Refined uniform estimates at blow-up and applications for nonlinear heat equations

Refined uniform estimates at blow-up and applications for nonlinear heat equations

Frank Merle

Universit´e de Cergy-Pontoise

Hatem Zaag

Ecole Normale Sup´erieure and Universit´e de Cergy-Pontoise´

1 introduction

We are interested in the following nonlinear heat equation:

( u_t = ∆u+u^p

u(0) = u₀ ≥0, (1)

where u is defined for (x, t) ∈ ^RN ×[0, T), 1 < p, (N −2)p < N+ 2 and u₀ ∈H¹(^RN).

In this paper, we deal with blow-up solutions of equation (1)u(t) which blow-up in finite time T > 0: this means that u exists for all t ∈ [0, T),

tlim→Tku(t)kH¹ = +∞ and lim

t→Tku(t)kL^∞ = +∞. Let us consider such a solution. We aim at studying the blow-up behavior of u(t) as t → T. In particular, we are interested in obtaining uniform estimates on u(t) and deducing from these estimates the asymptotic shape of the singularities.

One can show that in this case,u(t) has at least one blow-up point, that is x0 ∈ ^RN such that there exists (xn, tn)n∈^N satisfying (xn, tn) → (x0, T) and|u(x_n, t_n)| →+∞ asn→+∞.

For each a∈^RN, we introduce the following self-similar transformation:

y = √^x−a T−t

s = −log(T−t) wa(y, s) = (T−t)^p−11 u(x, t).

(2)

Then, we see thatw_a=wsatisfies ∀s≥ −logT, ∀y ∈^RN:

∂w

∂s = ∆w−1

2y.∇w− w

p−1+w^p. (3)

The study ofu(t) near (x₀, T) wherex₀ is a blow-up point is equivalent to the study of the long time behavior of w_x0 ass→+∞.

Giga and Kohn prove in [10] that there exists₀>0 such that

∀s≥ −logT, ₀≤ kw_x0(s)kL^∞ ≤ 1 ₀ or equivalently:

∀t∈[0, T), ₀(T−t)^−p−11 ≤ ku(t)kL^∞ ≤ 1

₀(T−t)^−p−11 . At this level, no other uniform estimates were known.

In [16], we proved the following Liouville Theorem for equation (3):

Let w be a nonnegative solution of (3) defined for (y, s)∈^RN ×^R such that w∈L^∞(^RN ×^R). Then, necessarily one of the following cases occurs:

w≡0 or w≡κ or ∃s₀∈^Rsuch that w(y, s) =ϕ(s−s₀) (4) where ϕ(s) =κ(1 +e^s)^−p−11 and κ= (p−1)^−p−11 .

From this theorem we derived in [16] the following uniform estimates of order zero:

Consider a solution w of (3) defined for s ≥ −logT (such that u(t) blows- up at time T). Then,

kw(s)kL^∞ →κ andk∇w(s)kL^∞ +k∆w(s)kL^∞ →0 as s→+∞. (5)

We also derived from this result the following localization theorem:

∀ >0, ∃C>0 such that ∀t∈[^T₂, T), ∀x∈^RN,

∂u

∂t −u^p

≤u^p+C. (6) These estimates are still insufficient to yield precise estimates on blow- up profile. But, we have a compactness property on wa(s) uniformly with respect to a∈ ^RN, which allows us to claim the following result from lin- earization around the limit set ass→+∞:

Theorem 1 (Refined L^∞ estimates for w(s) and u(t) at blow-up) There exist positive constants C₁, C₂ and C₃ such that if u is a solution of

(1) which blows-up at timeT >0and satisfies u(0)∈H¹(^RN), then∀ >0, there existss₀()≥ −logT such that

i) ∀s≥s₀, ∀a∈^RN,

kw_a(s)kL^∞ ≤ κ+ (^{N κ}_2p +)¹_s, k∇w_a(s)kL^∞ ≤ ^√C1_s, k∇²w_a(s)kL^∞ ≤ ^C_s², k∇³w_a(s)kL^∞ ≤ _s^C3/2³ , where κ= (p−1)^−p−11 ,

ii) ∀t≥T −e^−s0,

ku(t)kL^∞ ≤ κ+ (^{N κ}_2p +)_|log(T¹_−t)|(T −t)^−p−11 , k∇ⁱu(t)kL^∞ ≤ C_i^(T−t)−(

p−1 +1 i 2 )

|log(T−t)|^i/2

for i= 1,2,3.

Remark: Ifv :^RN → ^Ris regular, ∇ⁱv stands for the differential of order i of v. For all y ∈ ^RN, we define |∇v(y)|² =

N

X

j=1

(∂_jv(y))², |∇²v(y)| =

sup

z∈^RN

z^T∇²v(y)z

|z|² and |∇³v(y)|= sup

α,β,γ∈^RN

X

i,j,k

αi

|α| βj

|β| γ_k

|γ|∂_i,j,k³ v(y) . In addition,kvk^L∞ = sup

y∈^RN|v(y)| and k∇ⁱvk^L∞ = sup

y∈^RN|∇ⁱv(y)|.

In fact, we can see from the proof of Theorem 1 that s₀() depends only on the size of initial data. We have the following result:

Theorem 1’ (Compactness)Consider (u_n)_n∈N a sequence of nonnegative solutions of equation (1) such that for someT >0 and for all n∈^N, u_n is defined on [0, T) and blows-up at timeT. Assume also that ku_n(0)kH²(^RN)

is bounded uniformly in n. Then, ∀ > 0, there exists t₀() < T such that

∀t∈[t₀(), T), ∀n∈^N,

ku_n(t)kL^∞ ≤ κ+ (^{N κ}_2p +)_|log(T¹_−t)|(T −t)^−p−11 , k∇ⁱu_n(t)kL^∞ ≤ C_i^(T−t)−(

p−11 +i 2)

|log(T−t)|ⁱ²

where C_i are defined in Theorem 1.

Remark: In the case N = 1, Herrero and Vel´azquez [12] (Filippas and Kohn [6] also) prove some estimates related to Theorem 1, using a Sturm

property first used by Chen and Matano [4] (the space oscillations number is a decreasing function of time).

Remark: The constant ^{N κ}_2p appearing in the term of order one in the esti- mates on kw(s)kL^∞ and ku(t)kL^∞ is optimal. Indeed, there exist solutions of equation (3) such that kw(s)kL^∞ = κ+ ^{N κ}_2ps +o¹_s as s → +∞ (see Bricmont and Kupiainen [3], Filippas and Kohn [6], Merle and Zaag [17]).

Remark: From the local (in time) regularity of the solution to the Cauchy problem, we can obtain with the same proof an analogous compactness result when the blow-up times T_n are not the same. The assumptions are ku_n(0)kH²(^RN)+T_n is bounded uniformly in n. The conclusion is there is t⁰₀() such that∀n∈^N, ∀t∈[T_n−t⁰₀(), T_n), the inequalities hold.

Remark: Other compactness results can be shown considering for example equations of the type:

∂u

∂t = ∆u+b(x)u^p whereb∈C³(^RN) (see [16]).

These estimates are in fact crucial for the understanding of the solution at blow-up, especially, the shape of the singularity. Let us recall some results on this question.

Let us consider x₀ ∈ ^RN a blow-up point of u(t), a solution of (1), that is a pointx₀ ∈^RN such that there exists (x_n, t_n) →(x₀, T) such that u(x_n, t_n)→+∞asn→+∞. The question is to see whetheru(t) (orw_x0(s) defined in (2)) has a universal behavior ast→T (ors→+∞).

First, Giga and Kohn prove in [10] and [11] (see also [9]) that for a given blow-up point x₀ ∈^RN,

s→lim+∞w_x0(y, s) = lim

t→T(T −t)^p−11 u(x₀+y√

T−t, t) =κ (7) uniformly on compact subsets of^RN. The result is pointwise inx0. Besides, for a.e. y, lim

s→+∞∇w_x0(y, s) = 0.

Filippas and Liu [7] (see also Filippas and Kohn [6]) and Vel´azquez [18], [19]

(see also Herrero and Vel´azquez [12], [14]) classify the behavior of w(y, s) (= wx0(y, s)) for |y| bounded. They prove that one of the following cases occurs:

- Case 1: non degenerate rate of blow-up:

there exists k ∈ {0,1, ..., N −1} and a N ×N orthonormal matrixQ such

that

∀R >0, sup

|y|≤R

w_x0(y, s)−

κ+ κ 2ps

(N−k)−1 2y^TA_ky

=O

1 s^1+δ

(8) ass→+∞where δ >0,

A_k=Q I_N−k 0

0 0

!

Q⁻¹ (9)

andI_N−k is the (N−k)×(N−k) identity matrix,

- Case 2: degenerate rate of blow-up: ∀R > 0, sup

|y|≤R|w(y, s) −κ| ≤ C(R)e^−0s for some ₀>0.

This yields a blow-up behavior classification in a small range scale. In some sense and from a physical point of view, these results do not show the transition between the singular zone (w ≥ α where α > 0) and the regular one (w'0) well.

Using the renormalization theory, Bricmont and Kupiainen showed in [3]

the existence of a solution of (3) such that

∀s≥s0, ∀y∈^RN,

w(y, s)−f0

y

√s

≤ C

√s (10) where f0(z) = p−1 + ^(p−_4p¹⁾²|z|²⁻

1

p−1 (see also [1]). We show in [17] the same result through a reduction to a finite dimensional problem. We also obtained there a stability result of this behavior with respect to initial data.

This gives a result in an intermediate scalez= √^ys, which is more satisfactory since it separates the blow-up region (w > α > 0) and non-blow-up ones (w'0).

In [20], the second author showed that the behavior in the initial variablex is known in the case where (10) occurs. More precisely, u(x, t) → u^∗(x) as t→T uniformly on compact sets of^RN\{0} and

u^∗(x)∼

8p|log|x||

(p−1)²|x|² _p−1¹

asx→0. (11)

Therefore, except in the small range variable (which does not precise from a physical or analytical point of view the singular behavior), no result of classification was known.

In a first step, we use the estimates of Theorem 1 on∇w and∇²win a crucial way, and the results of Filippas and Liu, and Vel´azquez concerning

the classification of blow-up behaviors for|y|bounded to establish a blow-up profile classification theorem in the variablez= √^ys (which is the intermedi- ate scale that separates the regular and singular parts in the non degenerate case):

Theorem 2 (Existence of a blow-up profile in the intermediate scale for solutions of (1))

Let u(t) be a solution of (1) which blows-up at time T > 0 and satisfies u(0) ∈ H¹(^RN). Let x₀ be a blow-up point of u(t). Then, there exist k∈ {0,1, .., N} and an orthonormal N×N matrix Q such that

∀K₀ >0, sup

|z|≤K0

|w_x0(z√

s, s)−f_k(z)| →0 as s→+∞, (12) where

f_k(z) = p−1 +(p−1)² 4p z^TA_kz

!₋ ¹

p−1

(13) andA_k is defined in (9).

Remark: Vel´azquez in [19] obtained a related profile existence result. He extended the |y| bounded convergence of [18] to the larger set |y| ≤K₀√

s, by estimating the effect of the convective term −¹₂y.∇w in the equation (3), inL^p spaces with a Gaussian measure. However, the convergence that he obtains depends strongly on the considered blow-up point x₀. Let us point out that the convergence we have in Theorem 2 can be shown to be independent of x₀. Indeed, by using the uniform estimates of Theorem 1, we can give a uniform version of the result of [7] and [18], and obtain thanks to our techniques a convergence independent ofx₀ in Theorem 2. However, we use the result of [7] and [18] in this paper, since this shortens the proof.

We also notice that the proof yields that if the case (12) occurs, then (8) occurs with the same A_k (if k=N, then takeA_N = 0) and conversely. See also Theorem 3.

Remark: In the casek =N, this theorem yieldsκas asymptotic “profile” of w(s) in the variablez= √^y

s: this is a degenerate blow-up behavior. Indeed, in this case, the scale √^ys is not good for describing the blow-up behavior.

One must refine this scale and exhibit other blow-up profiles in different scalesy'exp^hk_2k⁻¹sⁱfork= 2,3, ...(see for instance [3], [18]). However, we suspect these profiles to be unstable with respect to initial data.

One interesting problem that follows from Theorem 2 is to find a re- lationship between the different notions of profile in the scales: |y| ≤ C,

z= √^|ys^| ≤C and |x−x0| small. We show in the following theorem that all these descriptions are equivalent in the case of a solution u(t) of (1) that blows-up at some pointx₀ ∈^RN in a non degenerate way (which is supposed to be the generic case):

k= 0 and A_k =I_N.

This answers many questions which were underlined on this problem in preceding works.

Theorem 3 (Equivalence of different notions of blow-up profiles) Let x₀ ∈ ^RN be an isolated blow-up point of u(t) solution of (1) such that u₀ ∈ H¹(^RN). The following blow-up behaviors of u(t) near x₀ or w(s) =w_x0(s) (defined in (2)) are equivalent:

(A) ∀R > 0, sup

|y|≤R

w(y, s)−

κ+ κ

2ps(N −1 2|y|²)

= o

1 s

as s → +∞ where κ= (p−1)^−p−11 ,

(B) ∃₀ >0 such that w(y, s)−f₀(√^y s)

L^∞(|y|≤0e^s/2) → 0 as s→+∞ with f₀(z) = (p−1 +^(p−_4p¹⁾²|z|²)^−p−11 ,

(C) ∃₀ >0 such that if |x−x₀| < ₀, then u(x, t) → u^∗(x) as t → T andu^∗(x)∼^h_(p^8p₋^|₁₎^log2_|^|_x^x₋⁻_x^x₀⁰_|^|2i

1

p−1 as x→x₀.

Remark: In [19], Vel´azquez shows that (A) =⇒ (B) =⇒(C) by estimating the local effect to the term −¹₂y.∇w in equation (3) in L^p with Gaussian measure. The classification of [19] also yields that (C) =⇒(A). Let us point that the estimates in our proof are quite elementary and rely on localization effect and uniform estimates. In addition, one can show from our proof and our uniform techniques that the convergence speeds in (A), (B) and (C) depend only on each other and on a bound on theC² norm of initial data ( and not on the initial data itself).

Remark: In fact, (A) (or (B) or (C)) imply that x₀ is an isolated blow- up point. It is conjectured that the equivalence holds (in the case of the (supposed to be) generic blow-up rate).

Remark: The techniques we introduce in the proof of Theorem 3 allow us to obtain the same results as Vel´azquez in the case where (8) occurs with k < N.

Section 2 is devoted to the proof of the uniform estimates on w (The- orems 1 and 1’). Section 3 deals with results on profiles (Theorems 2 and 3).

2 L^∞ estimates of order one for solutions of (3)

2.1 Formulation and reduction of the problem

We prove Theorems 1 and 1’ in this section. Let us first show Theorem 1.

Theorem 1’ follows from similar arguments.

Proof of Theorem 1: We consider u(t) a blow-up solution of (1) which blows-up at timeT >0.

We can assume from regularizing effect of the heat flow that T < 1, u₀ ∈C³(^RN)∩H¹(^RN). We are interested in findingL^∞estimates of order one forw₀(=w) defined in (2). In [16], we have already provedL^∞estimates of order zero forwstated in (5). Note that with obvious simple adaptations of the proof of (5), we have the following result:

kw(s)kL^∞ →κ and k∇w(s)kL^∞+k∇²w(s)kL^∞+k∇³w(s)kL^∞ →0 (14) ass→+∞.

We now want to refine the estimates (14). More precisely, we want to show that there exist positive constantsC₁, C₂ and C₃ depending only on psuch that ∀ >0,∃s₀() such that∀s≥s₀(),

kw(s)k^L∞ ≤ κ+ (^{N κ}_2p + (N+ 1))¹_s, k∇w(s)k^L∞ ≤ ^C√1_s

k∇²w(s)kL^∞ ≤ ^C_s², k∇³w(s)kL^∞ ≤ _s^C3/2³ . (15) For this purpose, we take an arbitrary ∈ (0, ₀) (where ₀ ≤ 1 is small enough) that we consider as fixed now, and introduce the following definitions:

Definition 2.1 For all A > 0 and s ≥ −logT, we define V_A(s) as being the set of all w∈W^3,∞(^RN) satisfying:

kwkL^∞ ≤ κ+^c_s⁰, k∇wkL^∞ ≤ ^√c1_s k∇²wk^L∞ ≤ ^A_s, k∇³wk^L∞ ≤ ^A_s3/2^5/4. and

∀a∈^RN, −c2

sI_N ≤ Z

RN∇²w(y+a)ρ(y)dy

in the sense of symmetricN ×N matrices, where the norms are introduced in the remark after Theorem 1,

c₀() = N κ

2p + (N+ 1), c₁() = κ

√p + 2√p, c₂() = κ

2p +, (16)

I_N is theN ×N identity matrix and ρ(y) = e^−|y|

2 4

(4π)^N/2. (17) Definition 2.2 For alls≥ −logT, we define

Vˆ_A(s) ={w∈C([−logT, s), W^3,∞(^RN)) | ∀τ ∈[−logT, s), w(τ)∈V_A(τ)}. Let us remark that condition (16) is in some sense a lower bound on∇²w(a).

Indeed, ifw∈VA(s), then we have ∀a∈^RN,

| Z

RN∇²w(y+a)ρ(y)dy− ∇²w(a)| ≤C^∗(N)k∇³wkL^∞ (18)

and A

sI_N ≥ ∇²w(a)≥ −

"

c₂

s +C^∗(N)A^5/4 s^3/2

#

I_N (19)

whereC^∗(N) =^R |y|ρ(y)dy.

Proof of (18) and (19): Using a Taylor expansion, we have: ∀y ∈ ^RN,

∇²w(y+a)− ∇²w(a) =^R₀¹∇³w(a+ty)(y)dt. Hence,

|∇²w(y+a)− ∇²w(a)| ≤ |y|k∇³wkL^∞ ≤ |y|^A_s3/2^5/4. This yields (18) and (19) by integration (use^R ρ(y)dy= 1).

Notice that the lower bound on ∇²w(a) is (consider the order ¹_s) inde- pendent ofA, which will be crucial in the proof.

Theorem 1 is in fact a consequence of the following proposition:

Proposition 2.1 (Reduction) There exist A(p) > 0 and ₀(p) ∈ (0,1) such that for all∈(0, 0), there existsS(A, )so that the following property is true:

Assume that w is a solution of (3) defined for all time s ≥ −logT and satisfying w(−logT) ∈ H¹(^RN). Assume in addition that w ∈ VˆA(ˆs) for somesˆ≥S(A, ), then:

i)w(ˆs)6∈∂V_A(ˆs),

ii)∀s≥ −logT, w(s)∈VˆA(s).

Proposition 2.1 implies Theorem 1:

Let ∈ (0, ₀), A = A(p) and S(A, ) defined in Proposition 2.1. Our strategy is to findn₀() =n₀ ∈^Nsuch that∀s≥ −logT,w(s+n₀)∈V_A(s).

Indeed, one can easily check the following result:

Lemma 2.1 Assume for all ∈ (0,1), there exists n₀() ∈ ^N such that

∀s≥ −logT, w(s+n₀)∈V_A(s). Then, (15) is satisfied with C₁ = κ

2p + 4√p, C₂ = 2A(p) and C₃ = 2A(p)^5/4.

Let us considerW =w(.+n). Then, W satisfies (3) for all s≥ −logT and W(−logT) = w(n−logT) ∈ H¹(^RN) from the solving of the initial value problem forw.

We claim the following: for n large, we have w(.+n) ∈ Vˆ_A(S(A, )).

Indeed, let

δ= 1

4(1 +C^∗(N))min c₀

S(A, ), c₁

pS(A, ), c₂

S(A, ), A

S(A, ), A^5/4 S(A, )^3/2

!

(20) where C^∗(N) is defined in (19). (14) implies that there exists n₀ ∈ ^N such that∀n≥n₀, ∀s∈[−logT, S(A, )], kw(s+n)kL^∞ ≤κ+δ ≤κ+^c_4s⁰, k∇w(s+n)kL^∞ ≤δ≤ ₄^c√1_s,k∇²w(s+n)kL^∞ ≤δ≤ _4s^A andk∇³w(s+n)kL^∞ ≤ δ≤ _4s^A5/43/2.

Let s ∈ [−logT, S(A, )] and a ∈ ^RN. According to (18), we have R

RN∇²w(y+a, s+n)ρ(y)dy

≥ − |∇²w(a, s+n)|+C^∗(N)k∇³w(s+n)kL^∞

I_N ≥ −(δ+C^∗(N)δ)I_N ≥

−_4s^c2₂I_N. Thus, w(.+n₀) ∈ Vˆ_A(S(A, )). Applying Proposition 2.1, we see fromii) that

∀s∈[−logT,+∞), w(s+n₀)∈V_A(s).

This concludes the proof of Theorem 1.

Proof of Theorem 1’:

For all n ∈ ^N, we introduce wn = wn,0 defined from un by (2). Then, by simple obvious adaptations of the proof of Theorem 1’ in [16], we claim that sup

n∈^Nkw_n(s)kL^∞ →κand sup

n∈^Nk∇ⁱw_n(s)kL^∞ →0 ass→+∞ fori= 1,2 and 3.

Hence, there exists n₀ ∈ ^N such that ∀n ∈ ^N, ∀s ∈ [−logT, S(A, )], kw_n(s+n₀)kL^∞ ≤ κ+δ and k∇ⁱw_n(s+n₀)kL^∞ ≤ δ for i = 1,2,3 where δ is defined in (20). Hence, as for the proof of Theorem 1, we get ∀n∈^N, w_n(.+n₀)∈Vˆ_A(S(A, )). Thus,

∀n∈^N, ∀s∈[−logT,+∞), w_n(s+n₀)∈V_A(s) byii) of Proposition (2.1). This concludes the proof of Theorem 1’.

Therefore, the question reduces to prove Proposition 2.1.

Proof of Proposition 2.1:

i) =⇒ ii): By contradiction, we assume that there exists s ≥ −logT such that w(s) 6∈ V_A(s). Let s⁰ be the lowest s satisfying this. Then, s⁰ ≥sˆ≥S(A, ), w∈VˆA(s⁰) andw(s⁰)∈∂VA(s⁰). This contradicts i).

Proof of i): Let us argue by contradiction. We suppose that for all A > 0, there is a sequence s_n → +∞ and a solution of (3) w_n defined for all s ≥ −logT such that w_n(−logT) ∈ H¹(^RN), ∀s ∈ [−logT, s_n], w_n(s)∈V_A(s) and w_n(s_n)∈∂V_A(s_n).

Let us denote w_n by w to simplify the notations. We claim the following Proposition 2.2 (Characterization of ∂V_A(s_n)) There exists y_n ∈ ^RN such that one of the following cases must occur:

Case 1: w(y_n, s_n) =κ+ ^c_s⁰

n, Case 2: |∇w(y_n, s_n)|= √^cs¹n,

Case 3: there exists a unitary ϕ_n∈^RN such that ϕ^T_n^RRN∇²w(y+y_n, s_n)ρ(y)dyϕ_n=−_s^c_n²,

Case 4: |∇²w(y_n, s_n)|= _s^A

n, Case 5: |∇³w(y_n, s_n)|= ^A5/4

s^3/2_n . Proof:

Let us remark that sincew(−logT)∈H¹(^RN), we can assume from the regularizing effect of the heat flow thatw(−logT, y)→0 and

∇ⁱw(−logT, y) → 0 as |y| → +∞ for i = 1,2 and 3. Hence, we have by classical estimatesw(y, s)→0 and∇ⁱw(y, s)→0 as|y| →+∞uniformly in s∈[s_n, s_n+ 1]. Hence, by Lebesgue’s Theorem, ^R∇²w(y+a, s)ρ(y)dy→0 as|a| →+∞.

This insures that one of the five cases of Proposition 2.2 occurs.

We now use the classification of Proposition 2.2 and consider in the following subsection all the five cases in order to reach a contradiction.

Let us notice that we reduce to the case y_n= 0.

Indeed, from (2) and the translation invariance of (1), we define for all y∈^RN and s≥ −logT:

W(y, s) =w(y+y_ne^s−sn2 , s). (21) We still have:

- W is solution of (3) defined fors∈[−logT,+∞), - W(s)∈V_A(s) for alls∈[−logT, s_n],

- W(s_n)∈∂V_A(s_n).

We will denote W by wand ϕ_n by ϕ.

We now claim that there exist ₀(p)>0 and A(p)>0 such that for all ∈ (0, ₀), there is S(A, ) such that all the cases 1, 2, 3, 4 and 5 do not occur ifsn≥S(A, ), which will conclude the proof of Proposition 2.1.