Asymptotic Stability of solitons for the ZK equation

# Asymptotic Stability of high-dimensional Zakharov-Kuznetsov solitons

Raphaël Côte Raphaël Côte, CNRS and École polytechnique Centre de Mathématiques Laurent Schwartz UMR 7640, Route de Palaiseau, 91128 Palaiseau cedex, France Claudio Muñoz Claudio Muñoz, CNRS and Laboratoire de Mathématiques d’Orsay UMR 8628, Bât. 425 Faculté des Sciences d’Orsay, Université Paris-Sud F-91405 Orsay Cedex France Didier Pilod Didier Pilod, Instituto de Matemática, Universidade Federal do Rio de Janeiro, Caixa Postal 68530, CEP: 21945-970, Rio de Janeiro, RJ, Brazil  and  Gideon Simpson Gideon Simpson, Department of Mathematics, Drexel University, 33rd and Market Streets, Philadelphia, PA 19104, USA
September 8th, 2015
###### Abstract.

We prove that solitons (or solitary waves) of the Zakharov-Kuznetsov (ZK) equation, a physically relevant high dimensional generalization of the Korteweg-de Vries (KdV) equation appearing in Plasma Physics, and having mixed KdV and nonlinear Schrödinger (NLS) dynamics, are strongly asymptotically stable in the energy space. We also prove that the sum of well-arranged solitons is stable in the same space. Orbital stability of ZK solitons is well-known since the work of de Bouard [10]. Our proofs follow the ideas by Martel [30] and Martel and Merle [35], applied for generalized KdV equations in one dimension. In particular, we extend to the high dimensional case several monotonicity properties for suitable half-portions of mass and energy; we also prove a new Liouville type property that characterizes ZK solitons, and a key Virial identity for the linear and nonlinear part of the ZK dynamics, obtained independently of the mixed KdV-NLS dynamics. This last Virial identity relies on a simple sign condition, which is numerically tested for the two and three dimensional cases, with no additional spectral assumptions required. Possible extensions to higher dimensions and different nonlinearities could be obtained after a suitable local well-posedness theory in the energy space, and the verification of a corresponding sign condition.

Partially supported by the project ERC 291214 BLOWDISOL
C.M. would like to thank the Laboratoire de Mathématiques d’Orsay for their kind hospitality
Partially supported by CNPq/Brazil, grants 302632/2013-1 and 481715/2012-6.

## 1. Introduction

We are interested in the the Zakharov-Kuznetsov (ZK) equation

 (1.1) ∂tu+∂x1(Δu+u2)=0,

where is a real-valued function, , , denotes the laplacian. The ZK equation is a particular case of the generalized Zakharov-Kuznetsov (gZK) equation

 (1.2) ∂tu+∂x1(Δu+up)=0,

where is such that if and if . We observe that when the spatial dimension is equal to , equation (1.2) becomes the well-known generalized Korteweg- de Vries (gKdV) equation.

The ZK equation was introduced by Zakharov and Kuznetsov in [22] to describe the propagation of ionic-acoustic waves in uniformly magnetized plasma in the two dimensional and three dimensional cases. The derivation of ZK from the Euler-Poisson system with magnetic field in the long wave limit was carried out by Lannes, Linares and Saut in [25]. The ZK equation was also derived by Han-Kwan [19] from the Vlasov-Poisson system in a combined cold ions and long wave limit. Moreover, the following quantities are conserved by the flow of ZK,

 (1.3) M(u)=∫u(x,t)2dx,

and

 (1.4)

The well-posedness theory for ZK and gZK has been extensively studied in the recent years. In the two dimensional case, Faminskii proved that the Cauchy problem associated to the ZK equation is globally well-posed in the energy space [13]. The local well-posedness result was pushed down to for by Linares and Pastor [27] and to by Grünrock and Herr [18] and Molinet and the third author [41]. The best result for the ZK equation in the three dimensional case was obtained last year by Ribaud and Vento [44]. They proved local well-posedness in for . Those solutions were extended globally in time in [41]. Note however that it is still an open problem to obtain well-posedness in and for the ZK equation. Finally, we also refer to [27, 28, 14, 45, 17] for more well-posedness results for the gZK equation with and to [42, 6, 7] for unique continuation results concerning ZK.

Note that if solves (1.2) with initial data , then is also a solution to (1.2) with initial data for any . Hence, , so that the scale-invariant Sobolev space for the gZK equation is , where . In particular, the gZK equation is -critical (or simply critical) if . In the sequel, we will say that the problem is subcritical if and supercritical if .

### 1.1. The elliptic problem

For , equation (1.2) admits special solutions of the form

 (1.5) u(x,t)=Qc(x1−ct,x2,⋯,xd)withQc(x)⟶0|x|→+∞,

where and satisfies

 (1.6) −ΔQ+Q−Qp=0.

Observe that .

We recall the following theorem on the elliptic PDE (1.6), which follows, for example, from the results of Berestycki and Lions [3] and Kwong [23].

###### Theorem.

Assume that if and if . Then there exists a unique positive radially symmetric solution to (1.6) in , which is called a ground state. In addition, , for all and there exists such that

 (1.7) |∂αQ(x)|≲αe−δ|x|∀x∈Rd, ∀α∈Zd+.

The solutions of (1.2) of the form (1.5) with are called solitary waves or solitons. They were proved by de Bouard in [10] to be orbitally stable in if and unstable for . In other words, the solitary waves associated to (1.2) are orbitally stable in the subcritical case and unstable in the supercritical case.

In the following, for any , we will denote by the operator which linearizes (1.6) around , i.e.,

 (1.8) Lc=−Δ+c−pQp−1c.

In the case , we also denote .

Next, we gather some well-known facts about the operator (see Weinstein [48]).

###### Theorem.

Assume that if and if . Then, the following assertions are true.

(i) is a self-adjoint operator and

 (1.9) σess(L)=[λess,+∞),for some λess>0.

(ii)

 (1.10) kerL=span{∂xjQ : j=1,⋯,d}.

(iii) has a unique single negative eigenvalue (with ) associated to a positive radially symmetric eigenfunction . Without loss of generality, we choose such that . Moreover, there exists such that , for all .

(iv) Let us define

 (1.11) ΛQ:=(ddcQc)c=1=1p−1Q+12x⋅∇Q.

Then,

 (1.12) LΛQ=−Q,

and

 (1.13) ∫QΛQ=cp,d∥Q∥2L2wherecp,d=1p−1−d4.

### 1.2. Statement of the results

As already mentioned, de Bouard proved in [10] that the solitary waves of the gZK equation are stable in the subcritical case in the following sense.

###### Theorem (Stability).

Assume that and that the Cauchy problem associated to (1.2) is well-posed in . Let . Then, there exists and such that if satisfies , the solution of (1.2) with satisfies

 supt∈Rinfτ∈Rd∥u(⋅,t)−Qc0(⋅−τ)∥H1≤K0ϵ.

The main result of this paper is the asymptotic stability of the family of solitons of (1.1) in the case . Then, we consider the stability of the multi-soliton case (see Theorem 1.7 below).

###### Theorem 1.1 (Asymptotic stability).

Assume . Let . For any , there exists such that if and is a solution of (1.1) satisfying

 (1.14) infτ∈R2∥u(⋅,t)−Qc0(⋅−τ)∥H1≤ϵ,∀t∈R,

then the following holds true.

There exist with , for some positive constant independent of , and such that

 (1.15) u(⋅,t)−Qc+(⋅−ρ(t))⟶t→+∞0in H1(x1>βt),
 (1.16) ρ′1(t)⟶t→+∞c+and% ρ′2(t)⟶t→+∞0.
###### Remark 1.1.

It will be clear from the proof that the convergence in (1.15) can also be obtained in regions of the form

 (1.17) AS(t,θ):={(x1,x2)∈R2 : x1−βt+(tanθ)x2>0}, where θ∈(−π3,π3).

Note that the maximal angle of improvement must be strictly less than on each side of the vertical line (see Figure 1). We also refer to Lemma 4.3 for more details and a relation with the nonlinear dynamics of the equation. Moreover, we expect the range of for which asymptotic stability occurs in to be sharp. Indeed, it will be shown in Appendix C that linear plane waves of ZK exist if and only if the velocity group vector has a negative -component and forms an angle with (as in Figure 1) satisfying .

###### Remark 1.2.

The angle is also related to the linear part of ZK. In [8], Carbery, Kenig and Ziesler proved that

 ∥∥|K(D)|18e−t∂x1Δφ∥∥L4xyt≲∥φ∥L2,

where is the Fourier multiplier associated to the symbol . This Strichartz estimate was used in [41] to improve the well-posed results for ZK at low regularity. Note that the multiplier cancels out along the cone . We also refer to Apendix C for an interesting relation between the angle and the linear plane waves of ZK.

###### Remark 1.3.

Our proof does not rely on the structure of the nonlinearity of (1.1) (i.e. ) neither on the dimension . Actually, our main theorem could be extended to (1.1) in dimension or to the following generalization of gZK

 (1.18) ∂tu+∂x1(Δu+|u|p−1u)=0,

where is a real number under the following conditions:

• The Cauchy problem associated to (1.1) with or to (1.18) is well-posed in .

• The spectral condition holds true. (Note that makes sense since is radial and orthogonal to , and we choose orthogonal to .)

This spectral condition was shown in the appendix to be true in dimension for , where is a real number satisfying .

On the other hand, in dimension , it is shown in the appendix that . Note however that in this case, one could try to verify the more general property: the operator restricted to the space is positive definite.

###### Remark 1.4.

The case in dimension is critical, so that solitons should be unstable (see [32] for example), and the validity of Theorem 1.1 is not clear at that level.

The proof of Theorem 1.1 is based on the following rigidity result for the solutions of (1.1) in spatial dimension around the soliton which are uniformly localized in the direction .

###### Theorem 1.2 (Nonlinear Liouville property around Qc0).

Assume . Let . There exists such that if and is a solution of (1.1) satisfying for some function and some positive constant

 (1.19) ∥u(⋅+ρ(t))−Qc0∥H1≤ϵ,∀t∈R,

and

 (1.20) ∫x2u2(x1+ρ1(t),x2+ρ2(t),t)dx2≲e−σ|x1|,∀(x1,t)∈R2,

then, there exist (close to ) and such that

 (1.21) u(x1,x2,t)=Qc1(x1−c1t−ρ01,x2−ρ02).
###### Remark 1.5.

Due to the stability result of de Bouard [10], Theorems 1.1 and 1.2 still hold true if we assume that

 (1.22) ∥u0−Qc0∥H1≤ϵ,

###### Remark 1.6.

Theorem 1.2 still holds true if we replace assumption (1.20) by the weaker assumption that the solution is -compact in the direction, i.e.:

 ∀ϵ>0, ∃A>0 such thatsupt∈R∫|x1|>Au2(x+ρ(t),t)dx≤ϵ.

We also prove a rigidity theorem for the solutions of the linearized gZK (or (1.18)) equation in spatial dimension around which are uniformly localized in the direction .

###### Theorem 1.3.

(Linear Liouville property around ) Assume . There exists such that for all , the following holds true. Let and be a solution to

 (1.23) ∂tη=∂x1Lc0ηon R2×R,

where is defined in (1.8). Moreover, assume that there exists a constant such that

 (1.24) ∫x2η2(x1,x2,t)dx2≲e−σ|x1|,∀(x1,t)∈R2.

Then, there exists such that

 (1.25) η(x,t)=a1∂x1Qc0(x)+a2∂x2Qc0(x),∀(x,t)∈R3.
###### Remark 1.7.

It will be clear from the proof (c.f. Remark 2.2) that Theorem 1.3 still holds true if we replace assumption (1.24) by the weaker assumption that the solution is -compact in the direction, i.e.: and

 ∀ϵ>0, ∃A>0 such thatsupt∈R∫|x1|>Aη2(x,t)dx≤ϵ.
###### Remark 1.8.

By using the scaling invariance of (1.1), it is enough to prove Theorems 1.3, 1.2 and 1.1 in the case where .

Recall that the first result of asymptotic stability of solitons for generalized KdV equations was proved by Pego and Weinstein [43] in weighted spaces (see also [39] for some refinements on the weights). In [31], Martel and Merle have given the first asymptotic result for the solitons of gKdV in the energy space . They improved their result in [34] and generalized it to a larger class of nonlinearities than the pure power case in [35].

Their proof relies on a Liouville type theorem for -compact solutions around a soliton (similar to Theorem 1.2 in one dimension). Then, it is proved that a solution near a soliton converges (up to subsequence) to a limit object, whose emanating solution satisfies a good decay property. Due to the rigidity result, this limit object has to be a soliton.

It is worth noting that this technique of proof was also adapted to prove asymptotic stability in the energy space for other one dimensional models such as the BBM equation [12] and the BO equation [20].

We also refer to [38, 1, 40] for stability results for KdV and mKdV in and to [4, 16] for asymptotic results for the Gross-Pitaevskii equation in one dimension. For other results on asymptotic stability for nonlinear Schrödinger and wave equations, see [47, 5, 24, 21] and references therein.

About the proofs. Comparison with previous results. When proving Theorems 1.1, 1.2 and 1.3, we generalize the ideas of Martel and Merle [31, 34, 35] and Martel [30] to a multidimensional model. However, compared with these previous results, the higher dimensional case describing the ZK dynamics presents new challenges, that we explain in the following lines.

First of all, as far as we know, our results represent the first two dimensional model where asymptotic stability is proved, in the energy space, and with no nonstandard spectral assumptions on the linearized dynamical operator. As we expressed before, we only need to check the numerical condition

 (1.26) ∫L−1ΛQΛQ<0.

Obtaining a direct proof of this result seems far from any reasonable approach because the soliton , and therefore, the function , have no closed and explicit forms. This is the first difference with respect to the one dimensional case: we work with a solitary wave that is not explicit at all.

We will see through the proofs that ZK behaves as a KdV equation in the direction, and as a nonlinear Schrödinger (NLS) equation in the variable. In particular, we are able to prove monotonicity properties (see Lemma 3.3) along the direction and along a slighted perturbed cone around the direction (Lemma 4.3). This last result is, to our knowledge, new in the literature and makes use of the geometrical properties of the nonlinear ZK dynamics around a solitary wave. Remark 1.1 and the asymptotic stability result inside the set (see (1.17))

 AS(t,θ):={(x1,x2)∈R2 : x1−βt+(tanθ)x2>0}, where θ∈(−π3,π3),

are deep consequences of these geometrical properties. Recall that such rich foliations are not present in the one dimensional case. We also complement our results by a simple linear analysis leading to the same formal conclusions, carried out in Appendix C.

Another barrier that appears in the higher dimensional case is the lack of control on the solution if we only assume bounds. We need such a control to ensure pointwise exponential decay around solitons at infinity for a compact part of the solution. In the one dimensional case, the proof of this fact is direct from the Sobolev embedding. However, since is not contained in in , we must prove new monotonicity properties at the level (cf. Lemma 3.6), which are obtained by proving new energy estimates.

No monotonicity property seems to hold for the direction, mainly because of the conjectured existence of trains of small solitons moving to the right in but without restrictions on the coordinate. From the point of view associated to the variable, such solutions represent movement of mass along the direction without a privileged dynamics. In particular, no asymptotic stability result is expected for a half-plane involving the variable only (see Fig. 2). This is the standard situation in many models like KP-I and NLS equations. However, here we are able to prove the asymptotic stability of ZK solitons because the KdV dynamics is exactly enough to control the movement of mass along the direction.

The second ingredient in the proofs of Theorems 1.1, 1.2 and 1.3 is a new virial identity in higher dimensions (cf. (2.36)), which holds only for the half space and for and slightly larger. Compared with the previous works by Martel and Merle, the additional dimensions make things harder because they induce transversal variations that seem to destroy any virial-type inequality. In order to overcome this difficulty, we use a different orthogonality condition for the function employed in the virial (see Lemma 2.3):

 (1.27) ∫vΛQ=0.

We emphasize that this condition is somehow natural and necessary if we want to get full control of the perturbations appearing from the variations of the virial terms. Without using this modified condition, any form of two-dimensional virial identity is no longer true.

Here is when the nonstandard spectral condition (1.26) appears: under the orthogonality condition (1.27), the virial identity holds provided (1.26) is satisfied. We prove that (1.26) holds for the case and , as expressed by some numerical computations obtained in Appendix A. This condition in fact generalizes the Martel one in [30] and seems to be the natural one for the case, as described when proving the nonlinear stability result (see (3.60)-(3.61) for instance). It is worth noting that this condition has already been used by Kenig and Martel in the Benjamin-Ono context [20], for different reasons. For powers of the nonlinearity which are definitely larger than , or just the three dimensional case for , we have a strong instability effect at the level of the previous spectral theory, probably associated to the dynamics around the soliton in the variable, and the virial identity seems no longer to hold.111In the one dimensional case, this instability condition does not appear, see Martel [30]. Once again, a good understanding of the dynamics for powers close the the critical case or supercritical as in [30] needs a deep extension of (2.36) by incorporating now the dynamics in the variable, which could be very complicated, in view of some results by del Pino et al. [11]. The extension of the ideas introduced by Martel [30] to any power of seems a very interesting problem.

Finally, we mention that another crucial application of the monotonicity formula on perturbed cones, needed in the higher dimensional case, is given in Lemma 4.4. Here, a new compact region of the plane is introduced, outside of which we prove exponential decay. This set is constructed in order to prove the strong convergence of sequences of bounded solutions, thanks to the use of the Sobolev embedding theorem.

One can also ask for the nonlinear dynamics in the remaining part of the plane, namely the region , see (1.17). We believe that in addition to radiation, one can find small solitons moving to the right in a very slow fashion. No finite energy solitary waves with speed along the direction are present, as shows the following (general) definition and result. As usual, we define the symbol by using its corresponding Fourier representation .

###### Definition 1.4.

We say that , is the profile of a solitary wave of speeds if

 u(x1,x2,⋯,xd,t):=v(x1−c1t,x2−c2t,⋯,xd−cdt),v≢0,

is solution of (1.2).

Note that such a must satisfy the equation in

 (1.28) Δv−c1v+vp−d∑j=2cj∂−1x1∂xjv=0.
###### Theorem 1.5.

Assume that for some . Then (1.28) has no finite energy solutions.

We prove this result in Appendix D, using adapted Pohozahev identities.

Finally, as a consequence of the monotonicity properties associated to the linear part of the dynamics, in particular, using Lemma 2.1, we are able to prove the stability of the sum of essentially non-colliding solitons.

###### Definition 1.6.

Let be an integer and . Consider solitons with scalings and centers , where . We say that these solitons are -decoupled if

 (1.29) inf{∣∣((c0k−c0j)t,0)+ρk,0−ρj,0∣∣∣j≠k, t≥0}≥L,

that is, the solitons centers remains separated by a distance of at least for positive times. (See Fig. 3 below.)

-decoupled solitons can be characterized by a condition on the initial data only, at least up to a constant in : indeed, one can check that if, for all , we have either:

• , or

• and ,

then the solitons are -decoupled.

###### Theorem 1.7 (Stability of the sum of N decoupled solitons).

Assume . Consider a set of solitons of the form

 Qc01(x−ρ1,0),Qc02(x−ρ2,0),…,Qc0N(x−ρN,0),

where each is a fixed positive scaling, for all , and . Assume that the solitons are -decoupled, in the sense of Definition 1.6. Then there are , and depending on the previous parameters such that, for all , and for every , the following holds. Suppose that satisfies

 (1.30) ∥u0−N∑j=1Qc0j(x−ρj,0)∥H1<ε.

Then there are fixed and defined for all such that , solution of (1.1) with initial data satisfies

 (1.31) supt≥0∥u(t)−N∑j=1Qc0j(x−ρj(t))∥H1

The proof of this result is obtained by adapting the ideas by Martel, Merle and Tsai [36] for the generalized, one dimensional KdV case. Note that we do not need strictly well-prepared initial data as in [36]. Instead, from (1.29) we just need sufficient well-separated solitons in the variable (no particular order at the beginning), and in the case where solitons have the same coordinate, we ask for well-ordered solitons to avoid multi-collisions.

The rest of the paper is organized as follows. The linear and nonlinear Liouville properties (Theorems 1.3 and 1.2) are proved respectively in Section 2 and 3. The nonlinear Liouville property is used to show Theorem 1.1 in Section 4. Section 5 is devoted to the proof of Theorem 1.7. Finally, in Appendix A, we present some numerical computations which establish the negativity of a scalar product in the case . Recall that this condition is a crucial element in the proofs of the rigidity results (Theorems 1.2 and 1.3). We also make an interesting observation about the plane wave solutions of the linear part of ZK in Appendix C and give the proof of Theorem 1.5 in Appendix D.

## 2. Linear Liouville property

This section is devoted to the proof of Theorem 1.3. According to Remark 1.8, we will assume in this section that .

### 2.1. Monotonicity

In this subsection, we prove a monotonicity formula for the solutions of (1.23) satisfying (1.24).

Let denote a positive number such that . We define by

 (2.1) ψL(y)=2πarctan(ey/L),

so that and . Note also that

 (2.2) ψ′L(y)=1πLcosh(y/L)and|ψ′′′(y)|≤1L2ψ′L(y)≤116ψ′L(y).
###### Lemma 2.1.

Let , and for . Let be a solution of (1.23) satisfying condition (1.24). Define , where is the positive number in (1.7). Then

 (2.3) ∫(∂αη)2(x,t0)ψL(x1−y0)dx+∫t0−∞∫(|∇∂αη|2+(∂αη)2)(x,t)ψ′L(~x1)dxdt≲Le−y0/L,

for all multi-index and all positive number .

###### Remark 2.1.

The implicit constant appearing in (2.3) depends only on . In particular it does not depend on and .

###### Remark 2.2.

It will be clear from the proof below that Lemma 2.1 and thus Theorem 1.3 still hold true if we replace (1.24) by the weaker assumption that the solution is -compact in the direction, i.e.: and

 ∀ϵ>0, ∃A>0 such thatsupt∈R∫|x1|>Aη2(x,t)dx≤ϵ.
###### Proof.

First, we prove (2.3) for . Fix . Observe from condition (1.24) that

 (2.4) supt∈R∫η(x,t)2dx≤C.

By using the equation (1.23), integrations by parts and the inequality in (2.2), we compute that

 (2.5) ddt∫η2ψL(~x1)dx=2∫η∂tηψL(~x1)dx+12∫η2ψ′L(~x1)dx≤−∫(3(∂x1η)2+(∂x2η)2)ψ′L(~x1)dx−14∫η2ψ′L(~x1)dx+p∫η2(−∂x1(Qp−1)ψL(~x1)+Qp−1ψ′L(~x1))dx.

To deal with the last term appearing on the right-hand side of (2.5), let us define

 (2.6) T0(η)=p∫η2(−∂x1(Qp−1)ψL(~x1)+Qp−1ψ′L(~x1))dx.

We claim that

 (2.7) ∣∣T0(η)∣∣≤Ce−(12(t0−t)+y0)/L+18∫η2ψ′L(~x1)dx.

To prove (2.7), we argue as in Lemma 5 in [30]. Recall from (1.7) and (2.1) that

 ∣∣∂x1(Qp−1)ψL(~x1)∣∣+∣∣Qp−1ψ′L(~x1)∣∣≲e−δ(p−1)|x|ψL(~x1)≲e−δ(p−1)|x1|ψL(~x1),

where is the positive constant appearing in (1.7). Let to be fixed later. We consider the three following cases.

Case: . Then , so that

 e−δ(p−1)|x1|ψL(~x1)≲ψL(~x1)≤e~x1/L≤e(R1−12(t0−t)−y0)/L.

Case: . Then, we get that

 e−δ(p−1)|x1|ψL(~x1)≲e−δ(p−1)x1ψ′L(~x1)≤e−δ(p−1)R1ψ′L(~x1),

since for .

Case: . In this case

 e−δ(p−1)|x1|ψL(~x1)≲e−δ(p−1)(12(t0−t)+y0)≤e−(12(t0−t)+y0)/L,

since .

We deduce then that

 ∣∣T0(η)∣∣≤Ce(R1−12(t0−t)−y0)/L∫η2dx+Ce−δ(p−1)R1∫η2ψ′L(~x1)dx,

which yields estimate (2.7) by using (2.4) and fixing large enough so that .

Thus, we conclude gathering (2.5)–(2.7) and integrating between and that

 (2.8) ∫η2(x,t0)ψL(x1−y0)dx+18∫t0t∫(|∇η|2+η2)(x,s)ψ′L(~x1)dxds≲e−y0/L+∫η2(x,t)ψL(x1−12(t0−t)−y0)dx,

for all and . To handle the second term of the right-hand side of (2.8), we use the fact that satisfies condition (1.24). Given , there exists such that

 (2.9) ∫|x1|>Aη2(x,t)dx≤∫|x1|>Ae−σ|x1|dx1≤ϵ.

On the other hand, it follows from (2.4) that

 (2.10) ∫|x1|≤Aη2(x,t)ψL(~x1)dx≤ψL(A−12(t0−t)−y0)∫η2(x,t)dx⟶t→−∞0.

Therefore, we conclude the proof of (2.3) in the case by using (2.9)–(2.10) and sending to in (2.8).

Next, we prove (2.3) in the general case by induction on . Let be such that . Assume that estimate (2.3) is true for all such that . Let be such that . Arguing as in (2.5), we get that

 (2.11) ddt∫(∂αη)2ψL(~x1)dx≤−∫(|∇∂αη|2+14(∂αη)2)ψ′L(~x1)dx+Tα(η),

where

 (2.12) Tα(η)=−2p∫∂x1∂α(Qp−1η)∂αηψL(~x1)dx.

By using the Leibniz rule and integrations by parts, we get that

 (2.13) Tα(η)=∑0≤β≤αCβ∫∂x1(∂β(Qp−1)