1 Introduction
We are interested in the following semilinear heat equation:
(1.1)
{
∂
t
u
=
Δ
u
+
|
u
|
p
-
1
u
,
u
(
0
)
=
u
0
∈
L
∞
(
ℝ
N
)
,
where u(t):x∈ℝN→u(x,t)∈ℝ, Δ denotes the Laplacian in ℝN, and p>1 or 1<p<N+2N-2 if N≥3. It is well known that for each initial data u0 the Cauchy problem (1.1) has a unique solution u∈𝒞([0,T),L∞(ℝN)) for some 0<T≤+∞, and that either T=+∞ or
T
<
+
∞
and
lim
t
→
T
∥
u
(
t
)
∥
L
∞
=
+
∞
.
In the latter case we say that the solution blows up in finite time, and T is called the blow-up time. In such a blow-up case, a point a^∈ℝN is called a blow-up point if u(x,t) is not locally bounded in some neighborhood of (a^,T), this means that there exists (xn,tn)→(a^,T) such that |u(xn,tn)|→+∞ when n→+∞. We denote by S the blow-up set, that is, the set of all blow-up points of u.
Given a^∈S, we know from Velázquez [15] (see also Filippas and Kohn [5], Filippas and Liu [6], Herrero and Velázquez [9], Merle and Zaag [12]) that up to replacing u by -u, one of following two cases occurs:
Case 1 (non-degenerate rate of blow-up). For all K0>0, there is an orthonormal (N×N)-matrix Qa^ and ℓa^∈{1,…,N} such that
(1.2)
sup
|
ξ
|
≤
K
0
|
(
T
-
t
)
1
p
-
1
u
(
a
^
+
Q
a
^
ξ
(
T
-
t
)
|
log
(
T
-
t
)
|
,
t
)
-
f
ℓ
a
^
(
ξ
)
|
→
0
as
t
→
T
,
where
(1.3)
f
ℓ
a
^
(
ξ
)
=
(
p
-
1
+
(
p
-
1
)
2
4
p
∑
i
=
1
ℓ
a
^
ξ
i
2
)
-
1
p
-
1
.
Case 2 (degenerate rate of blow-up). For all K0≥0, there exists an even integer m≥4 such that
(1.4)
sup
|
ξ
|
≤
K
0
|
(
T
-
t
)
1
p
-
1
u
(
a
^
+
ξ
(
T
-
t
)
1
m
,
t
)
-
(
p
-
1
+
∑
|
α
|
=
m
c
α
ξ
α
)
-
1
p
-
1
|
→
0
as
t
→
T
,
where ξα=∏i=1Nξiαi, |α|=∑i=1Nαi if α=(α1,…,αn)∈ℕN and ∑|α|=mcαξα≥0 for all ξ∈ℝN.
According to Velázquez [15], if case 1 occurs with ℓa^=N or case 2 occurs with ∑|α|=mcαξα>0 for all ξ≠0, then a^ is an isolated blow-up point. Herrero and Velázquez [8, 7] prove that the profile (1.3) with ℓa^=N is generic in the case N=1, and they announced the same for N≥2, but they never published it. Bricmont and Kupiainen [1] and Merle and Zaag [10] show the existence of initial data for (1.1) such that the corresponding solutions blow up in finite time T at only one blow-up point a^ and verify the behavior (1.2) with ℓa^=N. The method of [10] also gives the stability of the profile (1.3) (ℓa^=N) with respect to perturbations in the initial data (see also Fermanian Kammerer, Merle and Zaag [3, 4] for other proofs of the stability). Ebde and Zaag [2] and Nguyen and Zaag [13] prove the stability of the profile (1.3) (ℓa^=N) with respect to perturbations in the initial data and also in the nonlinearity, in some class allowing lower order terms in the solution and also in the gradient. All the other asymptotic behaviors are suspected to be unstable.
When
ℓ
a
^
≤
N
-
1
in (1.2), we do not know whether a^ is isolated or not, or whether S is continuous near a^. In this paper, we assume that a^ is a non-isolated blow-up point and that S is continuous locally near a^, in a sense that we will describe precisely later. Our main concern is the regularity of S near a^. The first relevant result is due to Velázquez [16] who shows that the Hausdorff measure of S is less than or equal to N-1. No further results on the description of S were known until the contributions of Zaag [18, 17, 20] (see also [19] for a summarized note). In [18], he proves that if S is locally continuous, then S is a 𝒞1 manifold. He also obtains the first description of the singularity near a^. More precisely, he shows in [18, Theorems 3 and 4] that for some t0<T and δ>0, for all K0>0, t∈[t0,T) and x∈B(a^,2δ) such that d(x,S)≤K0(T-t)|log(T-t)|, one has
(1.5)
|
(
T
-
t
)
1
p
-
1
u
(
x
,
t
)
-
f
1
(
d
(
x
,
S
)
(
T
-
t
)
|
log
(
T
-
t
)
|
)
|
≤
C
(
K
0
)
log
|
log
(
T
-
t
)
|
|
log
(
T
-
t
)
|
,
where f1 is defined in (1.3) (ℓa^=1). Moreover, for all x∈ℝN∖S, one has u(x,t)→u*(x) as t→T with
(1.6)
u
*
(
x
)
∼
U
(
d
(
x
,
S
)
)
=
(
8
p
(
p
-
1
)
2
|
log
d
(
x
,
S
)
|
d
2
(
x
,
S
)
)
1
p
-
1
as
d
(
x
,
S
)
→
0
and
x
∈
B
(
a
^
,
2
δ
)
.
If
ℓ
a
^
=
1
,
Zaag [17] further refines the asymptotic behavior (1.5) and gets error terms of order (T-t)μ for some μ>0. This way, he obtains more regularity on the blow-up set S. The key idea is to replace the explicit profile f1 in (1.5) by a non-explicit function, say u~(x1,t), then go beyond all logarithmic scales through scaling and matching. In fact, for u~(x1,t), Zaag takes a symmetric, one-dimensional solution of (1.1) that blows up at the same time T only at the origin, and behaves like (1.2) with ℓa^=1. More precisely, he abandons the explicit profile function f1 in (1.5) and chooses a non-explicit function u~σ(d(x,S),t) as a first-order description of the singular behavior, where u~σ is defined by
(1.7)
u
~
σ
(
x
1
,
t
)
=
e
-
σ
p
-
1
u
~
(
e
-
σ
2
x
1
,
T
-
e
-
σ
(
T
-
t
)
)
.
He shows that for each blow-up point a near a^, there is an optimal scaling parameter σ=σ(a) so that the difference (T-t)1p-1(u(x,t)-u~σ(a)(d(x,S),t)) along the normal direction to S at a is minimized. Hence, if the function u~σ(a)(d(x,S),t) is chosen as a first-order description for u(x,t) near (a,T), we avoid logarithmic scales. More precisely, for all t∈[t0,T) and x∈B(a^,2δ) such that d(x,S)≤K0(T-t)|log(T-t)|, one has
(1.8)
(
T
-
t
)
1
p
-
1
|
u
(
x
,
t
)
-
u
~
σ
(
a
)
(
d
(
x
,
S
)
,
t
)
|
≤
C
(
T
-
t
)
μ
,
for some μ>0. Note that any other value of σ≠σ(a) in (1.8) gives an error of logarithmic order of the variable (T-t) (the same as in (1.5)). Exploiting estimate (1.8) yields geometric constraints on S which imply the 𝒞1,12-η-regularity of S for all η>0. A further refinement of (1.8) given in [20] yields better estimates in the expansion of u(x,t) near (a,T). Moreover, some terms following in the expansion of u(x,t) near (a,T) contain geometrical information about S, resulting in more regularity of S, namely the 𝒞2-regularity.
In this work, we want to know whether the 𝒞2-regularity near a^ proven in [20] for ℓa^=1 would hold in the case where u behaves like (1.2) near (a^,T) with
(1.9)
ℓ
a
^
∈
{
2
,
…
,
N
-
1
}
.
Since Zaag obtains the result in [18, 20] only when ℓa^=1, this corresponds to an (N-1)-dimensional blow-up set (the codimension of the blow-up set is one, according to [18]). In our opinion, in those papers the major obstacle towards the case (1.9) lays in the fact that Zaag could not refine the asymptotic behavior (1.2) with ℓa^∈{2,…,N-1} to go beyond all logarithmic scales and get a smaller error term in polynomial orders of the variable (T-t). It happens that a similar difficulty was already encountered by Fermanian Kammerer and Zaag in [4], when they wanted to find a sharp profile in the case (1.2) with ℓa^=N, which corresponds to an isolated blow-up point, as we have pointed out right after estimate (1.4). Such a sharp profile could be obtained in [4] only when N=1 (which corresponds also to ℓa^=1): unsurprisingly it was u~σ(x1,t), the dilated version of u~(x1,t), the one-dimensional blow-up solution mentioned between estimates (1.6) and (1.7). As a matter of fact, the use of u~(x1,t) was first used in [4] for the isolated blow-up point in one space dimension (N=1 and ℓa^=1), then later in higher dimensions with an (N-1)-dimensional blow-up surface (N≥2 and still ℓa^=1) in [17].
The interest of u~(x1,t) is that it provides a one-parameter family of blow-up solutions, thanks to the scaling parameter in (1.7), which enables us to get the sharp profile by suitably choosing the parameter.
Handling the case ℓa^≥2 remained open, both for the case of an isolated point (ℓa^=N≥2) and a non-isolated blow-up point (ℓa^=2,…,N-1). From the refinement of the expansion around the explicit profile in fℓa^ in (1.2), it appeared that one needs a ℓa^(ℓa^+1)/2-parameter family of blow-up solutions obeying (1.2).
Such a family was constructed by Nguyen and Zaag in [14], and successfully used to derive a sharp profile in the case of an isolated blow-up point (ℓa^=N≥2), by fine-tuning the ℓa^(ℓa^+1)/2=N(N+1)/2 parameters.
In this paper, we aim at using that family to handle the case of a non-isolated blow-up point (N≥2 and ℓa^=2,…,N-1), in order to generalize the results of Zaag in [18, 17, 20], proving in particular the C2-regularity of the blow-up set, under the hypothesis that it is merely continuous.
The main result in this paper is the following.
Theorem 1.1
Theorem 1.1 (C2-Regularity of the Blow-Up Set Assuming C1-Regularity)
Take N≥2 and ℓ∈{1,…,N-1}. Consider u, a solution of (1.1) that blows up in finite time T on a set S. Take a^∈S where u behaves locally as stated in (1.2) with ℓa^=ℓ. If S is locally a C1 manifold of dimension N-ℓ, then it is locally C2.
Remark 1.2
Theorem 1.1 was already proved by Zaag [20] only when ℓ=1. Thus, the novelty of our contribution lays in the case ℓ∈{2,…,N-1} and N≥3.
Under the hypotheses of Theorem 1.1, Zaag [18] already proved that S is a 𝒞1 manifold near a^, assuming that S is continuous. Therefore, Theorem 1.1 can be restated under a weaker assumption. Before stating this stronger version, let us first clearly describe our hypotheses and introduce some terminology borrowed from [18] (see also [17, 20]). According to Velázquez [15, Theorem 2], we know that for all ϵ>0, there is δ(ϵ)>0 such that
S
∩
B
(
a
^
,
2
δ
)
⊂
Ω
a
^
,
ϵ
≡
{
x
∈
ℝ
N
∣
|
P
a
^
(
x
-
a
^
)
|
≥
(
1
-
ϵ
)
|
x
-
a
^
|
}
,
where Pa^ is the orthogonal projection over πa^, where
π
a
^
=
a
^
+
span
{
Q
a
^
T
e
ℓ
a
^
+
1
,
…
,
Q
a
^
T
e
N
}
is the so-called “weak” tangent plane to S at a^. Roughly speaking, Ωa^,ϵ is a cone with vertex a^ and shrinks to πa^ as ϵ→0. In some “weak” sense, S is (N-ℓa^)-dimensional. In fact, here comes our second hypothesis: we assume there is Γ∈𝒞((-1,1)N-ℓa^,ℝN) such that Γ(0)=a^ and ImΓ⊂S, where ImΓ is at least (N-ℓa^)-dimensional, in the sense that
(1.10)
for all
b
∈
Im
Γ
, there are
(
N
-
ℓ
a
^
)
independent vectors
v
1
,
…
,
v
N
-
ℓ
a
^
in
ℝ
N
and
functions
Γ
1
,
…
,
Γ
N
-
ℓ
a
^
in
𝒞
1
(
[
0
,
1
]
,
S
)
such that
Γ
i
(
0
)
=
b
and
Γ
i
′
(
0
)
=
v
i
.
Hypothesis (1.10) means that b is actually non-isolated in (N-ℓa^) independent directions. We assume in addition that a^ is not an endpoint in ImΓ in the sense that
(1.11)
for all
ϵ
>
0
, the projection of
Γ
(
(
-
ϵ
,
ϵ
)
N
-
ℓ
a
^
)
on the “weak” tangent plane
π
a
^
at
a
^
contains an open ball centered at
a
^
.
This is the stronger version of our result:
Theorem 1.1${}^{\prime}$
Take N≥2 and ℓ∈{1,…,N-1}. Consider u, a solution of (1.1) that blows up in finite time T on a set S. Take a^∈S where u behaves locally as stated in (1.2) with ℓa^=ℓ. Consider Γ∈C((-1,1)N-ℓ,RN) such that a^=Γ(0)∈ImΓ⊂S and ImΓ is at least (N-ℓ)-dimensional (in the sense of (1.10)). If a^ is not an endpoint (in the sense of (1.11)), then there are δ>0, δ1>0 and γ∈C2((-δ1,δ1)N-ℓ,Rℓ) such that
S
δ
=
S
∩
B
(
a
^
,
2
δ
)
=
graph
(
γ
)
∩
B
(
a
^
,
2
δ
)
=
Im
Γ
∩
B
(
a
^
,
2
δ
)
,
and the blow-up set S is a C2-hypersurface locally near a^.
Let us now briefly give the main ideas of the proof of Theorem 1.1. The proof is based on techniques developed by Zaag in [17, 20] for the case when the solution of equation (1.1) behaves like (1.2) with ℓ=1. As in [17, 20], the proof relies on two arguments:
The derivation of a sharp blow-up profile of u(x,t) near the singularity, in the sense that the difference between the solution u(x,t) and this sharp profile goes beyond all logarithmic scales of the variables (T-t). This is possible thanks to the recent result in [14].
The derivation of a refined asymptotic profile of u(x,t) near the singularity linked to geometric constraints on the blow-up set. In fact, we derive an asymptotic profile for u(x,t) in every ball B(a,K0T-t) for some K0>0 and a blow-up point a close to a^. Moreover, this profile is continuous in a and the speed of convergence of u to the profile in the ball B(a,K0T-t) is uniform with respect to a. If a and b are in S and 0<|a-b|≤K0T-t, then the balls B(a,K0T-t) and B(b,K0T-t) intersect each other, leading to different profiles for u(x,t) in the intersection. However, these profiles have to coincide, up to the error terms. This creates a geometric constraint which gives more regularity for the blow-up set near a^.
Let us explain the difficulty raised in [17, 20] for the case ℓ≥2. Consider a∈S∩B(a^,2δ) for some δ>0 and introduce the following self-similar variables:
(1.12)
W
a
(
y
,
s
)
=
(
T
-
t
)
1
p
-
1
u
(
x
,
t
)
,
y
=
x
-
a
T
-
t
,
s
=
-
log
(
T
-
t
)
.
Then, we see from (1.1) that for all (y,s)∈ℝN×[-logT,+∞),
(1.13)
∂
W
a
∂
s
=
Δ
W
a
-
1
2
y
⋅
∇
W
a
-
W
a
p
-
1
+
|
W
a
|
p
-
1
W
a
.
Under the hypotheses stated in Theorem 1.1, Zaag proved in [18, Proposition 3.1 and pp. 530–533, Section 6.1] that for all a∈Sδ≡S∩B(a^,2δ) for some δ>0 and s≥-logT, there exists an (N×N) orthogonal matrix Qa such that
(1.14)
∥
W
a
(
Q
a
y
,
s
)
-
{
κ
+
κ
2
p
s
(
ℓ
-
|
y
¯
|
2
2
)
}
∥
L
ρ
2
≤
C
log
s
s
2
,
where κ=(p-1)-1p-1, y¯=(y1,…,yℓa), Qa is continuous in terms of a such that {QaTej∣j=ℓ+1,…,N} spans the tangent plane πa to S at a and QaTei, i=1,…,ℓ are the normal directions to S at a, Lρ2 is the weighted L2 space associated with the weight ρ=1(4π)N/2e-|y|2/4. Note that estimate (1.14) implies (1.5) (see [18, Appendix C]).
When ℓ=1, in order to refine estimate (1.14), Zaag in [17] subtracts from Wa a one-dimensional solution with the same profile. Let us do the same when ℓ=2,…,N-1, and explain how Zaag succeeds in handing the case ℓ=1 and gets stuck when ℓ≥2. To this end, we consider u^(x¯,t) with x¯=(x1,…,xℓ) a radially symmetric solution of (1.1) in ℝℓ which blows up at time T only at the origin with the profile (1.2) with ℓa^=ℓ (see [14, Appendix A.1] for the existence of such a solution). If the ℓ-dimensional solution u^ is considered in ℝN, then it blows up on the (N-ℓ)-dimensional vector space {x¯=0} in ℝN. In particular, if we introduce
(1.15)
w
^
(
y
¯
,
s
)
=
(
T
-
t
)
1
p
-
1
u
^
(
x
¯
,
t
)
,
y
¯
=
x
¯
T
-
t
,
s
=
-
log
(
T
-
t
)
,
then w^ is a radially symmetric solution of (1.13) which satisfies
(1.16)
∥
w
^
(
y
¯
,
s
)
-
{
κ
+
κ
2
p
s
(
ℓ
-
|
y
¯
|
2
2
)
}
∥
L
ρ
2
≤
C
log
s
s
2
.
Noting that u^ and w^ may be considered as solutions defined for all y∈ℝN (and independent of yℓ+1,…,yN), and given that w^(y¯,s) and Wa(Qay,s) have the same behavior up to the first order (see (1.14) and (1.16)), we may try to use w^ as a sharper (though non-explicit) profile for Wa(Qay,s). In fact, we have the following classification (see Corollary 2.2 below):
Case 1. There is a symmetric, real (ℓa×ℓa)-matrix ℬ=ℬ(a)≠0 such that
(1.17)
W
a
(
Q
a
y
,
s
)
-
w
^
(
y
¯
,
s
)
=
1
s
2
(
1
2
y
¯
T
ℬ
y
¯
-
tr
(
ℬ
)
)
+
o
(
1
s
2
)
as
s
→
+
∞
in
L
ρ
2
.
Case 2. There is a positive constant C0 such that
(1.18)
∥
W
a
(
Q
a
y
,
s
)
-
w
^
(
y
¯
,
s
)
∥
L
ρ
2
=
𝒪
(
e
-
s
2
s
C
0
)
as
s
→
+
∞
.
If ℓ=1 (ℬ(a)∈ℝ), Zaag in [17] noted the following property:
(1.19)
w
^
(
y
1
,
s
+
σ
0
)
-
w
^
(
y
1
,
s
)
=
2
κ
σ
0
p
s
2
(
1
2
y
1
2
-
1
)
+
o
(
1
s
2
)
in
L
ρ
2
.
Therefore, choosing σ0(a) such that 2κσ0p=ℬ(a), we see from (1.17) and (1.19) that
W
a
(
Q
a
y
,
s
)
-
w
^
(
y
1
,
s
+
σ
0
(
a
)
)
=
o
(
1
s
2
)
as
s
→
+
∞
in
L
ρ
2
.
From the classification given in (1.17) and (1.18), only (1.18) holds and
(1.20)
∥
W
a
(
Q
a
y
,
s
)
-
w
^
(
y
1
,
s
+
σ
0
(
a
)
)
∥
L
ρ
2
=
𝒪
(
e
-
s
2
s
C
0
)
as
s
→
+
∞
.
If we return to the original variables u(x,t) and u^(x1,t) through (1.12) and (1.15), then (1.8) follows from the transformation (1.7) together with estimate (1.20) (see [17, Appendix C]). In other words, w^(y1,s+σ0(a)) serves as a sharp (though non-explicit) profile for Wa(Qay,s) in the sense of (1.20). Using estimate (1.20) together with some geometrical arguments, we are able to prove the 𝒞1,12-η-regularity of the blow-up set, for any η>0. Then, a further refinement of (1.20) up to order of e-s2/s together with a geometrical constraint on the blow-up set S results in more regularity for S, which yields the 𝒞2-regularity.
If ℓ≥2, the matrix ℬ(a) in (1.17) has ℓ(ℓ+1)2 real parameters. Therefore, applying the trick of [17] (see (1.19) above) only allows us to control one parameter; there remain ℓ(ℓ+1)2-1 real parameters to be handled. This is the major reason which prevents Zaag in [17, 20] from deriving a similar estimate to (1.20), hence, the refined regularity of the blow-up set. Fortunately, we can overcome this obstacle thanks to a recent result by Nguyen and Zaag (see Proposition 2.4 below) who show in [14] that for any symmetric, real (ℓ×ℓ)-matrix 𝒜, there is a solution w𝒜 of equation (1.13) in ℝℓ such that
(1.21)
w
𝒜
(
y
¯
,
s
)
-
w
^
(
y
¯
,
s
)
=
1
s
2
(
1
2
y
¯
T
𝒜
y
¯
-
tr
(
𝒜
)
)
+
o
(
1
s
2
)
as
s
→
+
∞
in
L
ρ
2
.
Hence, choosing 𝒜=ℬ(a), we see from (1.21), (1.17) and (1.18) that
(1.22)
∥
W
a
(
Q
a
y
,
s
)
-
w
ℬ
(
a
)
(
y
¯
,
s
)
∥
L
ρ
2
≤
C
e
-
s
2
s
C
0
for s large enough. Exploiting estimate (1.22) and adapting the arguments given in [17, 20], we are able to prove the 𝒞2-regularity of the blow-up set.
The next result shows how the 𝒞2-regularity is linked to the refined asymptotic behavior of Wa. More precisely, we link in the following theorem the refinement of the asymptotic behavior of Wa to the second fundamental form of the blow-up set at a.
Theorem 1.3
Theorem 1.3 (Refined Asymptotic Behaviors Linked to the Geometrical Description of the Blow-Up Set)
Under the hypotheses of Theorem 1.1, there exist s~0≥-logT and δ>0 such that for all a∈Sδ=S∩B(a^,2δ), there exists a symmetric (ℓ×ℓ) matrix B(a) such that for all s≥s~0,
(1.23)
∥
W
a
(
Q
a
y
,
s
)
-
w
ℬ
(
a
)
(
y
¯
,
s
)
-
κ
e
-
s
2
2
p
s
∑
i
=
1
ℓ
y
i
∑
k
,
j
=
ℓ
+
1
N
Λ
k
,
j
(
i
)
(
a
)
1
+
δ
k
,
j
(
y
k
y
j
-
2
δ
k
,
j
)
∥
L
ρ
2
≤
C
e
-
s
2
s
3
2
-
ν
,
for some ν∈(0,12), where a→{Λk,j(i)(a)}ℓ+1≤j,k≤N is a continuous symmetric matrix representing the second fundamental form of the blow-up set at the blow-up point a along the unitary normal vector QaTei. Moreover,
(1.24)
Λ
k
,
j
(
i
)
(
a
)
=
p
4
κ
lim
s
→
+
∞
s
e
s
2
∫
ℝ
N
W
a
(
Q
a
y
,
s
)
y
i
(
y
k
y
j
-
2
δ
k
,
j
)
ρ
(
y
)
𝑑
y
.
In Section 2, we give the main steps of the proofs of Theorems 1.1 and 1.3. We leave all long and technical proofs to Section 3.
2 Setting of the Problem and Strategy of the Proof of the 𝒞2-regularity of the Blow-Up Set
In this section we give the main steps of the proofs of Theorems 1.1 and 1.3. All long and technical proofs will be left to the next section. We proceed in three parts corresponding to three separate subsections. For the reader’s convenience, we briefly describe these parts as follows:
Part 1: We derive a sharp blow-up behavior for solutions of equation (1.1) having the profile (1.2) with ℓa^∈{1,…,N-1} such that the difference between the solution and this sharp blow-up behavior goes beyond all logarithmic scales of the variable T-t. The main result in this step is stated in Proposition 2.5.
Part 2: Through the introduction of a local chart, we give a geometrical constraint on the expansion of the solution linked to the asymptotic behavior (see Proposition 2.7). This geometrical constraint is a crucial point which is the bridge between the asymptotic behavior and the regularity of the blow-up set.
Part 3: Using the sharp blow-up behavior derived in Part 1, we first get the 𝒞1,12-η-regularity of the blow-up set S (see Proposition 2.8), then together with the geometrical constraint, we achieve the 𝒞1,1-η-regularity of S (see Proposition 2.9). With this better regularity and the geometric constraint, we further refine the asymptotic behavior (see Proposition 2.10) and use again the geometric constraint to get 𝒞2-regularity of S, which yields the conclusion of Theorems 1.1 and 1.3.
We remark that Parts 1 and 2 are independent, whereas Part 3 is a combination of the first two. Throughout this paper, we work under the hypotheses of Theorem 1.1. Since S is locally near a^ a manifold of dimension N-ℓ, we may assume that there is a 𝒞1 function γ such that
S
δ
≡
S
∩
B
(
a
^
,
2
δ
)
=
graph
(
γ
)
∩
B
(
a
^
,
2
δ
)
,
for some δ>0 and γ∈𝒞1((-δ1,δ1)N-ℓ,ℝℓ) with δ1>0.
In what follows, ℓ∈{1,…,N-1} is fixed, and for all z=(z1,…,zN)∈ℝN, we denote by z¯ the first ℓ coordinates of z, namely z¯=(z1,…,zℓ), and by z~ the last (N-ℓ) coordinates of z, namely z~=(zℓ+1,…,zN). We usually use indices i, m for the range 1,…,ℓ and indices j, k, n for the range ℓ+1,…,N.
2.1 Part 1: Blow-Up Behavior Beyond All Logarithmic Scales of (T-t)
In this subsection, we use the ideas given by Fermanian Kammerer and Zaag [4] together with a recent result by Nguyen and Zaag in [14] in order to derive a sharp (though non-explicit) profile for blow-up solutions of (1.1) in the sense that the first order in the expansion of the solution around this sharp profile goes beyond all logarithmic scales of (T-t) and reaches polynomial scales of (T-t). In fact, we replace the 1-scaling parameter σ in (1.8) by a ℓ(ℓ+1)2-parameters family, which generates a substitution for u~σ defined in (1.7) and serves as a sharp profile for solutions having the behavior (1.2) with ℓa^∈{1,…,N-1}. The main result in this part is Proposition 2.5 below.
Consider a∈Sδ. If Wa(y,s) and w^(y¯,s) are defined as in (1.12) and (1.15), then we know from [18] that
(2.1)
∥
W
a
(
Q
a
y
,
s
)
-
{
κ
+
κ
2
p
s
(
ℓ
-
|
y
¯
|
2
2
)
}
∥
L
ρ
2
≤
C
log
s
s
2
and
(2.2)
∥
w
^
(
y
¯
,
s
)
-
{
κ
+
κ
2
p
s
(
ℓ
-
|
y
¯
|
2
2
)
}
∥
L
ρ
2
≤
C
log
s
s
2
.
The first step is to classify all possible asymptotic behaviors of Wa(Qay,s)-w^(y¯,s) as s goes to infinity. To do so, we shall use the following result which is inspired by Fermanian Kammerer and Zaag [4].
Proposition 2.1
Proposition 2.1 (Classification of the Difference Between Two Solutions of (1.13) Having the Same Profile)
Assume that W1 and W2 are two solutions of (1.13) verifying
(2.3)
∥
W
i
(
y
,
s
)
-
{
κ
+
κ
2
p
s
(
ℓ
-
|
y
¯
|
2
2
)
}
∥
L
ρ
2
≤
C
log
s
s
2
,
i
=
1
,
2
,
where y¯=(y1,…,yℓ) for some ℓ∈{1,…,N-1}. Then, one of the two following cases occurs:
Case 1. There is a symmetric, real
(
ℓ
×
ℓ
)
-matrix
ℬ
≠
0
such that
(2.4)
W
1
(
y
,
s
)
-
W
2
(
y
,
s
)
=
1
s
2
(
1
2
y
¯
T
ℬ
y
¯
-
tr
(
ℬ
)
)
+
o
(
1
s
2
)
as
s
→
+
∞
in
L
ρ
2
.
Case 2.
There is
C
0
>
0
such that
∥
W
1
(
y
,
s
)
-
W
2
(
y
,
s
)
∥
L
ρ
2
=
𝒪
(
e
-
s
2
s
C
0
)
as
s
→
+
∞
.
Proof.
The proof follows from the strategy given in [4] for the difference of two solutions with the radial profile (ℓ=N). Note that the case when ℓ=1 was treated in [17]. Since some technical details are straightforward, we briefly give the main steps of the proof in Section 3.1 and just emphasize the novelties. ∎
An application of Proposition 2.1 with W1(y,s)=Wa(Qay,s) and W2(y,s)=w^(y¯,s) yields the following corollary directly.
Corollary 2.2
As s goes to infinity, one of the two following cases occurs:
Case 1. There is a symmetric, real
(
ℓ
×
ℓ
)
-matrix
ℬ
=
ℬ
(
a
)
≠
0
continuous as a function of
a
such that
(2.5)
W
a
(
Q
a
y
,
s
)
-
w
^
(
y
¯
,
s
)
=
1
s
2
(
1
2
y
¯
T
ℬ
y
¯
-
tr
(
ℬ
)
)
+
o
(
1
s
2
)
in
L
ρ
2
.
Case 2. There is
C
0
>
0
such that
(2.6)
∥
W
a
(
Q
a
y
,
s
)
-
w
^
(
y
¯
,
s
)
∥
L
ρ
2
=
𝒪
(
e
-
s
2
s
C
0
)
.
Remark 2.3
Note that the continuity of ℬ comes from the continuity of Wa with respect to a, where Wa behaves as in (2.1). In particular, Zaag [18] showed the stability of the blow-up behavior (2.1) with respect to blow-up points (see [18, Proposition 3.1 and Section 6.1]).
In the next step, we recall a recent result by Nguyen and Zaag [14], which gives the construction of solutions for equation (1.13) with some prescribed behavior.
Proposition 2.4
Proposition 2.4 (Construction of Solutions for (1.13) with Some Prescribed Behavior)
Let ℓ∈{1,…,N-1}. For all A∈Mℓ(R), where Mℓ(R) is the set of all symmetric, real (ℓ×ℓ)-matrices, there exists a solution wA(y,s) of (1.13) defined on RN×[s0(A),+∞) such that
(2.7)
w
𝒜
(
y
¯
,
s
)
-
w
^
(
y
¯
,
s
)
=
1
s
2
(
1
2
y
¯
T
𝒜
y
¯
-
tr
(
𝒜
)
)
+
o
(
1
s
2
)
as
s
→
+
∞
in
L
ρ
2
,
where w^ is the radially symmetric, ℓ-dimensional solution of (1.13) satisfying (2.2).
Proof.
See [14, Theorem 3]. Although that result is stated for the case ℓ=N, we can extend it to the case when ℓ≤N-1 by considering solutions of (1.13) as ℓ-dimensional solutions, those artificially generated by adding irrelevant space variables (yℓ+1,…,yN) to the domain of definition of the solutions. ∎
The following result is a direct consequence of Corollary 2.2 and Proposition 2.4.
Proposition 2.5
Proposition 2.5 (Sharp (Non-Explicit) Profile for Solutions of (1.1) Having the Behavior (1.2) with ℓ≤N-1)
There exist s0>0 and a continuous matrix B:Sδ→Mℓ(R), such that for all a∈Sδ and s≥s0,
(2.8)
∥
W
a
(
Q
a
y
,
s
)
-
w
ℬ
(
a
)
(
y
¯
,
s
)
∥
L
ρ
2
≤
C
e
-
s
2
s
C
0
,
where wB is the solution constructed as in Proposition 2.4, C0>0 is given in Proposition 2.1. Moreover, we have the following:
For all
s
≥
s
0
+
1
,
(2.9)
sup
|
y
|
≤
K
s
|
W
a
(
y
,
s
)
-
w
ℬ
(
a
)
(
y
¯
a
,
s
)
|
≤
C
(
K
)
e
-
s
2
s
3
2
+
C
0
,
where
y
¯
a
=
(
y
⋅
Q
a
e
1
,
…
,
y
⋅
Q
a
e
ℓ
)
.
For all
t
∈
[
T
-
e
-
s
0
-
1
,
T
)
,
(2.10)
sup
|
x
-
a
|
≤
K
(
T
-
t
)
|
log
(
T
-
t
)
|
|
(
T
-
t
)
1
p
-
1
u
(
x
,
t
)
-
w
ℬ
(
a
)
(
y
¯
a
,
x
,
-
log
(
T
-
t
)
)
|
≤
C
(
K
)
(
T
-
t
)
1
2
|
log
(
T
-
t
)
|
3
2
+
C
0
,
where
y
¯
a
,
x
=
1
T
-
t
(
(
x
-
a
)
⋅
Q
a
e
1
,
…
,
(
x
-
a
)
⋅
Q
a
e
ℓ
)
.
Proof.
From (2.5) and (2.7), we have for any symmetric (ℓ×ℓ)-matrix 𝒜,
W
a
(
Q
a
y
,
s
)
-
w
𝒜
(
y
¯
,
s
)
=
1
s
2
(
1
2
y
¯
T
(
ℬ
-
𝒜
)
y
¯
-
tr
(
ℬ
-
𝒜
)
)
+
o
(
1
s
2
)
in
L
ρ
2
.
Choosing 𝒜=ℬ(a), we get
(2.11)
∥
W
a
(
Q
a
y
,
s
)
-
w
ℬ
(
a
)
(
y
¯
,
s
)
∥
L
ρ
2
=
o
(
1
s
2
)
as
s
→
+
∞
.
Note that an alternative application of Proposition 2.1 with W1=Wa and W2=wℬ(a) yields either (2.5) or (2.6). However, the case (2.5) is excluded by (2.11). Hence, (2.8) follows. Since we showed in Corollary 2.2 that a↦ℬ(a) is continuous, the same holds for a↦𝒜(a).
As for (2.9), it is a direct consequence of the following lemma which allows us to carry estimate (2.8) from compact sets |y|≤K to sets |y|≤Ks.
Lemma 2.6
Lemma 2.6 (Extension of the Convergence from Compact Sets to Sets |y|≤Ks)
Assume that Z satisfies
(2.12)
∂
s
Z
≤
Δ
Z
-
1
2
y
⋅
∇
Z
+
Z
+
C
1
s
Z
,
0
≤
Z
(
y
,
s
)
≤
C
1
for all
(
y
,
s
)
∈
ℝ
N
×
[
s
^
,
+
∞
)
,
for some C1>0. Then for all s′≥s^ and s≥s′+1 such that e(s-s′)/2=s, we have
sup
|
y
|
≤
K
s
Z
(
y
,
s
)
≤
C
(
C
1
,
K
)
e
s
-
s
′
∥
Z
(
s
′
)
∥
L
ρ
2
.
Proof.
This lemma is a corollary of [15, Proposition 2.1] and it is proved in the course of the proof of [4, Proposition 2.13] (in particular, pp. 1203–1205). ∎
Let us derive (2.9) from Lemma 2.6. If we define G(y,s)=Wa(Qay,s)-wℬ(a)(y¯,s), straightforward calculations based on (1.13) yield
(2.13)
∂
s
G
=
Δ
G
-
1
2
y
⋅
∇
G
+
G
+
α
G
for all
(
y
,
s
)
∈
ℝ
N
×
[
-
log
T
,
+
∞
)
,
where
α
(
y
,
s
)
=
|
W
a
|
p
-
1
W
a
-
|
w
ℬ
|
p
-
1
w
ℬ
W
a
-
w
ℬ
-
p
p
-
1
=
p
|
w
~
(
y
,
s
)
|
p
-
1
-
p
p
-
1
if
W
a
≠
w
ℬ
,
for some w~(y,s)∈(Wa(Qay,s),wℬ(a)(y¯,s)).
From [11, Theorem 1], we know that for s large enough,
∥
w
~
(
s
)
∥
L
∞
≤
κ
+
C
s
,
which implies
(2.14)
α
(
y
,
s
)
≤
p
(
κ
+
C
s
)
p
-
1
-
p
p
-
1
≤
C
1
s
.
If Z=|G|, then we use Kato’s inequality ΔG⋅sgn(G)≤Δ(|G|) to derive equation (2.12) from (2.13) and (2.14). Applying Lemma 2.6 together with estimate (2.8) yields
sup
|
y
|
≤
K
s
Z
(
y
,
s
)
≤
C
e
s
-
s
′
e
-
s
′
2
(
s
′
)
C
0
≤
C
e
-
s
2
s
3
2
+
C
0
for all s′≥s1 and s≥s′+1 for some s1>0 large such that e(s-s′)/2=s. This yields (2.9). Estimate (2.10) directly follows from (2.9) by the transformation (1.12). This ends the proof of Proposition 2.5. ∎
2.2 Part 2: A Geometric Constraint Linked to the Asymptotic Behaviors
In this subsection, we follow the idea of [20] to introduce local 𝒞1,α*-charts of the blow-up set, and get a geometric constraint mechanism on the blow-up set (see Proposition 2.7 below) which is a crucial step in linking refined asymptotic behaviors of the solution to geometric descriptions of the blow-up set.
Consider a∈Sδ and ℓ∈{1,…,N-1}. We introduce the local 𝒞1,α*-chart of the blow-up set at the point a as follows:
ℝ
N
-
ℓ
→
ℝ
N
,
ξ
~
↦
(
γ
a
,
1
(
ξ
~
)
,
…
,
γ
a
,
ℓ
(
ξ
~
)
,
ξ
~
)
,
where ξ~=(ξℓ+1,…,ξN) and γa,i∈𝒞1,α*((-ϵa,ϵa)N-ℓ) for some α*∈(0,12) and ϵa>0. Then the set Sδ is locally near a defined by
(2.15)
{
a
+
∑
i
=
1
ℓ
γ
a
,
i
(
ξ
~
)
η
i
(
a
)
+
∑
j
=
ℓ
+
1
N
ξ
k
τ
k
(
a
)
|
|
ξ
~
|
<
ϵ
a
}
,
where η1(a),…,ηℓ(a) and τℓ+1(a),…,τN(a) are of norm 1 and, respectively, normal and tangent to Sδ at a. By definition, we have
γ
a
,
i
(
0
)
=
0
and
∇
γ
a
,
i
(
0
)
=
0
for all
i
=
1
,
…
,
ℓ
.
Let Qa be the orthogonal matrix whose columns are ηi(a) and τj(a), namely
(2.16)
η
i
(
a
)
=
Q
a
e
i
and
τ
j
(
a
)
=
Q
a
e
j
.
Define
(2.17)
w
a
(
y
,
s
)
=
(
T
-
t
)
1
p
-
1
u
(
x
,
t
)
,
y
=
Q
a
T
(
x
-
a
T
-
t
)
,
s
=
-
log
(
T
-
t
)
.
Then we see from (1.12) that wa satisfies (1.13) and
(2.18)
w
a
(
y
,
s
)
=
W
a
(
Q
a
y
,
s
)
for all
(
y
,
s
)
∈
ℝ
N
×
[
-
log
T
,
+
∞
)
.
Note from (2.16) that the point (y,s) in the domain of wa becomes the point (x,t) in the domain of u, where
x
=
a
+
e
-
s
2
Q
a
y
=
a
+
e
-
s
2
(
∑
i
=
1
ℓ
y
i
η
i
(
a
)
+
∑
j
=
ℓ
+
1
N
y
j
τ
j
(
a
)
)
,
t
=
T
-
e
-
s
.
Now, fix a∈Sδ and consider an arbitrary b∈Sδ. From (2.17), we have
(2.19)
w
a
(
y
,
s
)
=
w
b
(
Y
,
s
)
,
where
Y
=
Q
b
T
(
Q
a
y
+
e
s
2
(
a
-
b
)
)
.
If we differentiate (2.19) with respect to yk with k∈{ℓ+1,…,N}, we get
(2.20)
(
T
-
t
)
1
p
-
1
+
1
2
∂
u
∂
τ
k
(
a
)
(
x
,
t
)
=
∂
w
a
∂
y
k
(
y
,
s
)
=
∑
i
=
1
ℓ
τ
k
(
a
)
⋅
η
i
(
b
)
∂
w
b
∂
y
i
(
Y
,
s
)
+
∑
j
=
ℓ
+
1
N
τ
k
(
a
)
⋅
τ
j
(
b
)
∂
w
b
∂
y
j
(
Y
,
s
)
.
If we fix b as the projection of x=a+e-s2Qay on the blow-up set in the orthogonal direction to the tangent space to the blow-up set at a, then b has the same components on the tangent space spanned by {τℓ+1(a),…,τN(a)} as x. In particular,
(2.21)
b
=
b
(
a
,
y
,
s
)
=
a
+
∑
i
=
1
ℓ
γ
a
,
i
(
e
-
s
2
y
~
)
η
i
(
a
)
+
∑
j
=
ℓ
+
1
N
e
-
s
2
y
j
τ
j
(
a
)
,
y
~
=
(
y
ℓ
+
1
,
…
,
y
N
)
.
The following proposition gives a geometric constraint on the expansion of wa, which is the bridge linking the refined asymptotic behavior to the refined regularity of the blow-up set.
Proposition 2.7
Proposition 2.7 (A Geometric Constraint on the Expansion of wa)
Assume that
γ
a
∈
𝒞
1
,
α
*
(
(
-
ϵ
a
,
ϵ
a
)
N
-
ℓ
,
ℝ
ℓ
)
for some
α
*
∈
(
0
,
1
2
)
and
ϵ
a
>
0
.
Then, there exists s1≥max{-logT,s0} (s0 is introduced in Proposition 2.5) such that for all a∈Sδ, |y|≤1, s≥s1 and k=ℓ+1,…,N, it holds that
|
∂
w
a
∂
y
k
(
y
,
s
)
-
{
∂
w
b
∂
y
k
(
y
¯
,
0
,
…
,
0
,
s
)
+
κ
2
p
s
∑
i
=
1
ℓ
∂
γ
a
,
i
∂
ξ
k
(
e
-
s
2
y
~
)
y
i
}
|
(2.22)
≤
C
∑
i
=
1
ℓ
|
∂
γ
a
,
i
∂
ξ
k
(
e
-
s
2
y
~
)
|
[
|
y
¯
|
log
s
s
2
+
1
s
e
-
α
*
s
2
+
e
-
s
2
s
C
0
]
+
C
e
-
(
1
+
α
*
)
s
2
s
C
0
,
where y¯=(y1,…,yℓ), y~=(yℓ+1,…,yN) and b is defined by (2.21).
Proof.
Note that the proof of Proposition 2.7 was given in [20] only when ℓ=1. Of course, that proof naturally extends to the case when ℓ∈{2,…,N-1}. Since our paper is relevant only when ℓ≥2 and Proposition 2.7 presents an essential link between the asymptotic behavior of the solution and a geometric constraint of the blow-up set, we felt we should give the proof of this proposition for the completeness and for the reader’s convenience. As said earlier, this section just gives the main steps of the proof of Theorem 1.1, and because the proof is long and technical, we leave it to Section 3.3. ∎
2.3 Part 3: Refined Regularity of the Blow-Up Set and Conclusion of Theorem 1.1
In this subsection, we give the proof of the 𝒞2-regularity of the blow-up set (Theorems 1.1 and 1.3). We proceed in two steps:
Step 1: We derive from Proposition 2.5 that γa is 𝒞1,12-η for all η>0. Then we apply Proposition 2.7 with α*=α∈(0,12) to improve the regularity of γa which reaches 𝒞1,1-η for all η>0.
Step 2: Using the 𝒞1,1-η-regularity and the geometric constraint in Proposition 2.7, we refine the asymptotic behavior given in Proposition 2.5, which involves terms of order 1se-s2. Exploiting this refined asymptotic behavior together with the geometric constraint (2.22), we derive that γa is of class 𝒞2, which is the conclusion of Theorem 1.1. From the information obtained on the 𝒞2-regularity, we calculate the second fundamental form of the blow-up set, which concludes the proof of Theorem 1.3.
Step 1: Deriving 𝒞1,1-η-Regularity of the Blow-Up Set.
We first derive the 𝒞1,12-η-regularity of the blow-up set for all η>0 from Proposition 2.5. Then we apply Proposition 2.7 with α*=α∈(0,12) to get 𝒞1,1-η-regularity for all η>0. In particular, we claim the following:
Proposition 2.8
Proposition 2.8 (C1,12-η-Regularity for S)
Under the hypotheses of Theorem 1.1, S is the graph of a vector function γ∈C1,12-η((-δ1,δ1)N-ℓ,Rℓ) for any η>0, locally near a^. More precisely, there is an h0>0 such that for all |ξ~|<δ1 and |h~|<h0 such that |ξ~+h~|<δ1, one has for all i∈{1,…,ℓ},
(2.23)
|
γ
i
(
ξ
~
+
h
~
)
-
γ
i
(
ξ
~
)
-
h
~
⋅
∇
γ
i
(
ξ
~
)
|
≤
C
|
h
~
|
3
2
|
log
|
h
~
|
|
1
2
+
C
0
2
.
Proof.
The proof is mainly based on the derivation of the sharp asymptotic profile given in Proposition 2.5. In fact, we exploit the estimate (2.10) to find out a geometric constraint on the blow-up set S, which implies some more regularity on S. Since the argument follows the same lines as in [17, Section 4] for the case ℓ=1, and no new ideas are needed for the case ℓ≥2, we will just sketch the proof by underlying the most relevant aspects in Section 3.2 for the sake of convenience. ∎
The next proposition shows the 𝒞1,1-η-regularity of the blow-up set.
Proposition 2.9
Proposition 2.9 (C1,1-η-Regularity for Sδ)
There exists ξ0>0 such that for each a∈Sδ, the local chart defined in (2.15) satisfies for all k=ℓ+1,…,N and |ξ~|<ξ0,
∑
i
=
1
ℓ
|
∂
γ
a
,
i
∂
ξ
k
(
ξ
~
)
|
≤
C
|
ξ
~
|
|
log
|
ξ
~
|
|
1
+
μ
for some
μ
>
0
.
Proof.
Note that the case ℓ=1 was already proven in [20, p. 516, Lemma 3.4]. Here we use again the argument of [20] for the case ℓ≥2. Using the estimate given in Proposition 2.5 and parabolic regularity, we see that for all k≥ℓ+1 and s≥s0+1,
sup
a
∈
S
δ
,
|
y
|
<
2
|
∂
w
a
∂
y
k
(
y
,
s
)
|
≤
C
e
-
s
2
s
μ
for some
μ
>
0
.
Consider a∈Sδ and y=(y¯,y~), where y¯=(y1,…,yℓ) is such that yi*=1 for some i*∈{1,…,ℓ}, yj=0 for 1≤j≠i*≤ℓ, and y~=(yℓ+1,…,yN) is arbitrary in ∂BN-ℓ(0,1). For s≥max{s0+1,s1}, we consider b=b(a,y,s) defined as in (2.21). Since γa is 𝒞1,12-η for any η>0, we use (2.22) with α*=α∈(0,12) to write for k∈{ℓ+1,…,N},
κ
2
p
s
|
∂
γ
a
,
i
*
∂
ξ
k
(
e
-
s
2
y
~
)
|
≤
C
log
s
s
2
∑
i
=
1
ℓ
|
∂
γ
a
,
i
∂
ξ
k
(
e
-
s
2
y
~
)
|
+
C
e
-
s
2
s
μ
.
Since i* is arbitrary in {1,…,ℓ}, we get
κ
2
p
s
∑
i
=
1
ℓ
|
∂
γ
a
,
i
∂
ξ
k
(
e
-
s
2
y
~
)
|
≤
C
log
s
s
2
∑
i
=
1
ℓ
|
∂
γ
a
,
i
∂
ξ
k
(
e
-
s
2
y
~
)
|
+
C
e
-
s
2
s
μ
,
which gives
∑
i
=
1
ℓ
|
∂
γ
a
,
i
∂
ξ
k
(
e
-
s
2
y
~
)
|
≤
C
e
-
s
2
s
1
+
μ
.
If ξ~=e-s2y~, then |ξ~|=e-s2 and |log|ξ~||=s2 since |y~|=1. Therefore,
∑
i
=
1
ℓ
|
∂
γ
a
,
i
∂
ξ
k
(
e
-
s
2
y
~
)
|
≤
C
|
ξ
~
|
|
log
|
ξ
~
|
|
1
+
μ
.
Since y~ is arbitrary in ∂BN-ℓ(0,1), ξ~=e-s2y~ covers a whole neighborhood of 0, namely B(0,ξ0), where ξ0=e-12max{s0+1,s1}. This concludes the proof of Proposition 2.9. ∎
Step 2: Further Refined Asymptotic Behavior and Deriving 𝒞2-Regularity of S.
In this part, we shall use the 𝒞1,1-η-regularity of the blow-up set together with the geometric constraint (2.22) in order to refine further the asymptotic behavior (2.8). In particular, we claim the following:
Proposition 2.10
Proposition 2.10 (Further Refined Asymptotic Behavior (2.8))
There exist s2>0, d∈(0,12) and continuous functions a→λβ(a) for all β∈NN with |β|=3 and |β¯|=1, where β¯=(β1,…,βℓ), |β¯|=∑i=1ℓβi, such that for all a∈Sδ and s≥s2,
(2.24)
∥
W
a
(
Q
a
y
,
s
)
-
w
ℬ
(
a
)
(
y
¯
,
s
)
-
e
-
s
2
s
∑
|
β
|
=
3
,
|
β
¯
|
=
1
λ
β
(
a
)
h
β
(
y
)
∥
L
ρ
2
≤
C
e
-
s
2
s
d
-
3
2
,
where hβ is defined in (3.2).
Proof.
The proof of this proposition is based on ideas of [20] where the case ℓ=1 was treated. As in [20], the geometric constraint given in Proposition 2.7 plays an important role in deriving (2.24). Since the proof is long and technical, we leave it to Section 3.4. ∎
Let us derive Theorem 1.1 from Propositions 2.10 and 2.7. In particular, Theorem 1.1 is a direct consequence of the following result.
Proposition 2.11
For all a∈Sδ, we have for all i∈{1,…,ℓ}, j,k∈{ℓ+1,…,N},
Λ
j
,
k
(
i
)
(
a
)
=
∂
2
γ
a
,
i
∂
ξ
j
ξ
k
(
0
)
=
2
p
κ
(
1
+
δ
j
,
k
)
λ
e
i
+
e
j
+
e
k
(
a
)
,
where a→λβ(a) is introduced in Proposition 2.10, ei is the i-th vector of canonical base of RN, and δi,k is the Kronecker symbol.
Proof.
From (2.18), (2.24) and the fact that estimate (2.24) also holds in W2,∞(|y|<2) by parabolic regularity, we derive for all k≥ℓ+1 and s≥s2+1,
(2.25)
sup
a
∈
S
δ
,
|
y
|
<
2
|
∂
w
a
∂
y
k
(
y
,
s
)
-
e
-
s
2
s
∑
|
β
|
=
3
,
|
β
¯
|
=
1
λ
β
(
a
)
∂
h
β
∂
y
k
(
y
)
|
≤
C
e
-
s
2
s
d
-
3
2
,
for some d∈(0,12). Note that if |β¯|=1, then there is a unique index i*∈{1,…,ℓ} such that βi*=1 and βm=0 for m∈{1,…,ℓ}, m≠i*. Note also from the definition of hβ (see (3.2) below) that
∂
h
β
∂
y
k
(
y
)
=
β
k
h
β
k
-
1
(
y
k
)
∏
j
=
1
,
j
≠
k
N
h
β
j
(
y
j
)
,
and that h0=1. Therefore, (2.25) yields
(2.26)
|
∂
w
a
∂
y
k
(
y
,
s
)
-
e
-
s
2
s
∑
i
=
1
ℓ
∑
|
β
|
=
3
,
β
i
=
1
λ
β
(
a
)
h
1
(
y
i
)
β
k
h
β
k
-
1
(
y
k
)
∏
j
=
ℓ
a
+
1
,
j
≠
k
h
β
j
(
y
j
)
|
≤
C
e
-
s
2
s
d
-
3
2
.
Take i*∈{1,…,ℓ} arbitrarily and y=ei*+ϵej where ϵ=±1 and j≥ℓ+1. Since hm(0)=0 if m is odd, and βi*=1 if |β|=3, we have either β=ei*+ej*+ek* or β=ei*+2ej* for some j*,k*∈{ℓ+1,…,N}. Using (2.26) yields
(2.27)
|
∂
w
a
∂
y
k
(
e
i
*
+
ϵ
e
j
,
s
)
-
ϵ
e
-
s
2
s
(
1
+
δ
k
,
j
)
λ
e
i
*
+
e
k
+
e
j
(
a
)
|
≤
C
e
-
s
2
s
d
-
3
2
.
Similarly, we have
(2.28)
|
∂
w
a
∂
y
k
(
e
i
*
,
s
)
|
≤
C
e
-
s
2
s
d
-
3
2
.
Now using Proposition 2.7, we write for y=ei*+ϵej and s≥max{s2+1,s1},
|
∂
w
a
∂
y
k
(
e
i
*
+
ϵ
e
j
,
s
)
-
∂
w
a
∂
y
k
(
e
i
*
,
s
)
-
κ
2
p
s
∂
γ
a
,
i
*
∂
ξ
k
(
e
-
s
2
ϵ
e
j
)
|
≤
C
log
s
s
2
∑
i
=
1
ℓ
|
∂
γ
a
,
i
∂
ξ
k
(
e
-
s
2
ϵ
e
j
)
|
+
C
e
-
(
1
+
α
*
)
s
2
s
C
0
+
C
e
-
s
s
C
0
+
1
.
Using this estimate together with (2.27) and (2.28), we obtain
(2.29)
|
ϵ
e
-
s
2
(
1
+
δ
k
,
j
)
λ
e
i
*
+
e
k
+
e
j
(
a
)
-
κ
2
p
∂
γ
a
,
i
*
∂
ξ
k
(
e
-
s
2
ϵ
e
j
)
|
≤
C
log
s
s
∑
i
=
1
ℓ
|
∂
γ
a
,
i
∂
ξ
k
(
e
-
s
2
ϵ
e
j
)
|
+
C
e
-
s
2
s
d
-
1
2
.
From Proposition 2.10, we see that
∥
W
a
(
Q
a
y
,
s
)
-
w
ℬ
(
a
)
(
y
¯
,
s
)
∥
L
ρ
2
≤
C
s
-
1
e
-
s
2
for all
s
≥
s
2
.
Using this estimate and noticing that the same proof of Proposition 2.9 holds with μ=-1, we derive
∑
i
=
1
ℓ
|
∂
γ
a
,
i
∂
ξ
k
(
e
-
s
2
ϵ
e
j
)
|
≤
C
e
-
s
2
.
Putting this estimate into (2.29) and noticing that ∂γa,i*∂ξk(0)=0, we find that
(2.30)
∂
2
γ
a
,
i
*
∂
ξ
k
∂
ξ
j
(
0
)
=
lim
s
→
+
∞
∂
γ
a
,
i
*
∂
ξ
k
(
e
-
s
2
ϵ
e
j
)
ϵ
e
-
s
2
=
2
p
κ
(
1
+
δ
k
,
j
)
λ
e
i
*
+
e
k
+
e
j
(
a
)
.
Since i* is taken arbitrarily belonging to {1,…,ℓ}, identity (2.30) holds for all i*∈{1,…,ℓ}. This concludes the proof of Proposition 2.11. ∎
Proof of Theorem 1.1.
From the definition of the local chart (2.15), we have γa,i(0)=∇γa,i(0)=0 for all i∈{1,…,ℓ}. Hence, we deduce from (2.30) the expression of the second fundamental form of the blow-up set at the point a along the unitary basic vector QaTei: for all k,j∈{ℓ+1,…,N},
(2.31)
Λ
k
,
j
(
i
)
(
a
)
=
∂
2
γ
a
,
i
∂
ξ
k
∂
ξ
j
(
0
)
=
2
p
κ
(
1
+
δ
k
,
j
)
λ
e
i
+
e
k
+
e
j
(
a
)
.
In addition, since a→λβ(a) is continuous, we conclude that the blow-up set is of class 𝒞2. This completes the proof of Theorem 1.1. ∎
Proof of Theorem 1.3.
The estimate (1.23) directly follows from Propositions 2.10 and 2.11. Indeed, the sum in estimate (2.24) can be indexed as
{
β
∈
ℕ
N
∣
|
β
|
=
3
,
|
β
¯
|
=
1
}
=
{
e
i
+
e
j
+
e
k
∣
1
≤
i
≤
ℓ
,
ℓ
+
1
≤
j
,
k
≤
N
}
,
where ek is the k-th canonical basis vector of ℝN. By (2.31) and the definition of hβ (see (3.2) below), we write
∑
|
β
|
=
3
,
|
β
¯
|
=
1
λ
β
(
a
)
h
β
(
y
)
=
∑
i
=
1
ℓ
∑
j
,
k
=
ℓ
+
1
N
λ
e
i
+
e
j
+
e
k
h
e
i
+
e
j
+
e
k
(
y
)
=
κ
2
p
∑
i
=
1
ℓ
y
i
∑
j
,
k
=
ℓ
+
1
N
Λ
j
,
k
(
i
)
(
a
)
1
+
δ
j
,
k
(
y
j
y
k
-
2
δ
j
,
k
)
,
which yields (1.23).
As for (1.24), we note from (2.24) that for all |β|=3 with |β¯|=1, one has
|
g
a
,
β
(
s
)
-
e
-
s
2
s
λ
β
(
s
)
|
≤
C
e
-
s
2
s
d
-
3
2
(recall that ga(y,s)=Wa(Qay,s)-wℬ(a)(y¯,s)). Hence, we write from (2.31),
Λ
j
,
k
(
i
)
(
a
)
=
2
p
κ
(
1
+
δ
j
,
k
)
λ
e
i
+
e
j
+
e
k
(
a
)
=
2
p
κ
(
1
+
δ
j
,
k
)
lim
s
→
+
∞
s
e
s
2
g
a
,
e
i
+
e
j
+
e
k
(
s
)
=
2
p
κ
(
1
+
δ
j
,
k
)
lim
s
→
+
∞
s
e
s
2
∫
ℝ
N
g
a
(
y
,
s
)
h
e
i
+
e
j
+
e
k
(
y
)
∥
h
e
i
+
e
j
+
e
k
∥
L
ρ
2
2
ρ
(
y
)
𝑑
y
.
Using again the definition of hβ (see (3.2) below), we see that
h
e
i
+
e
j
+
e
k
=
y
i
(
y
j
y
k
-
δ
j
,
k
)
and
∥
h
e
i
+
e
j
+
e
k
∥
L
ρ
2
2
=
8
(
1
+
δ
j
,
k
)
.
Recall that w𝒜 does not depend on yj for j≥ℓ+1. Hence, for all j,k≥ℓ+1,
Λ
j
,
k
(
i
)
(
a
)
=
p
4
κ
lim
s
→
+
∞
s
e
s
2
∫
ℝ
N
W
a
(
Q
a
y
,
s
)
y
i
(
y
j
y
k
-
2
δ
j
,
k
)
ρ
(
y
)
𝑑
y
,
which is (1.24). This concludes the proof of Theorem 1.3. ∎
3 Proof of Propositions 2.1, 2.7, 2.8 and 2.10
3.1 Classification of the Difference of Two Solutions of (1.13) Having the Same Asymptotic Behavior
In this subsection, we give the proof of Proposition 2.1. The formulation is the same as given in [4] for the difference of two solutions with the radial profile (ℓ=N). Therefore, we sketch the proof and emphasize only the novelties. Note also that the case ℓ=1 was treated in [17].
Let us define
g
(
y
,
s
)
=
W
1
(
y
,
s
)
-
W
2
(
y
,
s
)
,
where Wi, i=1,2 are the solutions of equation (1.13) and behave like (2.3). We see from (1.13) and (2.3) that for all (y,s)∈ℝN×[-logT,+∞),
(3.1)
∂
s
g
=
ℒ
g
+
α
g
,
∥
g
(
s
)
∥
L
ρ
2
≤
C
log
s
s
2
,
where
ℒ
=
Δ
-
1
2
y
⋅
∇
+
1
and
α
(
y
,
s
)
=
|
W
1
|
p
-
1
W
1
-
|
W
2
|
p
-
1
W
2
W
1
-
W
2
-
p
p
-
1
if
W
1
≠
W
2
,
in particular,
α
(
y
,
s
)
=
p
|
W
0
(
y
,
s
)
|
p
-
1
-
p
p
-
1
for some
W
0
(
y
,
s
)
∈
(
W
1
(
y
,
s
)
,
W
2
(
y
,
s
)
)
.
The operator ℒ is self-adjoint on 𝒟(ℒ)⊂Lρ2(ℝN). Its spectrum consists of eigenvalues
spec
(
ℒ
)
=
{
λ
n
=
1
-
n
2
∣
n
∈
ℕ
}
.
The eigenfunctions corresponding to 1-n2 are
(3.2)
h
β
(
y
)
=
h
β
1
(
y
1
)
⋯
h
β
N
(
y
N
)
,
β
1
+
⋯
+
β
N
=
|
β
|
=
n
,
where
h
m
(
ξ
)
=
∑
i
=
0
[
m
/
2
]
m
!
i
!
(
m
-
2
i
)
!
(
-
1
)
i
ξ
m
-
2
i
,
m
∈
ℕ
satisfy
∫
ℝ
h
m
(
ξ
)
h
n
(
ξ
)
ρ
(
ξ
)
𝑑
ξ
=
2
m
m
!
δ
m
,
n
.
The component of g on hβ is given by
g
β
(
s
)
=
∫
ℝ
N
k
β
(
y
)
g
(
y
,
s
)
ρ
(
y
)
𝑑
y
,
where
k
β
(
y
)
=
h
β
(
y
)
∥
h
β
∥
L
ρ
2
2
.
If we denote by Pn the orthogonal projector of Lρ2 over the eigenspace of ℒ corresponding to the eigenvalue 1-n2, then
P
n
g
(
y
,
s
)
=
∑
|
β
|
=
n
g
β
(
s
)
h
β
(
y
)
.
Since the eigenfunctions of ℒ span the whole space Lρ2, we can write
g
(
y
,
s
)
=
∑
n
∈
N
P
n
g
(
y
,
s
)
=
∑
β
∈
ℕ
N
g
β
(
s
)
h
β
(
y
)
=
∑
β
∈
ℕ
N
,
|
β
|
≤
k
g
β
(
s
)
h
β
(
y
)
+
R
k
+
1
g
(
y
,
s
)
,
where Rkg=∑n≥kPng. We also denote
I
(
s
)
2
=
∥
g
(
s
)
∥
L
ρ
2
2
=
∑
n
∈
ℕ
l
n
2
(
s
)
=
∑
n
≤
k
l
n
2
(
s
)
+
r
k
+
1
2
(
s
)
,
where
(3.3)
l
n
(
s
)
=
∥
P
n
g
(
s
)
∥
L
ρ
2
,
r
k
(
s
)
=
∥
R
k
g
(
s
)
∥
L
ρ
2
.
As for α, we have the following estimates.
Lemma 3.1
Lemma 3.1 (Estimates on α)
For all y∈RN and s≥-logT, we have
α
(
y
,
s
)
≤
C
s
,
|
α
(
y
,
s
)
|
≤
C
s
(
1
+
|
y
|
2
)
,
|
α
(
y
,
s
)
+
1
4
s
∑
i
=
1
ℓ
h
2
(
y
i
)
|
≤
C
s
3
2
(
1
+
|
y
|
3
)
.
Proof.
The proof follows the same lines as the proof of [4, Lemma 2.5] where the case ℓ=N was treated. ∎
In the following lemma, we project equation (3.1) on the different modes to get estimates for I(s), ln(s) and rn(s). More precisely, we claim the following:
Lemma 3.2
Lemma 3.2 (Evolution of I(s), ln(s) and rn(s))
There exist s3≥-logT and s*>0 such that for all s≥s3, n∈N and β∈NN, one has
(3.4)
|
l
n
′
(
s
)
+
(
n
2
-
1
)
l
n
(
s
)
|
≤
C
(
n
)
I
(
s
)
s
,
(3.5)
I
′
(
s
)
≤
(
1
-
n
+
1
2
+
C
0
s
)
I
(
s
)
+
∑
k
=
0
n
1
2
(
n
+
1
-
k
)
l
k
(
s
)
,
(3.6)
|
g
β
′
(
s
)
+
(
-
1
+
|
β
|
2
+
1
s
∑
i
=
1
ℓ
β
i
)
g
β
(
s
)
|
≤
C
(
β
)
(
1
s
3
2
I
(
s
)
+
1
s
(
l
|
β
|
-
2
(
s
)
+
l
|
β
|
+
2
)
)
,
(3.7)
r
n
′
(
s
)
≤
(
1
-
n
2
)
r
n
(
s
)
+
C
s
I
(
s
-
s
*
)
.
Proof.
See [4, Lemma 2.7] for (3.4) and (3.5). See [17, p. 545, Appendix B.1] for a calculation similar to (3.6). For (3.7), see [20, p. 523], where the calculation is mainly based on the following regularizing property of equation (3.1) by Herrero and Velázquez [9] (control of the Lρ4-norm by the Lρ2-norm up to some delay in time, see [9, Lemma 2.3]):
(
∫
g
4
(
y
,
s
)
ρ
𝑑
y
)
1
4
≤
C
(
∫
g
2
(
y
,
s
-
s
*
)
ρ
𝑑
y
)
1
2
for some
s
*
>
0
.
This ends the proof of Lemma 3.2. ∎
In the next step, we use Lemma 3.2 to show that either the null mode or a negative mode of ℒ will dominate as s→+∞. In particular, we have the following:
Proposition 3.3
Proposition 3.3 (Dominance of a Mode and Its Description)
Either
l
n
(
s
)
=
𝒪
(
I
(
s
)
s
)
for all
n
∈
ℕ
, and there exist
σ
n
, Cn>0 and Cn′>0 such that
I
(
s
)
≤
C
n
s
C
n
′
exp
(
(
1
-
n
2
)
s
)
for all
s
≥
σ
n
;
or there is
n
0
≥
2
such that
(3.8)
I
(
s
)
∼
l
n
0
(
s
)
𝑎𝑛𝑑
l
n
(
s
)
=
𝒪
(
I
(
s
)
s
)
as
s
→
+
∞
for all
n
≠
n
0
.
Moreover,
if
n
0
=
2
, namely
I
(
s
)
∼
l
2
(
s
)
, then for all
|
β
|
=
2
,
(3.9)
{
|
g
β
(
s
)
|
≤
C
log
s
s
5
2
if
∑
i
=
1
ℓ
β
i
≠
2
,
|
g
β
(
s
)
-
c
β
s
2
|
≤
C
log
s
s
5
2
if
∑
i
=
1
ℓ
β
i
=
2
,
if
n
0
=
3
, namely
I
(
s
)
∼
l
3
(
s
)
, then
(3.10)
I
(
s
)
≤
C
0
e
-
s
2
s
C
0
for some
C
0
>
0
.
Proof.
See [4, Proposition 2.6] for the existence of a dominating component, where the proof relies on (3.4) and (3.5). If case (ii) occurs with n0=2, by (3.6) we write for all β∈ℕN with |β|=2,
|
g
β
′
(
s
)
+
g
β
s
∑
i
=
1
ℓ
β
i
|
≤
C
(
β
)
(
I
(
s
)
s
3
2
+
l
0
(
s
)
+
l
4
(
s
)
s
)
≤
C
(
β
)
I
(
s
)
s
3
2
≤
C
(
β
)
log
s
s
7
2
,
where we used (3.8) and (3.1) from which we have l0(s)+l4(s)=𝒪(I(s)s) and I(s)=𝒪(logss2). Since ∑i=1ℓβi is only equal to 0,1 or 2 if |β|=2, estimate (3.9) follows after integration. Estimate (3.10) immediately follows from (3.4). This ends the proof of Proposition 3.3. ∎
Let us now derive Proposition 2.1 from Proposition 3.3. Indeed, we see from Proposition 3.3 that if case (i) occurs, we already have exponential decay for I(s). If case (ii) occurs with n0≥3, by (3.4) we write
|
l
n
0
′
(
s
)
+
(
n
0
2
-
1
)
l
n
0
|
≤
C
s
l
n
0
.
Since ln0≠0 in a neighborhood of infinity, this gives
l
n
0
(
s
)
≤
C
0
s
C
0
e
(
1
-
n
0
2
)
s
≤
C
0
s
C
0
e
-
s
2
,
which yields (2.4). If case (ii) occurs with n0=2, by definition of P2, we derive from (3.9) that there is a symmetric, real (ℓ×ℓ)-matrix ℬ such that
P
2
g
(
y
,
s
)
=
1
s
2
(
1
2
y
¯
T
ℬ
y
¯
-
tr
(
ℬ
)
)
+
o
(
1
s
2
)
,
which is (2.3). This concludes the proof of Proposition 2.1.
3.2
𝒞
1
,
1
2
-
η
-Regularity of the Blow-Up Set
We give the proof of Proposition 2.8 in this section. The proof uses the argument given in [17] treated for the case ℓ=1. Here we shall exploit the refined estimate (2.10) to obtain a geometric constraint on the blow-up set. Without loss of generality, we assume a^=0 and Qa^=Id. Under the hypotheses of Proposition 2.8, we know that γ∈𝒞1((-δ1,δ1)N-ℓ,ℝℓ) with ℓ∈{1,…,N-1}. If we introduce
Γ
(
x
~
)
=
(
γ
1
(
x
~
)
,
…
,
γ
ℓ
(
x
~
)
,
x
~
)
,
x
~
=
(
x
ℓ
+
1
,
…
,
x
N
)
,
then
Im
Γ
∩
B
(
0
,
2
δ
)
=
graph
(
γ
)
∩
B
(
0
,
2
δ
)
=
S
δ
.
Consider x~ and h~ in ℝN-ℓ such that x~ as well as x~+h~ are in B(0,δ1) and Γ(x~) as well as Γ(x~+h~) are in Sδ. For all t∈[T-e-s0-1,T) such that |Γ(x~)-Γ(x~+h~)|≤(T-t)|log(T-t)|, we use (2.10) with x=a=Γ(x~+h~), then with x=Γ(x~+h~) and a=Γ(x~) to find that
(3.11)
{
|
(
T
-
t
)
1
p
-
1
u
(
Γ
(
x
~
+
h
~
)
,
t
)
-
w
ℬ
(
Γ
(
x
~
+
h
~
)
)
(
0
,
s
)
|
≤
C
e
-
s
2
s
3
2
+
C
0
,
|
(
T
-
t
)
1
p
-
1
u
(
Γ
(
x
~
+
h
~
)
,
t
)
-
w
ℬ
(
Γ
(
x
~
)
)
(
y
¯
Γ
(
x
~
)
,
Γ
(
x
~
+
h
~
)
,
s
,
s
)
|
≤
C
e
-
s
2
s
3
2
+
C
0
,
where y¯Γ(x~),Γ(x~+h~),s is defined as
(3.12)
y
¯
a
1
,
a
2
,
s
=
e
s
2
(
(
a
1
-
a
2
)
⋅
Q
a
1
e
1
,
…
,
(
a
1
-
a
2
)
⋅
Q
a
1
e
ℓ
)
.
Since Γ is 𝒞1, we have
|
Γ
(
x
~
+
h
~
)
-
Γ
(
x
~
)
|
≤
C
|
h
~
|
.
Let us fix t=t~(x~,h~) such that
(3.13)
|
Γ
(
x
~
+
h
~
)
-
Γ
(
x
~
)
|
=
(
T
-
t
~
)
|
log
(
T
-
t
~
)
|
,
and take h~∈BN-ℓ(0,h1(s0)) for some h1(s0)>0. Then we have t~≥T-e-s0-1. Hence, if s~=-log(T-t~), by (3.11) we have
(3.14)
|
w
ℬ
(
Γ
(
x
~
+
h
~
)
)
(
0
,
s
~
)
-
w
ℬ
(
Γ
(
x
~
)
)
(
y
¯
Γ
(
x
~
)
,
Γ
(
x
~
+
h
~
)
,
s
~
,
s
~
)
|
≤
C
e
-
s
~
2
s
~
3
2
+
C
0
.
Similarly, by changing the roles of x~ and x~+h~, we get
(3.15)
|
w
ℬ
(
Γ
(
x
~
)
)
(
0
,
s
~
)
-
w
ℬ
(
Γ
(
x
~
+
h
~
)
)
(
y
¯
Γ
(
x
~
+
h
~
)
,
Γ
(
x
~
)
,
s
~
,
s
~
)
|
≤
C
e
-
s
~
2
s
~
3
2
+
C
0
,
where y¯Γ(x~+h~),Γ(x~),s~ is defined as in (3.12).
From a Taylor expansion for wℬ(y¯,s~) near y¯=0, we write
(3.16)
w
ℬ
(
y
¯
,
s
~
)
=
w
ℬ
(
0
,
s
~
)
+
y
¯
⋅
∇
w
ℬ
(
0
,
s
~
)
+
1
2
y
¯
T
∇
2
w
ℬ
(
0
,
s
~
)
y
¯
+
𝒪
(
|
y
¯
|
3
|
∇
3
w
ℬ
(
z
,
s
~
)
|
)
,
for some z between 0 and y¯.
Since (2.2) and (2.7) also hold in 𝒞lock by parabolic regularity, we deduce that
|
∇
w
ℬ
(
0
,
s
~
)
|
=
𝒪
(
log
s
~
s
~
2
)
,
∇
2
w
ℬ
(
0
,
s
~
)
=
-
κ
4
p
s
~
I
ℓ
×
ℓ
+
𝒪
(
log
s
~
s
~
2
)
.
From [11, Theorem 1], we know that
∥
∇
3
w
ℬ
(
s
~
)
∥
L
∞
≤
C
3
s
~
3
2
.
Substituting all these above estimates into (3.16) yields
w
ℬ
(
y
¯
,
s
~
)
≤
w
ℬ
(
0
,
s
~
)
-
κ
8
p
s
~
|
y
¯
|
2
+
C
3
|
y
¯
|
3
6
s
~
3
2
+
C
log
s
~
s
~
2
.
Therefore, we have
(3.17)
w
ℬ
(
y
¯
,
s
~
)
≤
w
ℬ
(
0
,
s
~
)
-
κ
16
p
s
~
|
y
¯
|
2
for all
|
y
¯
|
≤
3
κ
8
C
3
p
s
~
.
We claim from (3.14), (3.15) and (3.17) the following:
(3.18)
|
w
ℬ
(
Γ
(
x
~
)
)
(
0
,
s
~
)
-
w
ℬ
(
Γ
(
x
~
+
h
~
)
)
(
0
,
s
~
)
|
≤
C
e
-
s
~
2
s
~
3
2
+
C
0
.
Indeed, if wℬ(Γ(x~))(0,s~)-wℬ(Γ(x~+h~))(0,s~)≥0, then by (3.17) and (3.15) we have
0
≤
w
ℬ
(
Γ
(
x
~
)
)
(
0
,
s
~
)
-
w
ℬ
(
Γ
(
x
~
+
h
~
)
)
(
0
,
s
~
)
≤
w
ℬ
(
Γ
(
x
~
)
)
(
0
,
s
~
)
-
w
ℬ
(
Γ
(
x
~
+
h
~
)
)
(
y
¯
Γ
(
x
~
+
h
~
)
,
Γ
(
x
~
)
,
s
~
,
s
~
)
≤
C
e
-
s
~
2
s
~
3
2
+
C
0
.
If wℬ(Γ(x~))(0,s~)-wℬ(Γ(x~+h~))(0,s~)≤0, then we do as above and use (3.14) instead of (3.15) to obtain (3.18).
From (3.18), (3.14) and (3.17), we get
κ
16
p
s
~
|
y
¯
Γ
(
x
~
)
,
Γ
(
x
~
+
h
~
)
,
s
~
|
2
≤
w
ℬ
(
Γ
(
x
~
)
)
(
0
,
s
~
)
-
w
ℬ
(
Γ
(
x
~
)
)
(
y
¯
Γ
(
x
~
)
,
Γ
(
x
~
+
h
~
)
,
s
~
,
s
~
)
≤
C
e
-
s
~
2
s
~
3
2
+
C
0
.
Hence, we obtain
(3.19)
|
y
¯
Γ
(
x
~
)
,
Γ
(
x
~
+
h
~
)
,
s
~
|
2
≤
C
e
-
s
~
2
s
~
5
2
+
C
0
.
From the definition (3.12), we have
(3.20)
|
y
¯
Γ
(
x
~
)
,
Γ
(
x
~
+
h
~
)
,
s
~
|
=
e
s
~
2
d
(
Γ
(
x
~
)
,
π
Γ
(
x
~
+
h
~
)
)
,
where we recall πΓ(x~+h~) is the tangent plan of S at Γ(x~+h~). On the other hand, we claim that
(3.21)
d
(
Γ
(
x
~
)
,
T
Γ
(
x
~
+
h
~
)
)
≥
|
γ
i
(
x
~
+
h
~
)
-
γ
i
(
x
~
)
-
h
~
⋅
∇
γ
i
(
x
~
)
|
1
+
|
∇
γ
i
(
x
~
)
|
2
,
where Si is the surface of equation xi=γi(x~), and Ti,Γ(x~+h~) is the tangent plan of Si at Γ(x~+h~). Indeed, we note that
d
(
Γ
(
x
~
)
,
T
i
,
Γ
(
x
~
+
h
~
)
)
=
|
γ
i
(
x
~
+
h
~
)
-
γ
i
(
x
~
)
-
h
~
⋅
∇
γ
i
(
x
~
)
|
1
+
|
∇
γ
i
(
x
~
)
|
2
,
and ImΓ⊂Si, hence, (3.21) follows from d(Γ(x~),TΓ(x~+h~))≥d(Γ(x~),Ti,Γ(x~+h~)).
Combining (3.19), (3.20), (3.21) together with the relation s~=-log(T-t~) yields
|
γ
i
(
x
~
+
h
~
)
-
γ
i
(
x
~
)
-
h
~
⋅
∇
γ
i
(
x
~
)
|
2
≤
C
(
T
-
t
~
)
3
2
|
log
(
T
-
t
~
)
|
5
2
+
C
0
.
If we denote A=|Γ(x~+h~)-Γ(x~)|≤C|h~|, then by relation (3.13) we have
|
log
(
T
-
t
~
)
|
∼
2
|
log
A
|
,
T
-
t
~
∼
A
2
2
|
log
A
|
as
A
→
0
.
Hence,
|
γ
i
(
x
~
+
h
~
)
-
γ
i
(
x
~
)
-
h
~
⋅
∇
γ
i
(
x
~
)
|
2
≤
C
A
3
|
log
A
|
1
+
C
0
≤
C
|
h
~
|
3
|
log
|
h
~
|
|
1
+
C
0
,
which yields (2.23). This concludes the proof of Proposition 2.8.
3.3 A Geometric Constraint Linking the Blow-Up Behavior of the Solution to the Regularity of the Blow-Up Set
This section is devoted to the proof of Proposition 2.7. The proof follows ideas given in [20]. By the hypothesis, we have γa∈𝒞1,α*((-ϵa,ϵa)N-ℓ,ℝℓ) for some α*∈(0,12) and ϵa>0, and γa,i(0)=∇γa,i(0)=0. Thus, for all |ξ~|<ϵa,
(3.22)
|
γ
a
,
i
(
ξ
~
)
|
≤
C
|
ξ
~
|
1
+
α
*
and
|
∇
γ
a
,
i
(
ξ
~
)
|
≤
C
|
ξ
~
|
α
*
.
In what follows, k∈{ℓ+1,…,N} is fixed, and we use indexes i and m for the range 1,…,ℓ, index j for the range ℓ+1,…,N.
We now use (3.22) to approximate all the terms appearing in (2.20).
(a) Term τk(a)⋅ηi(b). From the local coordinates (2.21), we have
η
i
(
b
)
=
1
1
+
|
∇
γ
a
,
i
(
e
-
s
2
y
~
)
|
2
(
η
i
(
a
)
-
∑
j
=
ℓ
+
1
N
∂
γ
a
,
i
∂
ξ
j
(
e
-
s
2
y
~
)
τ
j
(
a
)
)
.
Using (3.22) and the fact that τk(a)⋅ηi(a)=0 and τk(a)⋅τj(a)=δk,j, we obtain
|
τ
k
(
a
)
⋅
η
i
(
b
)
+
∂
γ
a
,
i
∂
ξ
k
(
e
-
s
2
y
~
)
|
=
|
(
1
-
1
1
+
|
∇
γ
a
,
i
(
e
-
s
2
y
~
)
|
2
)
∂
γ
a
,
i
∂
ξ
k
(
e
-
s
2
y
~
)
|
≤
|
∂
γ
a
,
i
∂
ξ
k
(
e
-
s
2
y
~
)
|
|
∇
γ
a
,
i
(
e
-
s
2
y
~
)
|
2
(3.23)
≤
|
∂
γ
a
,
i
∂
ξ
k
(
e
-
s
2
y
~
)
|
e
-
α
*
s
.
(b) Term τk(a)⋅τj(b). From (2.21) and (3.22), we have
|
b
-
a
|
≤
|
∑
i
=
1
ℓ
γ
a
,
i
(
e
-
s
2
y
~
)
|
+
e
-
s
2
|
y
~
|
≤
C
e
-
s
2
.
Since ηi and τj are 𝒞α*, it holds that
|
η
i
(
a
)
-
η
i
(
b
)
|
+
|
τ
j
(
a
)
-
τ
j
(
b
)
|
≤
C
|
a
-
b
|
α
*
≤
C
e
-
α
*
s
2
.
It follows that
(3.24)
{
|
η
i
(
a
)
⋅
η
m
(
b
)
-
δ
i
,
m
|
+
|
τ
k
(
a
)
⋅
τ
j
(
b
)
-
δ
k
,
j
|
≤
C
e
-
α
*
s
2
,
|
η
i
(
a
)
⋅
τ
j
(
b
)
|
+
|
η
i
(
b
)
⋅
τ
j
(
a
)
|
≤
C
e
-
α
*
s
2
.
(c) The point Y(a,y,s). Using (2.16), (2.19) and (2.21), we write
Y
m
=
Y
⋅
e
m
=
(
Q
a
y
+
e
s
2
(
a
-
b
)
)
⋅
Q
b
e
m
=
{
∑
i
=
1
ℓ
y
i
η
i
(
a
)
+
∑
j
=
ℓ
+
1
N
y
j
τ
j
(
a
)
-
e
s
2
[
∑
i
=
1
ℓ
γ
a
,
i
(
e
-
s
2
y
~
)
η
i
(
a
)
+
∑
j
=
ℓ
+
1
N
e
-
s
2
y
j
τ
j
(
a
)
]
}
⋅
Q
b
e
m
=
{
∑
i
=
1
ℓ
[
y
i
-
e
s
2
γ
a
,
i
(
e
-
s
2
y
~
)
]
η
i
(
a
)
}
⋅
Q
b
e
m
.
From (2.16), we write for m∈{1,…,ℓ},
Y
m
-
y
m
=
{
(
y
m
-
e
s
2
γ
a
,
m
(
e
-
s
2
y
~
)
)
η
m
(
a
)
⋅
η
m
(
b
)
-
y
m
η
m
(
a
)
⋅
η
m
(
a
)
}
+
∑
i
=
1
,
i
≠
m
ℓ
(
y
i
-
e
s
2
γ
a
,
i
(
e
-
s
2
y
~
)
)
η
i
(
a
)
⋅
η
m
(
b
)
,
and for n∈{ℓ+1,…,N},
Y
n
=
∑
i
=
1
ℓ
(
y
i
-
e
s
2
γ
a
,
i
(
e
-
s
2
y
~
)
)
η
i
(
a
)
⋅
τ
n
(
b
)
.
Using (3.24) yields
|
Y
m
-
y
m
|
≤
C
e
-
α
*
s
2
and
|
Y
k
|
≤
C
e
-
α
*
s
2
.
Hence, if we write
Y
¯
=
(
Y
1
,
…
,
Y
ℓ
)
and
Y
~
=
(
Y
ℓ
+
1
,
…
,
Y
N
)
,
then
(3.25)
|
y
¯
-
Y
¯
|
≤
C
e
-
α
*
s
2
and
|
Y
~
|
≤
C
e
-
α
*
s
2
.
(d) Term ∂wb∂yi(Y,s). From Proposition 2.5 and the parabolic regularity, we have that
(3.26)
sup
s
≥
s
′
∥
w
b
(
y
,
s
)
-
w
ℬ
(
b
)
(
y
¯
,
s
)
∥
W
loc
2
,
∞
(
|
y
¯
|
<
2
)
≤
C
e
-
s
2
s
C
0
.
This implies
(3.27)
|
∂
w
b
∂
y
i
(
Y
,
s
)
-
∂
w
ℬ
(
b
)
∂
y
i
(
y
¯
,
s
)
|
+
∑
m
=
ℓ
+
1
N
|
∂
w
b
∂
y
m
(
Y
,
s
)
|
+
sup
|
z
|
<
2
,
(
m
,
n
)
≠
(
i
,
i
)
,
i
≥
ℓ
+
1
|
∂
2
w
b
∂
y
m
∂
y
n
(
z
,
s
)
|
≤
C
e
-
s
2
s
C
0
.
Similarly, from (2.1) and (2.18),
(3.28)
sup
s
≥
-
log
T
∥
w
a
(
y
,
s
)
-
{
κ
+
κ
2
p
s
(
ℓ
-
|
y
¯
|
2
2
)
}
∥
W
loc
2
,
∞
(
|
y
¯
|
<
2
)
≤
C
log
s
s
2
.
From (3.26) and (3.28), we deduce that
(3.29)
sup
s
≥
s
′′
∥
w
ℬ
(
a
)
(
y
,
s
)
-
{
κ
+
κ
2
p
s
(
ℓ
-
|
y
¯
|
2
2
)
}
∥
W
loc
2
,
∞
(
|
y
¯
|
<
2
)
≤
C
log
s
s
2
.
Using (3.29), we have for |z|≤2,
|
∂
2
w
ℬ
(
b
)
∂
y
i
2
(
z
,
s
)
+
κ
2
p
s
|
≤
C
log
s
s
2
and
|
∂
2
w
ℬ
(
b
)
∂
y
i
∂
y
m
(
z
,
s
)
|
≤
C
log
s
s
2
,
m
≠
i
.
Note that ∂wℬ(b)/∂yi(0,s)=0. We then take the Taylor expansion of ∂wℬ(b)/∂yi(y¯,s) near y¯=0 up to the first order to get
|
∂
w
ℬ
(
b
)
∂
y
i
(
y
¯
,
s
)
+
Y
i
κ
2
p
s
|
≤
C
|
y
¯
|
log
s
s
2
.
Using (3.27) and (3.25) yields
(3.30)
|
∂
w
b
∂
y
i
(
Y
,
s
)
+
y
i
κ
2
p
s
|
≤
C
e
-
s
2
s
C
0
+
C
|
y
¯
|
log
s
s
2
+
C
s
e
-
α
*
s
2
.
(e) Term ∂wb∂yj(Y,s). We just use (3.27) and (3.25) to get
(3.31)
|
∂
w
b
∂
y
j
(
Y
,
s
)
-
∂
w
b
∂
y
j
(
y
¯
,
0
,
…
,
0
,
s
)
|
≤
C
e
-
(
1
+
α
*
)
s
2
.
Estimate (2.22) then follows by substituting (3.31), (3.30), (3.27), (3.23) and (3.24) into (2.20). This concludes the proof of Proposition 2.7.
3.4 Further Refined Asymptotic Behavior
We prove Proposition 2.10 in this subsection. We first refine estimate (2.8) and find the following terms in the expansion which is of order e-s2. Using the geometric constraint, we show that all terms of order e-s2 must be identically zero, which gives a better estimate for ∥Wa(Qay,s)-wℬ(a)(y¯,s)∥Lρ2. We then repeat the process and use again Proposition 2.7 in order to get the term of order 1se-s2 and conclude the proof of Proposition 2.10.
Let us define
(3.32)
g
a
(
y
,
s
)
=
W
a
(
Q
a
y
,
s
)
-
w
ℬ
(
a
)
(
y
¯
,
s
)
and
I
a
(
s
)
2
=
∥
g
a
(
s
)
∥
L
ρ
2
2
,
l
a
,
n
(
s
)
=
∥
P
n
g
a
(
s
)
∥
L
ρ
2
,
r
a
,
k
(
s
)
=
∥
∑
n
≥
k
P
n
g
a
(
s
)
∥
L
ρ
2
.
From (2.8), we have
(3.33)
I
a
(
s
)
=
𝒪
(
e
-
s
2
s
μ
)
for some
μ
>
0
.
Note that Lemma 3.2 also holds with W1=Wa and W2=wℬ. We claim the following:
Lemma 3.4
Assume that Ia(s)=O(e-s2sμ0) for some μ0∈R. There exists s4>0 such that for all s≥s4,
(3.34)
∑
n
=
0
2
l
a
,
n
(
s
)
+
r
a
,
4
(
s
)
≤
C
e
-
s
2
s
μ
0
-
1
and
(3.35)
|
d
d
s
(
g
a
,
β
(
s
)
e
s
2
s
|
β
¯
|
)
|
≤
C
s
|
β
¯
|
+
μ
0
-
3
2
for all
β
∈
ℕ
N
,
|
β
|
=
3
,
where β¯=(β1,…,βℓ), |β¯|=∑i=1ℓβi.
Proof.
By (3.4) and (3.7), we can write for all s≥s3,
|
d
d
s
(
l
a
,
n
(
s
)
e
(
n
/
2
-
1
)
s
)
|
≤
C
e
(
n
/
2
-
3
2
)
s
s
μ
0
-
1
,
n
=
0
,
1
,
2
,
and
|
d
d
s
(
r
a
,
4
(
s
)
e
s
)
|
≤
C
e
s
2
s
μ
0
-
1
.
Estimate (3.34) then follows after integration of the above inequalities. As for (3.35), we just use (3.6) and (3.34) (note that la,5≤ra,4 by definition (3.3)). ∎
Using (3.33) and applying Lemma 3.4 a finite number of steps, we obtain the following:
Lemma 3.5
There exist s5>0 and continuous functions a→λβ(a) for all β∈NN with |β|=3 and |β¯|=∑i=1ℓβi=0 such that for all a∈Sδ and s≥s5,
∥
g
a
(
y
,
s
)
-
e
-
s
2
∑
|
β
|
=
3
,
|
β
¯
|
=
0
λ
β
(
a
)
h
β
(
y
)
∥
L
ρ
2
≤
C
e
-
s
2
s
d
-
1
2
,
for some d∈(0,12), where hβ is defined by (3.2).
Proof.
We first show that there is s5>0 such that
(3.36)
I
a
(
s
)
≤
C
e
-
s
2
s
d
for some
d
∈
(
0
,
1
2
)
for all
s
≥
s
5
.
From (3.33), if μ∈(0,12), we are done. If μ≥12, we apply Lemma 3.4 with μ0=μ to get
∑
n
=
0
2
l
a
,
n
(
s
)
+
r
a
,
4
(
s
)
≤
C
e
-
s
2
s
μ
-
1
and
|
g
a
,
β
(
s
)
|
≤
C
e
-
s
2
s
μ
-
1
2
for all
|
β
|
=
3
.
Hence,
I
a
(
s
)
≤
C
e
-
s
2
s
μ
-
1
2
.
Estimate (3.36) then follows by repeating this process a finite number of steps.
Now using (3.36) and Lemma 3.4 with μ0=d, we distinguish the following two cases:
If |β|=3 and |β¯|≥1, we integrate (3.35) on [s,+∞) to derive
|
g
a
,
β
(
s
)
|
≤
C
e
-
s
2
s
d
-
1
2
for all
|
β
|
=
3
,
|
β
¯
|
≥
1
.
If |β|=3 and |β¯|=0, by integrating (3.35) on [s5,s], we deduce that there exist continuous functions a→λβ(a) such that
|
g
a
,
β
(
s
)
-
λ
β
(
a
)
e
-
s
2
|
≤
C
e
-
s
2
s
d
-
1
2
for all
|
β
|
=
3
,
|
β
¯
|
=
0
.
This concludes the proof of Lemma 3.5. ∎
Now we shall use the geometric constraint on the asymptotic behavior of the solution given in Proposition 2.7 to show that all the coefficients λβ(a) with |β|=3 and β¯=0 in Lemma 3.5 have to be identically zero. In particular, we claim the following:
Lemma 3.6
There exists s6>0 such that for all s≥s6,
∥
g
a
(
s
)
∥
L
ρ
2
≤
C
e
-
s
2
s
d
-
1
2
for some
d
∈
(
0
,
1
2
)
and all
a
∈
S
δ
.
Proof.
Consider a∈Sδ. We aim at proving that
λ
β
(
a
)
=
0
for all
β
∈
ℕ
N
,
|
β
|
=
3
,
|
β
¯
|
=
0
,
where λβ(a) is introduced in Lemma 3.5 and |β¯|=∑i=1ℓβi.
From (2.18), (3.32) and the fact that the estimate given in Lemma 3.5 also holds in W2,∞(|y|<2) by parabolic regularity, we write for all k≥ℓ+1 and s≥s5+1,
(3.37)
sup
a
∈
S
δ
,
|
y
|
<
2
|
∂
w
a
∂
y
k
(
y
,
s
)
-
e
-
s
2
∑
|
β
|
=
3
,
β
¯
=
0
λ
β
(
a
)
∂
h
β
∂
y
k
(
y
)
|
≤
C
e
-
s
2
s
d
-
1
2
.
Take y=(y¯,y~), where y¯=(y1,…,yℓ)=(0,…,0) and y~∈BN-ℓ(0,1). Then we use Proposition 2.9 and (2.22) to obtain
(3.38)
|
∂
w
a
∂
y
k
(
y
,
s
)
-
∂
w
b
∂
y
k
(
0
,
s
)
|
≤
C
e
-
(
1
+
α
*
)
s
2
s
C
0
+
C
e
-
s
s
C
0
+
1
,
for some α*∈(0,12).
From (3.37) and (3.38), we get
(3.39)
|
∑
|
β
|
=
3
,
|
β
¯
|
=
0
λ
β
(
a
)
∂
h
β
∂
y
k
(
y
)
-
∑
|
β
|
=
3
,
|
β
¯
|
=
0
λ
β
(
b
)
∂
h
β
∂
y
k
(
0
)
|
≤
C
s
d
-
1
2
.
From (2.21) and Proposition 2.9, we see that b→a as s→+∞. Since a→λβ(a) is continuous, d∈(0,12), hβ1(0)=⋯=hβℓ(0)=h0(0)=1 from definition (3.2), and
∂
h
β
∂
y
k
(
y
)
=
β
k
h
β
k
-
1
(
y
k
)
∏
j
=
1
,
j
≠
k
N
h
β
j
(
y
j
)
,
we derive, by passing to the limit in (3.39),
∑
|
β
|
=
3
,
|
β
¯
|
=
0
λ
β
(
a
)
β
k
h
β
k
-
1
(
y
k
)
∏
j
=
ℓ
+
1
,
j
≠
k
N
h
β
j
(
y
j
)
=
∑
|
β
|
=
3
,
|
β
¯
|
=
0
λ
β
(
a
)
β
k
h
β
k
-
1
(
0
)
∏
j
=
ℓ
+
1
,
j
≠
k
N
h
β
j
(
0
)
.
By the orthogonality of the polynomials hi, this yields
β
k
λ
β
(
a
)
=
0
for all
k
≥
ℓ
+
1
and
|
β
|
=
3
with
|
β
¯
|
=
0
.
Take β arbitrary with |β|=3 and |β¯|=0, then there exists k≥ℓ+1 such that βk≥1, which implies that λβ(a)=0. This ends the proof of Lemma 3.6. ∎
Proof of Proposition 2.10.
From Lemmas 3.6 and 3.4, we see that for all s≥s7=max{s4,s5,s6},
∑
n
=
0
2
l
a
,
n
(
s
)
+
r
a
,
4
(
s
)
≤
C
s
-
s
2
s
d
-
3
2
and
(3.40)
|
d
d
s
(
g
a
,
β
(
s
)
s
s
2
s
|
β
¯
|
)
|
≤
C
e
|
β
¯
|
+
d
-
2
for all
|
β
|
=
3
,
for some d∈(0,12). Integrating (3.40) between s and +∞ if |β¯|=0 and between s7 and s if |β¯|≥1, we get
|
g
a
,
β
(
s
)
|
≤
C
e
-
s
2
s
d
-
1
for all
|
β
|
=
3
.
Hence,
I
a
(
s
)
=
∥
g
a
(
s
)
∥
L
ρ
2
≤
C
e
-
s
2
s
d
-
1
for all
s
≥
s
7
.
With this new estimate, we use again Lemma 3.4 with μ0=d-1 to show that there exists s8>0 such that for all s≥s8,
∑
n
=
0
2
l
a
,
n
(
s
)
+
r
a
,
4
(
s
)
≤
C
e
-
s
2
s
d
-
2
and
|
d
d
s
(
g
a
,
β
(
s
)
e
s
2
s
|
β
¯
|
)
|
≤
C
s
|
β
¯
|
+
d
-
5
2
for all
|
β
|
=
3
.
This new inequality implies that for all |β|=3 and s≥s8,
if |β¯|=0 or |β¯|≥2, then |ga,β(s)|≤Ce-s2sd-32,
if |β¯|=1, then we obtain the existence of continuous functions a→λβ(a) such that
|
g
a
,
β
(
s
)
-
e
-
s
2
s
λ
β
(
a
)
|
≤
C
e
-
s
2
s
d
-
3
2
.
This concludes the proof of Proposition 2.10. ∎
