A Proof of Proposition 3.2

Optimal estimates for blow-up rate and behavior for nonlinear heat equations

Frank Merle

Institute for Advanced Study and Universit´e de Cergy-Pontoise

Hatem Zaag

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

Abstract: We first describe all positive bounded solutions of

∂w

∂s = ∆w−1

2y.∇w− w

p−1+w^p.

where (y, s) ∈^RN ×^R, 1 < p and (N −2)p ≤N + 2. We then obtain for blow-up solutionsu(t) of

∂u

∂t = ∆u+u^p

uniform estimates at the blow-up time and uniform space-time comparison with solutions ofu⁰ =u^p.

1 Introduction

We consider the following nonlinear heat equation:

∂u

∂t = ∆u+|u|^p−1u in Ω×[0, T)

u = 0 on ∂Ω×[0, T) (1)

whereu(t)∈H¹(Ω) and Ω =^RN (or Ω is a convex domain).

We assume in addition that

1< p, (N−2)p < N + 2 and u(0)≥0.

In this paper, we are interested in blow-up solutionsu(t) of equation (1):

u(t) blows-up in finite time T if u exists for t∈[0, T) and lim

t→Tku(t)kH¹ = +∞. In this case, one can show thatu has at least one blow-up point, that is a∈ Ω such that there exists (an, tn)n∈^N satisfying (an, tn) → (a, T) and

|u(a_n, t_n)| → +∞. We aim in this work at studying the blow-up behavior of u(t). In particular, we are interested in obtaining uniform estimates on u(t) at or near the singularity, that is estimates “basically” independent of initial data.

We will give two types of uniform estimates: the first one holds especially at the singular set (Theorem 1) and the other one consists in surprising global estimates in space and time (Theorem 3). It will be deduced from the former by some strong control of the interaction between regular and singular parts of the solution. Various applications of this type of estimates will be given in [12].

For the first type of estimates, we introduce for eacha∈Ω (amay be a blow-up point ofu or not) the following similarity variables:

y = √^x−a T−t

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

(2) w_a (=w) satisfies ∀s≥ −logT, ∀y∈D_a,s:

∂w

∂s = ∆w−1

2y.∇w− w

p−1 +|w|^p−1w (3) where

D_a,s ={y∈^RN | a+ye^−s/2 ∈Ω}. (4) We introduce also the following Lyapunov functional:

E(w) = 1 2

Z

|∇w|²ρdy+ 1 2(p−1)

Z

|w|²ρdy− 1 p+ 1

Z

|w|^p+1ρdy (5) whereρ(y) = e^−|y|2/4

(4π)^N/2 (6)

and the integration is done over the definition set ofw.

The study of u(t) near (a, T) where a is a blow-up point is equivalent to the study of the long time behavior of w_a. Note that D_a,s 6=^RN in the case Ω 6= ^RN. This in fact is not a problem since we know from [8] that a6∈ ∂Ω in the case Ω is C^2,α, and therefore, for a given a∈Ω, D_a,s → ^RN ass→+∞. Let a∈Ω be a blow-up point ofu.

If Ω is a bounded convex domain in^RN or Ω =^RN, then Giga and Kohn prove in [7] that:

∀s≥ −logT, kw_a(y, s)kL^∞(Da,s) ≤ C or equivalently

∀t∈[0, T), ku(x, t)kL^∞(Ω) ≤ C(T−t)^−p−11 . (7)

They also prove in [7] and [8] (see also [6]) that for a given blow-up point a∈Ω,

s→lim+∞w_a(y, s) = lim

t→T(T−t)^p−11 u(a+y√

T−t, t) =κ

whereκ= (p−1)^−p−11 , uniformly on compact subsets of^RN. The result is pointwise ina. Besides, for a.ey, lim

s→+∞∇w_a(y, s) = 0.

Let us denote L^∞(D_a,s) byL^∞.

In this paper, we first obtain uniform (on a and in some sense on u(0)) sharp estimates on wa, and we find a precise long time behavior for kw_a(s)kL^∞, k∇w_a(s)kL^∞ and k∆w_a(s)kL^∞ (global estimates).

Theorem 1 (Optimal bound onu(t) at blow-up time)Assume that Ω is a convex bounded C^2,α domain in ^RN or Ω =^RN. Consider u(t) a blow- up solution of equation (1) which blows-up at time T. Assume in addition u(0)≥0 andu(0)∈H¹(Ω). Then

(T −t)^p−11ku(t)kL^∞(Ω)→κ= (p−1)^−p−11 as t→T and

(T −t)^{p−11 +1}k∆u(t)kL^∞(Ω)+ (T−t)^{p−11 +12}k∇u(t)kL^∞(Ω)→0 as t→T, or equivalently for anya∈Ω,

kw_a(s)kL^∞ →κ as s→+∞ and

k∆wa(s)k^L∞ +k∇wa(s)k^L∞ →0 as s→+∞.

Remark: We can point out that we do not consider local norm inwvariable such asL²(dµ) withdµ=e^−|y|2/4dyas a center manifold theory for equation (3) would suggest. Instead, we use L^∞ norm which yields results uniform with respect to∈Ω. Indeed, we have from (2) that∀a, b∈Ω,∀(y, s)∈D_b,s,

w_b(y, s) =w_a(y+ (b−a)e^s2, s),

which yields kwak^L∞ = kw_bk^L∞, k∇wak^L∞ = k∇w_bk^L∞ and k∆wak^L∞ = k∆w_bkL^∞).

One interest of Theorem 1 is that in fact, its proof yields the following compactness result:

Theorem 1’ (Compactness of blow-up solutions of (1))Assume that Ωis a convex bounded C^2,α domain in ^RN or Ω =^RN. Consider (u_n)_n∈N a sequence of nonnegative solutions of equation (1) such that for some T >0 and for all n∈^N, u_n is defined on [0, T) and blows-up at time T. Assume also thatku_n(0)kH²(Ω) is bounded uniformly inn. Then

sup

n∈^N(T −t)^p−11ku_n(t)kL^∞(Ω)→κ as t→T and

sup

n∈^N

(T−t)^p−11+1k∆u_n(t)kL^∞(Ω)+ (T −t)^p−11+12k∇u_n(t)kL^∞(Ω)

→0 ast→T.

Remark: The same results can be proved for the following heat equation:

∂u

∂t =∇.(a(x)∇u) +b(x)f(u), u(0)≥0

wheref(u)∼u^p asu→+∞, (a(x)) is a symmetric, bounded and uniformly elliptic matrix,b(x) is bounded, and a(x) and b(x) areC¹.

Let us point out that this result is optimal. One way to see it is by the following Corollary which improves the local lower bound on the blow-up solution given in [8] by Giga and Kohn.

Corollary 1 (Lower bound on the blow-up behavior for equation (1)) Assume that Ω is a convex bounded C^2,α domain in ^RN or Ω = ^RN. Then for all nonnegative solutionu(t)of (1) such thatu(0)∈H¹(Ω)andu(t) blows-up at time T, and for all ₀ ∈(0,1), there exists t₀ = t₀(₀, u₀) < T such that if for some a∈Ωand some t∈[t₀, T) we have

0≤u(a, t)≤(1−₀)κ(T −t)^−p−11 , (8) then ais not a blow-up point of u(t).

Remark: The result is still true for a sequence of nonnegative solutionsu_n blowing-up atT >0 and satisfying the assumptions of Theorem 1’, with a t₀ independent ofn.

Remark: κ is the optimal constant giving such a result. The result of [8]

was the same except that (1−₀)κ was replaced by ₀ small and it was required that (8) is true for all (x, t) ∈B(a, r)×[T−r², T) for somer >0 (no sign condition was required there).

The proof of Theorem 1 relies strongly on the caracterization of all con- nections between two critical points of equation (3) in L^∞_loc. Due to [6], the only bounded global nonnegative solutions of the stationary problem asso- ciated to (3) in^RN are 0 andκ, provided that (N −2)p≤N + 2. Here we classify the solutionsw(y, s) of (3) defined on^RN×^Rand connecting two of the cited critical points between them, and we obtain the surprising result:

Theorem 2 (Classification of connections between critical points of (3)) Assume that 1 < p and (N −2)p < N + 2 and that w is a global nonnegative solution of (3) defined for(y, s)∈^RN×^Rbounded inL^∞. Then necessarily one of the following cases occurs:

i) w≡0 or w≡κ,

or ii) there exists s₀ ∈ ^R such that ∀(y, s) ∈ ^RN ×^R, w(y, s) = ϕ(s−s₀) where

ϕ(s) =κ(1 +e^s)^−p−11 . (9) Note thatϕ is the unique global solution (up to a translation) of

ϕ_s =− ϕ

p−1+ϕ^p

satisfying ϕ→κ as s→ −∞ and ϕ→0 as s→+∞.

Remark: This result is in the same spirit as the result of Berestycki and Nirenberg [1], and Gidas, Ni and Nirenberg [5]. Here, the moving plane technique is not used, even though the proof uses some elementary geomet- rical transformations. It is unclear whether the result holds without a sign condition or not. The assumptionw is bounded in L^∞ and is defined fors up to +∞ is not really needed, in the following sense:

Corollary 2 Assume that 1 < p and (N −2)p < N + 2 and that w a nonnegative solution of (3) defined for (y, s) ∈ ^RN ×(−∞, s^∗) where s^∗ is finite ors^∗ = +∞. Assume in addition that there is a constant C₀ such that

∀a∈^RN, ∀s≤s^∗,Ea(w(s))≤C0, where

E_a(w(s)) =E(w(.+ae^2s, s)) (10) andE is defined in (5). Then, one of the following cases occurs:

i) w≡0 or w≡κ,

or ii)∃s₀∈^Rsuch that ∀(y, s)∈^RN ×(−∞, s^∗), w(y, s) =ϕ(s−s₀) where ϕ(s) =κ(1 +e^s)^−p−11,

or iii)∃s₀≥s^∗ such that∀(y, s)∈^RN×(−∞, s^∗),w(y, s) =ψ(s−s₀)where ψ(s) =κ(1−e^s)^−p−11 .

Theorem 2 has an equivalent formulation for solutions of (1):

Corollary 3 (A Liouville theorem for equation (1))Assume that1<

p and (N −2)p < N+ 2 and that u is a nonnegative solution inL^∞ of (1) defined for (x, t) ∈ ^RN ×(−∞, T). Assume in addition that 0 ≤ u(x, t) ≤ C(T −t)^−p−11 . Then u ≡ 0 or there exist T₀ ≥ T such that ∀(x, t) ∈ R^N ×(−∞, T), u(x, t) =κ(T0−t)^−p−11.

Remark: u≡0 oru blows-up in finite time T₀ ≥T.

The third main result of the paper shows that near blow-up time, the solutions of equation (1) behave globally in space like the solutions of the associated ODE:

Theorem 3 Assume that Ω is a convex bounded C^2,α domain in ^RN or Ω =^RN. Consider u(t) a nonnegative solution of equation (1) which blows- up at time T > 0. Assume in addition that u(0) ∈ H¹(^RN) if Ω = ^RN. Then∀ >0, ∃C>0 such that ∀t∈[^T₂, T), ∀x∈Ω,

|∂u

∂t − |u|^p−1u| ≤|u|^p+C. (11) Remark: (11) is true until the blow-up time. Let us point out that the result is global in time and in space. The same result holds for a sequence u_n as before (Theorem 1’). For clear reasons, the result is optimal.

Remark: Let us note that the result is still true for equation

∂u

∂t =∇.(a(x)∇u) +b(x)f(u)

wheref(u)∼u^p asu→+∞, (a(x)) is a symmetric, bounded and uniformly elliptic matrix,b(x) is bounded, and a(x) and b(x) areC¹.

The conclusion in this case is

|∂u

∂t −b(x)f(u)| ≤|f(u)|+C.

It is unclear whether Theorems 1, 2 and 3 hold without a sign condition.

Remark: u⁰ = u^p is a reversible equation. Therefore the non reversible equation behaves like a reversible equation near and at the blow-up time.

Theorem 3 localizes the equation. In particular, it shows that the interac- tions between two singularities or one singularity and the “regular” region are bounded up to the blow-up time.

Note that Theorem 3 has obvious corollaries. For example:

Ifx₀ is a blow-up point, then

- u(x, t) → +∞ as (x, t) → (x0, T) (In other words, u is a continuous function in ¯^Rof (x, t)∈Ω×(0, T)).

- ∃₀ > 0 such that for all x ∈ B(x₀, ₀) and t ∈ (T −₀, T), we have

∂u

∂t(x, t)>0.

Let us notice that theorems 1 and 3 have interesting applications in the understanding of the asymptotic behavior of blow-up solutions u(t) of (1) near a given blow-up pointx0. Various points of view has been adopted in the literature ([8], [2], [9], [14]) to describe this behavior. In [12], we sharpen these estimates and put them in a relation.

In the second section, we see how Theorems 1 and 3 are proved using Theorem 2. The third section is devoted to the proof of Theorem 2.

2 Optimal blow-up estimates for equation (1)

In this section, we assume that Theorem 2 holds and prove Theorems 1 and 1’, Corollary 1 and Theorem 3. The first three are mainly a consequence of compactness procedure and Theorem 2. Theorem 3 follows from Theorem 1 and scaling properties of equation (1) used in a suitable way.

2.1 L∞ estimates for the solution of (1)

We prove Theorems 1 and 1’ and Corollary 1 in this subsection.

Proof of Theorem 1: Let u(t) be a nonnegative solution of equation (1) defined on [0, T), which blows-up at timeT and satisfies u(0)∈H¹(Ω). It is clear that the estimates onwa for all a∈Ω follow from the estimates onu by (2). In addition, the estimates onu follow from the estimates onw_a for a particulara∈Ω still by (2). Hence, we considera∈Ω a blow-up point of u and prove the estimates on this particularwa defined by

w_a(y, s) =e^−p−1s u(a+ye^−s2, T −e^−s).

Note that we have∀a, b∈Ω, ∀(y, s)∈D_b,s,

w_b(y, s) =w_a(y+ (b−a)e^s2, s).

We proceed in three steps: in a first step, we show thatw_a, ∇w_a and ∆w_a are uniformly bounded (without any precision on the bounds). Then, we show in Step 2 that blow-up for equation (1) must occur inside a compact set K ⊂ Ω and that u, ∇u and ∆u are bounded in Ω\K. We finally find the optimal bounds onw_a through a contradiction argument.

Let us recall the expression of the energyE(w) introduced in (5), since it will be useful for further estimates:

E(w_a) = 1 2

Z

|∇w_a|²ρdy+ 1 2(p−1)

Z

|w_a|²ρdy− 1 p+ 1

Z

|w_a|^p+1ρdy (12) whereρ is defined in (6) and integration is done over the definition set ofw.

By means of the transformation (2), (12) yields a local energy for equation (1):

Ea,t(u) = t^{p−12 −N2+1} Z 1

2|∇u(x)|²− 1

p+ 1|u(x)|^p+1

ρ(x−a

√t )dx

+ 1

2(p−1)t^{p−12 −N2} Z

|u(x)|²ρ(x−a

√t )dx. (13) Without loss of generality, we can supposea= 0. We recall that the notation L^∞ stands forL^∞(D_0,s).

Step 1: Preliminary estimates on w

Lemma 2.1 (Giga-Kohn, Uniform estimates on w) There exists a positive constant M such that ∀s≥ −logT + 1, ∀y ∈D_0,s,

|w₀(y, s)|+|∇w₀(y, s)|+|∆w₀(y, s)|+|∇∆w₀(t, s)| ≤M and |∂w

∂s(y, s)| ≤M(1 +|y|).

Let us recall the main steps of the proof:

Since u(0) ≥ 0, we know from Giga and Kohn [8] that there exists B > 0 such that

∀t∈[0, T), ∀x∈Ω, |u(x, t)| ≤B(T−t)^−p−11 . (14) In order to prove this, they argue by contradiction and construct by scaling properties of equation (3) a solution of







0 = ∆v+v^p in^RN v ≥ 0

v(0) ≥ ¹₂

which does not exist if (N −2)p < N+ 2 and p >1.

The estimate on w₀ is equivalent to (14).

For s₀ ≥ −logT + 1 and y₀ ∈ D_0,s0, consider W(y⁰, s⁰) = w₀(y⁰ + y₀e^s2, s₀+s⁰). Then W(0,0) =w₀(y₀, s₀) andW satisfies also (3). Ify₀e^−s20 (which is in Ω) is not near the boundary, then we have|W(y⁰, s⁰)| ≤M for all (y⁰, s⁰)∈B(0,1)×[−1,1]. By parabolic regularity (see lemma 3.3 in [7]

for a statement), we obtain|∇W(0,0)|+|∆W(0,0)|+|∇∆W(0,0)| ≤M⁰ = M⁰(M). Ify₀e^−s02 is near the boundary, then lemma 3.4 in [7] allows to get the same conclusion. Since this is true for all (y0, s0), we have the bound for∇w₀, ∆w₀ and ∇∆w₀.

The estimate on ^∂w_∂s⁰ follows then by equation (3).

Step 2: No blow-up for u outside a compact

Proposition 2.1 (Uniform boundedness of u(x, t) outside a com- pact) Assume that Ω = ^RN and u(0) ∈ H¹(^RN), or that Ω is a convex boundedC^2,α domain. Then there existC >0, t₁ < T and K a compact set ofΩsuch that∀t∈[t1, T),∀x∈Ω\K,|u(x, t)|+|∇u(x, t)|+|∆u(x, t)| ≤C.

Proof: Case Ω =^RN and u(0)∈H¹(^RN): Giga and Kohn prove in [8] that uniform estimates onEa,t (13) give uniform estimates inL^∞_locon the solution of (1). More precisely,

Proposition 2.2 (Giga-Kohn) Let u be a solution of equation (1).

i) If for all x ∈ B(x₀, δ), Ex,T−t0(u(t₀)) ≤σ, then ∀x ∈ B(x₀,^δ₂), ∀t ∈ (^t0+T₂ , T), |u(t, x)| ≤η(σ)(T−t)^−p−11 where η(σ)≤cσ^θ,θ >0, and c andθ depend only on p.

ii) Assume in addition that ∀x ∈ B(x0, δ), |u(^t0+T₂ , x)| ≤ M. There exists σ0 = σ0(p) > 0 such that if σ ≤ σ0, then ∀x ∈ B(x0,^δ₄), ∀t ∈ (^t0+T₂ , T), |u(t, x)| ≤M^∗ where M^∗ depends only on M, δ, T andt0. Proof: see Proposition 3.5 and Theorem 2.1 in [8].

Now, since u(0) ∈ H¹(^RN), we have u(t) ∈ H¹(^RN) for all t ∈ [0, T).

Therefore, for fixedt₀and σ≤σ₀, (13), (6) and the dominated convergence theorem yield the existence of a compactK₀⊂^RN such that∀x∈^RN\K₀, Ex,T−t0(u(t₀))≤σ.

Hence, ii) of Proposition 2.2 applied to u(.+x₁, .) for x₁ ∈ K₀ and with δ= 1, asserts the existence of a compact K₁⊂^RN such that∀x∈^RN\K₁,

∀t∈(^t0+T₂ , T),|u(x, t)| ≤M^∗.

Parabolic regularity (see lemma 3.3 in [7] for a statement) implies the estimates on∇u and ∆u on Ω\K with a compactK containingK1.

Case Ωis a bounded convexC^2,α domain: The main feature in the proof of the estimate on|u(x, t)| is the result of Giga and Kohn which asserts that blow-up can not occur at the boundary (Theorem 5.3 in [8]). The bounds on ∇u and ∆u follow from a similar argument as before (see lemma 3.4 in [7]).

Step 3: Conclusion of the proof

The result has been proved pointwise. Therefore, the question is in some sense to prove it uniformly.

We want to prove that kw0(s)k^L∞ →κ ass→+∞.

From [7] and [8], we know that|w_b(0, s)| →κass→+∞ifbis a blow-up point. Sincekw₀(s)kL^∞ ≥ |w₀(ae^s2, s)|=|w_a(0, s)|, this implies that

lim inf

s→+∞kw₀(s)kL^∞ ≥κ and lim inf

s→+∞kw₀(s)kL^∞+k∇w₀(s)kL^∞+k∆w₀(s)kL^∞ ≥κ. (15) The conclusion will follow if we show that

lim sup

s→+∞ kw0(s)k^L∞+k∇w0(s)k^L∞+k∆w0(s)k^L∞ ≤κ. (16) Let us argue by contradiction and suppose that there exists a sequence (s_n)_n∈N such thats_n→+∞ asn→+∞ and

n→lim+∞kw0(sn)k^L∞+k∇w0(sn)k^L∞+k∆w0(sn)k^L∞ =κ+ 30 where0 >0.

We claim that (up to extracting a subsequence), we have either lim

n→+∞kw₀(s_n)kL^∞ = κ+₀

or lim

n→+∞k∇w₀(s_n)kL^∞ = ₀

or lim

n→+∞k∆w₀(s_n)kL^∞ = ₀.

(17)

From Proposition 2.1 and the scaling (2), we deduce fornlarge enough the existence ofyn⁽⁰⁾,y⁽¹⁾n andy⁽²⁾n inD0,sn such that

kw₀(s_n)kL^∞ = |w₀(yn⁽⁰⁾, s_n)|, ork∇w₀(s_n)kL^∞ = |∇w₀(y⁽¹⁾n , s_n)|, ork∆w₀(s_n)kL^∞ = |∆w₀(y⁽²⁾_n , s_n)|.

(18)

Let y_n =y⁽ⁱ⁾n where i is the number of the case which occurs. Since y_n ∈ D_0,sn, (4) implies that y_ne^−sn/2 ∈Ω. Therefore, we can use (2) and define for eachn∈^N

v_n(y, s) = w_yne−sn/2(y, s+s_n)

= e^−s

+sn

p−1 u(ye^−s+2sn +y_ne^−sn/2, T −e^−(s+sn))

= w0(y+yne^s/2, s+sn) (19) We claim that (v_n) is a sequence of solutions of (3) which is compact in C_loc³ (^RN ×^R). More precisely,

Lemma 2.2 (v_n)_n∈N is a sequence of solutions of (3) with the following properties:

i) lim

n→+∞|vn(0,0)| =κ+0 or lim

n→+∞|∇vn(0,0)| =0

or lim

n→+∞|∆v_n(0,0)|=₀.

ii) ∀R >0, ∃n₀ ∈^N such that ∀n≥n₀,

-v_n(y, s) is defined for (y, s)∈B(0, R)¯ ×[−R, R],

-v_n≥0 andkv_nkL^∞( ¯B(0,R)×[−R,R]) ≤B where B is defined in (14).

-∃m(R)>0 such that kv_nkC³( ¯B(0,R)×[−R,R]) ≤m(R).

Proof: i)v_n satisfies (3) sincew_yne−sn/2 does the same. From (19), (17) and (18), we obtain i): lim

n→+∞|vn(0,0)| =κ+0 or lim

n→+∞|∇vn(0,0)|=0

or lim

n→+∞|∆v_n(0,0)|=₀. ii) LetR >0.

If Ω = ^RN, then it is obvious form (19) that v_n is defined for (y, s) ∈ B(0, R)¯ ×[−R, R] for large n.

If Ω is bounded, then we can suppose that up to extracting a subse- quence, y_ne^−sn/2 converges to y_∞ ∈ Ω as¯ n → +∞. In fact y_∞ ∈ Ω.

Indeed, sinceu(yn⁽⁰⁾e^−sn/2, T −e^−sn) =e^p−sn1vn(0,0)→+∞ asn→+∞ (or

|∇u(y⁽¹⁾n e^−sn/2, T−e^−sn)|=e^{sn(p−11 +12)}|∇v_n(0,0)| →+∞, or

|∆u(y⁽²⁾_n e^−sn/2, T −e^−sn)| =e^sn(p−11+1)|∆v_n(0,0)| →+∞), in all cases, y_∞ is a blow-up point of u. Therefore, Step 2 implies that y_∞ ∈ K and that B(y_∞, δ₀) ⊂Ω for some δ₀ >0. Together with (19), this implies that v_n is defined for (y, s)∈B¯(0, R)×[−R, R] for large n.

From (19), (14) and the fact thatu≥0, it directly follows thatv_n(y, s)≥ 0 andkv_nkL^∞( ¯B(0,R)×[−R,R]) ≤B.

From lemma 2.1 and (19), it directly follows that ∀(y, s) ∈ B¯(0, R)× [−R, R],|vn(y, s)|+|∇vn(y, s)|+|∆vn(y, s)|+|∇∆vn(y, s)| ≤M and|^∂v∂sⁿ| ≤

M×(1+R). Sincew≥0, parabolic estimates and strong maximum principle imply that kv_nkC³( ¯B(o,R)×[−R,R]) ≤ m(R) for some m(R) > 0. Just take m(R) =M×(1 +R).

Now, using the compactness property of (v_n) shown in lemma 2.2, we find v ∈C²(^RN ×^R) such that up to extracting a subsequence, v_n →v as n→+∞inC_loc² (^RN ×^R). From lemma 2.2, it directly follows that

i)v satisfies equation (3) for (y, s)∈^RN ×^R ii) v≥0 and kvkL^∞(^RN×^R) ≤B

iii) |v(0,0)| =κ+0 or|∇v(0,0)| =0 or|∆v(0,0)| =0 with 0 >0.

By Theorem 2, i) andii) imply v≡0 orv ≡κ orv =ϕ(s−s₀) where ϕ(s) =κ(1 +e^s)^−p−11 . In all cases, this contradictsiii). Thus, Theorem 1 is proved.

Proof of Theorem 1’: The proof of Theorem 1’ is similar to the proof of Theorem 1. Let us sketch the main differences.

Step 1: One can remark that a uniform estimate onE(w_n,a(s₀)) where s0 =−logT is needed. Since ku0kH²(Ω) is uniformly bounded, we have the conclusion.

Step 2: One can use a uniform version of Giga and Kohn’s estimates, as they are stated (for example) in [11].

Step 3: Same proof.

Proof of Corollary 1: Let us prove Corollary 1 now. We argue by con- tradiction and assume that for some ₀ > 0, there is t_n → T and (a_n)_n a sequence of blow-up points ofu in Ω such that

∀n∈^N, 0≤u(an, tn)≤(1−0)κ(T−tn)^−p−11. Let us give two different proofs:

Proof 1: Consider the following solution of equation (3):

v_n(y, s) =w_an(y, s−log(T −t_n)).

From Proposition 2.1, a_n ∈K, since it is a blow-up point of u. As before, we can use a compactness procedure on v_n to get a nonnegative bounded solutionvof (3) defined for (y, s)∈^RN×^Rsuch that|v(0,0)| ≤(1−₀)κand v_n→ v in C_loc² . Therefore, Theorem 2 implies thatv ≡0 or v =ϕ(s−s₀) for somes₀ ∈^R. In particular, E(v(0))< E(κ). Since E(v_n(0))→E(v(0)) asn→+∞, we have fornlargeE(wan(−log(T−tn))) =E(vn(0))< E(κ),

and in particulara_n can not be a blow-up point of u (we have from [6], for any blow-up pointa of u, E(w_a(s))≥E(κ) for all s≥ −logT). From this fact, a contradiction follows.

Proof 2: It is a more elementary proof based on Theorem 3. Since a_n is a blow-up point and that the blow-up set is closed and bounded (see Proposition 2.1), we can assume thatan→a_∞wherea_∞is a blow-up point.

We know from Theorem 3 that for some C²

0 2

, we have ∀x ∈ Ω, ∀t ∈ [^T₂, T),

∂u

∂t(x, t)−u^p(x, t) ≤ ²₀

2|u(x, t)|^p+C2 0 2

. (20)

In particular,u(x, t) → +∞ as (x, t) → (a_∞, T) (see next subsection for a proof of Theorem 3 and this fact(22)-(23)). Letη >0 such that

∀(x, t)∈B(0, η)×(T −η, T), C²

0 2

< ²₀

2u^p(x, t). (21) For large n, a_n ∈ B(a_∞, η) and t_n ∈ [T −η, T). Therefore (20) and (21) yield

∀t∈[tn, T), ∂u

∂t(an, t)≤(1 +²₀)u^p(an, t).

Since 0< u(a_n, t_n) ≤κ(1−₀)(T −t_n)^−p−11 , we get by direct integration:

∀t∈[t_n,min(T, T^∗(₀))), 0≤u(an, t)≤κ

T−t_n

(1−₀)^p−1 −(1 +²₀)(t−tn) _−p−1¹

withT^∗(₀) =t_n+ ₍₁₊2^T−tn

0)(1−0)^p−1 > T if₀ < ₁(p) for some positive ₁(p).

Thus,a_n is not a blow-up point and a contradiction follows.

2.2 Global approximated behavior like an ODE

We prove Theorem 3 in this subsection. It follows from Theorem 1 and propagation of flatness (through scaling arguments) observed in [14].

Let us first show how to derive the consequences of Theorem 3 announced in the introduction:

Ifx₀ is a blow-up point ofu(t), then

u(x, t)→+∞as (x, t)→(x₀, T) (22)

and ∃₀>0 such that ∀(x, t)∈B(x₀, ₀)×(T −₀, T), ∂u

∂t(x, t)>0. (23) Proof of (22) and (23):

From Theorem 3 applied with > 0, there exists C such that ∀(x, t) ∈ Ω×[^T₂, T)

∂u

∂t(x, t)≥(1−)u^p(x, t)−C. (24) LetA be an arbitrary large positive number satisfying

(1−)A^p−C>0. (25)

From the continuity of u(x, t), there exist ₁ > 0 and ₂ > 0 such that

∀x∈B(x₀, ₁),

u(x, T −₂)> A. (26)

From (24) and (25), we have ∀x ∈ B(x₀, ₁), ^∂u_∂t(x, T −₂) > 0. Now we claim that ∀(x, t) ∈ B(x₀, ₁)×(T −₂, T), u(x, t) > A (which yields (22) and (23) also, by (24) and (25)). Indeed, if not, then there exists (x₁, t₁)∈B(x₀, ₁)×(T−₂, T) such thatu(x₁, t₁)≤A. From the continuity of u, we get t₂ ∈ (T −₂, t₁] such that ∀t ∈ (T −₂, t₂), u(x₁, t) > A and u(x₁, t₂) =A. From (24) and (25), we have ∀t∈(T−₂, t₂), ^∂u_∂t(x₁, t)>0, therefore, u(x₁, t₂) > u(x₁, T −₂) > A by (26). Thus, a contradiction follows, and (22) and (23) are proved.

We now prove Theorem 3.

Proof of Theorem 3: Let us argue by contradiction and suppose that for some 0 > 0, there exist (xn, tn)n∈^N a sequence of elements of Ω×[^T₂, T) such that∀n∈^N,

|∆u(xn, tn)| ≥0|u(xn, tn)|^p+n. (27) Since k∆u(t)kL^∞(Ω) is bounded on compact sets of [^T₂, T), we have that tn→T asn→+∞. We can also assume the existence ofx_∞∈Ω such that x_n → x_∞ as n → +∞. Indeed, if not, then either d(x_n, ∂Ω) → 0 (if Ω is bounded) or |x_n| → +∞ (if Ω =^RN) as n→ +∞, and in both cases, (27) is no longer satisfied for large n, thanks to Proposition 2.1.

We claim that x_∞is a blow-up point ofu. Indeed, if not, then parabolic regularity implies the existence of a positiveδ such that

ku(., t)kW^2,∞(B(x∞,δ)) ≤C for some positive C, which is a contradiction by (27).

Theorem 1 implies thatu(x_n, t_n)(T−t_n)^p−11 is uniformly bounded, there- fore, we can assume that it converges asn→+∞.

Let us consider two cases:

- Case 1: u(x_n, t_n)(T −t_n)^p−11 → κ⁰ > 0 ((x_n, t_n) is in some sense in the singular region “near” (x_∞, T)). From (27), it follows that k∆u(t_n)kL^∞ ≥

|∆u(x_n, t_n)| ≥₀^κ₂^0p(T −t_n)^−p−1p with t_n→T, which contradicts Theo- rem 1.

- Case 2: u(x_n, t_n)(T −t_n)^p−11 → 0 ((x_n, t_n) is in the transitory region be- tween the singular and the regular sets).

Let us first define (t(x_n))_n such thatt(x_n)≤t_n, t(x_n)→T and

u(x_n, t(x_n))(T −t(x_n))^p−11 =κ₀ (28) whereκ₀ ∈(0, κ) satisfies∀t >0,∀a∈Ω,Ea,t(κ₀t^−p−11 )≤ _2(p^κ₋²⁰₁₎−^κp

+1 0

p+1 ≤ ^σ₂⁰ andσ₀ is defined in Proposition 2.2.

Step 1: Existence of t(xn)

Since x_∞ is a blow-up point ofu, lim

t→Tu(x_∞, t)(T−t)^p−11 =κ. It follows that for any δ > 0 small enough, there exists a ball B(x_∞, δ⁰) such that

∀x∈B(x_∞, δ⁰),δ^p−11 u(x, T −δ)≥ ^3κ+κ₄ ⁰. Since x_n→x_∞ asn→+∞, this implies that

∀n≥n₁, δ^p−11 u(x_n, T −δ)≥ κ+κ₀

2 (29)

for some n1 = n1(δ) ∈ ^N. Since u(xn, tn)(T −tn)^p−11 → 0, we have the existence of t_δ(x_n) ∈ [T −δ, t_n] ⊂ [T −δ, T) such that u(x_n, t_δ(x_n))(T − t_δ(x_n))^p−11 =κ₀, for all n≥n₂(δ), wheren₂(δ) ∈^N. Sinceδ was arbitrarily small, it follows from a diagonal extraction argument that there exists a subsequencet(x_n)→T asn→+∞ such thatt(x_n)≤t_n and

u(x_n, t(x_n))(T−t(x_n))^p−11 =κ₀.

Now, we claim that a contradiction follows if we prove the following Proposition:

Proposition 2.3 Let

vn(ξ, τ) = (T−t(xn))^p−11u(xn+ξ^qT−t(xn), t(xn) +τ(T −t(xn))). (30) Then,v_nis a solution of (1) for τ ∈[0,1), and there existsn₀∈^Nsuch that

∀n≥n0,

∀τ ∈[0,1), |∆vn(0, τ)| ≤ ₀

2|vn(0, τ)|^p. (31)

Indeed, from (31) and (30), we obtain: ∀n≥n₀, ∀t∈[t(x_n), T),

|∆u(x_n, t)|= (T−t(x_n))^{−(p−11 +1)}|∆_ξv_n(0, τ(t, n))|

≤ ₂⁰(T −t(xn))^−p−p1|vn(0, τ(t, n))|^p = ₂⁰|u(xn, t)|^p with τ(t, n) = _T^t−₋^t(x_t(xⁿ_n⁾₎, which contradicts (27), sincet_n≥t(x_n). Thus, Theorem 3 is proved.

Step 2: Flatness of v_n

In this Step we prove Proposition 2.3.

We claim that the following lemma concludes the proof of Proposition 2.3:

Lemma 2.3 i)∀δ0 >0, ∀A >0, ∃n3(δ0, A)∈^N such that ∀n≥n3(δ0, A), for all |ξ| ≤ A and τ ∈ [0, ³₄], |v_n(ξ,0) −κ₀| ≤ δ₀, |∇ξv_n(ξ, τ)| ≤ δ₀ and

|∆_ξv_n(ξ, τ)| ≤δ₀.

ii) ∀ > 0, ∀A > 0, ∃n4(, A) ∈ ^N such that ∀n ≥ n4, ∀τ ∈ [0,1), for

|ξ| ≤ ^A₄, |v_n(ξ, τ)−v(τˆ )| ≤ , |∇v_n(ξ, τ)| ≤ and |∆v_n(ξ, τ)| ≤ where ˆ

v(τ) =κ

κ κ0

p−1

−τ _−p−1¹

is a solution of ^dˆ_dτ^v = ˆv^p with v(0) =ˆ κ₀. Indeed, ifis small enough andnis large enough, then∀τ ∈[0,1),v_n(0, τ)≥

1

2ˆv(0) = ^κ₂⁰ and|∆v_n(0, τ)| ≤ ^κ₂^0p ₂⁰ ≤ ₂⁰|v_n(0, τ)|^p.

Proof of lemma 2.3: i) Let δ₀ > 0 andA >0. From (28) and (30), we have: for all|ξ| ≤A and τ ∈[0, ³₄]:

v_n(0,0) =κ₀,

|v_n(ξ,0)−v_n(0,0)| ≤(T −t(x_n))^{p−11 +12}Ak∇u(t(x_n))kL^∞(Ω),

∇v_n(ξ, τ) = (T−t(x_n))^{p−11 +12}∇ux_n+ξ^pT−t(x_n), t(x_n) +τ(T −t(x_n))

=₁₋¹_τ

1 p−1+¹₂

(T−(t(x_n) +τ(T−t(x_n))))^{p−11 +12} ×

∇ux_n+ξ^pT−t(x_n), t(x_n) +τ(T −t(x_n))and

∆v_n(ξ, τ) = (T−t(x_n))^{p−11 +1}∆ux_n+ξ^pT−t(x_n), t(x_n) +τ(T −t(x_n))

=₁₋¹_τ

1 p−1+1

(T −(t(x_n) +τ(T−t(x_n))))^p−11+1×

∆ux_n+ξ^pT−t(x_n), t(x_n) +τ(T −t(x_n)).

Since τ ≤ ³₄, t(x_n)→T asn→+∞, and (T −t)^{p−11 +12}k∇u(t)kL^∞(Ω)+ (T −t)^{p−11 +1}k∆u(t)kL^∞(Ω)→0 as t→T (Theorem 1),i) is proved.

ii) From i) and continuity arguments, it follows that for all |ξ| ≤ A, Eξ,1(v_n(0))≤2Eξ,1(κ₀)≤σ₀ forn large enough, by definition ofκ₀. There- fore, from Proposition 2.2 (applied withδ = 1 and using translation invari- ance), we have∀τ ∈[¹₂,1),∀|ξ| ≤ ^A₂, |vn(ξ, τ)| ≤M(p).

By classical parabolic arguments, we get

∀τ ∈[3

4,1), ∀|ξ| ≤ A

3, |v_n|+|∇v_n|+|∆v_n| ≤M(p). (32) Now, using i), (32) and classical estimates for the heat flow, we get for all > 0: ∀|ξ| ≤ ^A₄, ∀τ ∈ [0,1), |∇v_n(ξ, τ)| ≤ and |∆v_n(ξ, τ)| ≤ if n≥n₅(, A).

Since v_n is a solution of equation (1), combining this withi) and ODE estimates yields for all > 0: ∀|ξ| ≤ ^A₄, ∀τ ∈[0,1), |v_n(ξ, τ)−ˆv(τ)| ≤ if n≥n₆(, A). This concludes the proof ofii).

3 Classification of connections between critical points of equation (3) in L

loc

We prove Theorem 2 and Corollaries 2 and 3 in this section.

We first prove Theorem 2, and then we show how Corollaries 2 and 3 can be deduced from Theorem 2.

Proof of Theorem 2: We assume that 1 < p and (N −2)p < N + 2, and consider w(y, s) a nonnegative global bounded solution of (3) defined for (y, s)∈^RN ×^R. Our goal is to show thatw depends only on times.

We proceed in 5 steps.

In Step 1, we show that w has a limitw_±∞ as s→ ±∞, where w_±∞ is a critical point of (3), that isw_±∞≡0 or w_±∞≡κ. We focus then on the non trivial case, that isw_−∞≡κ and w_+∞≡0.

In Step 2, we investigate the linear problem around κ, ass→ −∞, and show that wwould behave at most in three ways.

In Step 3, we show that among these three ways we have the situation w(y, s) =ϕ(s−s₀) with ϕ(s) =κ(1 +e^s)^−p−11 . We then show (respectively in Step 4 and in Step 5) that the two other ways actually can not occur, we find in fact a contradiction through a blow-up argument forw(s) using the geometrical transformation:

a→w_a defined by w_a(y, s) =w(y+ae^s2, s) (33) (w_a is also a solution of (3)) and a blow-up criterion for equation (3).

Step 1: Behavior of w ass→ ±∞

This step can be found in Giga and Kohn [6]. The results are mainly consequences of parabolic estimates and the gradient structure of equation (3). Let us recall them briefly. We first restate lemma 2.1 of section 2:

Lemma 3.1 (Parabolic estimates) There is a positive constant M such that ∀(y, s)∈^RN ×^R,

|w(y, s)|+|∇w(y, s)|+|∆w(y, s)| ≤M and

∂w

∂s(y, s)

≤M(1 +|y|).

Lemma 3.2 (Stationary solutions) Assume p ≤ (N + 2)/(N −2) or N ≤2. Then the only nonnegative bounded global solutions in ^RN of

0 = ∆w− 1

2y.∇w− w

p−1+|w|^p−1w (34) are the trivial ones: w≡0 and w≡κ.

Proof: The following Pohozaev identity can be derived for each bounded solutions of equation (3) in ^RN (see Proposition 2 in [6]):

(N + 2−p(N −2)) Z

|∇w|²ρdy+p−1 2

Z

|y|²|∇w|²ρdy= 0.

Hence, for (N −2)p≤N+ 2, wis constant. Thus, w≡0 orw≡κ.

Lemma 3.3 (Gradient structure) Assumep <(N+ 2)/(N−2)or N ≤ 2. We define for eachw solution of (3)

E(w) = 1 2

Z

RN|∇w|²ρdy+ 1 2(p−1)

Z

RN|w|²ρdy− 1 p+ 1

Z

RN|w|^p+1ρdy (35) where ρ(y) = e^−|y|2/4

(4π)^N/2. (36)

Then,∀s₁, s₂ ∈^R, Z s2

s1

Z

RN

∂w

∂s

2

ρdyds=E(w(s₁))−E(w(s₂)) (37) Outline of the proof: (seeProposition 3 in [6] for more details).

One may multiply equation (3) by ^∂w_∂sρ and integrate over the ball B(0, R) with R > 0. Then, using lemma 3.1 and the dominated conver- gence theorem yields the result.

Proposition 3.1 (Limit of w ass→ ±∞) Assume p <(N+ 2)/(N−2) or N ≤2. Let w be a bounded nonnegative global solution of (3) in ^RN+1. Thenw_+∞(y) = lim

s→+∞w(y, s) exists and equals0 or κ. The convergence is uniform on every compact subset of^RN. The corresponding statements hold also for the limit w_−∞(y) = lim

s→−∞w(y, s).

Outline of the proof: (seePropositions 4 and 5 in [6] for more details).

Let (s_j) be a sequence tending to +∞, and let w_j(y, s) = w(y, s+ s_j). From lemma 3.1, (w_j) converges uniformly on compact sets to some w_+∞(y, s) and∇w_j → ∇w_+∞ a.e. Assuming thats_j+1−s_j →+∞, one can use lemma 3.3 to show thatw_j does not depend ons. Therefore, w_+∞≡0 orw+∞=κby lemma 3.2. The continuity of wthen asserts thatw+∞does not depend on the choice of the subsequence (s_j). The analysis in −∞ is completely parallel.

According to (37) (with s₁ → −∞ and s₂ → +∞), there are only two cases:

-E(w_−∞)−E(w+∞) = 0: hence, ^∂w_∂s ≡0. Therefore,wis a bounded global solution of (34). Thus,w≡0 or w≡κ according to lemma 3.2. This case has been treated by Giga and Kohn in [6].

-E(w_−∞)−E(w+∞)>0: sinceE(κ) = (¹₂−_p+1¹ )κ^p+1R ρdy >0 =E(0), we have (w_−∞, w_+∞) = (κ,0). It remains to treat this case in order to finish the proof of Theorem 2.

In the following steps, we consider the case (w_−∞, w_+∞) = (κ,0).

Step 2: Classification of the behavior of w ass→ −∞:

Since w is globally bounded in L^∞ and w → κ as s→ −∞, uniformly on compact subsets of ^RN, we have lim

s→−∞kw−κkL²_ρ = 0 where L²_ρ is the L²-space associated to the Gaussian measureρ(y)dyandρis defined in (36).

In this part, we classify the L²_ρ behavior of w−κ as s→ −∞. Let us introducev =w−κ. From (3), v satisfies the following equation: ∀(y, s)∈ R^N+1,

∂v

∂s =Lv+f(v) (38)

whereLv= ∆v−1

2y.∇v+v and f(v) =|v+κ|^p−1(v+κ)−κ^p−pκ^p−1v. (39) Sincew is bounded inL^∞, we can assume |v(y, s)| ≤M, and then|f(v)| ≤ Cv² withC=C(M).

L is self-adjoint onD(L)⊂L²_ρ. Its spectrum is spec(L) ={1− m

2|m∈^N},

and it consists of eigenvalues. The eigenfunctions of L are derived from Hermite polynomials: