Existence and stability of a solution with a new prescribed behavior for a heat equation with or without a critical nonlinear gradient term

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Existence and stability of a solution with a new prescribed behavior for a heat equation with or without a critical nonlinear

gradient term

Hatem ZAAG

CNRS and LAGA Université Paris 13

Tunis, July 22-23, 2015

Joint work with:

M.A. Ebde,

F. Merle(Université de Cergy-Pontoise and IHES), S. Tayachi(Faculté des Sciences de Tunis).

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Introduction: The equation

We consider the following PDE:

∂tu = ∆u+µ|∇u|^q+|u|^p−1u, u(·,0) =u₀

where:

p >3,µ ∈R, 0≤ q≤ qc = _p+1^2p , u(t) :x ∈R^N →u(x,t) ∈R, u₀ ∈W^1,∞(R^N).

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History of the equation µ = 0

This is the simplest case of a heat equation which may exhibit blow-up.

Indeed, ifu₀≡ α₀ >0, then

u(x,t) = u(t) =κ(T −t)⁻^p−1¹ → ∞as t→T, with

κ= (p−1)⁻^p−1¹ andT = 1 (p−1)α^p−1₀ is calledthe blow-up time.

This is alab modelto develop new techniques for the study of blow-up in more physical situations(Ginzburg-Landau, Navier-Stokes, etc...),

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History of the equation µ 6= 0

Introduction: Chipot-Weissler (1989), mathematical motivation (µ < 0).

Population dynamics interpretation: Souplet (1996).

Mathematical analysis: Chipot, Weissler, Peletier, Kawohl, Fila, Quittner, Deng, Alfonsi, Tayachi, Souplet, Snoussi, Galaktionov, Vázquez, Ebde, Z., Nguyen, ...

Elliptic version: Chipot, Weissler, Serrin, Zou, Peletier, Voirol, Fila, Quittner, Bandle ...

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Two limiting cases

- Whenµ= 0, this is the well-knownsemilinear heat equation:

∂tu= ∆u+|u|^p−1u.

- Whenµ= +∞, we recover (after rescaling) theDiffusive Hamilton-Jacobi equation:

∂tu = ∆u+|∇u|^q.

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A critical case: q = q

_c

Whenµ ∈_Randw(y,s)is the similarity variables version ofu(x,t):

w(y,s) = (T −t)^p−1¹ u(x,t), y= x

√T −t ands =−log(T −t), we have for alls ≥ −logT andy∈ _R^N:

∂sw = ∆w− 1

2y· ∇w− w

p−1 +|w|^p−1w+µe^αs|∇w|^q. with

α= q(p+1)

2(p−1) − p

p−1 = (p+1)

2(p−1)(q−q_c)andq_c = 2p p+1. Therefore, we have 3 cases:

subcriticalwhenq <q_c: we have a “perturbation” of thesemilinear heat equation;

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Other indications for the criticality of q

_c

= 2p/(p + 1)

Scaling: Only whenq=qc, we have “usolution⇒u_λ solution”, where u_λ(x,t) = λ^2/(p−1)u(λx, λ²t), ∀λ >0, ∀t >0, x ∈R^N, as for the equation without gradient term (µ= 0).

Large time behavior: it depends on whetherq <qc,q =qc,q >qc; see Snoussi-Tayachi-Weissler (1999) and Snoussi-Tayachi (2007).

Blow-up behavior forµ <0: it depends also on whetherq <qc,q =qc,q >qc; see Souplet (2001, 2005), Chlebik, Fila and Quittner (2003) (Bounded Domain)

Also for the elliptic version.

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Cauchy problem and blow-up solutions

Cauchy problem:

forµ=0, wellposed inL^∞(R^N),

forµ6=0, wellposed inW^1,∞(R^N)(fixed point argument, see Alfonsi-Weissler (1993), Souplet-Weissler (1999)).

Blow-up solutions: IfT <∞, thenlimt→Tku(t)k_W^1,∞₍_R^N₎ = ∞.

Definition: x₀is a blow-up point if∃(t_n,x_n)→(T,x₀)s.t. |u(x,t)| → ∞asn→ ∞.

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Aim of the talk

Take

q= qc and eitherµ= 0 orµ6= 0.

We have 3 goals:

construct a blow-up solution, determine its blow-up profile,

prove its stability (with respect to perturbations in initial data).

Remark. The caseq<qcis treated as a perturbation of the caseµ=0, but we don’t mention that here (see Ebde and Zaag (2011)).

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Case µ = 0: the standard semilinear heat equation

The (generic) profile is given by (T −t)^1/(p−1)u(zp

(T −t)|log(T −t)|,t)∼ f₀(z)ast →T, where

f₀(x) = p−1+b₀|x|²−1/(p−1)

andb₀ = (p−1)²/(4p).

See Galaktionov-Posashkov (1985), Berger-Kohn (1988), Herrero-Velázquez (1993).

The constructive existence proofby Bricmont-Kupiainen (1994), Merle-Z. (1997) is based on:

- The reduction of the problem to a finite-dimensional one;

- The solution of the finite-dimensional problem thanks to the degree theory.

Other profiles are possible.

Remark. We will give the proof in this case.

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Subcritical case: µ 6= 0 and q < q

_c

=

_p+1^2p

Ebde and Z. (2011) could adapt the previous existence strategy andfind the same behavior as forµ=0, since the gradient term is subcritical in size in similarity variables:

∂sw = ∆w− 1

2y· ∇w− w

p−1 +|w|^p−1w+µe^αs|∇w|^q. with

α= (p+1)

2(p−1)(q−q_c)< 0.

Remark. We don’t mention the proof in this case.

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Critical case: q = q

_c

, with −2 < µ < 0 and p − 1 > 0 small

Exact self-similar blow-up solution by Souplet, Tayachi and Weissler (1996):

u(x,t) = (T −t)^−1/(p−1)W

|x|

√T −t

whereW satisfies the following elliptic equation:

W⁰⁰+ N−1

r W⁰− 1

2rW⁰− W

p−1 +W^p+µ|W⁰|^q^c =0.

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Critical case: µ 6= 0 and q = q

_c

; A numerical result

A similar profile to the caseµ=0was discoverednumericallyby Van Tien Nguyen (2014):

(T−t)^1/(p−1)u(zp

(T−t)|log(T −t)|,t)∼ f₀(z)ast →T, where

f_µ(x) = p−1+b_µ|x|²−1/(p−1)

with

b_µ > 0andb₀= (p−1)²/(4p), the same as forµ=0.

Remark: We initially wanted to confirm this result, and ended by finding anewtype of behavior.

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The blow-up profile Existence of the new profile (µ6=0andq=qc)

Critical case: µ > 0 and q = q

_c

; Our new profile

Theorem (Tayachi and Z.) There exists a solutionu(x,t)s.t.:

Simultaneous Blow-up: Bothuand∇ublow up ast→T >0only at the origin;

Blow-up Profile:

(T −t)^p−1¹ u(z√

T −t|log(T −t)|^2(p−1)^p+1 ,t)∼¯f_µ(z)ast →T

with

¯f_µ(z) = p−1+ ¯b_µ|z|²−_p−1¹

withb¯_µ = 1

2(p−1)^p−2^p−1 (4π)^N²(p+1)² pR

R^N|y|^qe^−|y|²^/4dy

!^p+1_p−1 µ⁻

p+1 p−1 >0.

Final profileWhenx6=0,u(x,t)→u(x,T)ast→T with

u(x,T) ∼ ¯b_µ 2

|x|²

|log|x||^p+1^p−1

!−_p−1¹

asx →0.

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Comments

The exhibited behavior is new in two respects:

The scaling law: √

T −t|log(T −t)|^2(p−1)^p+1 instead of the laws of the caseµ=0, p(T −t)|log(T −t)|or(T −t)^2m¹ wherem ≥ 2 is an integer;

The profile function:¯f_µ(z) = p−1+ ¯b_µ|z|²−_p−1¹

is different from the profile of the caseµ=0, namelyf₀(z) = p−1+b₀|z|²−_p−1¹

, in the sense that¯b_µ 6=b₀. Note in particular, that

¯b_µ → ∞as µ→0.

Remark: Our solution is different already in the scaling from the numerical solution of Van Tien Nguyen, which is in theµ= 0style.

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Idea of the proof

We followthe constructive existence proofused by Bricmont-Kupiainen (1994), Merle-Z.

(1997) for thestandard semilinear heat equation.

That method is based on:

- The reduction of the problem to a finite-dimensional one (N +1parameters);

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The blow-up profile The stablity result

Stability of the constructed solution for µ ≥ 0

Thanks to the interpretation of the(N+1)parameters of the finite-dimensional problem in terms of the blow-up time (inR) and the blow-up point (inR^N), the existence proof yields the following:

Theorem (Merle and Z. (µ=0), Tayachi and Z. (µ > 0): Stability)

The constructed solution is stable with respect to perturbations in inital data inW^1,∞(R^N).

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The blow-up profile The stablity result

Applications: Perturbed Hamilton-Jacobi Equation (µ > 0)

Corollary (Tayachi and Z.)

After an appropriate scaling, our results yield stable blow-up solutions for the following Viscous Hamilton-Jacobiequation:

∂tv= ∆v+|∇v|^q+ν|v|^p−1v;

with

ν >0, 3/2 <q <2, p = q 2−q.

The solution and its gradient blow up simultaneously, only at one point.

Of course, the blow-up profile is given after an appropriate scaling.

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The proofs (µ≥0)

A formal approach to find the ansatz (N = 1)

Following the standard semilinear heat equation case, we work in similarity variables:

w(y,s) = (T −t)^p−1¹ u(x,t), y= x

√T −t ands =−log(T −t).

We need tofind a solutionfor the following equation defined for alls≥ s₀andy∈ _R:

∂sw =∂_y²w− 1

2y∂yw− w

p−1 +|w|^p−1w+µ|∂yw|^q, such that

0< 0≤ kw(s)k_L^∞_(R) ≤ 1

₀ (type 1 blow-up).

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Simple ideas

Idea 1: Look for a (non trivial) stationary solution:

forµ =0 andp <p_S ≡ ^N+2_N−2, we know from Giga and Kohn that we have only 3 solutions: κ,−κ, 0, where

κ ≡(p−1)⁻^p−1¹ . Problem: These are “trivial” solutions.

µ= 0 andp≥ p_S ≡ ^N+2_N−2: there are some non trivial solutions.

forµ 6=0, there is a non trivial solution, by Souplet, Tayachi and Weissler (1996), forp close to 1(self-similar solution in the u(x,t)setting).

Idea 2: Sincew ≡κ ≡ (p−1)⁻^p−1¹ is a trivial solution, let us look for a solutionwsuch that w →κ, as s→ ∞.

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Inner expansion

We write

w= κ+w, andlook forwsuch that

w→0 ass → ∞.

The equation to be satisfied bywis the following:

∂sw =Lw+B(w) +µ|∇w|^q^c, whereµ∈ Randq_c= _p+1^2p ,

Lv=∂_y²v− 1

2y∂yv+v, and

B(w) =|w+κ|^p−1(w+κ)−κ^p−pκ^p−1w.

Note thatBis quadratic:

B(w)− p 2κw²

≤ C|w³|.

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The linear operator

Note thatLis self-adjoint inD(L)⊂ L²_ρ(R)where L²_ρ(R) =

f ∈L²_loc(R)

Z

R

(f(y))²ρ(y)dy< ∞

and

ρ(y) = e^{− |}^y|₄²

√4π .

The spectrum ofLis explicitly given by spec(L) = n

1− m 2

m∈ N o

.

All the eigenvalues are simple, and the eigenfunctionsh_m are (rescaled) Hermite polynomials, with

Lh =

1− m h .

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Naturally, we expandw(y,s)according to the eigenfunctions ofL: w(y,s) =

∞

X

m=0

wm(s)hm(y).

Sinceh_mform≥ 3correspond to negative eigenvalues ofL, assumingweven iny, we may consider that

w(y,s) = w₀(s)h₀(y) +w₂(s)h₂(y), with

w₀, w₂ →0as s→ ∞.

Plugging this in the equation to be satisfied byw:

∂sw =Lw+B(w) +µ|∂yw|^q^c,

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we first see that

µ|∂yw|^q^c = µ2^q^c|y|^q^c|w₂|^q^c,

then, projecting onh₀andh₂, we get the following ODE system:

w⁰₀= w₀+ p

2κ w²₀+8w²₂

+ ˜c₀|w₂|^q^c+O |w₀|³+|w₂|³ ,

w⁰₂= 0+ p

κ w₀w₂+4w²₂

+ ˜c₂|w₂|^q^c+O |w₀|³+|w₂|³ , where

1< qc = 2p

p+1 <2, ˜c₀ =µ2^q^c Z

R

|y|^q^cρ

, ˜c₂ =µqc2^q^c⁻² Z

R

|y|^q^cρ

.

Note that the sign of˜c₀and˜c₂is the same as forµ.

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(Case µ = 0) Looking at the equation to be satisfied by w

₂

Let us write it as follows:

w⁰₂= 4p

κ w²₂(1+O(w₂)) + p

κw₀w₂+O |w₀|³ .

Assuming that

|w₀| |w₂|, (H1) we end-up with

w⁰₂ ∼ 4p κ w²₂, which yields

w₂ ∼ − κ 4ps.

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(Case µ = 0) Looking at the equation to be satisfied by w

₀

Let us recall it

w⁰₀= w₀+ p

2κ w²₀+8w²₂

+O |w₀|³+|w₂|³ , then write it as follows:

w⁰₀ =w₀(1+O(w0)) + 4p

κ w²₂(1+O(w2)). Assuming that

|w⁰₀| |w₀|, |w⁰₀| |w₂|², (H2) we end-up with

w₀ ∼ −4p

κ w²₂ ∼ − κ

4ps² w₂∼ − κ 4ps.

Note that both hypotheses(H1)and(H2)are satisfied by the solution we have just found.

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Case µ 6= 0

Let us recall the system to be satisfied byw₀andw₂: w⁰₀= w₀+ p

2κ w²₀+8w²₂

+ ˜c₀|w2|^q^c+O |w0|³+|w2|³ ,

w⁰₂= 0+ p

κ w₀w₂+4w²₂

+ ˜c₂|w₂|^q^c+O |w₀|³+|w₂|³ , where

1< q_c = 2p

p+1 <2, ˜c₀ =µ2^q^c Z

R

|y|^q^cρ

, ˜c₂ =µqc2^q^c⁻² Z

R

|y|^q^cρ

.

Note that the sign of˜c₀and˜c₂is the same as forµ.

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(Case µ 6= 0) Looking at the equation to be satisfied by w

₂

w⁰₂ = ˜c₂|w₂|^q^c 1+O |w₂|^2−q^c + p

κ(w₀w₂) +O |w₀|³ .

Assuming that

|w₀w₂| |w₂|^q^c, |w₀|³ |w₂|^q^c, (H1) we end-up with

w⁰₂∼ sign(µ)|˜c₂||w₂|^q^c, which yields

w₂=−sign(µ) B s^qc¹⁻¹

, for someB >0.

2p

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(Case µ 6= 0) Looking at the equation to be satisfied by w

₀

w⁰₀= w₀(1+O(w₀)) + ˜c₀|w₂|^q^c 1+O |w₂|^2−q^c . Assuming that

|w⁰₀| w₀, |w⁰₀| |w₂|^q^c, (H2) we end-up with

w₀ ∼ −˜c₀|w2|^q^c ∼ −˜c₀B^q^c

s^qc−1^qc w₂ =−sign(µ) B s

1 qc−1

. Note that both hypotheses(H1)and(H2)are satisfied by the found solution.

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Conclusion for the inner expansion

Recalling the ansatz

w(y,s) =κ+w(y,s) = κ+w₀(s)h₀(y) +w₂(s)h₂(y)withh₂(y) =y²−2,

we end-up with:

w(y,s) =κ−˜by² s^2β + 2˜b

s^2β +o 1

s^2β

, with

forµ =0: ˜b = _4p^κ andβ = ¹₂;

forµ 6=0: ˜b =sign(µ)Bandβ = _2(q¹

c−1) = _2(p−1)^p+1 > ¹₂, with B =

2^q^c⁻²qc(qc−1)R

R|y|^q^cρ− ¹

qc−1µ⁻^p+1^p−1.

Remark: This expansion is valid inL²_ρand uniformly on compact sets by parabolic regularity.

However, forybounded,we see no shape: the expansion is asymptotically a constant.

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Outer expansion

To have ashape, following the inner expansion, (valid for|y|bounded), w(y,s) = κ−˜bz²+ 2b˜

s^2β +o 1

s^2β

withz= y s^β, let us look for a solution of the following form (valid for|z|bounded):

w(y,s) =f(z) + a

s^2β +O(1

s^ν), ν >2β, withz= _s^yβ,f(0) = κandf bounded.

Plugging this ansatz in the equation,

∂sw= ∂_y²w− 1

2y∂yw− w

p−1 +|w|^p−1w+µ|∂yw|^q^c, then, keeping only the main order, we get

−1

2zf⁰(z)− 1

p−1f(z) + (f(z))^p= 0,

hence,f(z) =

p−1+ ¯b|z|²−_p−1¹

, for some constant¯b >0.

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Matching asymptotics

Forybounded, both the inner expansion (valid for|y|bounded) w(y,s) =κ−˜by²

s^2β + 2˜b s^2β +o

1 s^2β

,

and the outer expansion (valid for|z|bounded) w(y,s) =f(z) + a

s^2β +O(1

s^ν) whereν >2β, z = y s^β and

f(z) =

p−1+ ¯b|z|²−_p−1¹

=κ− ¯bκ

(p−1)²z²+O(z⁴), have to agree. Therefore,

˜b=

¯bκ

anda=2˜b.

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Matching asymptotics (continued)

In particular:

Forµ= 0,˜b = _4p^κ, hence¯b= ^(p−1)_4p ² anda = _2p^κ; Forµ6= 0,˜b =sign(µ)BwithB=

2^q^c⁻²q_c(qc−1)R

R|y|^q^cρ− ¹

qc−1µ⁻

p+1 p−1. SinceB >0and¯b >0from the inner and the outer expansions, it follows that

µ > 0, b¯= (p−1)²

κ Banda= 2B, withB=

2^q^c⁻²qc(qc−1) Z

R

|y|^q^cρ − ¹

qc−1

µ⁻^p+1^p−1.

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Conclusion of the formal approach

We have just derived theblow-up profilefor|y| ≤Ks^β: ϕ(y,s) = ¯f

y s^β

+ a

s^2β =

p−1+ ¯b|y|² s^2β

−_p−1¹

+ a s^2β, where:

Forµ= 0,

β = 1

2, ¯b= (p−1)²

4p anda= κ 2p; Formµ6=0,

β = p+1

2(p−1), ¯b= (p−1)²

κ Banda =2B,

with − ¹

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The proofs (µ≥0) A sketch of the proof of the existence result

Strategy of the proof

We follow the strategy used by Bressan (1992), Bricmont and Kupiainen (1994) then Merle and Z. (1997) for the standard semilinear heat equation, based on:

- The reduction of the problem to a finite-dimensional one;

This strategy was later adapted for:

the present equation withsubcriticalgradient exponentq <q_cin Ebde and Z. (2011);

the Ginzburg-Landau equation:

∂tu= (1+iβ)∆u+ (1+iδ)|u|^p−1u−γu in Z. (1998) and Masmoudi and Z. (2008);

the supercritical gKdV and NLS in Côte, Martel and Merle (2011);

the semilinear wave equation

∂_t²u =∂_x²u+|u|^p−1u

in Côte and Z. (2013), for the construction of a blow-up solution showing multi-solitons.

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Construction of solutions of PDEs with prescribed behavior

More generally, we are in the framework of constructing a solution to some PDE with some prescribed behavior:

NLS: Merle (1990), Martel and Merle (2006);

KdV (and gKdV): Martel (2005), Côte (2006, 2007), water waves: Ming-Rousset-Tzvetkov (2013),

Schrödinger maps: Merle-Raphaël-Rodniansky (2013), etc....

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The strategy of the proof (N = 1)

We recall our aim: to construct a solutionw(y,s)of the equation in similarity variables:

∂sw= ∂_y²w− 1

2y∂yw− w

p−1 +|w|^p−1w+µ|∂yw|^q^c, such that

w(y,s)∼ ϕ(y,s)whereϕ(y,s) =

p−1+ ¯b|y|² s^2β

−_p−1¹

+ a s^2β. Idea: We linearize aroundϕ(y,s)by introducing

v(y,s) = w(y,s)−ϕ(y,s).

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In that case, our aim becomes to constructv(y,s)such that kv(s)k_L^∞₍_R₎ →0as s→ ∞, andv(y,s)satisfies for alls ≥s₀andy∈ R,

∂sv= (L+V)v+B(v) +G(∂yv) +R(y,s), where

Lv=∂_y²v− 1

2y∂yv+v,

V(y,s) = pϕ(y,s)^p−1− p p−1,

B(v) = |ϕ+v|^p−1(ϕ+v)−ϕ^p−pϕ^p−1v,

G(∂yv) = µ|∂yϕ+∂yv|^q^c−µ|∂yϕ|^q^c,

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Effect of the different terms

The linear term: Its spectrum is given by{1− ^m₂, |m ∈N}and its eigenfunctions are Hermite polynomials withLhm = (1− ^m₂)hm.

Note that we have two positive directionsλ=1, ¹₂ and a null directionλ=0.

The potential termV: it has two fundamental properties:

(i) V(.,s)→0inL²_ρ(R)ass→ ∞.In practice, the effect ofVin the blow-up area(|y| ≤Ks^β)is regarded as a perturbation of the effect ofL(except on the null mode).

(ii) V(.,s)→ −_p−1^p ass→ ∞.and ^|y|_sβ → ∞. Since−_p−1^p <−1and1is the largest eigenvalue of the operatorL, outside the blow-up area (i.e. for|y| ≥Ks^β), we may consider that the operatorL+V has negative spectrum, hence, easily controlled.

The nonlinear term inv: It is quadratic: |B(v)| ≤C|v|²,

The nonlinear term in∂yv: It is sublinear: kG(∂yv)k_L^∞₍_R₎ ≤ ^√^C_sk∂yvk_L^∞₍_R₎. The rest term: It is small: kR(.,s)kL^∞ ≤ ^C_s.

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Space localization

From the properties of the profile and the potential, the variable z= y

s^β

plays a fundamental role, and our analysis will be different in the regions

|z| >K and|z|< 2K.

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Decomposition of v(y, s) into ‘”inner” and “outer” parts

Consider a cut-off function

χ(y,s) = χ₀ |y|

K s^β

, whereχ₀ ∈C^∞ [0,∞),[0,1]

, s.t. supp(χ₀)⊂ [0,2]andχ₀≡ 1in[0,1]. Then, introduce v(y,s) =v_inner(y,s) +v_outer(y,s),

with

v_inner(y,s) = v(y,s)χ(y,s)andv_outer(y,s) = v(y,s) 1−χ(y,s) . Note that

suppv_inner(s) ⊂[−2Ks^β,2Ks^β], suppv_outer(s) ⊂(−∞,−Ks^β]∪[Ks^β,∞).

Remark: v_outer(y,s)is easily controlled, becauseL+V has a negative spectrum (less than 1− _p−1^p + <0).

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Decomposition of the “inner” part

We decomposevinner, according to the sign of the eigenvalues ofL: vinner(y,s) =

2

X

m=0

vm(s)hm(y) +v−(y,s),

wherevm is the projection ofvinner (and notvonhm, andv−(y,s) =P−(vinner)withP−being the projection on the negative subspaceE₋ ≡Span{hm | m≥ 3}of the operatorL.

Remark: v−(y,s)is easily controlled because the spectrum ofLrestricted toE− is less than

−¹₂.

It remains then to controlv₀,v₁andv₂.

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Control of v

₂

This is delicate, because it corresponds to the directionh₂(y), the null mode of the linear operatorL.

Projecting the equation

∂sv = (L+V)v+B(v) +G(∂yv) +R(y,s)

onh₂(y), and recalling thatLh₂ =0, we need to refine the contributions ofVvandG(∂yv)to the linear term (this is delicate), and write:

v⁰₂(s) =−2

sv₂(s) +O 1

s^4β

+O

kv(s)k²_W1,∞(R)

. Working in the slow variable τ = logs=log|log(T −t)|,

we see that

d

dτv₂= −2v2+O 1

s^4β−1

+O

skv(s)k²_W1,∞(R)

, which shows anegativeeigenvalue.

Conclusion: v₂can be controlled as well.

We are left only with two componentsv₀andv₁: A finite dimensional problem.

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Dealing with v

₀

and v

₁

These remaining components correspond repectively to the projections alongh₀(y) = 1and h₁(y) = y, thepositivedirections ofL. Projecting the equation

∂sv = (L+V)v+B(v) +G(∂yv) +R(y,s) onhm(y)withm =0, 1, we write

v⁰₀(s) = v₀(s) +O 1

s^2β+1

+O

kv(s)k²_W1,∞(R)

, v⁰₁(s) = 1

2v₁(s) +O 1

s^2β+1

+O

kv(s)k²_W1,∞(R)

. Since all the other components are easy to control, we may assume that

v(y,s) =v₀(s)h₀(y) +v₁(s)h₁(s) = v₀(s) +v₁(s)y,

ending with a “baby” problem,which is two-dimensional, with initial data ats= s₀given by

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Solution of the baby problem

Recall the baby problem:

v⁰₀(s) = v₀(s) +O 1

s^2β+1

+O v₀(s)²

+O v₁(s)² , v⁰₁(s) = 1

2v₁(s) +O 1

s^2β+1

+O v₀(s)²

+O v₁(s)² ,

with initial data ats =s₀given by

v₀(s₀) =d₀, v₁(s₀) = d₁.

This problem can be easily solved by contradiction, based onIndex Theory:

There exist a particular value(d₀,d₁) ∈R²such that the “baby” problem has a solution (v₀(s),v₁(s))which converges to(0,0)ass → ∞.

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Conclusion for the full problem

For the full problem (which isinfinite-dimensional), recalling that v(y,s) = vinner(y,s) +vouter(y,s) =

2

X

m=0

vm(s)hm(y) +v−(y,s) +vouter(y,s),

and that all the three other components correspond tonegativeeigenvalues, hence easily converging to zero, we have the following statement:

Consider the equation

∂sv = (L+V)v+B(v) +G(∂yv) +R(y,s) equipped with initial data ats =s₀:

ψs0,d0,d1(y) =

d₀h₀(y) +d₁h₁(y)

χ(2y,s₀).

Then, there exists a particular value(d₀,d₁)such that the corresponding solutionv(y,s) exists for alls≥ s₀andy ∈R, and satisfies

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End of the proof of the existence proof

Introducing

T =e^−s⁰, and recalling that

v(y,s) =w(y,s)−ϕ(y,s)andu(x,t) = (T −t)⁻^p−1¹ w

x

√T −t,−log(T −t)

, and

ϕ(y,s) = ¯f_µ( y

s^β) + a s^2β, we derive the existence ofu(x,t), a solution to the equation

∂tu= ∆u+µ|∇u|^q^c +|u|^p−1u, such that

(T −t)^p−1¹ u(z√

T −t|log(T −t)|^2(p−1)^p+1 ,t)∼¯f_µ(z)ast →T. Using refined parabolic regularity estimates, we derive that:

u(x,t)blows up only at the origin;

the final profile satisfiesu(x,T) ∼

¯2

bµ|x|⁻²|log|x||^p+1^p−1_p−1¹

asx →0.

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The proofs (µ≥0) Proof of the stability result

Proof of the stability result (N = 1)

Let us recall the statement:

Theorem (Tayachi and Z.: Stability)

The constructed solution is stable with respect to perturbations in inital data inW^1,∞(R^N).

Proof: It follows from the existence proof, through the interpretation of the parameters ofthe finite-dimensional problemin terms of the blow-up time and the blow-up point.

Considerˆu(x,t)the constructed solution, with initial datauˆ₀, blowing up at timeTˆ only at one blow-up pointˆa(not necessarily0).

Consider nowu₀∈ W^1,∞(R)such that

u₀ = ˆu₀+₀ withk₀k_W^1,∞₍_R₎ small,

andu(x,t)the corresponding maximal solution and T(u0)≤ +∞its maximal existence time.

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Our finite dimensional parameters

At this stage, we don’t even know thatT(u0)<+∞, don’t mention the blow-up pointa(u0).

Since our goal is to show thatT(u₀)anda(u₀)are close toTˆ andˆarespectively, let us study ALLthe similarity variables versions ofu(x,t)considered witharbitrary(T,a)close to (ˆT,ˆa):

w_u₀_,T,a(y,s) =e⁻^p−1^s u(a+ye²^s,T −e^−s) and

vu0,T,a(y,s) = wu0,T,a(y,s)−ϕ(y,s) = e⁻^p−1^s u(a+ye^s²,T −e^−s)−ϕ(y,s), where the profile:

ϕ(y,s) = ¯f_µ( y

s^β) + a s^2β.

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The stability problem as an “existence” problem

Note thatfor any(T,a),vu₀,T,a(y,s)satisfies thesameequation as for the existence proof:

∂sv= (L+V)v+B(v) +G(∂yv) +R(y,s), with initial data, say at some times= s₀large enough, given by

vu₀,T,a(y,s₀) = ¯ψu₀,T,a(y)≡ e⁻

s0

p−1u(a+ye

s0

2,T −e^−s⁰)−ϕ(y,s₀).

Idea: These initial data depend ontwoparameters, exactly as in the existence proof, where initial data was

ψs0,d0,d1(y) =

d₀h₀(y) +d₁h₁(y)

χ(2y,s₀), depending also ontwoparameters.

It happens that the behaviors of

(d ,d ) 7→ψ and(T,a)7→ψ¯

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Statement for the “new” existence problem

For||u₀−ˆu₀k_W^1,∞ small ands₀large enough, there exists some(¯T(u₀),¯a(u₀))such that the solution of equation

∂sv= (L+V)v+B(v) +G(∂yv) +R(y,s),

with initial data ats =s₀given by ψ¯u₀,T,a(y)≡ e⁻

s0

p−1u(a+ye

s0

2,T−e^−s⁰)−ϕ(y,s₀) with(T,a) = (¯T(u₀),¯a(u₀)), converges to0ass → ∞.

But remember !

ψ¯_u₀_,T(u¯ 0),¯a(u0)(y) = v_u₀_,¯T(u0),¯a(u0)(y,s₀), so we know that solution: it is simplyv_u₀_,¯T(u0),¯a(u0)(y,s)!!!!

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Therefore, this is in reality the statement we have just proved:

There exists some(¯T(u₀),¯a(u₀))such thatkv_u₀_,T(u¯ 0),¯a(u0)k_W^1,∞ →0ass→ ∞.

Going back in the transformations, we see that for allt ∈ [0,T¯(u₀))andx∈ R, u(x,t) = (¯T(u₀)−t)⁻^p−1¹ w_u₀_,T¯(u0),¯a(u0)(y,s) = (¯T(u₀)−t)⁻^p−1¹

ϕ(y,s) +v_u₀_,T¯(u0),¯a(u0)(y,s)

where

y = x−¯a(u₀)

pT(u¯ ₀)−t ands =−log(¯T(u₀)−t).

From this identity, we see that:

The blow-up time ofu(x,t)is in factT¯(u₀);

u(x,t)blows up at the point¯a(u₀);

u(x,t)hasϕ(y,s)as blow-up profile, the same as forˆu(x,t),

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Thank you for your attention.

Existence and stability of a solution with a new prescribed behavior for a heat equation with or without a critical nonlinear gradient term