Openness of the set of non characteristic points and regularity of the blow-up curve for the 1 D

Openness of the set of non characteristic points and regularity of the blow-up curve for the 1 D

semilinear wave equation ^∗

Frank Merle

Universit´e de Cergy Pontoise, IHES and CNRS

Hatem Zaag

CNRS UMR 7539 LAGA Universit´e Paris 13

February 26, 2008

Abstract

We consider here the 1 D semilinear wave equation with a power nonlinearity and with no restriction on initial data. We first prove a Liouville Theorem for that equation. Then, we consider a blow-up solution, its blow-up curve x 7→ T(x) and I⁰⊂Rthe set of non characteristic points. We show thatI⁰is open and thatT(x) is C¹ onI⁰. All these results fundamentally use our previous result in [19] showing the convergence in selfsimilar variables forx∈I⁰.

AMS Classification: 35L05, 35L67

Keywords: Wave equation, Liouville Theorem, regularity, blow-up set.

1 Introduction

1.1 The problem and known results

We consider the one dimensional semilinear wave equation

∂

u = ∂

u + | u |

u,

u(0) = u

and u

(0) = u

, (1)

where p > 1, u(t) : x ∈

→ u(x, t) ∈

, u

∈ H

and u

∈ L

where k v k

= sup

a∈R

Z

|x−a|<1

| v(x) |

dx and k v k

loc,u

= k v k

loc,u

+ k∇ v k

loc,u

.

The Cauchy problem for equation (1) in the space H

× L

follows from the finite speed of propagation and the wellposedness in H

× L

(see Ginibre, Soffer and Velo [6]).

The existence of blow-up solutions for equation (1) follows from energy techniques (see

∗This work was supported by a grant from the french Agence Nationale de la Recherche, project ONDENONLIN, reference ANR-06-BLAN-0185.

Levine [9]). More blow-up results can be found in Caffarelli and Friedman [5], [4], Alinhac [1], [2] and Kichenassamy and Litman [7], [8].

Most of the previous literature considered blow-up for the wave equation from the point of view of prediction. Indeed, most of the papers gave sufficient conditions to have blow-up or constructed special solutions with a prescribed behavior (see [7] and [8] for example). As we did in our earlier work ([19], [17], [16] and [18]), we adopt in this paper a different point of view and aim at describing the blow-up behavior for any blow-up solution. More precisely, this paper is dedicated to the regularity of the blow-up curve.

If u is a blow-up solution of (1), we define (see for example Alinhac [1]) a continuous curve Γ as the graph of a function x 7→ T (x) such that u cannot be extended beyond the set

D

= { (x, t) | t < T (x) } . (2) The set D

is called the maximal influence domain of u. From the finite speed of propa- gation, T is a 1-Lipschitz function. Let ¯ T be the infimum of T (x) for all x ∈

. The time T ¯ and the surface Γ are called (respectively) the blow-up time and the blow-up surface of u. A point x

∈

is called a non characteristic point if

∃ δ

= δ

(x

) ∈ (0, 1) and t

(x

) < T (x

) such that u is defined on C

∩ { t ≥ t

} (3) where

C

= { (x, t) | t < ¯ t − δ ¯ | x − x ¯ |} . (4) We denote by I

the set of non characteristic points. So far, it was commonly thought that I

=

, for any blow-up solution. In a forthcoming paper [20], we prove that this is not the case.

Given some (x

, T

) such that 0 < T

≤ T (x

), we introduce the following self-similar change of variables:

w

(y, s) = (T

− t)

u(x, t), y = x − x

T

− t , s = − log(T

− t). (5) If T

= T (x

), then we simply write w

instead of w

. This change of variables transforms the backward light cone with vertex (x

, T

) into the infinite cylinder (y, s) ∈ B × [ − log T

, + ∞ ) where B = B (0, 1). The function w

(we write w for simplicity) satisfies the following equation for all y ∈ B and s ≥ − log T

:

∂

w = L w − 2(p + 1)

(p − 1)

w + | w |

w − p + 3

p − 1 ∂

w − 2y∂

w (6) where L w = 1

ρ ∂

ρ(1 − y

)∂

w

and ρ(y) = (1 − y

)

. (7) The Lyapunov functional for equation (6)

E(w(s)) =

−1

1

2 (∂

w)

+ 1

2 (∂

w)

(1 − y

) + (p + 1)

(p − 1)

w

− 1

p + 1 | w |

ρdy (8)

is defined in H =

q ∈ H

× L

( − 1, 1) | k q k

≡

−1

q

+ q

(1 − y

) + q

ρdy < + ∞

. (9) In [19], we find the behavior of w

(y, s) defined in (5) as s → ∞ where x

is a non characteristic or a characteristic point. More precisely, we proved this result (see Corollary 4 and Theorem 2 in [19]):

Blow-up profile near a non characteristic point There exist positive µ

and C

such that if u is a solution of (1) with blow-up curve Γ : { x 7→ T (x) } and x

∈

is non characteristic (in the sense (3)), then there exist d(x

) ∈ ( − 1, 1), θ(x

) = ± 1, s

(x

) ≥ − log T (x

) such that for all s ≥ s

(x

):

w

(s)

∂

w

(s)

− θ(x

)

κ(d(x

), .) 0

H

≤ C

e

, (10) where H and its norm are defined in (9), κ(d, y) is defined for all | d | < 1 and | y | ≤ 1 by

κ(d, y) = κ

(1 − d

)

(1 + dy)

where κ

=

2(p + 1) (p − 1)

_p−1¹

. (11)

Moreover, we have

E(w

(s)) → E(κ

) as s → ∞ (12)

and

k w

(s) − θκ(d(x

), y) k

+ k ∂

w

(s) − θ∂

κ(d(x

), y) k

+ k ∂

w

(s) k

→ 0 as s → ∞ .

(13)

Blow-up behavior near a characteristic point If x

∈

is characteristic, then, there exist k(x

) ∈

, θ

(x

) = ± 1 and continuous d

(s) = tanh ζ

(s) ∈ ( − 1, 1) for i = 1, ..., k such that

w

(s)

∂

w

(s)

−







k(x0)

X

i=1

θ

(x

)κ(d

(s), · ) 0





 H

→ 0,

| ζ

(s) − ζ

(s) | → ∞ for i 6 = j and E(w

(s)) → k(x

)E(κ

) as s → ∞ .

(14)

Remark: When k(x

) = 0, the sum in (14) has to be understood as 0. In [20], we prove the existence of solutions with characteristic points. Furthermore, we greatly improve estimate (14) and give a precise description of the set of characteristic points.

1.2 Statement of the results

With the result of [19], we are in a position to address the question of the regularity of the

blow-up set Γ for equation (1) and the notion of non characteristic points. Note that by

definition (see Alinhac [1]), Γ is the graph of a 1-Lipschitz function T (x). In [5], Caffarelli and Friedman proved that it is a C

function for N ≤ 3 under restrictive conditions on initial data that ensure that

for all x ∈

and t ≥ 0, u ≥ 0 and ∂

u ≥ (1 + δ

) |∇ u | for some δ

> 0.

Later on, they derived in [4] the same result in one dimension for p ≥ 3 and initial data in C

× C

(

). The techniques to prove these results are of elliptic type and based on the use of the maximum principle. There is no hope to generalize these techniques to the present situation. Furthermore, no results are available on the set of non characteristic points.

In this paper, we prove the following with no restrictions on the initial data:

Theorem 1 (C

regularity of the blow-up set and continuity of the blow-up profile on I

) Consider u a solution of (1) with blow-up curve Γ : { t = T (x) } .

Then, the set of non characteristic points I

is open and T (x) is C

on that set.

Moreover,

∀ x ∈ I

, T

(x) = d(x) ∈ ( − 1, 1) and θ(x) is constant on connected components of I

, where d(x) and θ(x) are such that (10) holds.

Remark: From the remark after Proposition 3.5 (with δ

=

), we get the existence of a non characteristic point. Thus, I

is never empty and Theorem 1 is always meaningful.

Unlike what was commonly thought until now, we prove in a forthcoming paper [20] the existence of blow-up solutions to (1) with

\ I

6 = ∅ .

Remark: From this theorem, the parameter d(x) related to the blow-up profile in selfsim- ilar variables (10), has a geometrical interpretation as the slope of T (x). In [21], Nouaili uses the results and techniques of [19] and this paper with a geometrical approach to get more regularity, namely C

regularity where the universal constant µ

is introduced before (10).

Remark: The techniques are based on a very good understanding of the behavior of the solution in selfsimilar variables in the energy space related to the selfsimilar variable, to- gether with a Liouville Theorem (see Theorem 2). For a similar approach of the blow-up problem, see Martel and Merle [11], [10] for the critical KdV equation; see also Merle and Rapha¨el [12], [13] for the NLS equation. Note that the main obstruction to extend the result in higher dimensions is the result of classification of all stationary solutions to (6).

The proof of this Theorem relies fundamentally on the convergence result (10) already obtained in [19] and on this Liouville Theorem in the u or w variable:

Theorem 2 (A Liouville Theorem for equation (1)) Consider u(x, t) a solution to equation (1) defined in the cone C

(4) such that for all t < T

,

(T

− t)

k u(t) k

δ∗ ))

(T

− t)

+(T

− t)

k u

(t) k

(T

− t)

+ k∇ u(t) k

(T

− t)

!

≤ C

(15)

for some (x

, T

) ∈

, δ

∈ (0, 1) and C

> 0.

Then, either u ≡ 0 or u can be extended to a function (still denoted by u) defined in { (x, t) | t < T

+ d

(x − x

) } ⊃ C

by u(x, t) = θ

κ

(1 − d

)

(T

− t + d

(x − x

))

, (16) for some T

≥ T

, d

∈ [ − δ

, δ

] and θ

= ± 1, where κ

is defined in (11).

Using the selfsimilar transformation (6), we have this equivalent formulation for Theorem 2:

Theorem 2’ (A Liouville Theorem for equation (6)) Consider w(y, s) a solution to equation (6) defined for all (y, s) ∈ ( −

,

∗

) ×

such that for all s ∈

, k w(s) k

δ∗,_δ¹_∗)

+ k ∂

w(s) k

δ∗,_δ¹_∗)

≤ C

(17)

for some δ

∈ (0, 1) and C

> 0. Then, either w ≡ 0 or w can be extended to a function (still denoted by w) defined in

{ (y, s) | − 1 − T

e

< d

y } ⊃

− 1 δ

, 1

δ

×

by w(y, s) = θ

κ

(1 − d

)

(1 + T

e

+ d

y)

, (18) for some T

≥ T

, d

∈ [ − δ

, δ

] and θ

= ± 1, where κ

defined in (11).

Remark: The limiting case δ

= 1 is still open. The case δ

= 0 trivially follows from the case δ

> 0. We expect the result to be valid in higher dimension using the same techniques. The main obstruction comes from the classification of stationary solutions to equation (6), which is only proved in one dimension (see [19]) and remains open in higher dimension.

Remark: From (16), we see that u is a particular solution to (1) which is defined in a half-space of equation t < T

+ d

(x − x

) (and not just the cone C

) and blows-up on a straight line of slope d

. Note that u ≡ 0 corresponds to the limiting case T

= + ∞ . Similarly, w given in (18) is a particular solution to (6). In particular, when T

= 0, we recover the stationary solution θ

κ(d

, y) defined in (11). Note also that up to a Lorentz transform, the solution given in (16) is space independent and given by

u(x, t) = θ

κ

(T

− t)

.

Remark: Note that deriving blow-up estimates through the proof of Liouville Theorems has been successful for different problems. For the case of the heat equation

∂

u = ∆u + | u |

u (19)

where u : (x, t) ∈

× [0, T ) →

, p > 1 and (N − 2)p < N + 2, the blow-up time T is

unique for equation (19). The blow-up set is the subset of

such that u(x, t) does not

remain bounded as (x, t) approaches (x

, T ). In [25], [22] and [23] (see also the note [24]),

the second author proved the C

regularity of the blow-up set under a non degeneracy

condition. A Liouville Theorem proved in [15] and [14] was crucially needed for the proof

of the regularity result in the heat equation. In the present work, we will see that the Liouville Theorem (Theorem 2) is crucial for the regularity of the blow-up set for the wave equation (Theorem 1).

Remark: The proof has a completely different structure from the proof for the heat equation (19). It is based on various energy arguments in selfsimilar variables (some of them holding even in the characteristic situation) and on a dynamical result again in selfsimilar variables obtained in [19] in the non characteristic case.

The paper is organized as follows:

- In section 2, we assume the Liouville Theorem and prove Theorem 1.

- In section 3, we prove the Liouville Theorem (Theorems 2 and 2’).

The authors wish to thank the referee for his meticulous reading and valuable sugges- tions which undoubtedly improved the paper.

2 C

regularity of the blow-up curve at non characteristic points

This section is devoted to the proof of Theorem 1 assuming Theorem 2.

Proof of Theorem 1: We consider a solution u of (1) with blow-up curve Γ : { t = T (x) } . We proceed in two parts, each making a separate subsection:

- In subsection 2.1, we consider a non characteristic point x

and show that T (x) is differentiable at x = x

with T

(x

) = d(x

) where d(x

) is such that (10) holds.

- In subsection 2.2, we conclude the proof of Theorem 1.

2.1 Differentiability of the blow-up curve at a given point In this subsection, we prove the proposition:

Proposition 2.1 (Differentiability of the blow-up curve at a given non charac- teristic point) If x

is a non characteristic point, then T (x) is differentiable at x

and T

(x

) = d(x

) where d(x

) is such that (10) holds.

Proof: We consider x

a non characteristic point. From (3), we have

C

∩ { t ≥ t

} ⊂ D

(20) for some δ

∈ (0, 1) and t

< T (x

).

From translation invariance, we can assume that x

= T (x

) = 0.

Using the convergence result of [19], we see that (10) holds for some d(0) ∈ ( − 1, 1) and θ(0) = ± 1. Up to replacing u(x, t) by − u(x, t) (also solution to equation (1)), we can assume that θ(0) = 1.

We proceed in 2 steps.

- In Step 1, we use the Liouville Theorem 2 to show that the convergence of w

(s) in (10) holds also in H

× L

−

0

,

where

δ

∈ (δ

, 1) is fixed. (21)

- In Step 2, we use energy and continuation arguments in selfsimilar variables to con- clude the proof by contradiction.

Step 1: Convergence of w

to κ(d(0), .) on larger sets We claim:

Lemma 2.2 (Convergence in selfsimilar variables on larger sets) It holds that

w

(s)

∂

w

(s)

−

κ(d(0), .) 0

H¹×L²

„

−_δ¹_′

0,_δ¹_′

0

«

→ 0 as s → ∞ .

Proof: For simplicity, we denote w

by w. Using the uniform bound on the solution at blow-up (Theorem 2’ in [17]) and the covering technique in that paper (Proposition 3.3 in [17]), we get for all s ≥ − log T(0) + 1,

w(s)

∂

w(s)

H¹×L²(−_δ¹_′

0

,_δ¹_′

0

)

≤ K (22)

for some constant K.

We proceed by contradiction and assume that for some ǫ

> 0 and some sequence s

→ ∞ , we have

∀ n ∈

,

w(s

)

∂

w(s

)

−

κ(d(0), .) 0

H¹×L²

„

−¹

δ′ 0

,¹

δ′ 0

«

≥ ǫ

> 0. (23) Let us introduce the sequence

w

(y, s) = w(y, s + s

). (24)

Using the uniform bound stated in (22), we can assume that w

(0) ⇀ z

in H

− 1 δ

, 1

δ

and ∂

w

(0) ⇀ v

in L

− 1 δ

, 1

δ

(25) as n → ∞ for some (z

, v

) ∈ H

× L

−

,

. Since we have from the convergence result (10), the definitions (9) and (24) of the norm in H and w

,

w

(0)

∂

w

(0)

−

κ(d(0), .) 0

H¹×L²(−1+ǫ,1−ǫ)

→ 0 as n → ∞ for any ǫ ∈ (0, 1), we deduce from (25) that

∀ y ∈ ( − 1, 1), z

(y) = κ(d(0), y) and v

(y) = 0 (26) (note that we still need to determine (z

, v

) for 1 < | y | <

0

). The following claim allows

us to conclude, thanks to the Liouville Theorem 2:

Claim 2.3 (Existence of a limiting object) There exists W (y, s) a solution to (6) de- fined for all (y, s) ∈

−

0

,

×

such that:

(i) W (0) = z

and ∂

W (0) = v

and the convergence is strong in (25).

(ii) For all s ∈

,

W (s)

∂

W (s)

H¹×L²

„

−_δ¹_′

0,_δ¹_′

0

«

≤ K (27)

where K is defined in (22).

Proof: See Appendix A.

Indeed, from this claim, we see that W (y, s) satisfies the hypothesis of Theorem 2’.

Therefore, either W ≡ 0 or there exists T

≥ 0, d

∈ [ − δ

, δ

] and θ

= ± 1 such that:

∀ (y, s) ∈

− 1 δ

, 1

δ

×

, W (y, s) = θ

κ

(1 − d

)

(1 + T

e

+ d

y)

(28) on the one hand, where κ

is defined in (11). On the other hand, using (26), (i) of Claim 2.3, and the definition (11) of κ(d, y), we see that

∀ y ∈ ( − 1, 1), W (y, 0) = z

(y) = κ(d(0), y) = κ

(1 − d(0)

)

(1 + d(0)y)

. (29)

Comparing (28) and (29) when y ∈ ( − 1, 1) and s = 0, we see that θ

= 1, d

= d(0) and T

= 0, hence, from (28),

∀ (y, s) ∈

− 1 δ

, 1

δ

×

, W (y, s) = κ(d(0), y).

In particular, from (24), (25) and (i) of Claim 2.3, this implies that

w(s

)

∂

w(s

)

−

κ(d(0), .) 0

H¹×L²

„

−¹

δ′ 0

,¹

δ′ 0

«

→ 0 as n → ∞ , which contradicts (23). Thus, Lemma 2.2 holds.

Step 2: Conclusion of the proof of Proposition 2.1

Our goal is to prove that T (x) is differentiable when x = 0 and that T

(0) = d(0). We proceed by contradiction. From the fact that T (x) is 1-Lipschitz, we assume that there is a sequence x

such that

x

→ 0 and T (x

)

x

→ d(0) + λ with λ 6 = 0 as n → ∞ . (30) Up to extracting a subsequence and to considering u( − x, t) (also solution to (1)), we can assume that x

> 0. Since 0 is non characteristic, we see from (20) and (21) that

λ + d(0) ≥ − δ

> − δ

. (31)

The following corollary transposing the convergence of w

(s) to w

(s) follows from Lemma

2.2:

Corollary 2.4 Let δ

=

. For σ

= − log

1(T(xn)+δ^′₀xn) δ1−δ₀^′

, we have

w

(σ

)

∂

w

(σ

)

−

w

(σ

)

∂

w

(σ

)

H¹×L²“

−_δ¹

1,_δ¹

1

”

→ 0 as n → ∞ where ± = − sgn λ,

σ

= log

| λ | (δ

− δ

) δ

(λ + d(0) + δ

)

and w

(y, s) = κ

(1 − d(0)

)

(1 ± e

+ d(0)y)

(32) is a solution to (6).

Proof:

We define for n large enough a time τ

as the largest t such that the section of the cone C

at time t is included in the cone C

. Note by definition of τ

that we have

B

x

, T (x

) − τ

δ

⊂ B

0, − τ

δ

(33) and τ

= T (x

) − δ

(ξ

− x

) = − δ

ξ

for some ξ

∈

, hence

τ

= − δ

δ

− δ

(T (x

) + δ

x

). (34) From (30), (31) and the fact that 0 < δ

< δ

< 1, note that τ

≤ 0.

Using the selfsimilar transformation (5), we write u(x, t) = ( − t)

w

(y, s) where y = x

− t , s = − log( − t), (35) u(x, t) = (T (x

) − t)

w

(z, σ) where z = x − x

T (x

) − t , σ = − log(T (x

) − t).

Therefore, we have

w

(z, σ) = (1 + e

T (x

))

w

(y, s), (36)

∂

w

(z, σ) = (1 + e

T (x

))

∂

w

(y, s), (37)

∂

w

(z, σ) = (1 + e

T (x

))

(1 + e

T (x

))∂

w

(y, s) + 2e

T (x

)

p − 1 w

(y, s) + e

(yT (x

) + x

)∂

w

(y, s)

(38) where

y = z + x

e

1 − e

T (x

) and s = σ − log(1 − e

T (x

)). (39) Using the same process as in (35) with κ(d(0), .) instead of w

(note that κ(d(0), .) is also solution to (6)), we define this solution to (6)

¯

w

(z, σ) = (1 + e

T (x

))

κ(d(0), y) = κ

(1 − d(0)

)

(1 + e

(d(0)x

− T (x

)) + d(0)z)

. (40)

Hence, estimates (36)-(38) hold when (w

, w

) is replaced by ( ¯ w

, κ

). If we take now σ = σ

≡ − log(T(x

) − τ

), then we see from (39) that s = σ

≡ − log( − τ

) and from (34) and (30) that as n → ∞ ,

e

(d(0)x

− T (x

)) → − λ(δ

− δ

) δ

(λ + d(0) + δ

) , e

x

→ δ

− δ

δ

(δ

+ d(0) + λ) and 1 + e

T (x

) → δ

(δ

+ d(0) + λ) δ

(δ

+ d(0) + λ) > 0

(41)

because of (31). Therefore, if n is large enough, we get from (36)-(38)

w

(σ

)

∂

w

(σ

)

−

w ¯

(σ

)

∂

w ¯

(σ

)

H¹×L²“

−_δ¹

1,_δ¹

1

”

(42)

≤ C

w

(σ

)

∂

w

(σ

)

−

κ(d(0), .) 0

H¹×L²(Jn)

where the interval J

⊂

−

,

by (33). Since we have from (32), (40) and (41)

w ¯

(σ

)

∂

w ¯

(σ

)

−

w

(σ

)

∂

w

(σ

)

H¹×L²

„

−¹

δ′ 0

,¹

δ′ 0

«

→ 0 as n → ∞

where w

(σ

) is defined in (32), we get the conclusion of (i) in Lemma 2.4 from Lemma 2.2 and (42).

Let us conclude the proof of Lemma 2.2. We consider two cases depending on the sign of λ and reach a contradiction in both cases. Using the selfsimilar transformation (5), we introduce the following solutions to equation (1):

v

(ξ, τ ) = (1 − τ )

w

(y, s) with y = ξ

1 − τ , s = σ

− log(1 − τ ), (43) and

¯

v

(ξ, τ ) = (1 − τ )

w

ξ

1 − τ , σ

− log(1 − τ )

= κ

(1 − d(0)

)

((1 − τ ) ± e

+ d(0)ξ)

. (44) From Corollary 2.4, we have

v

(0)

∂

v

(0)

−

¯ v

(0)

∂

v ¯

(0)

H¹×L²“

−_δ¹

1,_δ¹

1

”

→ 0 as n → ∞ . (45)

Case where λ < 0. Here, we will reach a contradiction using Corollary 2.4 and the

fact that u(x, t) cannot be extended beyond its maximal influence domain D

defined by

(2).

In this case, (45) holds with ± = +. Since v

and ¯ v

are solutions to (1) and ¯ v

is defined on { (ξ, τ ), | τ < 1 + e

+ d(0)ξ } which contains the closure of D

(τ

) ≡ { (ξ, τ ), | 0 ≤ τ < τ

− | ξ |} where τ

is fixed in (1, min

1

δ1

, 1 + e

), using (45) and the solution of the Cauchy problem for (1), we see that for some n

∈

and for all n ≥ n

, v

(ξ, τ ) is well defined in D

(τ

). Since we have from the selfsimilar transformation (5) and (43)

v

(ξ, τ ) = e

2σ′ n

p−1

u(x, t) with x = x

+ ξe

, t = T (x

) − e

(1 − τ ), this means that for all n ≥ n

, u(x, t) is well defined in the set

{ (x, t) | T(x

) − e

≤ t ≤ T (x

) + (τ

− 1)e

− | x − x

|} ,

which contains (x

, T (x

)). This is a contradiction by definition (2) of the maximal influence domain.

Case where λ > 0. Here, a contradiction follows from the fact that w

(y, s) exists for all (y, s) ∈ ( − 1, 1) × [ − log T (x

), + ∞ ) and satisfies a blow-up criterion at the same time.

In this case, (45) holds with ± = − . Since ¯ v

is defined on { (ξ, τ ), | τ < 1 − e

+ d(0)ξ } which contains the closure of D

(τ

) ≡ { (ξ, τ ), | 0 ≤ τ < min(τ

, 1 − | ξ | ) } for any τ

∈

0, 1 −

, using (45) and the solution of the Cauchy problem for (1), we see that for some n

(τ

) ∈

and for all n ≥ n

, v

(ξ, τ ) is well defined in D

(τ

). Moreover,

v

(τ

)

∂

v

(τ

)

−

v ¯

(τ

)

∂

¯ v

(τ

)

H¹×L²(−1+τ0,1−τ0)

→ 0 as n → ∞ .

Using back the transformation (43) and taking s = σ

− log(1 − τ

), we get for all s ∈ [σ

, log(1 − | d(0) | )),

w

(σ

− σ

+ s)

∂

w

(σ

− σ

+ s)

−

w

(s)

∂

w

(s)

H¹×L²(−1,1)

→ 0 as n → ∞ . (46) Since we have from (46)

E w

(σ

− σ

+ s)

→ E (w

(s)) as n → ∞ and

∀ s ∈ [s

, log(1 − | d(0) | )), E(w

(s)) < 0 for some s

< log(1 − | d(0) | ) (47) (see Appendix B for the proof), we see that for all n large enough, we have

E(w

(σ

− σ

+ s

)) ≤ 1

2 E(w

(s

)) < 0. (48)

Since by the definition (5), w

is defined for all (y, s) ∈ ( − 1, 1) × [ − log T (x

), + ∞ ), a

contradiction follows from the following claim:

Claim 2.5 (Blow-up criterion for equation (6), see Antonini and Merle[3]) Con- sider W (y, s) a solution to equation (6) such that W (y, s

) is defined for all | y | < 1 and E(W (s

)) < 0 for some s

. Then, W (y, s) cannot exist for all (y, s) ∈ ( − 1, 1) × [s

, ∞ ).

Proof: See Theorem 2 page 1147 in [3].

Thus, (30) does not hold and T (x) is differentiable at x = 0 with T

(0) = d(0). This concludes the proof of Proposition 2.1.

2.2 Proof of Theorem 1

Let x

be a non characteristic point. From (3), we know that (20) holds. One can assume that x

= T (x

) = 0 from translation invariance. From [19] and Proposition 2.1, we know (up to replacing u(x, t) by − u(x, t)) that (10) holds with some d(0) ∈ ( − 1, 1) and θ(0) = 1, and that T (x) is differentiable at 0 with

T

(0) = d(0). (49)

We proceed in 3 steps.

- In Step 1, we use [19] and the fact that 0 is non characteristic to derive that (10) holds in a small neighborhood of 0 for some d(x) ∈ ( − 1, 1) and θ(x) = 1 with d(x) → d(0) as x → 0.

- In Step 2, using a geometrical construction and the previous step, we show that in a small open interval containing 0, the Lipschitz constant of T (x) is less than

.

- In Step 3, using Steps 1 and 2, we conclude the proof of Theorem 1.

Step 1: Openness of the set of x such that (10) holds

We have from the dynamical study in selfsimilar variables (5) in [19]:

Lemma 2.6 (Convergence in selfsimilar variables for x close to 0 ) For all ǫ > 0, there exists η such that if | x | < η and x is non characteristic, then, (10) holds for w

with

| d(x) − d(0) | ≤ ǫ and θ(x) = 1.

Remark: Here, we don’t assume that all the points in some neighborhood of 0 are uni- formly non characteristic (that is, δ

(x) defined in (3) may have no positive lower bound in any neighborhood of 0). We use instead the fact that in [19], we have completely un- derstood the dynamical structure of equation (6) in H close to the stationary solution κ(d(0), y).

Proof:

- Since 0 is non characteristic, we have from (22), for all s ≥ s

for some s

∈

, k (w

(s), ∂

w

(s)) k

H¹×L²

„

−_δ¹_′

0

,_δ¹_′

0

«

≤ K for some constant K , where δ

∈ (δ

, 1) is fixed.

Again from the fact that 0 is non characteristic, note that (10) holds, hence (w

(s), ∂

w

(s)) converges to (κ(d(0), .), 0) as s → ∞ in the norm of H defined by (9).

- Since for fixed s, we have (w

(y, s), ∂

w

(y, s)) → (w

(y, s), ∂

w

(y, s)) in H from the continuity of solutions to equation (6) with respect to initial data, we know that for all ǫ > 0, there exists s

(ǫ) ≥ s

and η(ǫ) > 0 such that for all x ∈ ( − η(ǫ), η(ǫ)),

w

( · , s

(ǫ))

∂

w

( · , s

(ǫ))

−

κ(d(0), .) 0

H

≤ ǫ.

- From Theorem 3 in [19] (use in particular the first remark following Theorem 3), for a small enough fixed ǫ > 0, we have that for all x ∈ ( − η(ǫ), η(ǫ)), there exists d(x) such that

w

(y, s)

∂

w

(y, s)

−

κ(d(x), y) 0

H

→ 0 as s → ∞ and

| d(x) − d(0) | ≤ Cǫ.

This concludes the proof of Lemma 2.6.

Step 2: The Lipschitz constant of T (x) around 0 is less than (1 + | d(0) | )/2 Fix ǫ

small enough such that

ǫ

> 0 and d(0) + 2ǫ

< 1. (50) Using (49) and Lemma 2.6, we see that there exists η

> 0 such that

∀| x | ≤ η

, | T (x) − T (0) − d(0)x | ≤ ǫ

| x | , (51) and if in addition, x is non characteristic in the sense (3), then, (10) holds for w

with

| d(x) − d(0) | ≤ ǫ

. (52) We now claim:

Lemma 2.7 (The slope of T (x) around 0 is less than (1 + | d(0) | )/2) It holds that

∀ x, y ∈ [ − η

10 , η

10 ], | T (x) − T (y) | ≤ 1 + | d(0) |

2 | x − y | . Proof: We proceed by contradiction and assume that

for some x

and y

∈ [ − η

10 , η

10 ], | T (x

) − T (y

) | > 1 + | d(0) |

2 | x

− y

| . (53) Up to considering u( − x, t) (also a solution to equation (1)) and to renaming x

and y

, we can assume that

y

< x

, T (y

) ≤ T (x

) and 1 + | d(0) |

2 (x

− y

) ≤ T (x

) − T (y

). (54) We can also assume that y

is the minimum in [ − η

, x

] satisfying (54). Therefore, we have

∀ x ∈ [ − η

, y

), 1 + | d(0) |

2 (x

− x) ≥ T (x

) − T (x). (55) Let us define

x

∈ [y

, x

] such that T (x

) − d(0)x

= min

y0≤x≤x0

T (x) − d(0)x. (56)

Note from (54) that x

< x

. Indeed, if not, then T(x

) − d(0)x

≤ T (y

) − d(0)y

, hence,

T (x

) − T (y

) ≤ d(0)(x

− y

) which contradicts (54).

Considering a family of straight lines of slope d(0)+2ǫ

, there exists one which is “tangent”

to the blow-up curve on [x

, x

] at some point (m

, T (m

)) with

− η

≤ y

≤ x

≤ m

≤ x

, (57) in the sense that

∀ x ∈ [x

, x

], T (m

) + (d(0) + 2ǫ

)(x − m

) ≤ T (x). (58) We have the following:

Claim 2.8 (m

is a non characteristic point) There exists η

> 0 such that

∀ x ∈ [m

− η

, m

+ η

], T (m

) − 1 + | d(0) |

2 | x − m

| ≤ T (x). (59) Proof: We claim first that

− η

2 < y

< x

, (60)

which yields by minimality in (55) (1 + | d(0) | )

2 (x

− y

) = T (x

) − T (y

) (61) Note first from (54) and (51) that

x

− y

≤ 2

1 + | d(0) | (T (x

) − T (y

))

≤ 2

1 + | d(0) | [(T (x

) − T (0)) − (T(y

) − T (0))]

≤ 2d(0)

1 + | d(0) | (x

− y

) + 2ǫ

1 + | d(0) | ( | x

| + | y

| ) . (62) Since

< 1, this yields

x

− y

≤ C(d(0))ǫ

( | y

| + | x

| ) ≤ 2C(d(0))ǫ

η

≤ η

10 for ǫ

small enough. Since | x

| ≤

by(53), (60) follows.

Let us now prove (59). First note that m

< x

. Indeed, from (56), (61), (57) and (54), we have T (x

) ≤ T (y

) + d(0)(x

− y

) = T (x

) + d(0)(x

− x

) −

(1+|d(0)|)

2

− d(0)

(x

− y

) < T (x

) + (d(0) + 2ǫ

)(x

− x

) since | d(0) | < 1, so (58) cannot hold with m

= x

(it fails with x = x

).

- If m

∈ (y

, x

), then T (m

) is a local minimum for T (x) − (d(0) + 2ǫ

)(x − m

), hence, since d(0) + 2ǫ

< 1 by (50), m

is non characteristic.

- If m

= y

, then y

= x

= m

by (57). From (55) and the fact that T (y

) − d(0)y

is

the minimum of T(x) − d(0)x

for x ∈ [y

, x

], we see that m

is non characteristic, which

concludes the proof of Claim 2.8.