Construction of solutions to the subcritical gKdV equations with a given asymptotical behavior.

Construction of solutions to the subcritical gKdV equations with a given asymptotical

behavior.

Raphaël Côte

Abstract

We consider the sub-critical generalized Korteweg-de Vries equation : ut+ (uxx+u⁴)x= 0, t, x∈R.

LetRj(t, x) = Qc_j(x−xj−cjt) (j = 1, . . . , N) be N soliton solutions to this equation. DenoteU(t) the KdV linear group, and letV be in an adequate weighted Sobolev space.

We construct a solution u(t) to the generalized Korteweg-de Vries equation such that :

t→∞lim

‚

‚

‚u(t)−U(t)V −

N

X

j=1

Rj(t)

‚

‚

‚H¹

= 0.

1 Introduction

1.1 General setting

We consider the following sub-critical generalized Korteweg-de Vries equation : ut+ (uxx+u⁴)x= 0, t, x∈R. (1) It is a special case of the generalized Korteweg-de Vries equation :

ut+ (uxx+u^p)x= 0, t, x∈R, (2) wherep≥2. The casep= 2corresponds to the original equation introduced by Korteweg and de Vries [9] in the context of shallow water waves. For bothp= 2 and p= 3, this equation has many applications to Physics : see for example Miura [21], Lamb [11].

There are two formally conserved quantities for solutions to (2) : Z

u²(t) = Z

u²(0) (L² mass), (3)

E(u(t)) = 1 2

Z

u²_x(t)− 1 p+ 1

Z

u^p+1(t) =E(u(0)) (energy). (4)

2000 Mathematical Subject Classification : Primary 35Q53, Secondary 35B40, 35Q51.

Keywords : Generalized Korteweg-de Vries equations, wave operator, large data, solitons, linear-non linear interaction.

The local Cauchy problem for (2) has been intensively studied by many authors.

Kenig, Ponce and Vega [7] proved the following existence and uniqueness result in H¹(R) : for u0 ∈ H¹(R), there exist T = T(ku0k_H1) > 0 and a solution u∈C([0, T], H¹(R))to (1) satisfyingu(0) =u0, which is unique in some class YT ⊂C([0, T], H¹(R)). For such a solution, one has conservation of mass and energy. Moreover, ifT1denotes the maximal time of existence foru, then either T1 = +∞ (global solution) or T1 <∞ and ku(t)k_H1 → ∞as t↑ T1 (blow-up solution).

In the case2 ≤p < 5, all solutions to (2) in H¹ are global and uniformly bounded thanks to the conservation laws and the Gagliardo-Nirenberg inequal- ity :

∀v∈H¹(R), Z

|v|^p+1≤C(p) Z

v²

^p+3₄ Z v²_x

^p−1₄

. (5)

The casep= 5isL²-critical, in the sense that the mass remains unaffected by scaling. If

ut+ (uxx+u⁵)x= 0, t, x,∈R, (6) then u_λ(t, x) =λ^1/6u(λt, λ^1/3x)is also a solution to (6), andku_λk_L2 =kuk_L2. In this case, the local existence result of [7] is improved to initial data in L² (instead ofH¹). However, existence of finite time blow-up solutions was proved by Merle [20] and Martel and Merle [17]. Therefore p = 5 also appears as a critical exponent for the long time behavior of solution to (2).

A fundamental property of (2) is the existence of a family of explicit traveling wave solutions. IfQdenotes the only solution (up to translation) of :

Q >0, Q∈H¹(R), Qxx+Q^p=Q, i.e. Q(x) = p+ 1 2 cosh²(^p−1₂ x)

!_p−1¹ , then forc >0 the soliton

Rc,x₀ =c^p−11 Q(√

c(x−x0−ct))is a solution to (2).

For p= 2 and p = 3, equation (2) is completely integrable, and thus has very special features. The inverse scattering transform method allows to solve the Cauchy problem in an appropriate space (for example if u₀ ∈ H⁴ and xu₀ ∈ L¹) and the qualitative behaviour of solutions is well understood. For example, given u0 smooth and with rapid decay, there exist N solitons Rc_j,x_j

such that u(t)−

N

X

j=1

R_cj,xj(t)

_L∞(x≥−t^1/3)

≤ C

t^1/3 (as t→ ∞).

See for example Schuur [23], Eckhaus and Schuur [5], Miura [21].

However, if p 6= 2 or 3, the inverse scattering transform method does not longer apply, and the description of solutions in the general, non-integrable case is an open problem. It can be decomposed in two types of problems.

Problem 1 : Asymptotic behaviour. Given an initial datau₀, does the out coming solutionu(t)to (2) exists for all time ? If it does (for example in the subcritical case), can its behavior be described, as t → ∞ ? If blow up happens, can the blow up rate and profile be determined ?

Problem 2 : Non-linear wave operator. Given some reasonable behaviour as t → ∞, can we find a solution u(t) to (2) defined for large enough t, with this behaviour ? Is there uniqueness for u(t) ?

1.2 Recent results on Problems 1 and 2

Let us now develop some recent results which will be the base to our result. We denote U(t) the linear operator for KdV equation, i.e. v(t) = U(t)V satisfies v_t+v_xxx= 0,v(0) =V.

The first result deals with scattering for small initial data, a problem studied by many authors (see for example [24], [22], [2], [6]). Let us remind the result of Hayashi and Naumkin [6]. Introduce the following weighted Sobolev spaces : H^s,m={φ∈ S⁰| kφkH^s,m=k(1 +|x|²)^m/2(1−∂²_x)^s/2φk_L2 <∞}. (7) Scattering operator. Let p >3. Given u0 small enough inH^1,1, the out-coming solution u(t) to (2) is global in time, and there is scattering, in the sense that there exists a functionV ∈L² so that :

ku(t)−U(t)VkL² →0 as t→ ∞.

Furthermore,ku(t)kL^∞ ≤Ct^−1/3 (linear decay rate).

This is the description of solutions with initial data around 0 (in H^1,1), a result which can be understood as stability around 0.

The second type of results we want to focus on is that which describes the solutions around solitons or a sum of solitons. The following result of Martel, Merle, Tsai [18] solves the problem of stability in H¹ of a sum ofN decoupled solitons (see also Martel and Merle [14]).

Stability of the sum of N solitons. Suppose p = 2,3 or 4. Let N ∈ N, and 0< c1< . . . < cN. There existγ0 andα0 (small) andA,L0 (large), so that the following is true. Assume that there exist L≥L0, α < α0 and x⁰₁ < . . . < x⁰_N such that :

u(0)−

N

X

j=1

Qcj(· −x⁰_j) _H1

≤α, with x⁰_j ≥x⁰_j−1+L, forj= 2, . . . , N.

Then there existx₁(t), . . . , x_N(t)∈Rsuch that :

∀t≥0,

u(t)−

N

X

j=1

Qc_j(· −xj(t)−cjt) _H1

≤A(α+e^−γ0L).

These results are related to Problem 1. Let us now turn to results concerning Problem 2. First, Martel [12] proved the existence and uniqueness ofN-solitons in the casesp= 2,3,4or5 :

Existence and uniqueness of the N-soliton. Let p ∈ [2,5]. Let N ∈ N, 0 <

c1< . . . < cN, and x1, . . . , xN ∈R. There exist T0 ∈Rand a unique function

u∈C([T₀,+∞),R), which is aH¹ solution to (1), and such that :

u(t)−

N

X

j=1

Q_cj(· −x_j−c_jt) _H1

= 0 as t→ ∞.

Furthermore, u∈C^∞([T0,∞)×R) and convergence takes place inH^s for all s≥0, with an exponential decay :

∃γ >0, ∀s≥0,∃As /

u(t)−

N

X

j=1

Qc_j(· −xj−cjt) _H1

≤Ase^−γt.

This result appears as a development of monotonicity properties and a dy- namical argument, ideas which where used by Martel and Merle [14] and Martel, Merle and Tsai [18].

However, it is a surprise that the method could be adapted even to the critical casep= 5, although it is well known that solitons are unstable inH¹(R): there is in fact blow-up for a large class of initial data and the blow-up profile is stable, see [15], [17], [20], [16]. Another surprise is uniqueness of theN-soliton.

Notice that in view of this result, the stability of a sum ofN solitons can be interpreted as stability of theN-soliton (solution to (2)).

The last result solves the case of a linear behavior, that is the existence of a wave operator :

Large data wave operator. Let p > 3, and V ∈H^2,2. There exist T₀ ∈R and u∈C([T₀,∞), H¹)solution to (2) such that :

ku(t)−U(t)VkH¹ →0 as t→ ∞.

Furthermore uis unique in an adapted class.

In the same way that the result of Martel [12] was based on considerations of Martel, Merle and Tsai [18], this result strongly relies on the analysis of Hayashi and Naumkin [6].

1.3 Statement of the main result

Our goal is to construct solutions which behave like a sum of a linear term U(t)V, and of N solitons, in the subcritical p < 5 case. Notice that in [3]

such solutions are constructed in the critical case p= 5. More precisely, given 0< c₁< . . . < c_n andx₁, . . . x_N ∈R, we would like to construct solutionsu(t) to (2), defined for large enough times and such that

u(t)−U(t)V −

N

X

j=1

R_cj,xj(t)

_H1 →0 as t→ ∞.

In this article, we construct such solutions in the casep= 4(that is, for equation (1)), provided thatV is smooth enough, with sufficient decay on the right. From now on and throughout the rest of the article,

we focus on the sub-critical casep= 4. (8)

Let us first remind the functional setting which will be used throughout the proofs. Fix once for all the three constants :

γ∈(0,1/3), α=1

2 −γ∈(0,1/2) and δ= 1−2γ

3 >0. (9) (γ is arbitrary). These constants are those of [6] in the casep= 4.

Again as in [6], we will use the notation D = ∂x = _∂x^∂ for the partial differentiation with respect to the space variablex, and

D^αf =F⁻¹ξ^αe−(iπ/2)(1+α)f ,ˆ along with the two following operators

J^tf =U(t)xU(−t)f = (x−3t∂_x²)f, andI^tφ=xφ+ 3t Z x

−∞

∂_tf(t, y)dy.

We writeJ^tandI^tso as to emphasize that we will always consider norms at a fixed timet althoughJ^tandI^tare space-time operators.

Our working spaces will be defined through the time dependentM₀^t norm : H_t={f ∈L²(R)|M₀^t(f) =kfk_H1+kDJ^tfk_L2+kD^αJ^tfk_L2<∞}.

J^tonly appears in the norm, as it is convenient to do linear estimates (see [6], Lemma 2.3). But we introduced I^t because it is easier to handle when doing energy methods estimates. Notice thatM₀⁰is very similar tok · k_H1,1.

We will finally use the following notation for weighted spaces : for a positive functionh,

kfk²_Hs(h)= Z

|(Id−∆)^s/2f|²(x)h(x)dx.

Following a usual convention, different positive constants might be denoted by the same letterC.

Our main result is the following.

Theorem 1 (Nonlinear wave operator). LetV ∈H^5,1∩H^2,2 be such that : x^4/3₊ D⁵V ∈L², x⁸₊V ∈H¹,

(where x+= max{0, x}). LetN ∈N,0 < c1< . . . < cN andx1, . . . , xN ∈R. Denote Rj(t, x) =Qcj(x−xj−cjt)N solitons.

Then there existsu^∗∈C([T₀,+∞), H⁴∩ H^t₀), for someT₀∈R, solution to (1), such that if we introduce :

w^∗(t) =u^∗(t)−U(t)V −

N

X

j=1

Rj(t), we have

kw^∗(t)kH⁴+M₀^t(w^∗(t))→0 as t→ ∞.

Furthermore, we have the following decay rate :

kw^∗(t)kH⁴ ≤Ct^−1/3, M₀^t(w^∗(t))≤Ct^−δ.

Remark 1. This result allows to work with large data (V large inL²), which is both surprising and satisfactory. However, it deals with smooth and decaying data. A natural setting would be a result withV ∈H¹, and some decay on the right to ensure low interaction with the solitons. Theorem 1 should be viewed as a step in the solving process of Problem 2.

An important change in the method of proof when considering [12] is the following. Solitons have an exponential decay, and so integrability (in time) is always automatic. Here the linear termU(t)V will interfere with the solitons to produce a polynomial decay in time, and this will require taking care of.

Similarly, when handling the linear termU(t)V (following the framework of [4]), we will have to take care of the interference of the solitons.

Remark 2. This result is similar to [3], where a non-linear wave operator is constructed in theL² critical casep= 5.

In both cases, the scheme of proof first dwells on the interaction with the solitons, and in a second step uses arguments from the linear scattering theory to control the interaction with the linear term (along with the results obtained in the first step). The argument for the soliton interaction is very similar in the casep= 4 and in the casep= 5. However, the second step is very different.

For p = 5, the linear scattering theory of Kenig, Ponce and Vega [8] is available : it is done inL², and so requires much less smoothness and decay on V. The main difficulty is to mix both approaches, as the soliton theory relies on an analysis in C_t⁰H_x¹, and the natural space in the theory of [8] is L⁵_xL¹⁰_t : in particular, solitons do not belong to this space (nor to L⁵_xL¹⁰_t≥T for any T).

The problem is then to separate the linear analysis from the non-linear one, and when considering the interference of one over the other, to be able to interchange integrals in time and in space in an adequate way. This can be done with a small loss in the decay, with respect to the optimal result one can expect using this method.

In the non-critical case, the scattering analysis of [8] is no longer available, and we have to relie on the theory of Naumkin and Hayashi [6]. Their method break down at some point, when taking care of the interference between the solitons and the linear term. However, we manage to recover the leap by energy method arguments, and this is why we have to reinforce the assumptions onV, and obtain a stronger convergence (H⁴). Our method could be adapted also to the critical case, but would give a much less sharp result than what is obtained in [3].

The problem of the uniqueness of solutions behaving as the sum of a linear term and N soliton is an open question, in both the critical and sub-critical case.

Remind that ifV = 0, one has uniqueness inH¹ (see [12]) : this result is linked with very fast convergence of the constructed solution to its profile not only in H¹but inH⁴. However, it seems that one can not derive easily from this work a proof for V 6= 0.

Remark 3. Theorem 1 is valid only forp= 4 for two main reasons. First, it contains the existence of a scattering operator, so that p >3. Second, it also contains the existence of aN-soliton, which is only true forp≤5. The fact that our setting only deals with integerpcomes from our crucial use of the regularity of the non-linearity functionx7→x^pand also from better integrability properties (ifp≥4 instead ofp >3).

However, one can prove an analoguous result forp= 5, but that one would be much less precise than we is stated in [3].

Remark 4. There are some analogous results for the (critical) non-linear Schrö- dinger equation. See Bourgain and Wang [1], Krieger and Schlag [10], Merle [19].

In [1], a solution to the critical NLS equation with a given blow-up behaviour is constructed : due to the conformal transform, this is in fact equivalent to construct a solution to the critical NLS equation which behaves like the sum of a soliton and a linear term. High smoothness and low interaction with the soliton are required on the linear term.

In Section 2, we give a detailed outline of the proof of Theorem 1, decompos- ing it into steps : each of these step is summarized in a proposition. In Section 3, we give some preliminary results and each of the following sections is devoted to the proof of one of the propositions stated in Section 2.

Acknowledgment

I am grateful to my advisor Frank Merle for numerous discussions and his constant encouragements.

2 Outline of the proof

Let V ∈H^5,1∩H^2,2 such thatx⁸₊D⁵V ∈ L² and x^4/3₊ V ∈ H¹. Let 0 < c1 <

. . . < cN andx1, . . . , xN ∈R. Denote the soliton with speedcj and shiftxj : Rj(t, x) =Qc_j(x−xj−cjt).

Define alsoR(t) =PN

j=1Rj(t).

LetSn be an increasing sequence of time, so thatSn → ∞as n→ ∞. For n >0, we defineun(t), the solution to

u_nt+ (u_nxx+u⁴_n)x= 0,

un(Sn) =U(Sn) +R(Sn). (10) Equivalently, we introduce the error term

w_n(t) =u_n(t)−U(t)V −R(t), so thatwn(t)satisfies the equation

( w_nt+w_nxxx+

u⁴_n−PN j=1R_j⁴

x

= 0,

w_n(S_n) = 0. (11)

Asu_n(S_n)∈H¹, u_nn∈C_b(R,H¹); the same thing is true forw_n(t).

The heart of the proof of Theorem 1 is the following result.

Proposition 1 (Uniform estimates). There existsT0such that for allnsuch that S_n ≥T₀, the solution u_n(t) to (10) and the solutionw_n(t) to (11) belong toC([T₀, S_n],H^t₀∩H⁴). Furthermore, we have

∀t∈[T₀, S_n], kw_n(t)k_H4 ≤C₀t^−1/3, M₀^t(w_n(t))≤C₀t^−δ, (12) for some constant C0 not depending onn(recallδ >0 is introduced in (9)).

The proof of this proposition requires several steps.

The first remark allows us to further assume smallness onwn(t), in order to get the decay (12).

Proposition 1’ (Reduction of proof ). There exist ε0>0, C0, and T0 ≥1 with 2C0T₀^−δ ≤ε0 such that the following is true, for all n∈N. Suppose that there existsIn∈[T0, Sn]such that

∀t∈[In, Sn], kwn(t)kH⁴+M₀^t(wn(t))≤ε0. Then in fact

∀t∈[I_n, S_n], kwn(t)kH⁴≤C₀t^−1/3, M₀^t(w_n(t))≤C₀t^−δ.

Proof of Proposition 1 assuming Proposition 1’. LetT0= max{1, C₀^1/δε0}, and define

I_n^∗= inf

t^∗∈[1,Sn]{t^∗| ∀t∈[t^∗, Sn], kwn(t)k_H4+M₀^t(wn(t))≤ε0}.

As wn(Sn = 0), by upper semi-continuity of the norm of the flow (see [4, Ap- pendix B]), we obtain that the set on which we do the infimum is non-empty, so thatI_n^∗< Sn.

Then of course, for allt∈(I_n^∗, Sn],kwn(t)k_H4+M₀^t(wn(t))≤ε0. This allows us to apply Proposition 1’ so that

∀t∈(I_n^∗, Sn], kwn(t)kH⁴ ≤C0t^−1/3, M₀^t(wn(t))≤C0t^−δ. (13) If I_n^∗ >1, we also get that lim sup_t↓I∗

nkwn(t)kH⁴ +M₀^t(w_n(t))≥ε₀ (from the minimality ofI_n^∗). In particular, this gives

ε0≤lim sup

t↓I^∗_n

kwn(t)k_H4+M₀^t(wn(t))≤2C0I_n^∗−δ.

So thatI_n^∗ ≤ε0/(2C0))^−1/δ. In any case, we get that I_n^∗ ≤T0 : (13) allows us to conclude.

Thus, our goal is now to prove Proposition 1’.

Proof of Proposition 1’.

Step 1 : Monotonicity and non-linear tools. We obtain H¹ estimates on the right. Let us intoduce the cut-off speed

σ₀∈(0,min{c₁, c₂−c₁, . . . , c_N −c_N−1}), (14) to be determined in the proof of the following Proposition 2, and the cut-off function

ψ(x) = 2 πarctan

exp

−

√σ0

2 x

, ψ₀(t, x) =ψ(x−σ₀t−2|x₁|). (15) ψ0(t)allows us to separate the solitons interaction from theU(t)V interaction.

Proposition 2 (Interaction with the solitons). There existσ₁>0,ε₁>0, C₁ and T₁ such that the following is true. If σ₀ ≤ σ₁, ε₀ ≤ε₁ and T₀ ≥ T₁, then, for alln∈N and allt∈[In, Sn],

kwk_H1(1−ψ0(t))≤C1e^−σ0

√σ0

4 t+C1kU(t)Vk_H1(1−ψ0(t))

+C₁(S_n−t+ 1)kU(t)Vk_L2(1−ψ₀(S_n))+C₁ Z Sn

t

kU(t)Vk_H1(1−ψ₀(t))dt.

Observe that this proposition in fact holds for all p ∈ [2,5] ; however, we will only do it forp= 4.

Essentially we obtain a polynomial decay onkw_n(t)k_H1(1−ψ₀(t)) (instead of an exponential decay in the case of solely soliton). However the good point is that we can choose this polynomial decay to be as fast as we want by lowering the interaction ofU(t)V with the solitons, that is, by requiring sufficicient decay on the right forV.

Now we would like to complete theM₀^t estimate. But it happens that the construction of [6] relies on a very nice cancelation involving the operators J^t andI^t, which allows a bootstrap inH^t₀. Here, this nice clockwork breaks down because of the interaction with the solitonsRj (the precise term that arise will be treated in full detail in the proof of the final step 4). We therefore are forced to work inH³which is the more natural space where all the computations of [6]

are done (of course in H³, the bootstrap of [6] doesn’t work anymore because of a lack of information).

We need a good control on the interaction with the soliton at the H³ level : more precisely (this will be done in full detail in subsection, we need tkwnkH³(1−ψ0(t)) be integrable in time. This can not be achieved by improving Proposition 2 to H³, as its proof is done through considerations at H¹ level.

This is why we go up to H⁴ : with a weak control on kwnkH⁴, and a strong control on kwnkH¹(1−ψ0(t)), we obtain by interpolation the desired control on kw_nk_H3(1−ψ0(t)). Indeed, we have the following corollary to Proposition 2, in which we estimate some quantities which we will need later on.

Corollary 1. SupposeV ∈H^5,1∩H^2,2 is such that x^4/3₊ D⁵V ∈L², and x⁸₊V ∈H¹.

Then for someC₁⁰ >0, we have, for alln∈N and for allt∈[In, Sn], tkwn(t)k_H3(1−ψ0(t))+tkU(t)Vk_H2(1−ψ0(t))

+kU(t)VkH⁵(1−ψ0(t))+kU(t)(xVx)kH¹(1−ψ0(t))≤ C₁⁰ t^4/3. Proof. We combine the result of Proposition 2 and Lemma 3. First observe that from Lemma 3, our assumptions translate to

kD⁵U(t)Vk_L2(1−ψ0(t)) ≤Ct^−4/3, (16) kU(t)VkL²(1−ψ0(t))+kU(t)VxkL²(1−ψ0(t)) ≤Ct⁻⁸. (17) So that by interpolation of (16) and (17),

kU(t)Vk_H5(1−ψ0(t))≤Ct^−4/3.

Again by interpolation, we get

kU(t)Vk_H2(1−ψ0(t)) ≤ kU(t)Vk^3/4_H1(1−ψ0(t))kU(t)Vk^1/4_H5(1−ψ0(t))

≤ C t^34·8 · C

t^14·43 ≤ C t^19/3 ≤ C

t^7/3. Now, by Proposition 2 and (17), we get

kw_n(t)k_H1(1−ψ₀(t))≤ C t⁷. Now recall thatkwn(t)k_H4 ≤ε0, so that by interpolation

kwn(t)kH³(1−ψ0(t))≤Ckwn(t)k^1/3_H1(1−ψ0(t))kwn(t)k^2/3_H4(1−ψ0(t))

≤ C

t^7/3kwn(t)k^2/3_H4 ≤ C t^7/3. For thexVx estimate : first notice that

Z

V_xx²(x)x^14/3₊ dx= Z ∞

0

V_xx²x^14/3dx

=− Z ∞

0

VxxxVxx^14/3dx− Z ∞

0

VxxVxx^11/3dx

≤ Z ∞

0

V_xxx² x^8/3dx Z ∞

0

V_x²x^20/3dx 1/2

+ Z ∞

0

V_xx²x^8/3dx Z ∞

0

V_x²x^14/3dx ^1/2

≤ kx^4/3₊ Vxxxk_L2kx^10/3₊ Vxk_L2+kx^4/3₊ Vxxk_L2kx^7/3₊ Vxk_L2. AsV ∈H^2,2, xVx∈H¹, and moreover,

Z

(xV_x)²+|D(xV_x)|²

x^8/3₊ dx≤ Z

V_x²+V_xx²

(1 +x^14/3₊ )dx, so that

k(1 +x^7/3₊ )(xV_x)k²_H1 ≤ k(1 +x^10/3₊ )VkH¹k(1 +x^4/3₊ )VkH³. From ourH⁵estimate and(1 +x⁸₊)V ∈H¹, we get

kU(t)(xVx)k_H1(1−ψ0(t))≤Ct^−7/3k(1 +x^7/3₊ )(xVx)k_H1≤Ct^−7/3.

Step 2 : Energy method estimates. Now that we have assumed H⁴ control, we have to obtainH⁴ uniform decay.

Proposition 3 (Interaction with the linear term, H⁴ bounds). There existsC₂ such that∀n∈N,∀t∈[I_n, S_n],

kwn(t)kH⁴ ≤ C₂ t^1/3.

First considerL²andH¹estimates. We want to control what happens in the zonex < σ₀t, that is the interaction with the linear termU(t)V : we follow the framework of [4]. The crucial point is to use our a priori control onM₀^t(w(t)).

We have

|w_n(t, x)| ≤ C t^1/3

1 + |x|

√3

t −1/4

M₀^t(w_n(t)),

|w_nx(t, x)| ≤ C t^2/3

1 + |x|

√3

t ^1/4

M₀^t(w_n(t)).

These, along with Proposition 2 allow to obtain theH¹ decay estimate, in a very similar way to [4].

For the higher order estimates, i.e. H², H³ and H⁴, the pointwise control that we have onwnandw_nxis not enough. If we wanted to improve our control toM₀^t(w_nx), we would always face the same problem for the higher order deriva- tives. The path that we will follow to avoid this is to use almost conservation quantities at levelH² etc. For example, letube a solution to (1), then

d dt

Z

u²_xx−20 3

Z u²_xu³

= 2 Z

u⁵_x+ 80 Z

u³_xu⁵. Three elements are to be noticed. First, there is a corrective term R

u²_xu³ to prevent the apparition of R

u²_xxu_xu², which we could not control, as noted in [12]. Second,R

u³_xu⁵ comes from the corrective term, and will never be harmful, as it has a better integrability than the others (power 8 instead of 5). Third, R u⁵_xhas a more than quadratic term inux(whenuxappear less than twice, we can use directly our control on kuxk_L2 already obtained). To control this kind of terms, we use the Gagliardo-Nirenberg inequality :

∀q≥2, ∀v∈H¹, kvk^q_Lq ≤C(q)kvk

q+2 2

L² kvxk

q−2 2

L² . (5)

As the maximal exponent on the term with highest derivatives is 5 or less, exponent onkvxkL² will always be less than 2, which means that we will always be in the position to apply Lemma 4. Assume for now that, when estimating the derivative in time of the H^s+1 norm (squared) of wn(t), all terms have appropriate control except for (β∈[0,3])

Z

|D^swn|^2+β|D^s−1wn|^3−β.

Further assume that all previous estimates gave a decay kwnkH^s ≤ Ct^−1/3. Thus, as our term has power5, from (5) we would get a control :

d

dtkw_nk²_Hs+1

≤ kw_nk^5−β_Hs kw_nk^β_Hs+1≤ kwnk^β_Hs+1 t^(5−β)/3 .

Withµ=β/2,λ= (5−β)/3, Lemma 4 gives the decaykwk_Hs+1≤Ct^−ν, with ν =1

2

(5−β)/3−1 1−β/2 =1

3.

This means that the rate of decayt^−1/3 is likely to propagate as the regularity indexsincreases (in fact, forp≥4, similar computations show that the rate of decayt^−(p−3)/3propagates).pinteger is interesting regarding the regularity of the non-linearity function : to obtain theH² formula quoted, we already need aC⁴ regularity, which translates top≥4. In any case, our assumptionp= 4is now crucial. Of course we will need the estimate of Corollary 1 to handle some interaction terms.

Observe finally that this decay rate oft^−1/3 is the best one can expect, due to the slow decay of the linear term U(t)V.

Step 3 : Linear tools from scattering theory. We can now complete the decay estimate, by controling the remaining of theM₀^t norm.

Proposition 4 (Interaction with the linear term, M₀^t bound). There existsC₃ such that∀n∈N,t∈[I_n, S_n]

M₀^t(wn(t))≤ C3

t^δ .

Remind that M₀^t(w_n(t)) = kwn(t)kH¹ +kD^αJ^tw_n(t)kL² +kDJ^tw_n(t)kL². kwn(t)kH¹ has already been estimated, so we only need to focus on the last two terms. We follow the framework of [6] and [4]. First, we estimatekD^αI^tw_n(t)k_L2

and kI^tw_nx(t)k_L2. For this, we use the usual ¹_2dt^dkfk²_L2 = (Lf, f), and plug in Lf the equation satisfied byf : heref =D^αI^twn(t)orDI^twn(t).

When doing the computations on(LI^tw_nx(t), I^tw_nx(t)), we encounter a term of the type

Z

(I^tw_nx(t))²R². (18) This is localized term in space, but in H³ regularity instead of H¹ regularity.

This fact explains that we needed to get decay for higher regularity norms than justH¹. Ideally, we would try to obtain directlyH³on the right decay. However, this seems not to be possible. One easy way is to obtain low decay rate for the global space normsH^s, which we did up toH⁴. Corollary 1 allows us to bound this troublesome term (18).

This explains how to obtain

kD^αI^twn(t)kL²+kI^tw_nx(t)kL² ≤Ct^−δ.

It remains to go back toJ^t, which is done in a similar way as in [6] and [4], and does not raise more difficulties than those treated earlier.

This concludes the proof of Proposition 1’, and thus of Proposition 1.

We can now conclude : Proof of Theorem 1.

Step 1 : A compactness result. From Proposition 1, we dispose of a sequence un(t)defined on[T0, Sn], solution to (1), such that

u(Sn) =U(Sn)V +

N

X

j=1

Rj(Sn) =U(Sn) +R(Sn),

and that the uniform estimates hold (w_n(t) =u_n(t)−U(t)V −R(t)) :

∃T₀≥1,∃C₀>0, ∀n∈N, ∀t∈[T₀, S_n], kw_n(t)k_H4+M₀^t(w_n(t))≤ C0

t^δ . Let us prove the following compactness result on the sequenceun(T0).

Claim. We have

A→∞lim sup

n∈N

Z

|x|≥A

u²_n(T0, x)dx= 0.

Proof. Indeed, letε >0, andT(ε)such thatC0T(ε)^−δ ≤√ ε. Then Z

(un(T(ε))−U(T(ε))V −R(T(ε))²≤ε.

LetA(ε)be such thatR

|x|≥A(ε)(U(T(ε))V +R(T(ε)))²(x)dx≤ε; we get Z

|x|≥A(ε)

u²_n(T(ε), x)dx≤2ε.

Let g ∈ C³ a function such that g(x) = 0 if x ≤ 0, g(x) = 1 if x ≥ 2, and furthermore0≤g⁰(x)≤1,0≤g⁰⁰⁰(x)≤1.

Remind that iff ∈C³ does only depend onx, we have d

dt Z

u²_nf =−3 Z

u_n²_xf_x+ Z

u²_nf_xxx+8 5

Z u⁵_nf_x.

(See Lemma 7 and its proof). ForC(ε)to be determined later, we then have : d

dt Z

u²_n(t, x)g

x−A(ε) C(ε)

=− 3 C(ε)

Z un2

xg⁰

x−A(ε) C(ε)

+ 1

C(ε)³ Z

u²_ng⁰⁰⁰

x−A(ε) C(ε)

+ 8

5C(ε) Z

u⁵_ng⁰

x−A(ε) C(ε)

. Ast≥T0≥1,un satisfieskun(t)k_H1≤C0+kVk_H1+PN

j=1kQc_jk_H1 ≤C⁰. So that :

d dt

Z

u²_n(t, x)g

x−A(ε) C(ε)

≤ 1 C(ε)

3

Z un2

x(t) + Z

u²_n(t) +8

5kunk³_L∞

Z u²_n(t)

≤ 1 C(ε)

3C⁰²+8

52^3/2C⁰⁵

. Now choose C(ε) = maxn

1,^T(ε)−T_ε ⁰

3C⁰²+⁸₅2^3/2C⁰⁵o

, from which we de-

rive

d dt

Z

u²_n(t, x)g

x−A(ε) C(ε)

≤ ε

T(ε)−T0

. And after integration in time betweenT0 andT(ε):

Z

x≥2C(ε)+A(ε)

u²_n(T₀, x)≤ Z

u²_n(T₀, x)g

x−A(ε) C(ε)

≤3ε.

Now considering _dt^d R

u²_n(t, x)g_−A(ε)−x

C(ε)

, we get in a similar way Z

x≤−2C(ε)−A(ε)

u²_n(T0, x)≤3ε.

So that if we denoteAε= 2C(ε/6) +A(ε/6), we obtain :

∀n∈N, Z

|x|≥Aε

u²_n(T₀, x)≤ε, as claimed.

Step 2 : Construction of u^∗. un(T0) is a bounded sequence in H⁴ ∩ H^T₀⁰, so that it converges weakly toϕ₀ ∈H⁴(R)∩ H^T₀⁰(R)(up to a subsequence). The previous compactness result ensures that the convergence is strong in L²(R).

Indeed, letε >0, andAsuch thatR

|x|≥Aϕ²₀(x)dx≤εand

∀n∈N, Z

|x|≥A

u²_n(T0, x)≤ε.

The injectionH¹([−A, A]),→L²([−A, A])is compact, so thatR

|x|≤A|un(T0, x)−

ϕ0(x)|²dx→0. We thus derive that lim sup

n∈N kun(T0)−ϕ0k²_L2(R)≤4ε.

As this is true for all ε > 0, u_n(T₀)→ ϕ₀ in L²(R). By interpolation, u_n(T₀) converges strongly toϕ0 inH³. Denote u^∗(t)the solution to

u^∗_t + (u^∗_xx+u^∗4)_x= 0, u^∗(T0) =ϕ0.

The Cauchy problem being globally well-posed in H¹, u^∗ is well defined. Now the flow is continuous inH³, so that for allt∈R,un(t)→u^∗(t)inH³, and we can pass to the limit in theH³estimates, to get

∀t∈R, ku^∗(t)−U(t)V −R(t)k_H3≤C0t^−1/3.

Denotew^∗(t) =u^∗(t)−U(t)V −R(t).wn(t)→w^∗(t)inH¹ so thatw^∗(t)is the only possible weak limit ofw_n(t)in H⁴∩ H^t₀. In particular, the convergence is strong inH³ and

kw^∗(t)kH⁴≤lim inf

n→∞ kwn(t)kH⁴ ≤ C0

t^1/3, M₀^t(w^∗(t))≤lim inf

n→∞ M₀^t(wn(t))≤ C0

t^δ . This completes the proof of Theorem 1.

This scheme of proof is similar to that of [12], [4]. Steps 2, 3 and 4 of the proof of Proposition 1’ remain to be completed.

In Section 3, we present some preliminary results. In Section 4, we prove Proposition 2. In Section 5, we prove Proposition 3. Finally, in Section 6, we prove Proposition 4. This completes the proof of Proposition 1’, and thus, the proof of Theorem 1.

3 Preliminaries

3.1 Cut-off functions and notation for localized quantities

We already introduced σ0 ∈ (0,1/2 min{c1, c2−c1, . . . , cN −cN−1}), and the cut off function

ψ(x) = 2 πarctan

exp

−

√σ0

2 x

. (15)

We can check that lim_+∞ψ = 0, lim_−∞ψ = 1, ψ is decreasing. Furthermore, by direct computations :

ψ⁰(x) =−

√σ₀ 2πcosh^√_σ

0

2 x, ψ⁰⁰⁰=σ0

4 ψ⁰(x) 1− 2 cosh^√_σ

0

2 x

! , so that

|ψ⁰⁰⁰(x)| ≤ −σ₀

4 ψ⁰(x). (19)

We introduce, forj = 1, . . . , N−1, m_j(t) = c_j+c_j+1

2 t+x_j+x_j+1

2 , m₀(t) =σ₀t−2|x1|.

So that we can define, forj= 0, . . . , N−1,

ψ_j(t, x) =ψ(x−m_j(t)), ψ_N(t, x) = 1.

Then we set, for j= 1, . . . , N−1,

φ₀(t) =ψ₀(t), φ_j(t) =ψ_j(t)−ψ_j−1(t), φ_N(t) = 1−ψ_N−1(t).

By construction,Pj

k=1φk=ψj. Finally, we define some local quantities related to mass and energy :

M_j(t) = Z

u²_t(t)φ_j(t), E_j(t) = Z 1

2u²_x(t)−1 5u⁵(t)

φ_j(t), F_j(t) =E_j(t) + 1

100M_j(t).

3.2 H

estimates

Remind our notations γ∈

0,1

3

, α= 1

2−γ, δ=1−2γ

3 >0, (9)

the operatorJ^tf =xf−3t∂²_xf =U(t)xU(−t)f, and our working norm M₀^t(f) =kfk_H1+kD^αJ^tfk_L2+k∂J^tfk_L2.

First a few remarks on M₀^t. Of course M₀⁰(f) ≤ CkfkH^1,1. Second, note that J^tU(t)V =U(t)xV (and U(t)is aH^s isometry), so that ifV ∈H^1,1, we have the uniform control in t:

M₀^t(U(t)V)≤CkVkH^1,1. (20) We now remind the linear results obtained in [6] (Lemma 2.2), in a slightly improved form.

Lemma 1 ([6]). Lett >0andf be a function so thatM₀^t(f)is bounded. Then forr >4,

kfkL^r ≤ C

(1 +t)^1/3−1/(3r)M₀^t(f).

And one also has the point wise inequalities

|f(x)| ≤ CM₀^t(f) (1 +t)^1/3

1 +

x t^1/3

−¹₄

, |f_x(x)| ≤ CM₀^t(f) t^2/3

1 +

x t^1/3

¹₄ . As a simple consequence, forV ∈H^1,1, we have similar decay estimates on U(t)V.

Proof. See [6], Lemma 2.2 and its proof (especially inequalities (2.16), (2.17) and (2.18)). The proof of refinement can be found in [4], Appendix A.

We will also need the polarized version of Lemma 2.3 of [6] (in the case p= 4) :

Lemma 2. Let p≥3 and g, h : R →R. Then the following inequalities are hold if their right-hand side is bounded :

kD^αg^pkL² ≤Ckg^p−1kL²(kggxk^1/2_L∞+kgk^3γ_L∞kggxk^(1−3γ)/2_L∞ ), kD^α|g|^p−1hxk_L2 ≤C(kD^αhk_L2+khxk_L2)(kgk^p−3_L∞kggxkL^∞

+kgk^p−3−2γ_L∞ kgk^2γ_L2kggxkL^∞+kgk^p−3+2γ_L∞ kggxk^1−γ_L∞).

Proof. See [6], Lemma 2.3 and its proof (caseσ= 0).

3.3 Estimates of U (t)V on the right

Recall our definition ofψ₀(t)(15), givenσ₀>0. We will often need estimates of the type kU(t)Vk_H1(1−ψ₀(t)), as it is a measure of the interaction between the linear termU(t)V and the solitons.

Let us denotex+= max{x,0}.

Lemma 3 (U(t)V estimates on the right). Let f ∈L², then

kU(t)fk_L2(1−ψ0(t))≤ kfk_L2(1−ψ0(t/2))→0 as t→ ∞. (21) Assume in addition that (1 +x^q₊)f(x)∈L², for some q >0. Then there exists a constant C=C(σ₀, x₁)independent of f such that

∀t≥1, kU(t)fk_L2(1−ψ0(t))≤ C

t^qk(1 +x^q₊)f(x)k_L2. (22) We will apply this result toV and its derivatives (see Corollary 1).

Proof. The key remark is thatU(t)“pushes” the L²-mass on the left. We com- pute :

d dτ

Z

|U(2τ−t)f|²ψ₀(τ)

= 2 Z

(U(2τ−t)f)τU(2τ−t)f ψ0(τ) + Z

|U(2τ−t)f|²ψ0τ(τ)

=−4 Z

U(2τ−t)fxxxU(2τ−t)f ψ0(τ) + Z

|U(2τ−t)f|²ψ0τ(τ)

= 4 Z

U(2τ−t)fxxU(2τ−t)fxψ0(τ) + 4 Z

U(2τ−t)fxxU(2τ−t)f ψ0x(τ) +

Z

|U(2τ−t)f|²ψ0τ(τ)

=−6 Z

|U(2τ−t)f_x|²ψ_0x(τ)−4 Z

U(2τ−t)f_xU(2τ−t)f ψ_0xx(τ) +

Z

|U(2τ−t)f|²ψ_0τ(τ)

=−6 Z

|U(2τ−t)fx|²ψ0x(τ) + Z

|U(2τ−t)f|²(2ψ0xxx(τ) +ψ0τ(τ)).

Asψ_xxx≤ ^σ₄⁰|ψx|,ψ_0τ =−σ0ψ_0x, andψ_x<0, we have that, ψ0x(τ)<0 and 2ψ0xxx(τ) +ψ0τ(τ)≥0.

So that τ 7→R

U(2τ−t)f(x)²ψ0(τ, x)dx is an increasing function of τ. In par- ticular, when comparing for τ=tandτ =t/2 (t≥0), we have :

∀t≥0, Z

|U(t)f|²ψ₀(t)≥ Z

f²ψ₀(t/2).

As the flowU(t)preserves theL²-mass, we get Z

|U(t)f|²(x)(1−ψ0(t, x))dx≤ Z

f²(x)(1−ψ0(t/2, x))dx. (23) Suppose that for someq >0, (1 +x^q₊)f(x)∈L². Then fort≥1,

Z

f²(1−ψ0(t/2)) = Z

x≤σ0t/4

f²(1−ψ0(t/2)) + Z

x≥σ0t/4

f²(1−ψ0(t/2))

≤ sup

x≤σ₀t/4

(1−ψ0(t/2, x)) Z

f²+ σ₀t

4

−2qZ

x≥σ0t/4

x^2qf²

≤C(x₀)e^−σ0

√σ0

4 t

kfk²_L2+C(σ₀)t^−2qkx^q₊fk²_L2. And we get

∀t≥1, kU(t)fkL²(1−ψ0(t))≤ C

t^qk(1 +x₊)^qfkL², which is (22).

3.4 An ODE lemma

Lemma 4 (Booster). Let κ >0,λ > 1, µ∈ (0,1), and f ∈L^µ([a, b]) (0 <

a < b <+∞) be a non-negative upper semi-continuous function satisfying

∀t∈[a, b], f(t)≤ C t^κ +C

Z b t

f^µ(τ) τ^λ dτ,

Defineν = min{κ,^λ−1_1−µ}. Then there existsk=k(C, κ, λ, µ)not depending onb such that

∀t∈[a, b], f(t)≤kC t^ν .

Remark 5. Of course, if instead we have f(t)≤ C

t^κ+

I

X

i=1

Ci

Z b t

f^µi(τ) τ^λi dτ, the final decay estimate is still valid, with ν = min{κ,(_1−µ^λi−1

i)_i} being the least favorable exponent.

Proof. Letk >1to be determined later. Let us consider T = inf

τ≥a

∀t∈[τ, b], f(t)≤kC t^ν

.

Observe that T is in fact minimal for the property. Asb >0,f(b)≤ _t^Cν < ^kC_tν , so that by upper semi continuity,T < b. Then, ift∈[T, b], we have (t≥a >0)

f(t)≤ C

t^ν + C(kC)^µ (λ−1 +cν)t^λ−1+µν. Ifν =^λ−1_1−µ,λ−1 +µν = (λ−1)

1 + _1−µ^µ

= ^λ−1_1−µ =ν. Elseν=κ, ^λ−1_1−µ ≥κ=ν so thatλ−1≥(1−µ)ν andλ−1 +µν≥ν. In any case, we get

f(t)≤C1 + _λ−1+µν^(kC)µ t^ν . Let us now choose k such that 2

1 +_λ−1+µν^(kC)µ

≤ k, which is possible as µ <

1 (notice that k > 2). We get finally f(t) ≤ _2t^kCν. By a standard continuity argument, we deduce thatT =a.

4 Estimates on the right : proof of Proposition 2

We follow the framework of [12]. The hypothesis we will use in this section is :

∀t∈[In, Sn], kwn(t)k_H1≤ε0.

4.1 Modulation close to asymptotic profile

Let us remind thatQ_c(x) =c^p−11 Q(√ cx).

Lemma 5 (Modulation ofw_n(t)). There existT₂andε₂ such that ifI_n ≥T₂ andε₀≤ε₂, the following is true. For allt∈[I_n, S_n], there existy_j(t)andγ_j(t) such that if we denote

R˜j(t, x) =Qγ_j(t)(x−yj(t)), R(t, x) =˜

N

X

j=1

R˜j(t, x),

˜

wn(t) =un(t, x)−U(t)V −R(t, x),˜ we have for all j= 1, . . . , N,

Z

˜

wn(t, x) ˜R_{j x}(t, x)dx= 0 and Z

˜

wn(t, x) ˜Rj(t, x)dx= 0.