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Abstract

We consider the semilinear heat equation

 ∂tu=Δu+|u|p−1ulnα(u2+2),

in the whole space , where and . Unlike the standard case , this equation is not scaling invariant. We construct for this equation a solution which blows up in finite time only at one blowup point , according to the following asymptotic dynamics:

 u(x,t)∼ψ(t)(1+(p−1)|x−a|24p(T−t)|ln(T−t)|)−1p−1 as t→T,

where is the unique positive solution of the ODE

 ψ′=ψplnα(ψ2+2),limt→Tψ(t)=+∞.

The construction relies on the reduction of the problem to a finite dimensional one and a topological argument based on the index theory to get the conclusion. By the interpretation of the parameters of the finite dimensional problem in terms of the blowup time and the blowup point, we show the stability of the constructed solution with respect to perturbations in initial data. To our knowledge, this is the first successful construction for a genuinely non-scale invariant PDE of a stable blowup solution with the derivation of the blowup profile. From this point of view, we consider our result as a breakthrough.

Blowup solution, Blowup profile, Stability, Semilinear heat equation, non-scaling invariant heat equation

A non-scaling invariant semilinear heat equation]Construction of a stable blowup solution with a prescribed behavior for a non-scaling invariant semilinear heat equation G. K. Duong, V. T. Nguyen, H. Zaag ] \subjclassPrimary: 35K55, 35B40; Secondary: 35L65, 35K57. G. K. Duong]duong@univ-paris13.fr V. T. Nguyen]Tien.Nguyen@nyu.edu H. Zaag] Hatem.Zaag@univ-paris13.fr thanks: —————–
July 3, 2019

Giao Ky Duong 111 G. K. Duong is fully funded by the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 665850., Van Tien Nguyen and Hatem Zaag 222 H. Zaag is supported by the ANR projet ANAÉ ref. ANR -13-BS01-0010-03

Université Paris 13, Sorbonne Paris Cité, LAGA, CNRS (UMR 7539), F-93430, Villetaneuse, France.

New York University in Abu Dhabi, P.O. Box 129188, Abu Dhabi, United Arab Emirates.

1 Introduction.

We are interested in the semilinear heat equation

 {∂tu=Δu+F(u),u(0)=u0∈L∞(Rn), (1.1)

where stands for the Laplacian in and

 F(u)=|u|p−1ulnα(u2+2),p>1,α∈R. (1.2)

By standard results the model (1.1) is wellposed in thanks to a fixed-point argument. More precisely, there is a unique maximal solution on with . If , then the solution of (1.1) may develop singularities in finite time , in the sense that

 ∥u(t)∥L∞→+∞ as t→T.

In this case, is called the blowup time of . Given , we say that is a blowup point of if and only if there exists as such that as .

In the special case the equation (1.1) becomes the standard semilinear heat equation

 ∂tu=Δu+|u|p−1u. (1.3)

This equation is invariant under the following scaling transformation

 u↦uλ(x,t):=λ2p−1u(λx,λ2t). (1.4)

An extensive literature is devoted to equation (1.3) and no rewiew can be exhaustive. Given our interest in the construction question with a prescribed blowup behavior, we only mention previous work in this direction.

In [Bricmont and Kupiainen(1994)], Bricmont and Kupiainen showed the existence of a solution of (1.3) such that

 ∥(T−t)1p−1u(a+z√(T−t)|ln(T−t)|,t)−φ0(z)∥L∞(Rn)→0, as t→T, (1.5)

where

 φ0(z)=(p−1+(p−1)2z24p)−1p−1,

(note that Herrero and Velázquez [Herrero and Velázquez(1992)] proved the same result with a different method; note also that Bressan [Bressan(1992)] made a similar construction in the case of an exponential nonlinearity).

Later, Merle and Zaag [Merle and Zaag(1997)] (see also the note [Merle and Zaag(1996)]) simplified the proof of [Bricmont and Kupiainen(1994)] and proved the stability of the constructed solution verifying the behavior (1.5). Their method relies on the linearization of the similarity variables version around the expected profile. In that setting, the linearized operator has two positive eigenvalues, then a non-negative spectrum. Then, they proceed in two steps:

• Reduction of an infinite dimensional problem to finite dimensional one: they show that controlling the similarity variable version around the profile reduces to the control of the components corresponding to the two positive eigenvalues.

• Then, they solve the finite dimensional problem thanks to a topological argument based on index theory.

The method of Merle and Zaag [Merle and Zaag(1997)] has been proved to be successful in various situations. This was the case of the complex Ginzgburg-Landau equation by Masmoudi and Zaag [Masmoudi and Zaag(2008)] (see also Zaag [Zaag(1998)] for an ealier work) and also for the case of a complex semilinear heat equation with no variational structure by Nouaili and Zaag [Nouaili and Zaag(2015)]. We also mention the work of Tayachi and Zaag [Tayachi and Zaag(2016)] (see also the note [Tayachi and Zaag(2015)]) and the work of Ghoul, Nguyen and Zaag [Ghoul et al.(2017)Ghoul, Nguyen, and Zaag] dealing with a nonlinear heat equation with a double source depending on the solution and its gradient in a critical way. In [Ghoul et al.(2016)Ghoul, Nguyen, and Zaag], Ghoul, Nguyen and Zaag successfully adapted the method to construct a stable blowup solution for a non variational semilinear parabolic system.

In other words, the method of [Merle and Zaag(1997)] was proved to be efficient even for the case of systems with non variational structure. However, all the previous examples enjoy a common scaling invariant property like (1.4), which seemed at first to be a strong requirement for the method. In fact, this was proved to be untrue.

As matter of fact, Ebde and Zaag [Ebde and Zaag(2011)] were able to adapt the method to construct blowup solutions for the following non scaling invariant equation

 ∂tu=Δu+|u|p−1u+f(u,∇u), (1.6)

where

 |f(u,∇u)|≤C(1+|u|q+|∇u|q′), with q

These conditions ensure that the perturbation turns out to exponentially small coefficients in the similarity variables. Later, Nguyen and Zaag [Nguyen and Zaag(2016b)] did a more spectacular achievement by addressing the case of stronger perturbation of (1.3), namely

 ∂tu=Δu+|u|p−1u+μ|u|p−1ulna(2+u2), (1.7)

where and . When moving to the similarity variables, the perturbation turns out to have a polynomial decay. Hence, when is small, we are almost in the case of a critical perturbation.

In both cases addressed in [Ebde and Zaag(2011)] and [Nguyen and Zaag(2016b)], the equations are indeed non-scaling invariant, which shows the robustness of the method. However, since both papers proceed by perturbations around the standard case (1.3), it is as if we are still in the scaling invariant case.

In this paper, we aim at trying the approach on a genuinely non-scaling invariant case, namely equation (1.1). This is our main result.

Theorem 1.1 (Blowup solutions for equation (1.1) with a prescribed behavior).

There exists an initial data such that the corresponding solution to equation (1.1) blows up in finite time only at the origin. Moreover, we have

• For all , there exists a positive constant such that

 ∥∥ ∥∥ψ−1(t)u(x,t)−f0(x√(T−t)|ln(T−t)|)∥∥ ∥∥L∞(Rn)≤C0√|ln(T−t)|, (1.8)

where is the unique positive solution of the following ODE

 ψ′(t)=ψp(t)lnα(ψ2(t)+2),limt→Tψ(t)=+∞, (1.9)

(see Lemma A.1 for the existence and uniqueness of ), and the profile is defined by

 f0(z)=(1+(p−1)4p|z|2)−1p−1. (1.10)
• There exits such that uniformly on compact sets of , where

 u∗(x)∼[(p−1)2|x|28p|ln|x||]−1p−1(4|ln|x||p−1)−αp−1 as x→0, (1.11)
Remark 1.2.

From , we see that as , which means that the solution blows up in finite time at . From , we deduce that the solution blows up only at the origin.

Remark 1.3.

Note that the behavior in (1.8) is almost the same as in the standard case treated in [Bricmont and Kupiainen(1994)] and [Merle and Zaag(1997)]. However, the final profile has a difference coming from the extra multiplication of the size , which shows that the nonlinear source in equation (1.1) has a strong effect to the dynamic of the solution in comparing with the standard case .

Remark 1.4.

Item is in fact a consequence of (1.8) and Lemma A.4. Therefore, the main goal of this paper is to construct for equation (1.1) a solution blowing up in finite time and verifying the behavior (1.8).

Remark 1.5.

By the parabolic regularity, one can show that if the initial data , then we have for ,

 ∥∥ ∥∥ψ−1(t)(T−t)i2∇ixu(x,t)−(T−t)i2∇ixf0(x√(T−t)|ln(T−t)|)∥∥ ∥∥L∞≤C√|ln(T−t)|,

where is defined by (1.10).

From the technique of Merle [Merle(1992)], we can prove the following result.

Corollary 1.6.

For arbitrary given set of points . There exists initial data such that the solution of (1.1) with initial data blows up exactly at points . Moreover, the local behavior at each blowup point is also given by (1.8) by replacing by .

As a consequence of our technique, we prove the stability of the solution constructed in Theorem 1.1 under the perturbations of initial data. In particular, we have the following result.

Theorem 1.7 (Stability of the solution constructed in Theorem 1.1).

Consider the solution constructed in Theorem 1.1 and denote by its blowup time. Then there exists a neighborhood of such that for all , equation (1.1) with the initial data has a unique solution blowing up in finite time at a single point . Moreover, the statements and in Theorem 1.1 are satisfied by , and

 (T(u0),a(u0))→(^T,0) as ∥u0−^u0∥L∞(Rn)→0. (1.12)
Remark 1.8.

We will not give the proof of Theorem 1.7 because the stability result follows from the reduction to a finite-dimensional case as in [Merle and Zaag(1997)] with the same proof. Here we only prove the existence and refer to [Merle and Zaag(1997)] for the stability.

2 Formulation of the problem.

In this section, we first use the matched asymptotic technique to formally derive the behavior (1.8). Then, we give the formulation of the problem in order to justify the formal result.

2.1 A formal approach.

In this part, we follow the approach of Tayachi and Zaag [Tayachi and Zaag(2016)] to formally explain how to derive the asymptotic behavior (1.8). To do so, we introduce the following self-similarity variables

 u(x,t)=ψ(t)w(y,s),y=x√T−t,s=−ln(T−t), (2.1)

where is the unique positive solution of equation (1.9) and as . Then, we see from (1.1) that solves the following equation: for all

 ∂sw=Δw−12y.∇w−h(s)w+h(s)|w|p−1wlnα(ψ21w2+2)lnα(ψ21+2), (2.2)

where

 h(s)=e−sψp−11(s)lnα(ψ21(s)+2), (2.3)

and

 ψ1(s)=ψ(T−e−s). (2.4)

Note that admits the following asymptotic behavior as ,

 (2.5)

(see ii) of Lemma A.5 for the proof of (2.5)). From (2.1), we see that the study of the asymptotic behavior of as is equivalent to the study of the long time behavior of as . In other words, the construction of the solution , which blows up in finite time and verifies the behavior (1.8), reduces to the construction of a global solution for equation (2.2) satisfying

 0<ϵ0≤limsups→+∞∥w(s)∥L∞(Rn)≤1ϵ0,ϵ0>0, (2.6)

and

 ∥∥ ∥ ∥∥w(y,s)−(1+(p−1)y24ps)−1p−1∥∥ ∥ ∥∥L∞(Rn)→0 as s→+∞. (2.7)

In the following, we will formally explain how to derive the behavior (2.7).

2.1.1 Inner expansion

We remark that are the trivial constant solutions to equation (2.2). Since we are looking for a non zero solution, let us consider the case when as . We now introduce

 w=1+¯w, (2.8)

then from equation (2.2), we see that satisfies

 ∂s¯w=L(¯w)+N(¯w,s), (2.9)

where

 L =Δ−12y.∇+\textupId, (2.10) N(¯w,s) =h(s)|¯w+1|p−1(¯w+1)lnα(ψ21(¯w+1)2+2)lnα(ψ21+2)−h(s)(¯w+1)−¯w, (2.11)

is defined in (2.4) and behaves as in (2.5). Note that admits the following asymptotic behavior,

 N(¯w,s)=p¯w22+O(|¯w|lnss2)+O(|¯w|2s)+O(|¯w|3) as (¯w,s)→(0,+∞), (2.12)

(see Lemma A.6 for the proof of this statement).

Since as and the nonlinear term is quadratic in , we see from equation (2.9) that the linear part will play the main role in the analysis of our solution. Let us recall some properties of . The linear operator is self-adjoint in , where is the weighted space associated with the weight defined by

 ρ(y)=e−|y|24(4π)n2,

and

 \textupspec(L)={1−m2,m∈N}.

More precisely, we have

• When , all the eigenvalues of are simple and the eigenfunction corresponding to the eigenvalue is the Hermite polynomial defined by

 hm(y)=[m2]∑j=0(−1)jm!ym−2jj!(m−2j)!. (2.13)

In particular, we have the following orthogonality

 ∫Rhihjρdy=i!2iδi,j,∀(i,j)∈N2.
• When , the eigenspace corresponding to the eigenvalue is defined as follows

 Em={hβ=hβ1⋯hβn, for % all β∈Nn,|β|=m,|β|=β1+⋯+βn}. (2.14)

Since the eigenfunctions of is a basic of , we can expand in this basic as follows

 ¯w(y,s)=∑β∈Nn¯wβ(s)hβ(y).

For simplicity, let us assume that is radially symmetric in . Since with corresponds to negative eigenvalues of , we may consider the solution taking the form

 ¯w=¯w0+¯w2(s)(|y|2−2n), (2.15)

where and go to as . Injecting (2.15) and (2.12) into (2.9), then projecting equation (2.9) on the eigenspace with and

 ⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩¯w′0=¯w0+p2(¯w20+8n¯w22)+O((|¯w0|+|¯w2|)lnss2)+O(|¯w0|2+|¯w2|2s)+O(|¯w0|3+|¯w2|3),¯w′2=4p¯w22+p¯w0¯w2+O((|¯w0|+|¯w2|)lnss2)+O(|¯w0|2+|¯w2|2s)+O(|¯w0|3+|¯w2|3), (2.16)

as . we now assume that as , then (2.17) becomes

 ⎧⎪⎨⎪⎩¯w′0=¯w0+O(|¯w2|2)+O(|¯w2|lnss2),¯w′2=4p¯w22+o(|¯w2|2)+O(|¯w2|lnss2), as s→+∞. (2.17)

We consider the following cases:

- Case 1: Either or as , then the second equation in (2.17) becomes

 ¯w′2=O(|¯w2|lnss2) as s→+∞,

which yields

 ln|¯w2|=O(lnss) as s→+∞,

which contradicts with the condition as .

- Case 2: as , then (2.17) becomes

 {¯w′0=¯w0+O(|¯w2|2),¯w′2=4p¯w22+o(|¯w2|2), % as s→+∞.

This yields

 ⎧⎪⎨⎪⎩¯w0=O(1s2),¯w2=−14ps+o(1s), as s→+∞. (2.18)

Substituting (2.18) into (2.17) yields

from which we improve the error for as follows

 ⎧⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪⎩¯w0=O(1s2),¯w2=−14ps+O(ln2ss2), as s→+∞. (2.19)

Hence, from (2.8), (2.15) and (2.19), we derive

 w(y,s)=1−y24ps+n2ps+O(ln2ss2), (2.20)

in as . Note that the asymptotic expansion (2.20) also holds for all , K is an arbitrary positive number.

2.1.2 Outer expansion.

The asymptotic behavior of (2.20) suggests that the blowup profile depends on the variable

 z=y√s,

From (2.20), let us try to search a regular solution of equation (2.2) of the form

 w(y,s)=ϕ0(z)+n2ps+o(1s) in L∞loc as s→+∞, (2.21)

where is a bounded, smooth function to be determined. From (2.20), we impose the condition

 ϕ0(0)=1. (2.22)

Since is supposed to be bounded, we obtain from Lemma A.7 that

 ∣∣ ∣∣h(s)|w|p−1wlnα(ψ21w2+2)lnα(ψ21+2)−|w|p−1wp−1∣∣ ∣∣=O(1s),

Note also that

 ∣∣∣∣∣∣ϕ0(z)+O(1s)∣∣∣p−1(ϕ0(z)+O(1s))−|ϕ0(z)|p−1ϕ0(z)∣∣∣=O(1s).

Hence, injecting (2.21) into equation (2.2) and comparing terms of order for , we derive the following equation for ,

 −12z.∇ϕ0(z)−ϕ0(z)p−1+|ϕ0|p−1ϕ0(z)p−1=0,∀z∈Rn. (2.23)

Solving (2.23) with condition (2.22), we obtain

 ϕ0(z)=(1+c0|z|2)−1p−1, (2.24)

for some constant (since we want to be bounded for all ). From (2.21), (2.24) and a Taylor expansion, we obtain

 w(y,s)=1−c0y2(p−1)s+n2ps+o(1s),∀|y|≤K as s→+∞,

from which and the asymptotic behavior (2.20), we find that

 c0=p−14p.

In conclusion, we have just derived the following asymptotic profile

 w(y,s)∼φ(y,s) as s→+∞, (2.25)

where

 φ(y,s)=(1+(p−1)y24ps)−1p−1+n2ps. (2.26)

2.2 Formulation of the problem.

In this subsection, we set up the problem in order to justify the formal approach presented in the Section 2.1. In particular, we give a formulation to prove item of Theorem 1.1. We aim at constructing for equation (1.1) a solution blowing up in finite time only at the origin and verifying the behavior (1.8). In the similarity variables (2.1), this is equivalent to the construction of a solution for equation (2.2) defined for all and satisfying (2.7). The formal approach given in subsection 2.1 (see (2.25)) suggests to linearize around the profile function defined by (2.26). Let us introduce

 q(y,s)=w(y,s)−φ(y,s), (2.27)

where is defined by (2.26). From (2.2), we see that satisfies the equation

 ∂sq=Lq+Vq+B(q)+R(y,s)+D(q,s), (2.28)

where is the linear operator defined by (2.10) and

 V =pp−1[φp−1−1], (2.29) B(q) =|q+φ|p−1(q+φ)−φp−pφp−1qp−1, (2.30) R(y,s) =Δφ−12y∇φ−φp−1+φpp−1−∂sφ, (2.31) D(q,s) =(q+φ)((h(s)−1p−1)(|q+φ|p−1−1)+h(s)|q+φ|p−1(q+φ)L(q+φ,s)), (2.32) L(v,s) =2αψ21ln(ψ21+2)(ψ21+2)(v−1)+1lnα(ψ21+2)∫v1f′′(u)(v−u)du, (2.33)

with and being defined by (2.3), (2.4) and (2.26) respectively, and

 f(z)=lnα(ψ21z2+2),z∈R.

Hence, proving (1.8) now reduces to construct for equation (2.28) a solution such that

 lims→+∞∥q(s)∥L∞→0.

Since we construct for equation (2.28) a solution verifying as , and the fact that

 |B(q)|≤C|q|min(2,p),∥R(s)∥L∞+∥D(q,s)∥L∞≤Cs,

(see Lemmas A.8, A.9 and A.10 for these estimates), we see that the linear part of equation (2.28) will play an important role in the analysis of the solution. The property of the linear operator has been studied in previous section (see page 2.13), and the potential has the following properties:

Perturbation of effect of inside the blowup region :

 ∥V(s)∥L2ρ→0ass→+∞.

For each , there exist and such that

 supy√s≥Kϵ,s≥sϵ∣∣∣V(y,s)+pp−1∣∣∣≤ϵ.

Since is the biggest eigenvalue of , the operator behaves as one with with a fully negative spectrum outside blowup region , which makes the control of the solution in this region easily.

Since the behavior of the potential inside and outside the blowup region is different, we will consider the dynamics of the solution for and for separately for some K to be fixed large. We introduce the following function

 χ(y,s)=χ0(|y|K√s), (2.34)

where and

 χ0(x)={1 for x≤1,0 for x≥2,

and is a positive constant to be fixed large later. We now decompose by

 q=χq+(1−χ)q=qb+qe. (2.35)

(Note that and ). Since the eigenfunctions of span the whole space , let us write

 qb(y,s)=q0(s)+q1(s)⋅y+12yT⋅q2(s)⋅y−\textuptr(q2(s))+q−(y,s), (2.36)

where and

 ∀β∈Nn,qβ(s)=∫Rnqb(y,s)~hβ(y)ρdy,~hβ=hβ∥hβ∥2L2β, (2.37)

and

 q−(y,s)=∑β∈Nn,|β|≥3qβ(s)hβ(y). (2.38)

In particular, we denote and is a symmetric matrix defined explicitly by

 q2(s)=∫qbM(y)ρdy=(q2,i,j)1≤i,j≤n, (2.39)

with

 M={14yiyj−δi,j2}1≤i,j≤n. (2.40)

Hence, by (2.35) and (2.36), we can write

 q(y,s)=q0(s)+q1(s)⋅y+12yT⋅q2(s)⋅y−\textuptr(q2(s))+q−(y,s)+qe(y,s). (2.41)

Note that and are the components of , and not those of .

3 Proof of the existence assuming some technical results.

In this section, we shall describe the main argument behind the proof of Theorem 1.1. To avoid winding up with details, we shall postpone most of the technicalities involved to the next section. According to the transformations (2.1) and (2.27), proving of Theorem 1.1 is equivalent to showing that there exists an initial data at the time such that the corresponding solution of equation (2.28) satisfies

 ∥q(s)∥L∞(Rn)→0 as s→+∞.

In particular, we consider the following function

 ψd0,d1(y)=As20(d0+d1.y)χ(2y,s0), (3.1)

as the initial data for equation (2.28), where are the parameters to be determined, and are constants to be fixed large enough, and is the function defined by (2.34).

We aim at proving that there exists such that the solution of (2.28) with initial data satisfies

 ∥qd0,d1(s)∥L∞→0 as s→+∞.

More precisely, we will show that there exists such that the solution belongs to the shrinking set defined as follows:

Definition 3.1 (A shrinking set to zero).

For all we define being the set of all functions such that

 |q0|≤As2,|q1,i|≤As2,|q2,i,j|≤A2ln2ss2,∀1≤i,j≤n, ∥∥ ∥∥q−(y)1+|y|3∥∥ ∥∥L∞(Rn)≤As2,∥qe(y)∥L∞(Rn)≤A2√s,

where , , , and are defined as in (2.41).

We also denote by being the set

 ^SA(s)=[−As2,As2]×[−As2,As2]n. (3.2)
Remark 3.2.

For each , we have the following estimates for all :

 |q(y,s)|≤CA2ln2ss2(1+|y|3),∀y∈Rn, (3.3)
 ∥q(s)∥L∞({|y|≤2K√s})≤CA√s, (3.4)
 ∥q(s)∥L∞(Rn)≤CA2√s. (3.5)

We aim at proving the following central proposition which implies Theorem 1.1.

Proposition 3.3 (Existence of a solution trapped in SA(s)).

There exists such that for all there exists such that for all , there exists such that the solution of (2.28) with the initial data at the time given by , where is defined as in (3.1), satisfies

 q(s)∈SA(s),∀s∈[s0,+∞).

From (3.5), we see that once Proposition 3.3 is proved, item of Theorem 1.1 directly follows. In the following, we shall give all the main arguments for the proof of this proposition assuming some technical results which are left to the next section.

As for the initial data at time defined as in (3.1), we have the following properties.

Proposition 3.4 (Properties of the initial data (3.1)).

For each , there exists such that for all we have the following properties:

• There exists such that the mapping

 Φ1:R1+n → R1+n, (d0,d1) ↦