E. Note that {e iθ k : k ≥ 1} = E. Let (r n,k ) n,k≥1 be a two-indexed enumeration of (r n ) n≥1 . Then put

z n,k = r n,k e iθ k , n, k ≥ 1.
Since on each fixed ray θ = θ k , there are infinitely many zeros z n,k which tend to the point

e iθ k , we have {z n,k : n, k ≥ 1} ∩ T = E. Moreover, d(z n,k , E) = 1 -r n,k and ω((2d(z n,k , E)) 1 -r n,k = ω(2(1 -r n,k )) 1 -r n,k ≤ 2 2(1-r k,n ) ω(t) t 2 dt = 2 2d(z n,k ,E) ω(t) t 2 dt.
The result now follows from Lemma 2.1.

Remark 2.2. Note that since ω is increasing and ω(0) = 0, we have

δ 2 2δ ω(t) t 2 dt = δ √ δ 2δ ω(t) t 2 dt + δ 2 √ δ ω(t) t 2 dt ≤ ω( √ δ) + √ δ ω(2) -→ 0, δ -→ 0.
Hence, we there are (r n ) n≥1 which satisfy (1.4).

Some specific examples

Let E be a closed subset of T with Lebesgue measure zero, and write T \ E = ∪ n≥1 I n , where I n 's are the complementary intervals of E. Then we have

T | log d(ζ, E)| |dζ| < ∞ ⇐⇒ ∞ n=1 |I n | log 1/|I n | < ∞. (3.1)
See [START_REF] Carleson | Sets of uniqueness for functions regular in the unit circle[END_REF]. Such a set is called a Carleson set. The condition (3.1) gives the motivation to consider the closed sets E which satisfy

T | log σ(d(ζ, E))| |dζ| < ∞, (3.2)
where ω is a positive continuous function. In this situation, the well-known formula

f σ,E (z) = exp T ζ + z ζ -z log σ(d(ζ, E)) |dζ| 2π
provides an outer function such that whose boundary values

f * σ,E satisfy |f * σ,E (ζ)| = σ(d(ζ, E)),
a.e. on T.

There are several results which show that, under certain conditions on ω and E, this outer function belongs to a specific function space [START_REF] El-Fallah | On the Brown-Shields conjecture for ciclicity in the Dirichlet space[END_REF]. Some relevant cases are treated below for the generalized Cantor set.

Let us briefly describe the construction of the generalized Cantor set on T. Put E 0 = T and ℓ 0 = 2π. Let (λ n ) n≥1 be a decreasing sequence of positive numbers. We remove an interval of length λ 1 from the middle of E 0 . Denote the union of two remaining intervals by E 1 , and denote the length of each interval in E 1 by ℓ 1 . Then we remove two intervals, each of length λ 2 , from the middle of intervals in E 1 . Let E 2 denote the union of the resulting four pairwise disjoint intervals of equal length ℓ 2 . Continuing this procedure, we obtain in the n-th step a compact set E n which is the union of 2 n closed intervals of length equal to ℓ n . According to the construction procedure, we must have

2ℓ n + λ n = ℓ n-1 . (3.3)
2 n λ n log 1/λ n < ∞. (3.5)
The set E is called a perfect symmetric set with constant ratio ℓ ∈ (0, 1/2) if ℓ n = ℓ n . In this case, by (3.3), for some exponent α > 1/2, is a zero set for D.

λ n = ℓ n-1 (1 -2ℓ). ( 3
Let A ∞ = C ∞ (T) ∩ Hol(D). Compare this with [14, Theorem 3.2] which says, roughly, that if (3.7) holds then E is a Carleson set and, moreover, the same result holds if one replaces D by A. As a very specific situation of Corollary 3.1, let (r n ) n≥1 be any Blaschke sequence, and let (e iθn ) n≥1 be any subset of E. Put z n = r n e iθn , n ≥ 1. In particular, e iθn maybe the end point of a complementary interval of E. Then (3.7) is clearly satisfied and thus (r n e iθn ) n≥1 is a zero set for D.

Example 2: Let σ(t) = e -1/t γ , with 0 < γ < 1, and let E be a subset set of T such that

T |dζ| d(ζ, E) γ < ∞. (3.8)
This condition is also equivalent to |E| = 0 and ∞ n=1 |I n | 1-γ < ∞, where (I n ) n≥1 are the complementary intervals of E [START_REF] El-Fallah | Cyclicity in the Dirichlet space[END_REF]. In the case of generalized Cantor sets, (3.8) becomes equivalent to

∞ n=1 2 n λ 1-γ n < ∞.
(3.9)

If E is perfect symmetric set with the constant ratio ℓ, then, by (3.6), the condition (3.9) holds if and only if

0 < γ < 1 + log 2 log ℓ . Now if (3.8) holds, then f σ,E ∈ A ∞ ⊂ D.
The condition (3.8) shows that E is a special type of Carleson set. But, for such a sacrifice, we get more freedom for zeros. More precisely, we have the following result.

Corollary 3.2. Let E fulfill (3.8). Then each Blaschke sequence (z n ) n≥1 satisfying ∞ n=1 e -2/d(zn,E) γ < ∞ (3.10)
is a zero set for D.

The case E = {1} of this result was treated in [START_REF] Mashreghi | Admissible functions for the Dirichlet space[END_REF]. As a very specific case of Corollary 3.2, let ε n = n -1+γ 1-γ , and let

E = {e iεn } ∪ {1}. Then E satisfies (3.8). Put z n,k = r n,k e iθ n,k with 1 -r n,k = n -n k -k and θ n,k = ε n + ε n-1 -ε n 2(log k)
0 |E t | η ′ (t) 2 dt < ∞. (3.13)
For example, the generalized Cantor set E with ℓ n = e -2 n /n σ satisfies this property [9, Theorem Hence cap(E t ) -→ 0 as t -→ 0, cap(E) = 0. Therefore, by (2.4),

ψ -1 (cap(E t )) = η(t) = 2 log log log(1/t) and ω(t) = exp -e ψ -1 (cap(Et)) ≤ 1 (log 1/t) 2 . Hence, δ δ ω(t) t 2 dt ≤ c (log 1/δ) 2 = o 1 log 1/δ , δ -→ 0.
According to Theorem 1.2, there is a zero set (z n ) n≥1 which satisfies

∞ n=1 1 log 2 (1 -|z n |) < ∞ but ∞ n=1 1 | log(1 -|z n |)| = ∞,
and the set of its accumulation points on T is precisely equal to E. Therefore, (1.4) holds but the zero sequence does not fulfil the Shapiro-Shields condition (1.1).

Corollary 3.4. Let E be any closed subset of T satisfying (3.11). Then each Blaschke sequence

(z n ) n≥1 satisfying ∞ n=1 exp - 2 2d(zn,E) ds |E s | α < ∞,
for some α ∈ (0, 1/2), is a zero set for D.

Proof. Since (3.11) holds, for each β > 0, we have

0 ds |E s |( s dt |Et| ) 1+β < +∞. (3.14) Let η α,E (t) = t ds |E s | α and ω(t) = e -η α,E (t) .
It clear that η α,E is decreasing function and η α,E (t) -→ ∞ as t -→ 0. Since α ∈ (0, 1/2), by (3.12) and (3.14), we get

2 0 cap(E t ) dη 2 α,E (t) > -∞.
Arguing as in the proof of Theorem 1.

Exceptional set of Blaschke products

Let B be the Blaschke product formed with the sequence Z = (z n ) n≥1 . According to a result of Frostman, B has a finite angular derivative in the sense of Carathéodory at ζ ∈ T if and only if

∞ n=1 1 -|z n | 2 |ζ -z n | 2 < ∞.
See [START_REF] Frostman | Sur les produits de Blaschke[END_REF][START_REF] Cargo | Angular and tangential limits of Blaschke products and their successive derivative[END_REF] or [START_REF] Cima | The Cauchy transform[END_REF]Theorem 1.7.12]. Moreover, at such a point,

|B ′ (ζ)| = ∞ n=1 1 -|z n | 2 |ζ -z n | 2 .
That is why we define the exceptional set

E(Z) = ζ ∈ T: ∞ n=1 1 -|z n | 2 |ζ -z n | 2 = ∞ .
The exceptional set can be quiet large. As a matter of fact, Frostman provided an example so that E(Z) = T. However, if Z = (z n ) n≥1 is a zero set for the Dirichlet space D, then, by Carleson's formula (2.3) and Jensen's inequality, we have

T log ∞ n=1 1 -|z n | 2 |ζ -z n | 2 |dζ| < ∞. (4.1)
Hence, the exceptional set E(Z) must have the Lebesgue measure zero.

Given a measurable function f on T, the associated distribution function m f is given by

m f (λ) = |{ζ ∈ T : |f (ζ)| > λ}|, λ ≥ 0. If |f | ≥ 1 on T, then T log |f (ζ)| |dζ| = ∞ 1 m f (λ) λ dλ. (4.2)
See [START_REF] Mashreghi | Representation Theorems in Hardy Spaces[END_REF]Page 347]. Parallel to this definition and to better study E(Z), let us define

E λ (Z) = ζ ∈ T : ∞ n=1 1 -|z n | 2 |ζ -z n | 2 ≥ λ . Note that inf ζ∈T ∞ n=1 1 -|z n | 2 |ζ -z n | 2 ≥ 1 4 ∞ n=1 (1 -|z n | 2 ) = λ 0 > 0.
For simplicity, we assume that λ 0 ≥ 1. Moreover,

lim λ→+∞ |E λ (Z)| = |E(Z)|,
and thus the latter is zero provided that the limit is equal to zero. However, by (4.2),

T log ∞ n=1 1 -|z n | 2 |ζ -z n | 2 |dζ| = ∞ 1 |E λ (Z)| λ dλ.
This identity shows that, if (4.1) holds, then |E λ (Z)| must tend to zero, although perhaps slowly.

We can say more about the size of this set, if the zero set (z n ) n≥1 satisfies the Shapiro-Shields condition (1.1). Proof. Let E be a Cantor generalized set on T.

Write E = k≥1 E k , where E k = 2 k l=1 I k,l with |I k,l | = ℓ k > 0. We choose the interval length ℓ k such that ℓ = sup k≥1 ℓ k+1 ℓ k < 1 2 , (4.3) k≥1 2 -k log 1/ℓ k < ∞, (4.4) k≥1 1 log 1/ℓ k = ∞. (4.5)
For example, if E is the triadic Cantor set, then we can choose ℓ n = 1/3 n . However, in order to provide more examples we just assume that the sequence (ℓ k ) k≥1 satisfies the above three conditions. Let us see why the above conditions are imposed. In the first place, the condition (4.3) implies that In the light of Frostman's Theorem [START_REF] Frostman | Sur les produits de Blaschke[END_REF], see also [START_REF] Cargo | Angular and tangential limits of Blaschke products and their successive derivative[END_REF], Proposition 4.2 implies that there exists a Blaschke product which fails to have a finite angular derivative on a set of positive logarithmic capacity.

Theorem 1 . 1 . 2 )

 112 Let ω : [0, 2] -→ [0, ∞) be a continuous increasing function such that ω(0)Let E ⊂ T and suppose that there exists a function f ∈ D, f = 0, such that|f (ζ)| 2 ≤ ω d(ζ, E) ,a.e. on T.(1.3)

Hence, by [ 9 ,

 9 Theorem 5.1], a converse of strong-type estimate for capacity, there exists a function ϕ ∈ D such that Re ϕ(ζ) ≥ η(d(ζ, E)) and |Im ϕ(ζ)| < π/4,q.e. on T.

. 6 )

 6 The classical Cantor set corresponds to ℓ = 1/3.

Example 1 :

 1 Let E be any Carleson set, and let σ(t) = t α with α > 1/2. Then, according to [10, Corollary 4.2], we have f σ,E ∈ D. Since d(z, E) ≤ (1 -|z|) + d(z/|z|, E), we deduce the following result from Theorem 1.1. Corollary 3.1. Let E be any Carleson set. Then each Blaschke sequence (r n e iθn ) n≥1 satisfying ∞ n=1 d(e iθn , E) 2α < ∞, (3.7)

Proposition 4 . 1 . 2 . 2 |Proposition 4 . 2 .

 412242 If Z = (z n ) n≥1 satisfies the Shapiro-Shields condition (1.1), then the logarithmic capacity of E(Z) is zero. Proof. Let I z be the open arc of T with midpoint z/|z| and arclength|I z | = (1 -|z|) log 1/(1 -|z|) 1/We now verify that, for each N ≥ 1,E(Z) ⊂ n≥N |I zn |. As a matter of fact, ζ / ∈ |I z | implies 1 -|z| 2 |ζ -z| 2 ≤ 4 | log(1 -|z|)| . Thus, if ζ / ∈ |I zn | for all n ≥ N, then, by (1.1), we get ζ / ∈ E(Z). Hence, cap(E(Z)) ≤ cap n≥N log(1 -|z n |)| .with c 1 , c 2 independent of N. Now, let N -→ ∞. By the Shapiro-Shields condition, the last expression tends to zero. There exists a zero set (z n ) n≥1 in Dirichlet space D such that cap(E(Z)) > 0.

k≥1 2 k ℓ 2 k ≤ k≥1 2 k 2 =

 22 ℓ k log 1/ℓ k ≤ | log ℓ| k≥1 k(2ℓ) k < ∞ (4.6)and, by(3.3) and (4.3), λ k ≍ ℓ k-1 . Hence, by(3.5), E is a Carleson set. Secondly, the condition (4.4) is equivalent to cap(E) > 0 [6, §, IV Theorem 3]. Now we can construct our zero sequence. Let (z n ) n≥1 = (z k,l ) l,k≥1 be such that z k,l /|z k,l | is the center of the interval I k,l and(1 -|z k,l |) log 1/(1 -|z k,l |) 1/ℓ k .Since t ≤ t log 1/t, (4.6) guaranties that (z n ) n≥1 = (z k,l ) l,k≥1 is a Blaschke sequence. The condition (4.5) ensures that the sequence (z k,l ) l,k≥1 does not satisfy the Shapiro-Shields condition (1.1) and, moreover,E = k≥1 E k ⊂ E(Z).As a matter of fact, if ζ ∈ E, then, for each k ≥ 1, there exists a midpointz k,l /|z k,l | such |ζ -z k,l /|z k,l || ≤ ℓ k /2. Hence, 1 -|z k,l | 2 |ζ -z k,l | 2 ≤ 2 log 1/ℓ k .Therefore, by (4.5), we deduce ζ ∈ E(Z), and thus cap(E(Z)) ≥ cap(E) > 0. To verify that (z k,l ) l,k≥1 is zero set in Dirichlet space, note that since z k,l /|z k,l | is the center of the interval I k,l , we have d(z k,l /|z k,l |, E) ≤ ℓ k /2. Hence, with σ(t) = t, by (4.6), we have k≥1 2 k σ 2 (2d(z k,l /|z k,l |, E)) ≤ k≥1 2 k ℓ 2 k < ∞. Corollary 3.1 now shows that (z n ) n≥1 is a zero set for D.

  2/γ . Simple computation shows that (z n,k ) n,k≥2 is Blaschke sequence satisfying (3.10). Hence, (z n,k ) k,n≥2 is a zero set for D. But, (3.7) does not hold and thus this result cannot be deduced from Corollary 3.1.

	Example 3: Let E a be a closed subset of T such that
	2 where E s := {ζ ∈ T : d(ζ, E) ≤ s}. The closed subset E satisfying (3.11) has logarithmic 0 ds = +∞, (3.11) |E s | capacity zero, but the converse is not true. However, if E is a generalized Cantor set, then
	(3.11) holds if and only if cap(E) = 0 [6, Chapter IV, Theorems 2 and 3]. Note that this
	is also equivalent to (3.4). Moreover,
	cap(E t ) ≤	t	2	ds |E s |	-1	.	(3.12)

Remark 3.3. Let η(t) = 2 log log log(1/t) and let E be a closed subset of T such that

Acknowledgments: The authors are grateful to the referee for his valuable remarks and suggestions.

This work was supported by NSERC (Canada), NRS and ANRDynop (France).


Juan Haba-Rodríguez

Álvaro Rodríguez-Lescure

Amparo Ruiz

Emilio Alba

Lourdes Calvo

Eva Carrasco

Maria Jose Escudero

Miguel Martín

MD Juan De La Haba-Rodríguez

Regional and seasonal influence in patient's toxicity to adjuvant chemotherapy for early breast cancer

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Chemotherapy is an integral part in the therapeutic management of the majority of tumour types. Efficacy and safety are assessed in clinical trials which are often conducted in several countries. The communication and diffusion of the results are globalized, as are the therapeutic attitudes and treatment regimens. However, there are variables in different countries and geographical areas which can have an influence on trial outcomes and, as such, need to be taken into account in the data analyses. One of these variables is climate.

Variations in the incidence of diseases related to the yearly seasonal stages have been published [START_REF] Fleming | Annual and seasonal variation in the incidence of common diseases[END_REF]. These include the incidence of cardiovascular [START_REF] Manfredini | Seasonal variation in occurrence of vascular diseases[END_REF] and cerebrovascular disease with a peak incidence during the winter [START_REF] Tsementzis | Seasonal variation of cerebrovascular diseases[END_REF][START_REF] Tomari | Seasonal variation of mortality from cerebro-cardiovascular diseases--effect of ambient temperature on death][END_REF] and infectious disease [START_REF] Serban | The annual and seasonal variation in the bacteria and rotaviruses implicated in the etiology of diarrheal diseases in children][END_REF]. In oncology, there have been reports of seasonal differences in lymphoma [START_REF] Boniol | Seasonal variation in the occurrence of cutaneous melanoma in Europe: influence of latitude. An analysis using the EUROCARE group of registries[END_REF] breast cancer [START_REF] Chang | Seasonal variation in the diagnosis of Hodgkin lymphoma in Sweden[END_REF][START_REF] Langagergaard | Seasonal variation in month of birth and diagnosis in children and adolescents with Hodgkin disease and non-Hodgkin lymphoma[END_REF] and other tumour types [START_REF] Mcintyre | A seasonal variation in breast cancer[END_REF].

Relationships between the season of the year and the natural history of the disease have been also published. A Norwegian prostate cancer study showed better prognosis in patients who were diagnosed during the summer and the autumn [START_REF] Lagunova | Prostate cancer survival is dependent on season of diagnosis[END_REF]. Also in Norwegian patients, there were differences in lung cancer incidence; the authors postulated that the underlying cause of this prognostic variation was the physiological modifications produced as a result of the differences in the levels of exposure to vitamin D [START_REF] Porojnicu | Seasonal and geographical variations in lung cancer prognosis in Norway. Does Vitamin D from the sun play a role?[END_REF][START_REF] Robsahm | Vitamin D3 from sunlight may improve the prognosis of breast-, colon-and prostate cancer (Norway)[END_REF].

Similarly, there have been descriptions of seasonal changes in the activity of some enzymatic systems responsible for the metabolism of substances and toxins, which can result in toxicity variations to specific drugs. As an example, the hepatic toxicity induced by tetracyclines in an animal model was greater in winter and autumn due to a decrease in the levels of toxin conjugation and excretion by the liver [START_REF] Skakun | Seasonal characteristics of the hepatotoxicity of tetracycline[END_REF]. Also described in another murine model was the circadian and seasonal variation in the nephrotoxicity induced by amikacin and gentamicin, and the haematological toxicity induced by adriamycin and epirubicin [START_REF] Dorian | Demonstration of seasonal changes of circadian rhythms of amikacin chrononephrotoxicity in rats[END_REF][START_REF] Levi | Circadian and seasonal rhythms in murine bone marrow colony-forming cells affect tolerance for the anticancer agent 4'-Otetrahydropyranyladriamycin (THP)[END_REF][START_REF] Pariat | Seasonal effects on the daily variations of gentamicin-induced nephrotoxicity[END_REF][START_REF] Mormont | Circadian rhythm and seasonal dependence in the toxicological response of mice to epirubicin[END_REF].

Experimental carcinogenesis studies performed in animals showed that the success in the tumours induction varied as a function of the season of the year. For example, 56 and 61% of Sprague-Dawley rats developed breast cancer when exposed to DMBA in the spring and summer, while only 24% develop the tumors if induced in the autumn [18]. Cytostatic drugs used in the treatment of cancer are characterized by narrow therapeutic margins. Slight modifications can have an impact on the toxicity profile and, as such, could affect the conditions of the drugs' employment. To-date, there have not been any publications addressing the influence of climate on chemotherapy toxicity.

Patients and Methods

We analyzed the toxicities recorded in the GEICAM 9906 study (19). This trial included 1246 patients recruited in 65 Spanish hospitals. Patients were randomized to receive FEC90 (5-fluorouracil 600mg/m2, epirubicin 90 mg/m2, cyclophosphamide 600 mg/m2) for 6 cycles every 21 days; or sequential FEC90 for 4 cycles every 21 days followed by weekly paclitaxel (100 mg/m2) for 8 weeks. 10.780 cycles were administered, 6.191 of FEC90 and 4,589 of paclitaxel.

The hematological and non-hematological toxicities were assessed in relation to climate (seasonal) and geographical areas. The seasons were defined as: spring (from 21st March to 21st June), summer (from 22nd June to 23rd September), autumn (from 24th September to 21st December) and winter (from 22nd December to 20th March). The geography-affected climate areas were defined according to the information provided in the Spanish Education Ministry web page [20]. They were classified as: Oceanic (characterized by mild temperatures throughout the year [10-20ºC] and corresponding to the Regions of Galicia, Cantabria and Basque Country); Mediterranean (characterized as high temperatures in summer and mild in winter and corresponding to the Regions of Catalunya, Balearic Islands, Valencia, Murcia, Andalucia including Almeria, Malaga, Sevilla, Huelva and Cadiz); Continental (characterized by extremes temperatures, and corresponding to the Regions of Castilla la Mancha, Aragon, Navarra, Extremadura, Castilla-Leon and Andalucia including Granada, Cordoba and Jaen) and Sub-tropical (characterized by temperatures between 18-25ºC throughout the year, and corresponding to the Canary Islands). This last geographic area was excluded from the current analyses because the original study had very few patients recruited from there.

Recorded toxicities were segregated according to the different seasons and Geographicclimatic areas. Contingency tables were used to compare the toxicity vs the season as well as the toxicity vs the climate area in a separate analysis. The χ2 test was used to check the hypothesis that the two variables were independent in each analysis separatelly. Statistical significance was set at p<0.05.

Results

We analyzed all toxicities by grade for both treatment arms and distributed them by the seasons of the year (Table 1) and by Geographic-climate areas (Table 2). Toxicity items and grades were different between the regimens and the tables because only the statistically significant differences in toxicities were included.

In general, the statistically significant differences observed were only in low grade toxicities (except for neutropenia and vomiting).

The results showed statistically significant differences for some of the toxicities related to the season of the year and the Geographic-climate area where the patients were treated. We observed a higher hematological toxicity with both FEC90 and paclitaxel in warm seasons (spring and summer) and in Oceanic climate regions. Asthenia was a symptom that occurred with greater frequency in the summer period, as well as in the Mediterranean areas. Liver transaminase elevations were more frequent in the summer and in the Oceanic areas. Myalgias and secondary sensory neuropathy to paclitaxel were recorded more frequently during autumn.

Commentary:

Internationalization of clinical research has resulted in globalization of therapeutic approaches and regimens. Changes in the incidence and intensity of toxicities to several drugs in relation with the yearly seasons have been reported in animal models. However, seasonal and climate differences in chemotherapy toxicity in patients have not been described yet.

We have seen that the tolerability of a specific drug or treatment regimen could be affected by the yearly seasons and the climate conditions in which the treatment is administered. Differences in toxicity can lead to differences in dose reductions. Bonadonna et al showed a dose-response effect to chemotherapy when administered in the adjuvant setting to patients with breast cancer [21]. When CMF was administered at full or almost full doses (greater than or equal to 85%) patients had a five-year disease free survival of 77%, in contrast, only 67% of patients were disease free at five year when only 65% of the planned doses were administered. Based on the results observed in the current work, and knowing the slim therapeutic margin of chemotherapy, drug efficacy could potentially be affected by climatic conditions. Due to that, the toxicity analysis as a function of climate variables should be taken into account when data from clinical trials are reviewed.

[Text word count = 1125 words, without tables]

Table 1 .

 1 Patient's toxicities related to the season of the year*

	18. Loscher W, Mevissen M, Haussler B. Seasonal influence on 7,12-
	dimethylbenz[a]anthracene-induced mammary carcinogenesis in Sprague-
	Dawley rats under controlled laboratory conditions. Pharmacol Toxicol
	1997;81(6):265-270.
	19. Martin M, Rodriguez-Lescure A, Ruiz A, et al. Randomized phase 3 trial of
	fluorouracil, epirubicin, and cyclophosphamide alone or followed by paclitaxel
	for early breast cancer. J Natl Cancer Inst 2008;100(11):805-814.
	20. http://www.ite.educacion.es/w3/eos/MaterialesEducativos/primaria/conocimient
	o/climas/index.htm.
	21. Bonadonna G, Calagussa R. Dose-response effect of adjuvant chemotherapy in
	breast cancer. N Engl J Med 1981;304:100.
			Spring	Summer	Autumn	Winter	p
	Toxicity	Grade	N(%)	N(%)	N(%)	N(%)
	FEC90 every 21 days (5FU: 600mg/m 2 +4EPI 90mg/m 2 +cyclophosphamide 600mg/m 2 )
			N= 6191 cycles
			N=1864	N=1583	N=1211	N=1533
	Hemoglobin	G1	769(41.3) 675(42.6) 465(38.4) 587(38.3)	0.034
	Leukocytes	G1	322(17.3) 312(19.7) 202(16.7) 319(20.8) 0.0102
	Neutrophils	G4	24(1.3)	48(3.0)	39(3.2)	42(2.7)	0.001
	Asthenia	G1	283(15.2) 267(16.9) 156(12.9) 228(14.9) 0.0354
	Diarrhea	G1 G2	48(2.6) 19(1)	46(2.9) 20(1.3)	28(2.3) 15(1.2)	20(1.3) 4(0.3)	0.0186 0.0135
	Stomatitis	G2	158(8.5)	131(8.3)	96(7.9)	91(5.9)	0.0276
	Vomiting	G3	36(1.9)	49(3.1)	32(2.6)	22(1.4)	0.0093
	Alkaline phosphatase	G1	54(2.9)	40(2.5)	54(4.5)	46(3.0)	0.0248
	SGOT/AST	G2	18(1.0)	8(0.5)	22(1.8)	18(1.2)	0.0092
	SGPT/ALT	G1	362(19.4) 293(18.5) 268(22.1) 263 (17.2) 0.0098
	Infection without neutropenia	G1	30(1.6)	31(2)	36(3.0)	57(3.7)	0.0003
	Hypercholesterolemia	G1	15(0.8)	2(0.1)	3(0.2)	7(0.5)	0.0159
	Insomnia	G1	4(0.2)	9(0.6)	9(0.7)	15(1.0)	0.0321
	Mood alteration-anxiety	G1	9(0.5)	22(1.4)	8(0.7)	8(0.5)	0.0091
	Paclitaxel 100 mg/m2 weekly; N=4589 cycles
			N=1206	N=1375	N=1126	N=882	p
	Allergic	G1	9(0.7)	7(0.5)	5(0.4)	17(1.9)	0.0008
	reaction/hypersensitivity
	Hemoglobin	G1	778(64.5)	853(62)	682(60.6) 495(56.1) 0.0013
		G2	85(7.0)	167(12.1)	87(7.7)	63(7.1)	<0.0001
	Leukocytes	G1	271(22.5) 358(26.0) 294(26.1) 252(28.6) 0.0142
		G2	19(1.6)	18(1.3)	5(0.4)	17(1.9)	0.019
	Neutrophils	G3	24(2.0)	12(0.9)	7(0.6)	6(0.7)	0.0035
	Asthenia	G1	178(14.8) 210(15.3) 165(14.7)	97(11)	0.0263
	Injection site reaction	G1	24(2.0)	14(1.0)	16(1.4)	24(2.7)	0.015
	Nail changes	G1	74(6.1)	53(3.9)	65(5.8)	46(5.2)	0.0468
	Pruritos	G1	15(1.2)	16(1.2)	25(2.2)	27(3.1)	0.0026
	Rash/desquamation	G1	73(6.1)	39(2.8)	70(6.2)	55(6.2)	<0.0001
	Nauseas	G1	50(4.1)	42(3.1)	25(2.2)	18(2.0)	0.0132
	Epistaxis	G1	17(1.4)	5(0.4)	18(1.6)	27(3.1)	<0.0001
	Alkaline phosphatase	G1	19(1.6)	35 (2.5)	45(4.0)	11(1.2)	0.0001

Table 2 .

 2 Patient's toxicities related to Geographic-climate area*.

			Oceanic	Mediterranean Continental	P
	Toxicity	Grade	N(%)	N(%)	N(%)
	FEC90 every 21 days (5FU: 600mg/m 2 +4EPI 90mg/m 2 +cyclophosphamide 600mg/m 2 )
			N= 6033 cycles
			N=1294	N=2858	N=1881
	Hemoglobin	G1	614(47.4)	1155(40.4)	681(36.2)	<0.0001
		G2	63(4.9)	128(4.5)	57(3.0)	0.0147
	Leukocytes	G1	281(21.7)	506(17.7)	349(18.6)	0.0086
		G2	140(10.8)	194(6.8)	77(4.1)	<0.0001
		G3	106(8.2)	60(2.1)	18(1.0)	<0.0001
	Neutrophils	G1	175(13.5)	298(10.4)	234(12.4)	0.0081
		G2	130(10.0)	221(7.7)	117(6.2)	0.0004
		G3	130(10.0)	148(5.2)	74(3.9)	<0.0001
		G4	101(7.8)	30(1.0)	19(1.0)	<0.0001
	Platelets	G1	22(1.7)	27(0.9)	36(1.9)	0.0131
	Phlebitis	G2	16(1.2)	88(3.1)	34(1.8)	0.0003
	Asthenia	G1	170(13.1)	549(19.2)	206(11.0)	<0.0001
		G2	113(8.7)	231(8.1)	112(6.0)	0.0050
		G3	2(0.2)	21(0.7)	8(0.4)	0.0432
	Fever	G1	25(1.9)	87(3.0)	40(2.1)	0.045
	Rash/desquamation	G1	24(1.9)	96(3.4)	30(1.6)	0.0002
	Hot flashes/flushes	G1	15(1.2)	31(1.1)	7(0.4)	0.0174
	Constipation	G1	50(3.9)	169(5.9)	55(2.9)	<0.0001
		G2	10(0.8)	54(1.9)	12(0.6)	0.0002
	Dyspepsia/Heartburn	G1	48(3.7)	80(2.8)	41(2.2)	0.0371
		G2	1(0.1)	10(0.3)	15(0.8)	0.0064
	Nauseas	G1	307(23.7)	869(30.4)	452(24.0)	<0.0001
		G2	160(12.4)	394(13.8)	294(15.6)	0.0288
		G3	6(0.5)	57(2.0)	25(1.3)	0.0006
	Taste disturbance (dysgeusia)	G1	13(1.0)	98(3.4)	20(1.1)	<0.0001
	Vomiting	G1	166(12.8)	458(16.0)	230(12.2)	0.0003
		G2	224(17.3)	309(10.8)	263(14.0)	<0.0001
	SGOT/AST	G2	6(0.5)	39(1.4)	16(0.9)	0.0190
	SGPT/ALT	G1	297(23.0)	495(17.3)	365(19.4)	0.0001
	Paclitaxel 100 mg/m2 weekly; N=4461 cycles
			N=1010	N=2082	N=1369	P
	Allergic	G1	4(0.4)	12(0.6)	22(1.6)	0.0011
	reaction/hypersensitivity
	Hemoglobin	G2	67(6.6)	237(11.4)	94(6.9)	<0.0001
	Leukocytes	G2	150(14.9)	213(10.2)	109(8.0)	<0.0001
	Neutrophils	G1	156(15.4)	247 (11.9)	181(13.2)	0.0213
		G2	107(10.6)	130(6.2)	67(4.9)	<0.0001
		G3	19(1.9)	19(0.9)	11(0.8)	0.0241
	Edema	G1	31(3.1)	37(1.8)	9(0.7)	<0.0001
	Asthenia	G1	109(10.8)	410(19.7)	128(9.3)	<0.0001
		G2	44(4.4)	135(6.5)	74(5.4)	0.0494
	Injection site reaction	G1	9(0.9)	48(2.3)	21(1.5)	0.0146
	Nail changes	G1	69(6.8)	115(5.5)	48(3.5)	0.001
	Rash/desquamation	G1	70(6.9)	127(6.1)	39(2.8)	<0.0001
	Constipation	G1	14(1.4)	64(3.1)	22(1.6)	0.0020
	Diarrhea	G1	8(0.8)	51(2.4)	24(1.8)	0.0056

*Only statistically significant differences in toxicities are included


Margaretha Jurstrand
email: margareta.jurstrand@orebroll.se.

Per Olcén

Anders Magnuson

Lena Jakobsson

Hans Fredlund

Magnus Unemo

Emergence of the new variant of Chlamydia trachomatis (nvCT) in a defined area of Sweden before 2002?

Keywords: Chlamydia trachomatis, mutant, nvCT, Sweden, diagnostics, statistics 2

Objectives: To compare the proportions of Chlamydia trachomatis positive specimens detected by Cobas Amplicor CT/NG (CA PCR; Roche Diagnostics) to C trachomatis positives in cell culture from 1999 to 2006 in order to estimate when the new variant of C trachomatis (nvCT) with a deletion in the cryptic plasmid (in the target region for CA PCR that resulted in false negative results) emerged in Örebro County, Sweden.

Methods:

The annual number of specimens analysed using CA PCR in 1999-2006 ranged from 5077 to 11622 and using cell culture (McCoy cells) from 5201 to 7425. Logistic regression was applied to evaluate the change of proportion of C trachomatis positive tests over the years between the two methods. The statistical interaction effect of year and method was estimated using both unadjusted and adjusted (age, gender and clinic) models.

Results: From 2002, the proportion of C trachomatis positive specimens using CA PCR annually decreased, whereas the proportion of culture positive specimens annually increased.

The logistic regression showed a statistically significant interaction effect between periods (1999)(2000)(2001)(2002)(2003)(2004)(2005)(2006) and groups of specimens analysed with CA PCR or cell culture. A statistically significant association between the interaction of CA PCR/cell culture and period was observed in the unadjusted and adjusted models.

Conclusion:

This study indicates that in Örebro County, Sweden, the nvCT was already present before 2002, i.e. when the difference between the proportions of C trachomatis positive specimens in CA PCR compared to cell culture positive specimens began to show a statistically significant decline.

INTRODUCTION

In 2006, a new variant of Chlamydia trachomatis (nvCT) was reported in Sweden. [START_REF] Ripa | A variant of Chlamydia trachomatis with deletion in cryptic plasmid: implications for use of PCR diagnostic tests[END_REF] This variant had a 377 bp deletion in the cryptic plasmid that included the targets for the diagnostic systems from Roche Diagnostics and Abbott Laboratories, which at that time were widely used globally, and accordingly, resulted in thousands of false negative results. [START_REF] Herrmann | A new genetic variant of Chlamydia trachomatis[END_REF][START_REF] Ripa | A Chlamydia trachomatis strain with a 377-bp deletion in the cryptic plasmid causing false-negative nucleic acid amplification tests[END_REF] Both of the affected companies have now developed and CE mark-certified new dual-target assays capable of detecting the nvCT. [START_REF] Hadad | Evaluation of the new COBAS TaqMan CT Test v2.0 and impact on the proportion of new variant Chlamydia trachomatis by the introduction of diagnostics detecting new variant C. trachomatis in Örebro County, Sweden[END_REF] Several years ago in Sweden nearly all of the clinical laboratories replaced their cell culture methods and other established methods by nucleic acid amplification tests (NAATs) for diagnosis of C trachomatis. However, in Örebro County (approximately 275,000 inhabitants) C trachomatis specimens were, at the time of the first report of nvCT, diagnosed using either Cobas Amplicor CT/NG, Roche Diagnostics (CA PCR) or a highly optimised and quality assured cell culture method (detecting the nvCT), dependent on the request from the submitting physician. Routine diagnostics of all specimens in Örebro County is performed at the Department of Laboratory Medicine, Section Clinical Microbiology, Örebro University Hospital. During the recent decade, the incidence of notified C trachomatis cases in Örebro County has been relatively similar to the national incidence (see fig 1 in web appendix), in which the number of cases increased from 157 per 100,000 inhabitants in 1997 to 367 per 100,000 in 2005. However, in 2006 the incidences decreased, i.e. from 367 to 357 cases per 100,000 inhabitants nationally and from 336 (year 2005) to 311 cases per 100,000 inhabitants in Örebro County. [START_REF] Unemo | Experiences with the new genetic variant of Chlamydia trachomatis in Örebro county, Sweden -proportion, characteristics and effective diagnostic solution in an emergent situation[END_REF] This decrease could subsequently be explained by the wide spread of the nvCT, e. g., in Örebro County the nvCT proportion was 38% at the time of the first report of the nvCT, [START_REF] Unemo | Experiences with the new genetic variant of Chlamydia trachomatis in Örebro county, Sweden -proportion, characteristics and effective diagnostic solution in an emergent situation[END_REF] about 30%-35% during entire 2007 and mid-2008 [START_REF] Hadad | Evaluation of the new COBAS TaqMan CT Test v2.0 and impact on the proportion of new variant Chlamydia trachomatis by the introduction of diagnostics detecting new variant C. trachomatis in Örebro County, Sweden[END_REF] and 24% in the beginning of 2009. [START_REF] Klint | Spread of the new variant of Chlamydia trachomatis within Sweden over time[END_REF] The spread and proportions of the nvCT have differed in the Swedish counties, which leads to the question; When did the mutant emerge and reach detectable levels in the different populations? Unfortunately, in 2006 few archival collections of C trachomatis samples were available at the Swedish laboratories and appropriate retrospective longitudinal studies to answer the question regarding the emergence of nvCT have not been possible to perform.

The aim of the present study was to compare the proportion of C trachomatis positive specimens in the CA PCR method to the proportion of C trachomatis cell culture positive specimens from 1999 to 2006 in order to estimate when the nvCT first emerged in Örebro County, Sweden.

MATERIAL AND METHODS

Study population and laboratory methods

Urogenital or first void urine specimens were obtained from men and women attending the outpatient STD clinic at Örebro University Hospital, the youth clinics, the welfare centres, the women's clinic, or other physisians in Örebro County and were sent for routine diagnosis of C trachomatis at the Örebro University Hospital. The diagnostic methods used were CA PCR (could not detect the nvCT at that time) or cell culture (cycloheximide treated McCoy cells), with subsequent identification of C trachomatis using fluorescein-labelled monoclonal antibodies (Phadebact Chlamydia IF Test, Bactus AB, Sweden) that can detect the nvCT. The total number of specimens analysed using CA PCR for each year from 1999 to 2006 was 5077, 5178, 6877, 7811, 8702, 10070, 11111, and 11622, and using cell culture 7425, 6666, 7133, 7311, 6446, 6148, 5533 and 5201, respectively. A low number of specimens (<1%) were excluded in the statistical analyses because of diagnostic problems such as cytotoxicity in the cell culture method and inhibited or equivocal results in the CA PCR. No specimen was analysed by both methods.

The percentage of C trachomatis positive specimens from all clinics for each year (1999)(2000)(2001)(2002)(2003)(2004)(2005)(2006) and method (CA PCR or cell culture) was calculated.

Statistical analyses

Logistic regression with positive test (yes/no) as the outcome variable was used to evaluate the change of proportion of positive tests over the years by estimating the statistical interaction effect between the two methods over years. The analysis was done with variable year as a covariate to estimate the interaction as a linear effect. To visualize data the proportion positive tests supplemented with 95% confidence intervals (CIs) was plotted using the binomial distribution for each year and method in Figure 2a and the difference of proportions of positive tests between methods in Figure 2b supplemented with 95% normal approximation CIs. To statistically analyse in more detail which year(s) the significant change of proportion occurred, years were categorised into two-year intervals: 1999-2000, 2001-2002, 2003-2004 and 2005-2006. The interaction effect was analysed between 1999-2000 and 2001-2002, 2001-2002 and 2003-2004, and 2003-2004 and 2005-2006. All analyses were done unadjusted and adjusted. In the adjusted analysis age, gender and clinics were included as covariates, to compensate for potential confounding effects. The effect parameters are expressed as odds ratios (ORs) supplemented with 95% CIs.

RESULTS

The distributions of patients examined for C trachomatis using CA PCR or cell culture in two-year intervals from 1999-2006 are presented in Table 1. Among the specimens analysed using CA PCR, 51% were from patients attending the STD clinic and the youth clinic (highrisk group), in total, 59% were women (median age 20-24 years). In contrast, among the specimens analysed using the cell culture method only 33% were from patients attending these clinics while 50% were from welfare centres (low-risk group), in total, 95% were women (median age 20-24 years).

Estimating only the specimens analysed using CA PCR, the proportion of positive tests decreased from 1999 to 2006, with an average of 3.5% per year in the unadjusted model and 2). Comparing the years 1999-2000 with 2001-2002, no significant statistical interaction effect between methods and period was found in either the unadjusted (p=0.997) or adjusted model (p=0.945) (Table 3). However, the interaction effect was significant when comparing the The incidence of notified C trachomatis cases in Örebro County (see fig 1 in web appendix) also indicates that the increase in the incidence, which has been stable since the mid-1990s, showed a reduction already in 2002, which is in full concordance with our statistical analyses. Furthermore, already in 1999-2000 a suspected culture positive mutant, which was positive in conventional ompA (omp1) gene PCR but repeatedly negative in CA PCR, was identified in Örebro County. [START_REF] Jurstrand | Characterization of Chlamydia trachomatis omp1 genotypes among sexually transmitted disease patients in Sweden[END_REF] This specimen was determined as genotype E (i.e. as the present nvCT), but 47% of the specimens in that study were genotype E. At that time, the isolate was considered as an exceedingly rare plasmid-free C trachomatis strain. [START_REF] An | Infection with a plasmid-free variant Chlamydia related to Chlamydia trachomatis identified by using multiple assays for nucleic acid detection[END_REF][START_REF] Magbanua | Chlamydia trachomatis variant not detected by plasmid based nucleic acid amplification tests: molecular characterisation and failure of single dose azithromycin[END_REF] However, today it cannot be excluded that this was a nvCT sample already present in 1999-2000 in Örebro County, although it was not until 2002-2003 that the spread of the nvCT had reached an impact on the C trachomatis population, which was possible to identify with the statistical analyses used in the present study. We have previously described that all the examined nvCT isolates in Örebro County have been of genotype E with an identical ompA sequence and displayed an identical and unique MLST profile. [START_REF] Unemo | Experiences with the new genetic variant of Chlamydia trachomatis in Örebro county, Sweden -proportion, characteristics and effective diagnostic solution in an emergent situation[END_REF][START_REF] Jurstrand | Characterization of Chlamydia trachomatis by ompA sequencing and multilocus sequence typing (MLST) in a Swedish county before and after identification of the new variant[END_REF] This profile was also found in arbitrarily selected nvCT isolates (n = 48) from four countries, [START_REF] Herrmann | Emergence and spread of Chlamydia trachomatis variant, Sweden[END_REF] which strongly indicates a clonal spread of the nvCT.

In another Swedish study a trend analysis was done in 2004 to 2006 on national data from the Swedish Institute for Infectious Disease Control of the number of genital chlamydial cases detected and the number of C trachomatis tests performed. [START_REF] Herrmann | Emergence and spread of Chlamydia trachomatis variant, Sweden[END_REF] This analysis showed a decline in the proportion of positive chlamydia tests in the counties that used the Abbott or Roche methods from 2005, whereas the positivity rates in the counties using the BD ProbeTec ET (Becton Dickinson) tests did not change. The discrepancies between the present study and the previous national study [START_REF] Herrmann | Emergence and spread of Chlamydia trachomatis variant, Sweden[END_REF] regarding the year the nvCT emerged may be that in the national study only a few years (2004)(2005)(2006) and the first six months of each year were studied. Another difference is the nationwide coverage of the previous study, which included all the counties of Sweden that significantly differed in population characteristics, C trachomatis incidence and prevalence, diagnostic facilities and preventive and other interventions performed, all of which may affect the positivity rates of the C trachomatis diagnostics. Nevertheless, the county-specific differences in coverage in testing of the target population were adjusted for.

Since 2006 the nvCT has been detected in high numbers throughout Sweden. However, the proportions of nvCT now seem to converge in the Swedish counties after the selective diagnostic advantage for nvCT has disappeared in the Roche/Abbott counties. [START_REF] Klint | Spread of the new variant of Chlamydia trachomatis within Sweden over time[END_REF] This observation is also supported by the unaltered biological fitness of nvCT, which was identified in a recent study comparing the nvCT genome to all available C trachomatis genome sequences, and also comprehensively investigating the biological properties of nvCT. [START_REF] Unemo | The Swedish new variant of Chlamydia trachomatis: Genome sequence, morphology, cell tropism and phenotypic characterisation[END_REF] Astonishingly, although many studies have been performed worldwide, for the most part only sporadic nvCT cases have been identified outside the Nordic countries. [START_REF] Herrmann | Update on the new variant of Chlamydia trachomatis: prevalence and diagnostics[END_REF][START_REF] Savage | Results of a Europe-wide investigation to assess the presence of a new variant of Chlamydia trachomatis[END_REF] Nevertheless, knowledge of the presence and prevalence of nvCT in other countries is limited today for the following reasons: few recent studies have been conducted, many European laboratories still cannot detect the nvCT because they are using CA PCR or inappropriate in house PCRs and laboratories that can detect nvCT do not know it because no nvCT-specific or other distinguishing NAATs are used. [START_REF] Unemo | Can the Swedish new variant of Chlamydia trachomatis (nvCT) be detected by UK NEQAS participants from seventeen European countries and five additional countries/regions in 2009?[END_REF][START_REF] Reischl | The Swedish new variant of Chlamydia trachomatis (nvCT) remains undetected by many European laboratories as revealed in the recent PCR/NAT ring trial organised by INSTAND e.V[END_REF] Accordingly, ideally all laboratories should use NAATs that detect the nvCT, because a wider geographic spread of the nvCT cannot be excluded. Furthermore, greater emphasis should be given to the importance of regular local, national and international surveillance for unexplained declines in the C trachomatis incidence. Finally, recommendations for diagnostics and participation in appropriate external quality assessment schemes need to be emphasised.

In conclusion, we were able to compare the C trachomatis positivity rates from 1999-2006 using two methods (Cobas Amplicor CT/NG PCR and cell culture on McCoy cells) within the same laboratory in Örebro County, Sweden. Based on our statistical analyses, the nvCT seems to have emerged before 2002, i.e. when the difference in proportions of C trachomatis positive specimens in Cobas Amplicor PCR compared to cell culture positive specimens began to show a statistically significant decline.

3 .

 3 9% per year in the adjusted model. In contrast, estimating only the specimens examined using cell culture, the proportion of positive tests increased during the same period with an average of 2.3% per year in the unadjusted model and 3.7% per year in the adjusted model (fig2a). When analysing year as a linear effect from 1999-2006 the logistic regression analysis showed a statistically significant interaction effect between methods (CA PCR and cell culture) and years(1999)(2000)(2001)(2002)(2003)(2004)(2005)(2006) on the proportion of positive tests both in the unadjusted (p<0.001) and adjusted model (p<0.001), (Table

  Furthermore, the difference between the proportions of positive tests using the two divergent methods plotted each year (fig 2b) showed a visual change in the distribution of positive tests between methods beginning in 2002-2003.

  intervals 2001-2002 and 2003-2004, both in the unadjusted (p=0.030) and adjusted model (p=0.036). Moreover, this significant interaction of methods and period was observed when comparing the intervals 2003-2004 and 2005-2006 (unadjusted, p=0.010 and adjusted, p=0.001). This finding showed that the difference between the proportions of C trachomatis positive specimens in CA PCR compared with cell culture positive specimens, began to statistically significantly decline between the intervals 2001-2002 and 2003-2004.

  proportions of C trachomatis positive specimens in CA PCR and in cell culture began to statistically significantly decline. All specimens from patients attending the outpatient STD clinic at Örebro University Hospital, the youth clinics, the welfare centres, the women's clinic, or other physicians in Örebro County and analysed for C trachomatis between 1999 and 2006 were included. The specimens were diagnosed using either CA PCR (could not detect the nvCT at that time) or cell culture (detecting the nvCT) and the C trachomatis positivity rates were statistically compared for the eight-year period (1999-2006). The estimation was based on a comparison of the positivity proportions using two diagnostic methods (CA PCR and cell culture), with the nvCT seemingly present before 2002 in Örebro County (fig 2a and 2b). The proportion of C trachomatis positive specimens diagnosed by CA PCR was higher, because about 50% of the specimens were from patients attending the STD clinic and youth clinics (i.e. high-risk group). The corresponding figure analysed by cell culture was about 30%. No major alterations of these methods or any preventive interventions in the studied population were performed during the examined years that could explain the findings.

Figure 2b

 2b Figure 2bDifference in the percentage of C trachomatis positive tests using CA PCR and

Table 1 .

 1 The distribution of patients examined for Chlamydia trachomatis using Cobas Amplicor CT/NG, Roche Diagnostics (CA PCR) or cell 121 culture (in percentage) by clinics, gender and age in two-year intervals from 1999-2006 122

	CA PCR

Table 2 .

 2 Interaction of a Chlamydia trachomatis positive test using Cobas Amplicor CT/NG, Roche Diagnostics (CA PCR) or cell culture by year (1999 to 2006) as a linear effect.

* Adjusted by clinic, gender and age.

Table 3 .

 3 The interaction of a Chlamydia trachomatis positive test analysed using Cobas Amplicor CT/NG, Roche Diagnostics (CA PCR) or cell The present study demonstrated that the nvCT was prevalent at non-negligible levels already in 2002-2003 in Örebro County (Table 3, fig 2b), i.e. when the difference between the

		CA PCR		Cell culture	Interaction
						(CA PCR/cell culture)
		OR	p	OR	p	OR	p
		(95% CI)		(95% CI)		(95% CI)
		0.965	<0.001	1.023	0.011	0.943	<0.001
	Unadjusted	(0.952 -0.978)		(1.005 -1.041)		(0.923 -0.964)
		0.961	<0.001	1.037	<0.001	0.933	<0.001
	Adjusted*	(0.948 -0.974)		(1.018 -1.055)		(0.913 -0.954)

* The total number of specimens analysed using CA PCR from 1999-2006 was 66428, but in each year some specimens were excluded, because of inhibited or equivocal results. No specimen was analysed by both methods. † The total number of specimens analysed using the cell culture method from 1999-2006 was 51863, but in each year some specimens were excluded, because of cytotoxicity destroyed the McCoy cell monolayer. ‡ Notably, there were mostly cervical specimens analysed in the cell culture while the distributions of specimens from the men and women were similar in the CA PCR.

Funding: This work was supported by grants from the Örebro Medical Research Foundation, Örebro University Hospital, Örebro, Sweden.

Competing interests: None.