# The Initial Value Problem for the Quadratic Nonlinear Klein-Gordon Equation

Authors: Nakao Hayashi; Pavel I. Naumkin
Journal: Advances in Mathematical Physics
 (2010)

---

## Abstract

We study the initial value problem for the quadratic nonlinear Klein-Gordon equationℒu=〈i∂x〉-1u¯2, (t,x)∈R×R, u(0,x)=u0(x), x∈R, where ℒ=∂t+i〈i∂x〉 and 〈i∂x〉=1-∂x2̅. Using the Shatah normal forms method, we obtain a sharp asymptotic behavior of small solutions without the condition of a compact support on the initial data which was assumed in the previous works.


---

## Body

## 1. Introduction

Let us consider the Cauchy problem for the nonlinear Klein-Gordon equation with a quadratic nonlinearity in one dimensional case(1.1)ℒu=λ〈i∂x〉-1u¯2,(t,x)∈R×R,u(0,x)=u0(x),x∈R,
where λ∈C,ℒ=∂t+i〈i∂x〉, and 〈i∂x〉=1-∂x2.Our purpose is to obtain the large time asymptotic profile of small solutions to the Cauchy problem (1.1) without the restriction of a compact support on the initial data which was assumed in the previous work [1]. One of the important tools of paper [1] was based on the transformation of the equation by virtue of the hyperbolic polar coordinates following to paper [2]. The application of the hyperbolic polar coordinates implies the restriction to the interior of the light cone, and therefore, requires the compactness of the initial data. Problem (1.1) is related to the Cauchy problem(1.2)vtt+v-vxx=μv2,(t,x)∈R×R,v(0,x)=v0(x),vt(0,x)=v1(x),x∈R,
where v0 and v1 are the real-valued functions, and μ∈R. Indeed we can put u=(1/2)(v+i〈i∂x〉-1vt), then u satisfies(1.3)ℒu=i2μ〈i∂x〉-1(u+u¯)2,(t,x)∈R×R,u(0,x)=u0(x),x∈R,
where u0=(1/2)(v0+i〈i∂x〉-1v1).There are a lot of works devoted to the study of the cubic nonlinear Klein-Gordon equation(1.4)vtt+v-vxx=μv3,(t,x)∈R×R,v(0,x)=v0(x),vt(0,x)=v1(x),x∈R
with μ∈R. When μ<0, the global existence of solutions to (1.4) can be easily obtained in the energy space, which is, however, insufficient for determining the large-time asymptotic behavior of solutions. The sharp L∞- time decay estimates of solutions and nonexistence of the usual scattering states for (1.4) were shown in [3] by using hyperbolic polar coordinates under the conditions that the initial data are sufficiently regular and have a compact support.The initial value problem for the nonlinear Klein-Gordon equation with various cubic nonlinearities depending onv,vt,vx,vxx,vtx and having a suitable nonresonance structure was studied in [4–6], where small solutions were found in the neighborhood of the free solutions when the initial data are small and regular and decay rapidly at infinity. Hence the cubic nonlinearities are not necessarily critical; however the resonant nonlinear term v3 was excluded in these works. In paper [4], the nonresonant nonlinearities were classified into two types, one of them can be treated by the nonlinear transformation which is different from the method of normal forms [7] and the other reveals an additional time decay rate via the operator x∂t+t∂x which was used in [2]. This nonlinear transformation was refined in [8] and applied to a system of nonlinear Klein-Gordon equations in one or two space dimensions with nonresonant nonlinearities. It seems that the method of normal forms is very useful in the case of a single equation; however it does not work well in the case of a system of nonlinear Klein-Gordon equations. Some sufficient conditions on quadratic or cubic nonlinearities were given in [1], which allow us to prove global existence and to find sharp asymptotics of small solutions to the Cauchy problem including (1.2) with small and regular initial data having a compact support. Moreover it was proved that the asymptotic profile differs from that of the linear Klein-Gordon equation. See also [9, 10] in which asymptotic behavior of solutions to (1.4) was studied as in [1] by using hyperbolic polar coordinates. Compactness condition on the data was removed in [11] in the case of the cubic nonlinearity v3 and a real-valued solution. Final value problem with the cubic nonlinearity was studied in [12] for a real-valued solution. As far as we know the problem of finding the large-time asymptotics is still open for the case of the cubic nonlinearity v3 and the complex valued initial data. When the initial data are complex-valued, global existence and L∞-time decay estimates of small solutions to the Klein-Gordon equation with cubic nonlinearity |v|2v were obtained in paper [13] under the conditions that the initial data are smooth and have a compact support.The scattering problem and the time decay rates of small solutions to (1.4) with super-critical nonlinearities |v|p-1v and |v|p with p>3 were studied in papers of [14, 15]. Finally, we note that the Klein-Gordon equation (1.4) with quadratic nonlinearities in two space dimensions was studied in [16], where combining the method of the normal forms of [7] and the time decay estimate through the operator x∂t+t∂x of [17], it was shown that every quadratic nonlinearity is nonresonant.We denote the Lebesgue space byLp={ϕ∈𝒮′;∥ϕ∥Lp<∞}, with the norm ∥ϕ∥Lp=(∫R|ϕ(x)|pdx)1/p if 1≤p<∞ and ∥ϕ∥L∞=supx∈R|ϕ(x)| if p=∞. The weighted Sobolev space is(1.5)Hpm,s={ϕ∈Lp;∥〈x〉s〈i∂x〉mϕ∥Lp<∞},
for m,s∈R, 1≤p≤∞, where 〈x〉=1+x2. For simplicity we write Hm,s=H2m,s. The index 0 we usually omit if it does not cause a confusion. We denote by ℱϕ=ℱx→ξϕ≡ϕ̂=1/2π∫Re-ixξϕ(x)dx the Fourier transform of the function ϕ. Then the inverse Fourier transformation is ℱ-1ϕ=ℱξ→x-1ϕ=(1/2π)∫Reixξϕ(ξ)dξ.Our main result of this paper is the following.Theorem 1.1.
Letu0∈H3,1 and the norm ∥u0∥H3,1=ɛ. Then there exists ɛ0>0 such that for all 0<ɛ<ɛ0 the Cauchy problem (1.1) has a unique global solution
(1.6)u(t)∈C([0,∞);H3,1)
satisfying the time decay estimate
(1.7)∥u(t)∥H∞1,0≤Cɛ(1+t)-1/2.
Furthermore there exists a unique final state Ŵ+∈H∞0,1∩H0,1 such that
(1.8)∥u(t)-e-i〈i∂x〉tℱ-1Ŵ+e-2i|λ|2Ω|Ŵ+|2logt∥H1,0≤Cɛ3/2tγ-1/4,∥ℱei〈i∂x〉tu(t)-Ŵ+e-2i|λ|2Ω|Ŵ+|2logt∥H∞0,1≤Cɛ3/2tγ-1/4,
where γ∈(0,1/4), Ω(ξ)≡〈ξ〉2/〈2ξ〉(2〈ξ〉+〈2ξ〉).An important tool for obtaining the time decay estimates of solutions to the nonlinear Klein-Gordon equation is implementation of the operator(1.9)𝒥=〈i∂x〉e-i〈i∂x〉txei〈i∂x〉t=ℱ-1〈ξ〉e-i〈ξ〉ti∂ξei〈ξ〉tℱ=〈i∂x〉x+it∂x,
which is analogous to the operator x+it∂x=e(-it/2)∂x2xe(it/2)∂x2 in the case of the nonlinear Schrödinger equations used in [18]. The operator 𝒥 was used previously in paper of [15] for constructing the scattering operator for nonlinear Klein-Gordon equations with a supercritical nonlinearity. We have [x,〈i∂x〉α]=α〈i∂x〉α-2∂x; therefore the commutator [ℒ,𝒥]=ℒ𝒥-𝒥ℒ=0, where ℒ=∂t+i〈i∂x〉 is a linear part of (1.1). Since 𝒥 is not a purely differential operator, it is apparently difficult to calculate the action of 𝒥 on the nonlinearity in (1.1). So, instead we use the first-order differential operator(1.10)𝒫=t∂x+x∂t
which is closely related to 𝒥 by the identity 𝒫=ℒx-i𝒥 and acts easily on the nonlinearity. Moreover, it almost commutes with ℒ, since [ℒ,𝒫]=-i〈i∂x〉-1∂xℒ.Also we use the method of normal forms of [7] by which we transform the quadratic nonlinearity into a cubic one with a nonlocal operator. We multiply both sides of equation (1.1) by the free Klein-Gordon evolution group ℱ𝒰(-t)=ℱeit〈i∂x〉=eit〈ξ〉ℱ and put v(t,ξ)=eit〈ξ〉û to get(1.11)vt(t,ξ)=λeit〈ξ〉ℱ(〈i∂x〉-1u¯2)=λ2π〈ξ〉∫ReitAv(t,η-ξ)v(t,-η)¯dη,
where A(ξ,η)=〈ξ〉+〈ξ-η〉+〈η〉. Integrating (1.11) with respect to time, we find(1.12)v(t,ξ)=v(0,ξ)+λ2π〈ξ〉∫0tdτ∫ReiτAv(τ,η-ξ)v(τ,-η)¯dη.
Then we integrate by parts with respect to τ, taking into account (1.11),(1.13)v(t,ξ)+iλ∫ReitAv(t,η-ξ)v(t,-η)¯dη2π〈ξ〉A(ξ,η)=v(0,ξ)+iλ∫Rv(0,η-ξ)v(0,-η)¯dη2π〈ξ〉A(ξ,η)-2i|λ|2∫0tdτ∫Rdη2π〈ξ〉A(ξ,η)eiτ(A-〈η〉)v(τ,η-ξ)¯ℱx→η(〈i∂x〉-1u2).
Returning to the function u(t,x)=𝒰(t)ℱξ→x-1v=ℱξ→x-1(e-it〈ξ〉v(t,ξ)), we obtain the following equation:(1.14)ℒ(u+iλ𝒢(u¯,u¯))=-2i|λ|2𝒢(u¯,〈i∂x〉-1u2),
with the symmetric bilinear operator(1.15)𝒢(ϕ,ψ)=ℱξ→x-1∫Rĝ̂(ξ-η,η)ϕ̂(ξ-η)ψ̂(η)dη,
where(1.16)ĝ̂(ζ,η)=12π〈ζ+η〉(〈ζ+η〉+〈ζ〉+〈η〉),
and ζ=ξ-η. Our main point in this paper is to show that the right-hand side of (1.14) can be decomposed into two terms; one of them is a cubic nonlinearity(1.17)-2i|λ|21t𝒰(t)ℱ-1Ω(ξ)|ℱ𝒰(-t)u(t,ξ)|2ℱ𝒰(-t)u(t,ξ),
and the other one is a remainder term with an estimate like O(t-5/4∥ℱ𝒰(-t)u(t)∥H1,33).Remark 1.2.
We believe that all quadratic nonlinear termsu2,u¯2,|u|2 of problem (1.3) also could be considered by this approach. In the same way as in the derivation of (1.14) we get from (1.3)
(1.18)ℒ(u+i2μ(𝒢(u¯,u¯)+𝒢1(u,u)+2𝒢2(u,u¯)))=iμ2𝒢(u¯,〈i∂x〉-1(u+u¯)2)+iμ2𝒢1(u,〈i∂x〉-1(u+u¯)2)+iμ2𝒢2(u+u¯,〈i∂x〉-1(u+u¯)2),
where
(1.19)𝒢j(ϕ,ψ)=ℱξ→x-1∫Rĝ̂j(ξ-η,η)ϕ̂(ξ-η)ψ̂(η)dη,ĝ̂j(ζ,η)=12πAj(ζ+η,η)〈ζ+η〉
with A1(ξ,η)=〈ξ〉-〈ξ-η〉-〈η〉,A2(ξ,η)=〈ξ〉-〈ξ-η〉+〈η〉. Some more regularity conditions are necessary to treat the bilinear operators 𝒢j. Also we have to show that
(1.20)𝒢(u¯,〈i∂x〉-1(u¯2+2|u|2)),𝒢1(u,〈i∂x〉-1(u2+u¯2)),𝒢2(u,〈i∂x〉-1(u2+u¯2)),𝒢2(u¯,〈i∂x〉-1(u¯2+2|u|2))
are the nonresonant terms (i.e., remainders) and to remove the resonant terms
(1.21)𝒢(u¯,〈i∂x〉-1u2),𝒢1(u,〈i∂x〉-1|u|2),𝒢2(u,〈i∂x〉-1|u|2),𝒢2(u¯,〈i∂x〉-1u2)
by an appropriate phase function. We will dedicate a separate paper to this problem.We prove our main result in Section3. In the next section we prove several lemmas used in the proof of the main result.


## 2. Preliminaries

First we give some estimates for the symmetric bilinear operator(2.1)𝒢(ϕ,ψ)=ℱξ→x-1∫Rĝ̂(ξ-η,η)ϕ̂(ξ-η)ψ̂(η)dη,
where(2.2)ĝ̂(ζ,η)=12π〈ζ+η〉(〈ζ+η〉+〈ζ〉+〈η〉).
Denote the kernel as follows:(2.3)g(y,z)=14π2∬R2eiyζ+izη(〈ζ+η〉+〈ζ〉+〈η〉)〈ζ+η〉dηdζ.Lemma 2.1.
The representation is true(2.4)𝒢(ϕ,ψ)=∬R2g(y,z)ϕ(x-y)ψ(x-z)dydz,
where the kernel g(y,z) obeys the following estimate:
(2.5)|g(y,z)|≤C〈y〉-3〈z〉-3ln2(1+1|z-y|)
for all y,z∈R,y≠z. Moreover the following estimates are valid:
(2.6)∥𝒢(ϕ,ψ)∥Lp≤C∥ϕ∥Lq∥ψ∥Lr,∥𝒫𝒢(ϕ,ψ)∥Lp≤C∥𝒫ϕ∥Lq∥ψ∥Lr+C∥𝒫ψ∥Lq∥ϕ∥Lr+C∥∂tϕ∥Lq∥ψ∥Lr+C∥∂tψ∥Lq∥ϕ∥Lr
for 1≤p≤∞,1<q≤p/α,1<r≤p/(1-α),α∈[0,1], provided that the right-hand sides are bounded.Proof.
To prove representation (2.4), we substitute the direct Fourier transforms
(2.7)ϕ̂(ζ)=12π∫Re-i(x-y)ζϕ(x-y)dy,ψ̂(η)=12π∫Re-i(x-z)ηψ(x-z)dz
into the definition of the operator 𝒢. Then changing ζ=ξ-η, we find
(2.8)𝒢(ϕ,ψ)=(2π)-3/2∬R2ϕ(x-y)ψ(x-z)(∬R2ĝ̂(ζ,η)eiyζ+izηdηdζ)dydz=∬R2g(y,z)ϕ(x-y)ψ(x-z)dydz,
where the kernel g(y,z) is
(2.9)g(y,z)=(2π)-3/2∬R2ĝ̂(ζ,η)eiyζ+izηdηdζ=14π2∬R2eiyζ+izη(〈ζ+η〉+〈ζ〉+〈η〉)〈ζ+η〉dηdζ.
Changing the variables of integration ζ=ξ/2-η′ and η=ξ/2+η′ (the prime we will omit), we get
(2.10)g(y,z)=14π2∬R2eiyζ+izη(〈ζ+η〉+〈ζ〉+〈η〉)〈ζ+η〉dηdζ=14π2∫Rdξ〈ξ〉ei((y+z)/2)ξ∫RB(ξ,η)ei(z-y)ηdη,
where B(ξ,η)=1/(〈ξ〉+〈ξ/2-η〉+〈ξ/2+η〉). We change s=y+z/2 and ρ=(z-y) and denote
(2.11)g̃(s,ρ)=14π2∫Rdξ〈ξ〉eisξ∫RB(ξ,η)eiρηdη.
For the case of |ρ|≤1,|s|≤1 we integrate by parts using the identity eiρη=A(η)∂η(ηeiρη), where A(η)=1/(1+iρη). Then we get
(2.12)g̃(s,ρ)=-14π2∫Rdξ〈ξ〉eisξ∫Reiρηη∂η(A(η)B(ξ,η))dη.
Note that
(2.13)η∂ηB(ξ,η)=η(〈ξ〉+〈ξ/2-η〉+〈ξ/2+η〉)2(η-ξ/2〈η-ξ/2〉+η+ξ/2〈ξ/2+η〉).
Then
(2.14)|〈ξ〉-1η∂η(A(η)B(ξ,η))|≤C〈ξ〉(1+|ρ||η|)(〈ξ〉+〈η〉).
Hence we can estimate the kernel g̃(s,ρ) as follows:
(2.15)|g̃(s,ρ)|≤C∫Rdξ〈ξ〉∫Rdη(1+|ρ||η|)(〈ξ〉+〈η〉)≤C∫1∞dη1+|ρ|η∫1∞dξξ(η+ξ)=C∫1∞dη1+|ρ|η1η(lnξη+ξ)|1∞dξ≤C∫1∞lnη(1+|ρ|η)ηdη≤C∫11/|ρ|lnηdηη+C|ρ|∫1/|ρ|∞lnηdηη2≤Cln2(1+1|ρ|)
for the case of |ρ|≤1,|s|≤1. For the case of |ρ|≤1,|s|≥1 we integrate three times by parts with respect to ξ(2.16)g̃(s,ρ)=C|s|-3∫Rdξeisξ∫Reiρη∂ξ3(〈ξ〉-1η∂η(A(η)B(ξ,η)))dη.
Note that
(2.17)|∂ξ3(〈ξ〉-1η∂η(A(η)B(ξ,η)))|≤C〈ξ〉(1+|ρ||η|)(〈ξ〉+〈η〉).
Hence we can estimate the kernel  g(y,z) as follows:
(2.18)|g̃(s,ρ)|≤C|s|-3∫Rdξ〈ξ〉∫Rdη(1+|ρ||η|)(〈ξ〉+〈η〉)≤C|s|-3ln2(1+1|ρ|)
for all |ρ|≤1,|s|≥1. For the case |ρ|≥1,|s|≤1 we integrate by parts three times with respect to η(2.19)g̃(s,ρ)=C|ρ|-3∫Rdξ〈ξ〉eisξ∫Reiρη∂η3B(ξ,η)dη.
Note that |∂η3B(ξ,η)|≤C(〈ξ〉+〈η〉)-3. Hence
(2.20)|g̃(s,ρ)|≤C|ρ|-3∫Rdξ〈ξ〉2∫Rdη〈η〉2≤C|ρ|-3.
Finally for the case of |ρ|≥1,|s|≥1 we integrate by parts three times with respect to ξ and η(2.21)g̃(s,ρ)=C|sρ|-3∫Rdξeisξ∫Reiρη∂ξ3∂η3(〈ξ〉-1B(ξ,η))dη.
Since
(2.22)|∂ξ3∂η3(〈ξ〉-1B(ξ,η))|≤C〈ξ〉2(〈ξ〉+〈η〉)2,
then we can estimate the kernel  g̃(s,ρ) as follows:
(2.23)|g̃(s,ρ)|≤C|sρ|-3∫Rdξ〈ξ〉2∫Rdη〈η〉2≤C|sρ|-3
for all |ρ|≥1,|s|≥1. Hence estimate (2.5) is true.
By virtue of estimate (2.5) applying the Hölder inequality with 1/p=1/p1+1/p2 and the Young inequality with 1/p1+1=1/q+1/q1 and 1/p2+1=1/r+1/r1, we find
(2.24)∥𝒢(ϕ,ψ)∥Lp≤C∥∫R|ϕ(x-y)|dy〈y〉3∥Lxp1∥∫Rln2(1+1|z-y|)|ψ(x-z)|dz〈z〉3∥Lxp2Ly∞≤C∥ϕ∥Lq∥ψ∥Lr,
where 1≤p≤∞,1<q≤p/α,1<r≤p/(1-α),α∈[0,1].
We now estimate the operator𝒫=t∂x+x∂t as follows:
(2.25)𝒫𝒢(ϕ,ψ)=𝒢(𝒫ϕ,ψ)+∬R2yg(y,z)ψ(x-z)∂tϕ(x-y)dydz+𝒢(ϕ,𝒫ψ)+∬R2zg(y,z)ϕ(x-y)∂tψ(x-z)dydz.
Then by virtue of estimate (2.4) applying the Hölder and Young inequalities, we get
(2.26)∥𝒫𝒢(ϕ,ψ)∥Lp≤∥𝒢(𝒫ϕ,ψ)∥Lp+∥𝒢(ϕ,𝒫ψ)∥Lp+∥∬R2yg(y,z)ψ(x-z)∂tϕ(x-y)dydz∥Lp+∥∬R2zg(y,z)ϕ(x-y)∂tψ(x-z)dydz∥Lp≤C∥𝒫ϕ∥Lq∥ψ∥Lr+C∥𝒫ψ∥Lq∥ϕ∥Lr+C∥∂tϕ∥Lq∥ψ∥Lr+C∥∂tψ∥Lq∥ϕ∥Lr.
Lemma 2.1 is proved.We now decompose the free Klein-Gordon evolution group𝒰(t)=e-i〈i∂x〉t=ℱ-1E(t)ℱ, where E(t)=e-it〈ξ〉 similarly to the factorization of the free Schrödinger evolution group. We denote the dilation operator by(2.27)𝒟ωϕ=1iωϕ(xω),(𝒟ω)-1=i𝒟1/ω.
Define the multiplication factor M(t)=e-it〈ix〉θ(x), where θ(x)=1 for |x|<1 and θ(x)=0 for |x|≥1. We introduce the operator(2.28)ℬϕ=θ(x)〈ix〉3/2ϕ(x〈ix〉).
The inverse operator ℬ-1 acts on the functions ϕ(x) defined on (-1,1) as follows:(2.29)ℬ-1ϕ=1〈ξ〉3/2ϕ(ξ〈ξ〉)
for all ξ∈R, since ξ=x/〈ix〉∈R and x=ξ/〈ξ〉∈(-1,1). We now introduce the operators(2.30)𝒱(t)=ℬ-1M¯(t)𝒟t-1ℱ-1e-it〈ξ〉,𝒲(t)=M¯(t)(1-θ)𝒟t-1ℱ-1e-it〈ξ〉,
so that we have the representation for the free Klein-Gordon evolution group(2.31)𝒰(t)ℱ-1=e-it〈i∂x〉ℱ-1=ℱ-1e-it〈ξ〉=𝒟tM(t)(ℬ𝒱(t)+𝒲(t))=𝒟tM(t)ℬ+𝒟tM(t)ℬ(𝒱(t)-1)+𝒟tM(t)𝒲(t).
The first term 𝒟tM(t)ℬϕ of the right-hand side of (2.31) describes inside the light cone the well-known leading term of the large-time asymptotics of solutions of the linear Klein-Gordon equation ℒu=0 with initial data ϕ. The second term of the right-hand side of (2.31) is a remainder inside of the light cone, whereas the last term represents the large time asymptotics outside of the light cone which decays more rapidly in time. We also have(2.32)ℱ𝒰(-t)=ℱeit〈i∂x〉=eit〈ξ〉ℱ=𝒱-1(t)ℬ-1M¯(t)𝒟t-1+𝒲-1(t)𝒟t-1=ℬ-1M¯(t)𝒟t-1+(𝒱-1(t)-1)ℬ-1M¯(t)𝒟t-1+𝒲-1(t)𝒟t-1,
where the right-inverse operators are(2.33)𝒱-1(t)=E¯(t)ℱ𝒟tM(t)ℬ,𝒲-1(t)=E¯(t)ℱ𝒟t(1-θ),
where E(t)=e-it〈ξ〉.In the next lemma we state the estimates of the operators𝒱(t)=ℬ-1M¯(t)𝒟t-1ℱ-1e-it〈ξ〉.Lemma 2.2.
The estimates hold as follows:(2.34)∥𝒱(t)ϕ∥L2≤C∥ϕ∥L2,∥〈ξ〉ρ∂ξ𝒱(t)ϕ∥L2≤C∥〈ξ〉α∂ξϕ∥L2+C∥〈ξ〉α-1ϕ∥L2,
where ρ<α,α∈[1,2],  and
(2.35)∥〈ξ〉ρ∂ξ𝒱-1(t)ϕ∥L2≤C∥〈ξ〉1/4+α∂ξϕ∥L2+C∥〈ξ〉α-3/4ϕ∥L2,
where ρ<α,α∈[0,1], provided the right-hand sides are finite.Proof.
Changing the variable of integrationx=ξ〈ξ〉-1, we see that
(2.36)∥ℬ-1ϕ∥L22=∫R|ϕ(ξ〈ξ〉)|2dξ〈ξ〉3=∫-11|ϕ(x)|2dx=∥ϕ∥L2(-1,1)2.
Hence ∥𝒱(t)ϕ∥L2=∥ϕ∥L2.
Consider the estimate for the derivative∂ξ𝒱(t)ϕ.  Define S(ξ,η)=〈η〉-(ξη+1)/〈ξ〉. Note that ∂S/∂η=η/〈η〉-ξ/〈ξ〉=((1+〈η〉〈ξ〉-ηξ)/〈ξ〉〈η〉(〈η〉+〈ξ〉))(η-ξ) and ∂S/∂ξ=(ξ-η)/〈ξ〉3. Hence integrating by parts one time yields
(2.37)∂ξ∫Re-itSϕ(η)dη=-it〈ξ〉-3∫Re-itSϕ(η)(ξ-η)dη=〈ξ〉-3∫Re-itS(ϕ′(η)g(ξ,η)+ϕ(η)gη(ξ,η))dη,
where g(ξ,η)=(ξ-η)/(ξ/〈ξ〉-η/〈η〉)=〈ξ〉〈η〉(〈η〉+〈ξ〉)/(1+〈η〉〈ξ〉-ηξ). Also we have
(2.38)(1+〈η〉〈ξ〉-ηξ)-1≤C〈ξ〉〈η〉(〈η〉+〈ξ〉)2.
Hence the estimate is true
(2.39)g(ξ,η)=〈ξ〉〈η〉(〈η〉+〈ξ〉)1+〈η〉〈ξ〉-ηξ≤C〈ξ〉2〈η〉2〈η〉+〈ξ〉.
Since ∂ξ〈ξ〉a=aξ〈ξ〉a-2, we get
(2.40)〈ξ〉ρ∂ξ𝒱(t)ϕ=〈ξ〉ρit2π∂ξ〈ξ〉-3/2∫Re-itSϕ(η)dη=-32ξ〈ξ〉ρ-2𝒱(t)ϕ+〈ξ〉ρ-9/2it2π∫Re-itSϕ′(η)g(ξ,η)dη+〈ξ〉ρ-9/2it2π∫Re-itSϕ(η)gη(ξ,η)dη.
Consider the L2-estimate of the integral
(2.41)t〈ξ〉ρ-9/2∫Reit(ξη/〈ξ〉)ψ(η)g(ξ,η)dη.
We have
(2.42)∥t∫Reit(ξη/〈ξ〉)ψ(η)g(ξ,η)〈ξ〉ρ-9/2dη∥L22=t∫Rdηψ(η)∫Rdζψ(ζ)¯∫Reit(ξ/〈ξ〉)(η-ζ)g(ξ,η)g(ξ,ζ)〈ξ〉2ρ-9dξ=∫Rdηψ(η)∫Rdζψ(ζ)¯G(η,ζ),
where the kernel
(2.43)G(η,ζ)=t∫Reit(ξ/〈ξ〉)(η-ζ)g(ξ,η)g(ξ,ζ)〈ξ〉2ρ-9dξ=t〈η〉〈ζ〉∫Reit(ξ/〈ξ〉)(η-ζ)〈η〉+〈ξ〉(1+〈η〉〈ξ〉-ηξ)〈ζ〉+〈ξ〉(1+〈ζ〉〈ξ〉-ζξ)〈ξ〉2ρ-7dξ.
After a change ξ=x/〈ix〉, we find
(2.44)G(η,ζ)=t〈η〉〈ζ〉∫-11eitx(η-ζ)〈η〉+1/〈ix〉(1+〈η〉/〈ix〉-ηx/〈ix〉)〈ζ〉+1/〈ix〉(1+〈ζ〉/〈ix〉-ζx/〈ix〉)〈ix〉4-2ρdx.
We now change the contour z=x+iy∈Γ of integration in G as y=sign(η-ζ)〈ix〉2; then
(2.45)G(η,ζ)=t〈η〉〈ζ〉∫Γeitz(η-ζ)〈η〉+1/〈iz〉(1+〈η〉/〈iz〉-ηz/〈iz〉)〈ζ〉+1/〈iz〉(1+〈ζ〉/〈iz〉-ζz/〈iz〉)〈iz〉4-2ρdz.
Since  |〈iz〉|=((〈ix〉2+y2)2+4x2y2)1/4~〈ix〉, using (2.38) and the inequality (〈η〉+〈ξ〉)≥〈η〉α-1〈ξ〉2-α for α∈[1,2], we get |(〈η〉+1/〈iz〉)/(1+〈η〉/〈iz〉-ηz/〈iz〉)|≤C〈η〉α-1〈ix〉α-2; also we have
(2.46)|eitz(η-ζ)|=e-〈ix〉2t|η-ζ|≤C〈ix〉-2δ(1+t|η-ζ|)δ
for δ>1. Therefore we obtain the estimate
(2.47)|G(η,ζ)|≤Ct〈η〉α〈ζ〉α(1+t|η-ζ|)δ∫-11〈ix〉2(α-ρ-δ)dx≤Ct〈η〉α〈ζ〉α(1+t|η-ζ|)δ,
if we choose ρ<α,α∈[1,2] and 1<δ<1+α-ρ. Thus we get
(2.48)∥t∫Reit(ξη/〈ξ〉)ψ(η)g(ξ,η)〈ξ〉ρ-9/2dη∥L22=∫Rdηψ(η)∫Rdζψ(ζ)¯G(η,ζ)≤C∥〈η〉αψ(η)∥L2∥∫Rdζψ(ζ)¯〈ζ〉αt(1+t|η-ζ|)δ∥L2≤Ct∥〈·〉αψ∥L22∫Rdη(1+t|η|)δ≤C∥〈·〉αψ∥L22
since δ>1.
In the same manner we consider the estimate of the integral(2.49)∥t∫Reit(ξη/〈ξ〉)ψ(η)gη(ξ,η)〈ξ〉ρ-9/2dη∥L22=t∫Rdηψ(η)∫Rdζψ(ζ)¯∫Reit(ξ/〈ξ〉)(η-ζ)gη(ξ,η)gζ(ξ,ζ)〈ξ〉2ρ-9dξ=∫Rdηψ(η)∫Rdζψ(ζ)¯G̃(η,ζ),
where the kernel
(2.50)G̃(η,ζ)=t∫Reit(ξ/〈ξ〉)(η-ζ)gη(ξ,η)gζ(ξ,ζ)〈ξ〉2ρ-9dξ=t∫Reit(ξ/〈ξ〉)(η-ζ)(η+ξ+(η/〈η〉)〈ξ〉)(1+〈η〉〈ξ〉-ηξ)(ζ+ξ+(ζ/〈ζ〉)〈ξ〉)(1+〈ζ〉〈ξ〉-ζξ)〈ξ〉2ρ-7dξ,
since by a direct calculation
(2.51)∂η〈η〉(〈η〉+〈ξ〉)(1+〈η〉〈ξ〉-ηξ)=η+ξ+(η/〈η〉)〈ξ〉(1+〈η〉〈ξ〉-ηξ).
In the same way as in (2.47) we have
(2.52)|G̃(η,ζ)|≤Ct〈η〉α-1〈ζ〉α-1(1+t|η-ζ|)δ
if we choose ρ<α,α∈[1,2], and 1<δ<1+α-ρ. Therefore we get
(2.53)∥t∫Reit(ξη/〈ξ〉)ψ(η)gη(ξ,η)〈ξ〉ρ-9/2dη∥L22≤C∥〈η〉α-1ψ(η)∥L22∫Rtdη(1+t|η|)δ≤C∥〈·〉α-1ψ∥L22.
Hence ∥〈ξ〉ρ∂ξ𝒱(t)ϕ∥L2≤C∥〈ξ〉α∂ξϕ∥L2+C∥〈ξ〉α-1ϕ∥L2. Thus we have the second estimate of the lemma.
Consider the estimate for the derivative∂ξ𝒱-1(t)ϕ.  Note that ∂S(η,ξ)/∂ξ=ξ/〈ξ〉-η/〈η〉=((1+〈η〉〈ξ〉-ηξ)/〈ξ〉〈η〉(〈η〉+〈ξ〉))(ξ-η) and ∂S(η,ξ)/∂η=(η-ξ)/〈η〉3. Hence integrating by parts one time yields
(2.54)〈ξ〉ρ∂ξ𝒱-1(t)ϕ=-t2iπ〈ξ〉ρ-1∫R1+〈η〉〈ξ〉-ηξ(〈η〉+〈ξ〉)〈η〉1/2ϕ(η)deitS(η,ξ)=t2iπ〈ξ〉ρ-1∫ReitS(η,ξ)(ϕ′(η)g1(ξ,η)+ϕ(η)g1η(ξ,η))dη,
where S(η,ξ)=〈ξ〉-(ξη+1)/〈η〉 and g1(ξ,η)=(((1+〈η〉〈ξ〉-ηξ))/(〈η〉+〈ξ〉))〈η〉1/2. The estimate is true
(2.55)g1(ξ,η)=1+〈η〉〈ξ〉-ηξ〈η〉+〈ξ〉〈η〉1/2≤C〈η〉3/2〈ξ〉〈η〉+〈ξ〉.
Consider the L2-estimate of the integral
(2.56)〈ξ〉ρ-1t∫Re-it(ξη/〈η〉)ψ(η)g1(ξ,η)dη.
We have, changing η/〈η〉=y and ζ/〈ζ〉=z,
(2.57)∥〈ξ〉ρ-1t∫Re-it(ξη/〈η〉)ψ(η)g1(ξ,η)dη∥L22=t∫Rdηψ(η)∫Rdζψ(ζ)¯∫Re-itξ(η/〈η〉-ζ/〈ζ〉)g1(ξ,η)g1(ξ,ζ)〈ξ〉2ρ-2dξ=t∫-11dy〈iy〉3ψ(y〈iy〉)∫-11dz〈iz〉3ψ(z〈iz〉)¯×∫Re-itξ(y-z)g1(ξ,y〈iy〉)g1(ξ,z〈iz〉)〈ξ〉2ρ-2dξ=∫-11dy〈iy〉3ψ(y〈iy〉)∫-11dz〈iz〉3ψ(z〈iz〉)¯G1(y,z),
where the kernel
(2.58)G1(y,z)=t∫Re-itξ(y-z)g1(ξ,y〈iy〉)g1(ξ,z〈iz〉)〈ξ〉2ρ-2dξ=t〈iy〉-1/2〈iz〉-1/2∫Re-itξ(y-z)1+〈ξ〉/〈iy〉-yξ/〈iy〉1/〈iy〉+〈ξ〉1+〈ξ〉/〈iz〉-zξ/〈iz〉1/〈iz〉+〈ξ〉〈ξ〉2ρ-2dξ.
We now change the contour ξ=x̃+iỹ∈Γ of integration in G as ỹ=sign(y-z)〈x̃〉2; then
(2.59)G1(y,z)=t〈iy〉-1/2〈iz〉-1/2∫Γe-itξ(y-z)1+〈ξ〉/〈iy〉-yξ/〈iy〉1/〈iy〉+〈ξ〉1+〈ξ〉/〈iz〉-zξ/〈iz〉1/〈iz〉+〈ξ〉〈ξ〉2ρ-2dξ.
Since  |〈ξ〉|=((〈x̃〉2-ỹ2)2+4x̃2ỹ2)1/2~〈x̃〉 and by (2.55) |(1+〈ξ〉/〈iy〉-yξ/〈iy〉)/(1/〈iy〉+〈ξ〉)|≤C〈ξ〉1-α〈iy〉-α for α∈[0,1], and also
(2.60)|e-itξ(y-z)|≤C(1+t〈x̃〉|y-z|)2≤Ct〈x̃〉δ|y-z|δ〈y-z〉
for δ<1,we obtain the estimate
(2.61)|G1(y,z)|≤C〈iy〉-1/2-α〈iz〉-1/2-α|y-z|δ〈y-z〉∫R〈x̃〉-2α+2ρ-δdx̃≤C〈iy〉-1/2-α〈iz〉-1/2-α|y-z|δ〈y-z〉,
if we choose ρ<α, and 1>δ>1-2α+2ρ. Therefore we get
(2.62)∥t∫Reit(ξη/〈ξ〉)ψ(η)g(ξ,η)〈ξ〉ρ-9/2dη∥L22=∫-11dy〈iy〉5/2-αψ(y〈iy〉)∫-11dz〈iz〉5/2-αψ(z〈iz〉)¯C|y-z|δ〈y-z〉≤C∫-11dy〈iy〉5/2-2α|ψ(y〈iy〉)|2≤C∫-11dξ〈ξ〉1/2+2α|ψ(ξ)|2≤C∥〈·〉1/4+αψ∥L22.
In the same manner we consider the estimate of the integral withϕ(η)g1η(ξ,η). Hence
(2.63)∥〈ξ〉ρ∂ξ𝒱-1(t)ϕ∥L2≤C∥〈ξ〉1/4+α∂ξϕ∥L2+C∥〈ξ〉α-3/4ϕ∥L2,
where α>ρ. Thus we have the second estimate of the lemma. Lemma 2.2 is proved.In the next lemma we prove an auxiliary asymptotics for the integral∫0∞e-itz2Φ(ξ,z)dz.Lemma 2.3.
The estimate holds as follows:(2.64)|〈ξ〉α(∫0∞e-itz2Φ(ξ,z)dz-Φ(ξ,0)π4it)|≤Ct-3/4∥〈ξ〉αΦz(ξ,z)∥Lξ∞Lz2
provided the right-hand side is finite, where α∈R.Proof.
We represent the integral(2.65)∫0∞e-itz2Φ(ξ,z)dz=Φ(ξ,0)∫0∞e-itz2dz+R=Φ(ξ,0)π4it+R,
where the remainder term is
(2.66)R=∫0∞e-itz2(Φ(ξ,z)-Φ(ξ,0))dz.
In the remainder term we integrate by parts via identity e-itz2=(1/(1-2itz2))∂z(ze-itz2)(2.67)∥〈ξ〉αR∥Lξ∞≤C∥〈ξ〉α∫0∞11+tz2(Φ(ξ,z)-Φ(ξ,0))dz∥Lξ∞+C∥〈ξ〉α∫0∞z1+tz2Φz(ξ,z)dz∥Lξ∞≤C∥〈ξ〉αΦz(ξ,z)∥Lξ∞Lz2∥z(1+tz2)-1∥Lz1+C∥〈ξ〉αΦz(ξ,z)∥Lξ∞Lz2∥z(1+tz2)-1∥Lz2≤Ct-3/4∥〈ξ〉αΦz(ξ,z)∥Lξ∞Lz2,
since
(2.68)|Φ(ξ,z)-Φ(ξ,0)|≤∫0z|Φz(ξ,z)|dz≤C|z|∥Φz(ξ,z)∥Lξ∞Lz2.
Thus we have the estimate for the remainder in the asymptotic formula. Lemma 2.3 is proved.In the next lemma we obtain the asymptotics for the operator𝒱(t)=ℬ-1M¯(t)𝒟t-1ℱ-1e-it〈ξ〉 and the right-inverse operator 𝒱-1(t)=E¯(t)ℱ𝒟tM(t)ℬ.Lemma 2.4.
The estimates hold as follows:(2.69)∥〈ξ〉β(𝒱(t)-1)ϕ∥L∞≤Ct-1/4(∥〈ξ〉β+3/4∂ξϕ∥L2+∥〈ξ〉β-1/4ϕ∥L2),∥〈ξ〉β-3/4(𝒱-1(t)-1)ϕ∥L∞≤Ct-1/4(∥〈ξ〉3/2+β∂ξϕ∥L2+∥〈ξ〉1/2+βϕ∥L2),
for all t≥1, where 0≤β≤3/2, provided the right-hand sides are finite.Proof.
We have the identities(2.70)S(ξ,η)=〈η〉-〈ξ〉-ξ〈ξ〉(η-ξ)=1+〈η〉〈ξ〉-ηξ(〈η〉+〈ξ〉)2〈ξ〉(η-ξ)2,Sη(ξ,η)=η〈η〉-ξ〈ξ〉=1+〈η〉〈ξ〉-ηξ〈ξ〉〈η〉(〈η〉+〈ξ〉)(η-ξ),
and Sηη(ξ,η)=〈η〉-3. We now change z=S(ξ,η). We denote The inverse functions by ηj(z), so that η1(z):(0,∞)→(ξ,∞) and η2(z):(0,∞)→(-∞,ξ).
Thus the stationary pointz=0 transforms into η=ξ. Hence we can write the representation
(2.71)𝒱(t)ϕ=〈ξ〉-3/2it2π∫Re-itS(ξ,η)ϕ(η)dη=2〈ξ〉-3/2it2π∑j=12∫0∞e-itz2ϕ(ηj(z))S(ξ,ηj(z))|Sη(ξ,ηj(z))|dz.
By a direct calculation we find
(2.72)dzdη=|Sη(ξ,η)|S(ξ,η)=1+〈η〉〈ξ〉-ηξ〈η〉〈ξ〉1/2.
Therefore we obtain
(2.73)𝒱(t)ϕ=∫0∞e-itz2Φ(ξ,z)dz,
where
(2.74)Φ(ξ,z)=2〈ξ〉-1it2π∑j=12〈η〉ϕ(η)1+〈ξ〉〈η〉-ξη|η=ηj(z).
Application of Lemma 2.3 yields
(2.75)𝒱(t)ϕ=ϕ(ξ)+O(t-3/4∥Φz(ξ,z)∥Lξ∞Lz2).
By a direct calculation we have
(2.76)Φz(ξ,z)dzdη=it2π∑j=122〈η〉3/2ϕ′(η)〈ξ〉3/4(1+〈η〉〈ξ〉-ηξ)3/4+it2π∑j=12η+(η〈ξ〉+ξ〈η〉)/(〈η〉+〈ξ〉)〈ξ〉3/4〈η〉1/2(1+〈η〉〈ξ〉-ηξ)3/4ϕ(η).
By (2.38) we get the estimate
(2.77)∥〈ξ〉βΦz(ξ,z)∥Lξ∞Lz2=∥〈ξ〉βΦz(ξ,z)dzdη∥Lξ∞Lη2≤Ct∥〈ξ〉β〈η〉9/4(〈η〉+〈ξ〉)3/2ϕ′(η)∥Lξ∞Lη2+Ct∥〈ξ〉β〈η〉5/4(〈η〉+〈ξ〉)3/2ϕ(η)∥Lξ∞Lη2≤Ct∥〈η〉3/4+βϕ′∥L2+Ct∥〈η〉β-1/4ϕ∥L2
for 0≤β≤3/2 since (〈η〉+〈ξ〉)≥C〈ξ〉(2/3)β〈η〉1-(2/3)β. This yields the first estimate of the lemma.
We now prove the second estimate. We have(2.78)𝒱-1(t)ϕ=t2iπ∫-11eit(〈ξ〉-ξy-〈iy〉)〈iy〉-3/2ϕ(y〈iy〉)dy,
then changing y=η〈η〉-1, we find
(2.79)𝒱-1(t)ϕ(x)=t2iπ∫ReitS(η,ξ)ϕ(η)〈η〉-3/2dη,
where
(2.80)S(η,ξ)=〈ξ〉-ξη+1〈η〉=1+〈η〉〈ξ〉-ηξ(〈η〉+〈ξ〉)2〈η〉(η-ξ)2.
As above we change z=S(η,ξ) and represent
(2.81)𝒱-1(t)ϕ(x)=t2iπ∫ReitS(η,ξ)ϕ(η)〈η〉-3/2dη=2tiπ∑j=12∫0∞eitz2ϕ(ηj(z))〈ηj(z)〉-3/2S(ηj(z),ξ)|Sη(ηj(z),ξ)|dz.
By a direct calculation we find Sη(η,ξ)=(η-ξ)/〈η〉^3 and
(2.82)dzdη=|Sη(η,ξ)|2S(η,ξ)=〈η〉+〈ξ〉2〈η〉5/21+〈η〉〈ξ〉-ηξ.
Therefore we obtain
(2.83)𝒱-1(t)ϕ(x)=∫0∞eitz2Φ(ξ,z)dz,
where
(2.84)Φ(ξ,z)=22tiπ∑j=12〈η〉1+〈η〉〈ξ〉-ηξ〈η〉+〈ξ〉ϕ(η)|η=ηj(z).
Application of Lemma 2.3 yields
(2.85)𝒱-1(t)ϕ(x)=ϕ(ξ)+O(t-3/4∥Φz(ξ,z)∥Lξ∞Lz2).
By a direct calculation we have
(2.86)Φz(ξ,z)dzdη=22tiπ∑j=12(1+〈η〉〈ξ〉-ηξ)3/4(〈η〉+〈ξ〉)3/2〈η〉9/4ϕ′(η)+22tiπ∑j=12〈η〉5/4(1+〈η〉〈ξ〉-ηξ)1/4〈η〉+〈ξ〉(〈η〉1+〈η〉〈ξ〉-ηξ〈η〉+〈ξ〉)ηϕ(η),
Since
(2.87)|(〈η〉1+〈η〉〈ξ〉-ηξ〈η〉+〈ξ〉)η|〈η〉5/4(1+〈η〉〈ξ〉-ηξ)1/4〈η〉+〈ξ〉=|〈ξ〉η〈η〉1/4(1+〈η〉〈ξ〉-ηξ)3/4(〈η〉+〈ξ〉)5/2+〈η〉5/4(η〈ξ〉-ξ〈η〉)2(〈η〉+〈ξ〉)3/2(1+〈η〉〈ξ〉-ηξ)1/4|≤C〈ξ〉3/4〈η〉2(〈ξ〉+〈η〉)3/2,
therefore we get the estimate
(2.88)∥〈ξ〉β-3/4Φz(ξ,z)∥Lξ∞Lz2=∥〈ξ〉β-3/4Φz(ξ,z)dzdη∥Lξ∞Lη2≤Ct(∥〈ξ〉β〈η〉3(〈η〉+〈ξ〉)3/2ϕ′(η)∥Lξ∞Lη2+∥〈ξ〉β〈η〉2(〈ξ〉+〈η〉)3/2ϕ(η)∥Lξ∞Lη2)≤Ct(∥〈η〉3/2+βϕ′∥L2+∥〈η〉1/2+βϕ∥L2)
for 0≤β≤2/3, since (〈η〉+〈ξ〉)≥C〈ξ〉2/3β〈η〉1-2/3β. This yields the second estimate of the lemma. Lemma 2.4 is proved.In the next lemma we find the estimates for the operator𝒲(t)=(1-θ)M¯(t)𝒟t-1ℱ-1e-it〈ξ〉.Lemma 2.5.
The estimates hold as follows:(2.89)∥𝒲(t)ϕ∥Lr≤Ct-1/2+(1-δ)(1-1/r)(∥〈ξ〉(1+2δ)(1-1/r)∂ξϕ∥L1+∥〈ξ〉(1+2δ)(1-1/r)-1ϕ∥L1)
for all t≥1, where 1<r≤∞,δ∈[0,1], provided the right-hand side is finite.Proof.
Note that𝒲(t)ϕ=0 for |x|≤1. To prove the estimate, we integrate by parts via the identity eit(xξ-〈ξ〉)=(1/it(x-ξ/〈ξ〉))∂ξ(eit(xξ-〈ξ〉))(2.90)𝒲(t)ϕ=it2π∫Reit(xξ-〈ξ〉)ϕ(ξ)dξ=-12πit∫Reit(xξ-〈ξ〉)1x-ξ/〈ξ〉ϕ′(ξ)dξ-12πit∫Reit(xξ-〈ξ〉)〈ξ〉-3(x-ξ/〈ξ〉)2ϕ(ξ)dξ
for all |x|≥2. Since 1/|x-ξ/〈ξ〉|≤C〈x〉-1 for all |x|≥2, then we get
(2.91)|𝒲(t)ϕ|≤Ct-1/2〈x〉-1∫R(|ϕ′(ξ)|+〈ξ〉-3|ϕ(ξ)|)dξ≤Ct-1/2〈x〉-1(∥∂ξϕ∥L1+C∥〈ξ〉-3ϕ∥L1).
For the case of 1<|x|≤2 we integrate by parts via the identity eit(xξ-〈ξ〉)=(1/(ξ/〈ξ〉+it〈ξ〉(x-ξ/〈ξ〉)))∂ξ(〈ξ〉eit(xξ-〈ξ〉))(2.92)𝒲(t)ϕ=-it2π∫Reit(xξ-〈ξ〉)〈ξ〉ξ/〈ξ〉+it〈ξ〉(x-ξ/〈ξ〉)ϕ′(ξ)dξ+it2π∫Reit(xξ-〈ξ〉)1/〈ξ〉^2+it〈ξ〉(ξx/〈ξ〉-1)(ξ/〈ξ〉+it〈ξ〉(x-ξ/〈ξ〉))2ϕ(ξ)dξ.
Since |x-ξ/〈ξ〉|≥||x|-1|+|1-|ξ|/〈ξ〉| and |ξx/〈ξ〉-1|≤|ξ|/〈ξ〉||x|-1|+|1-|ξ|/〈ξ〉|≤|x-ξ/〈ξ〉| for 1<|x|, therefore
(2.93)|𝒲(t)ϕ|≤Ct∫R〈ξ〉|ϕ′(ξ)|+|ϕ(ξ)||ξ|/〈ξ〉+t〈ξ〉(||x|-1|+|1-|ξ|/〈ξ〉|)dξ≤Ct∫R〈ξ〉|ϕ′(ξ)|+|ϕ(ξ)|1+t〈ξ〉-2+t〈ξ〉||x|-1|dξ
for 1<|x|≤2. Hence we get
(2.94)∥𝒲(t)ϕ∥Lr≤Ct-1/2∥〈x〉-1∥Lr(|x|≥2)(∥∂ξϕ∥L1+C∥〈ξ〉-3ϕ∥L1)+Ct∫Rdξ〈ξ〉|ϕ′(ξ)|+|ϕ(ξ)|1+t〈ξ〉-2∥11+(t〈ξ〉/((1+t〈ξ〉^(-2))))||x|-1|∥Lr(1<|x|≤2)≤Ct-1/2(∥∂ξϕ∥L1+C∥〈ξ〉-3ϕ∥L1)+Ct1/2-1/r∫R(〈ξ〉1-1/r|ϕ′(ξ)|+〈ξ〉-1/r|ϕ(ξ)|)(1+t〈ξ〉-2)1/r-1dξ≤Ct-1/2+(1-δ)(1-1/r)(∥〈ξ〉(1+2δ)(1-1/r)∂ξϕ∥L1+∥〈ξ〉(1+2δ)(1-1/r)-1ϕ∥L1)
for δ∈[0,1]. This yields the estimate of the lemma. Lemma 2.5 is proved.We next prove the time decay estimate in terms of the operator𝒥.Lemma 2.6.
The estimate is valid(2.95)∥u∥L∞≤C〈t〉-1/2∥u∥H3/21/2(∥u∥H3/21/2+∥𝒥u∥H1/21/2)
for all t≥0, provided that the right-hand side is finite.Proof.
Since∥u∥L1≤C∥xu∥L21/2∥u∥L21/2, by applying the L∞-L1 time decay estimate of the free evolution group e-i〈i∂x〉t (see paper of [19, Lemma 1]), we get
(2.96)∥u∥L∞=∥e-i〈i∂x〉tei〈i∂x〉tu∥L∞≤Ct-1/2∥〈i∂x〉3/2ei〈i∂x〉tu∥L1≤Ct-1/2∥x〈i∂x〉3/2ei〈i∂x〉tu∥L21/2∥〈i∂x〉3/2ei〈i∂x〉tu∥L21/2≤Ct-1/2∥𝒥u∥H1/21/2∥u∥H3/21/2
for all t>0. Then by the Sobolev inequality we have ∥u∥L∞≤C∥u∥H1. Thus the desired estimate follows. Lemma 2.6 is proved.Next we obtain the asymptotics for the integral(2.97)∫ReitA(ξ,y)ϕ(ξ,y)dy,
where A(ξ,y)=〈ξ〉+〈ζ〉-2〈η〉, and ζ=ξ+2y,η=ξ+y.Lemma 2.7.
The following asymptotics is true:(2.98)∫ReitA(ξ,y)ϕ(ξ,y)dy=iπt〈ξ〉3/2ϕ(ξ,0)+O(t-3/4∥〈ζ〉3/2〈η〉3/2(〈ζ〉+〈η〉)3/4ϕy(ξ,y)∥Lξ∞Ly2)+O(t-3/4∥〈ζ〉1/2〈η〉1/2(〈ζ〉+〈η〉)1/4ϕ(ξ,y)∥Lξ∞Ly2)
for t≥1 uniformly with respect to ξ∈R, provided the right-hand side is finite.Proof.
Denoteζ=ξ+2y,η=ξ+y, so that y=ζ-η=η-ξ. We have the identities
(2.99)A(ξ,y)=21+〈η〉〈ζ〉-ηζ+〈ξ〉〈ζ〉-ξζ+〈ξ〉〈η〉-ξη(〈ξ〉+〈ζ〉)(〈ξ〉+〈η〉)(〈ζ〉+〈η〉)y2,Ay(ξ,y)=2(ζ〈ζ〉-η〈η〉)=21+〈η〉〈ζ〉-ηζ〈ζ〉〈η〉(〈ζ〉+〈η〉)y,
and Ayy(ξ,y)=2(2/〈ζ〉^3-1/〈η〉^3). We now change z=A(ξ,y). We denote the inverse functions by yj(z), so that y1(z):(0,∞)→(0,∞) and y2(z):(0,∞)→(-∞,0). Thus the stationary point z=0 transforms into y=0. Hence we can write the representation
(2.100)∫ReitA(ξ,y)ϕ(ξ,y)dy=2∑j=12∫0∞e-itz2ϕ(ξ,yj(z))A(ξ,yj(z))|Ay(ξ,yj(z))|dz.
By a direct calculation we find
(2.101)dzdy=|Ay(ξ,y)|2A(ξ,y)=1Q(ξ,ζ,η),
where
(2.102)Q(ξ,ζ,η)≡2〈ζ〉〈η〉〈ζ〉+〈η〉1+〈η〉〈ζ〉-ηζ+〈ξ〉〈ζ〉-ξζ+〈ξ〉〈η〉-ξη(1+〈η〉〈ζ〉-ηζ)(〈ξ〉+〈ζ〉)(〈ξ〉+〈η〉).
Therefore we obtain
(2.103)∫ReitA(ξ,y)ϕ(ξ,y)dy=∫0∞e-itz2Φ(ξ,z)dz,
where
(2.104)Φ(ξ,z)=2∑j=12ϕ(ξ,y)Q(ξ,ζ,η)|y=yj(z).
Since Q(ξ,ξ,ξ)=〈ξ〉3/2, then the application of Lemma 2.3 yields
(2.105)∫ReitA(ξ,y)ϕ(ξ,y)dy=iπt〈ξ〉3/2ϕ(ξ,0)+O(t-3/4∥Φz(ξ,z)∥Lξ∞Lz2).
By a direct calculation we have
(2.106)Φz(ξ,z)dzdy=∑j=12ϕy(ξ,y)(Q(ξ,ζ,η))3/2|y=yj(z)+∑j=12ϕ(ξ,y)Q(ξ,ζ,η)(2Qζ(ξ,ζ,η)+Qη(ξ,ζ,η))|y=yj(z).
By (2.38)
(2.107)Q(ξ,ζ,η)≤C〈ζ〉〈η〉(〈ζ〉+〈η〉)1/2,|Qζ(ξ,ζ,η)|≤C〈ζ〉-1Q(ξ,ζ,η)(1+|η-ξ|〈η〉+〈ξ〉+|ζ-ξ|〈ζ〉+〈ξ〉)≤C〈η〉(〈ζ〉+〈η〉)1/2,|Qη(ξ,ζ,η)|≤C〈ζ〉(〈ζ〉+〈η〉)1/2.
Therefore we get the estimate
(2.108)∥Φz(ξ,z)∥Lξ∞Lz2=∥Φz(ξ,z)dzdy∥Lξ∞Ly2≤C∥〈ζ〉3/2〈η〉3/2(〈ζ〉+〈η〉)3/4ϕy(ξ,y)∥Lξ∞Ly2+C∥〈ζ〉1/2〈η〉1/2(〈ζ〉+〈η〉)1/4ϕ(ξ,y)∥Lξ∞Ly2.
This yields the estimate of the lemma. Lemma 2.7 is proved.In the next lemma we obtain the asymptotics for the nonlinear termℱ𝒰(-t)𝒢(u¯,〈i∂x〉-1u2) in (1.14).Lemma 2.8.
Letv∈H1,3. Then the asymptotic formula holds as follows:
(2.109)|〈ξ〉(ℱ𝒰(-t)𝒢(u¯,〈i∂x〉-1u2)-1tΩ(ξ)|v(t,ξ)|2v(t,ξ))|≤Ct-5/4∥v∥H1,33
for t≥1 uniformly with respect to ξ∈R, where v(t,ξ)=eit〈ξ〉û(t,ξ),Ω(ξ)≡〈ξ〉^2/〈2ξ〉(2〈ξ〉+〈2ξ〉).Proof.
We first find the representation forℱ𝒰(-t)(〈i∂x〉-1u2(t)). We introduce the operator 𝒬(t)=ℬ𝒱(t)+𝒲(t)=M¯𝒟t-1ℱ-1e-it〈ξ〉, so that the representation for the free evolution group is true 𝒰(t)ℱ-1=e-it〈i∂x〉ℱ-1=ℱ-1e-it〈ξ〉=𝒟tM𝒬(t). We also have ℱeit〈i∂x〉=𝒬-1(t)M¯𝒟t-1, where the right-inverse operator 𝒬-1(t)=𝒱-1(t)ℬ-1+𝒲-1(t)=eit〈ξ〉ℱ𝒟tM.
Since𝒟ωt=i𝒟ω𝒟t,𝒟ω-1=i𝒟1/ω, and ℱ𝒟1/ω=𝒟ωℱ, we find
(2.110)𝒬-1(t)Mω-1=eit〈ξ〉ℱ𝒟tMω=ieit〈ξ〉𝒟ωℱ𝒟ωtMω=ieit〈ξ〉𝒟ωe-iωt〈ξ〉𝒬-1(ωt)
with ω>0. Applying this formula with ω=2 and putting u(t)=𝒰(t)ℱ-1v=𝒟tM𝒬(t)v, we get
(2.111)ℱ𝒰(-t)(〈i∂x〉-1u2(t))=〈ξ〉-1𝒬-1(t)M¯𝒟t-1(𝒟tM𝒬(t)v)2=1〈ξ〉it𝒬-1(t)M(𝒬(t)v)2=eit〈ξ〉t𝒟2e-2it〈ξ〉〈2ξ〉𝒬-1(2t)(𝒬(t)v)2.
Since ℱ𝒰(-t)=ℱei〈i∂x〉t=eit〈ξ〉ℱ, we can write
(2.112)ℱx→η(〈i∂x〉-1u2(t))=1t𝒟2e-2it〈η〉〈2η〉𝒬-1(2t)(𝒬(t)v)2=12ite-2it〈η/2〉Φ(t,η2),
where Φ(t,η)=〈2η〉-1𝒬-1(2t)(𝒬(t)v)2. Hence we obtain
(2.113)ℱ𝒰(-t)𝒢(u¯,〈i∂x〉-1u2)=eit〈ξ〉ℱ𝒢(u¯,〈i∂x〉-1u2)=eit〈ξ〉2π〈ξ〉∫Rû(t,η-ξ)¯ℱx→η(〈i∂x〉-1u2(t))dη〈ξ〉+〈ξ-η〉+〈η〉.
Since û(t,η-ξ)¯=eit〈η-ξ〉v(t,η-ξ)¯, we get
(2.114)ℱ𝒰(-t)𝒢(u¯,〈i∂x〉-1u2)=12πit〈ξ〉∫ReitÃv(t,η-ξ)¯Φ(t,η/2)〈ξ〉+〈ξ-η〉+〈η〉dη
with Ã=〈ξ〉+〈η-ξ〉-2〈η/2〉.
Since𝒬(t)=ℬ𝒱(t) for |x|≤1 and 𝒬(t)=𝒲(t) for |x|≥1, we then have
(2.115)(𝒬(t)v)2=(ℬ𝒱(t)v)2+(𝒲(t)v)2
for all x∈R. In the same manner applying the identity ℬ-1(ℬv)2=〈ξ〉3/2v2(ξ), we obtain
(2.116)Φ(t,η)=〈2η〉-1𝒱-1(2t)ℬ-1(ℬ𝒱(t)v)2+〈2η〉-1𝒲-1(2t)(𝒲(t)v)2=〈2η〉-1𝒱-1(2t)〈ξ〉3/2(𝒱(t)v)2+〈2η〉-1𝒲-1(2t)(𝒲(t)v)2=Φ1(t,η)+Φ2(t,η).
After a change η=2ξ+2y, we find
(2.117)ℱ𝒰(-t)𝒢(u¯,〈i∂x〉-1u2)=∫ReitA(ξ,y)ϕ1(ξ,y)dy+∫ReitA(ξ,y)ϕ2(ξ,y)dy,
where
(2.118)ϕ1(ξ,y)≡v(t,ξ+2y)¯Φ1(t,ξ+y)πit〈ξ〉(〈ξ〉+〈ξ+2y〉+〈2(ξ+y)〉),ϕ2(ξ,y)≡v(t,ξ+2y)¯Φ2(t,ξ+y)πit〈ξ〉(〈ξ〉+〈ξ+2y〉+〈2(ξ+y)〉),
and A(ξ,y)=〈ξ〉+〈ξ+2y〉-2〈ξ+y〉. Note that
(2.119)∥𝒲-1(2t)ϕ∥L2=∥E¯(2t)ℱ𝒟2t(1-θ)ϕ∥L2=∥𝒟2t(1-θ)ϕ∥L2=∥(1-θ)ϕ∥L2≤∥ϕ∥L2.
Hence by Lemma 2.5 with δ=5/6,r=4(2.120)∥𝒲(t)ϕ∥L42≤Ct-3/4(∥〈ξ〉2∂ξϕ∥L1+∥〈ξ〉ϕ∥L1)
for all t≥1, and
(2.121)∥𝒲-1(2t)(𝒲(t)v)2∥L2≤∥𝒲(t)v∥L42≤Ct-3/4(∥〈ξ〉2∂ξv∥L1+∥〈ξ〉v∥L1)2.
Thus
(2.122)∥〈ξ〉∫ReitA(ξ,y)ϕ2(ξ,y)dy∥L∞≤Ct-1/2∥v(t)∥L2∥〈ξ〉-2𝒲-1(2t)(𝒲(t)v)2∥L2≤Ct-5/4∥v∥H1,33.
We apply Lemma2.7 to find
(2.123)|〈ξ〉ℱ𝒰(-t)𝒢(u¯,〈i∂x〉-1u2)-t-1〈ξ〉-1/2Ω(ξ)v(t,ξ)¯𝒱-1(2t)〈ξ〉3/2(𝒱(t)v)2|≤Ct-5/4∥〈ξ+2y〉1/2〈ξ+y〉1/2(〈ξ+2y〉+〈ξ+y〉)1/4v(t,ξ+2y)¯Φ1(t,ξ+y)〈ξ+2y〉+〈2(ξ+y)〉∥Lξ∞Ly2+Ct-5/4∥〈ξ+2y〉3/2〈ξ+y〉3/2(〈ξ+2y〉+〈ξ+y〉)3/4∂yv(t,ξ+2y)¯Φ1(t,ξ+y)〈ξ〉+〈ξ+2y〉+〈2(ξ+y)〉∥Lξ∞Ly2+Ct-5/4∥v∥H1,33≤Ct-5/4(∥v∥L∞∥Φ1∥H0,1/4+∥Φ1∥L∞∥v∥H0,1/4+∥v∥H∞0,1∥Φ1∥H1,1/4+∥v∥H1,1∥Φ1∥H∞0,1/4+∥v∥H1,33)≤Ct-5/4(∥v∥H1,1∥Φ1∥H1,1/4+∥v∥H1,33)
since
(2.124)〈ξ〉ϕ1(ξ,y)≡v(t,ξ+2y)¯Φ1(t,ξ+y)πit(〈ξ〉+〈ξ+2y〉+〈2(ξ+y)〉).
By Lemma 2.2 we have
(2.125)∥Φ1(t)∥H1,1/4≤C∥〈ξ〉-3/4∂ξ(𝒱-1(2t)〈ξ〉3/2(𝒱(t)v)2)∥L2≤C∥〈ξ〉3/2+1/4+γ∂ξ(𝒱(t)v)2∥L2+C∥〈ξ〉3/2-3/4+γ(𝒱(t)v)2∥L2≤C∥v∥H1,22.
We now write the representation for Ψ(t,ξ) as
(2.126)Ψ(t,ξ)=〈ξ〉3/2〈2ξ〉(2〈ξ〉+〈2ξ〉)v(t,ξ)¯𝒱-1(2t)〈ξ〉3/2(𝒱(t)v)2=〈ξ〉Ω(ξ)|v|2v+R,
where the remainder is
(2.127)R=〈ξ〉-1/2〈2ξ〉Ω(ξ)v(t,ξ)¯〈2ξ〉-1×((𝒱-1(2t)-1)〈ξ〉3/2(𝒱(t)v)2+〈ξ〉3/2((𝒱(t)v)2-v2)).
By the second estimate of Lemma 2.4 with β=5/4 and the first estimate of Lemma 2.4 with β=3/4, we obtain
(2.128)∥〈2ξ〉-1(𝒱-1(2t)-1)〈ξ〉3/2(𝒱(t)v)2∥L∞≤Ct-1/4(∥〈ξ〉3/4𝒱(t)v∥L∞∥〈ξ〉2∂ξ𝒱(t)v∥L2+∥〈ξ〉𝒱(t)v∥L42)≤Ct-1/4∥〈ξ〉3/4(𝒱(t)-1)ϕ∥L∞∥v∥H1,2+Ct-1/4∥v∥H1,22≤Ct-1/4∥v∥H1,22
for all t≥1. Also
(2.129)∥〈ξ〉1/2((𝒱(t)v)2-v2)∥L∞≤C(∥〈ξ〉1/4(𝒱(t)-1)v∥L∞+∥〈ξ〉1/4v∥L∞)∥〈ξ〉1/4(𝒱(t)-1)v∥L∞≤Ct-1/4∥v∥H1,22.
Thus we find the estimate for the remainder R(t) as
(2.130)∥R(t)∥L∞≤C∥〈ξ〉-1/2〈2ξ〉Ω(ξ)v∥L∞∥〈2ξ〉-1(𝒱-1(2t)-1)〈ξ〉3/2(𝒱(t)v)2∥L∞+C∥〈ξ〉-1/2〈2ξ〉Ω(ξ)v∥L∞∥〈ξ〉1/2((𝒱(t)v)2-v2)∥L∞≤Ct-1/4∥v∥H1,23.
Hence
(2.131)Ψ(t,ξ)=〈ξ〉Ω(ξ)|v|2v+O(t-1/4∥v∥H1,23).
Therefore the asymptotics of the lemma is true. Lemma 2.8 is proved.


## 3. Proof of Theorem1.1

We introduce the function space(3.1)XT={ϕ∈C([0,T];L2);∥ϕ∥XT<∞},
where(3.2)∥ϕ∥XT=supt∈[0,T](〈t〉-γ∥ϕ(t)∥H3+〈t〉-γ∥𝒥ϕ(t)∥H1+〈t〉-3γ∥𝒥ϕ(t)∥H2+〈t〉1/2∥ϕ(t)∥H∞1),
and γ>0 is small.The local existence in the function spaceXT can be proved by a standard contraction mapping principle. We state it without a proof.Theorem 3.1.
Letu0∈H3,1 and the norm ∥u0∥H3,1=ɛ. Then there exist ɛ0>0 and T>1 such that for all 0<ɛ<ɛ0 the initial value problem (1.1) has a unique local solution u∈C([0,T];H3,1) with the estimate ∥u∥XT<ɛ.Let us prove that the existence timeT can be extended to infinity which then yields the result of Theorem 1.1. By contradiction, we assume that there exists a minimal time T>0 such that the a priori estimate ∥u∥XT<ɛ does not hold; namely, we have ∥u∥XT≤ɛ.We apply the energy method to (1.14) ℒ(u+iλ𝒢(u¯,u¯))=-2i|λ|2𝒢(u¯,〈i∂x〉-1u2), (i.e., multiplying both sides of the above equation by 〈i∂x〉6(u+iλ𝒢(u¯,u¯))¯, taking the real part, and integrating over the space) to obtain(3.3)ddt∥u+iλ𝒢(u¯,u¯)∥H3≤C∥𝒢(u¯,〈i∂x〉-1u2)∥H3≤C∥u∥H∞12∥u∥H3≤Cɛ3/2〈t〉γ-1
since by Lemma 2.1(3.4)∥𝒢(u¯,〈i∂x〉-1u2)∥H3≤C∥𝒢(∂x3u¯,u2)∥L2+C∥𝒢(u¯,∂x2u2)∥L2+C∥𝒢(∂x2u¯,u2)∥L2+C∥𝒢(∂xu¯,u2)∥L2≤C∥u∥H∞12∥u∥H3
and ∥u∥H∞1≤ɛ〈t〉-1/2, ∥u∥H3≤ɛ〈t〉γ by the estimate ∥u∥XT<ɛ. Hence by Theorem 3.1 we have ∥u+iλ𝒢(u¯,u¯)∥H3≤Cɛ〈t〉γ and(3.5)∥u∥H3≤∥u+i𝒢(u¯,u¯)∥H3+∥𝒢(u¯,u¯)∥H3≤Cɛ〈t〉γ+C∥u∥H∞1∥u∥H3≤Cɛ〈t〉γ.Next we use the commutator relations[x,〈i∂x〉α]=α〈i∂x〉α-2∂x,[𝒫〈i∂x〉α]=α〈i∂x〉α-2∂x∂t,∂tu=-i〈i∂x〉u+i〈i∂x〉-1(u¯2), and ℒ𝒫=(𝒫-i〈i∂x〉-1∂x)ℒ, to getℒ(u+i𝒢(u¯,u¯))=-2i|λ|2𝒢(u¯,〈i∂x〉-1u2). Hence
(3.6)ℒ𝒫(u+iλ𝒢(u¯,u¯))=-2i|λ|2𝒫𝒢(u¯,〈i∂x〉-1u2)+2|λ|2〈i∂x〉-1∂x𝒢(u¯,〈i∂x〉-1u2).
Then by the energy method (i.e., multiplying both sides of the above equation by 〈i∂x〉4𝒫(u-𝒢(u¯,u¯))¯, taking the real part, and integrating over the space), by Lemma 2.1, using ∂tu=-i〈i∂x〉u+i〈i∂x〉-1(u¯2), we obtain(3.7)∥𝒫𝒢(u¯,〈i∂x〉-1u2)∥H2≤C(∥𝒫u∥H2+∥∂tu∥H2)∥u∥H∞12≤C(∥𝒫u∥H2+∥u∥H3)∥u∥H∞12.
Hence(3.8)ddt∥𝒫(u+iλ𝒢(u¯,u¯))∥H2≤C∥𝒫𝒢(u¯,〈i∂x〉-1u2)∥H2+C∥𝒢(u¯,〈i∂x〉-1u2)∥H2≤C(∥𝒫u∥H2+∥u∥H3)∥u∥H∞12≤Cɛ〈t〉-1∥𝒫u∥H2+Cɛ2〈t〉γ-1.
Therefore by Theorem 3.1 it follows that(3.9)∥𝒫u∥H2≤Cɛ〈t〉γ.
The energy estimate and the identity ℒx=xℒ-i〈i∂x〉-1∂x imply that(3.10)ddt∥x(u+iλ𝒢(u¯,u¯))∥H2≤C∥x𝒢(u¯,〈i∂x〉-1u2)∥H2+∥u+i𝒢(u¯,u¯)∥H2≤C∥u∥H∞12∥xu∥H2+C∥u∥H∞12∥u∥H2+Cɛ〈t〉γ≤Cɛ〈t〉-1∥xu∥H2+Cɛ〈t〉γ,
which yields(3.11)∥xu∥H2≤Cɛ〈t〉γ+1.
Then by the identity 𝒥=i𝒫-ixℒ-〈i∂x〉-1∂x we obtain(3.12)∥𝒥(u+iλ𝒢(u¯,u¯))∥H2≤∥𝒫(u+iλ𝒢(u¯,u¯))∥H2+∥xℒ(u+iλ𝒢(u¯,u¯))∥H2+∥u+iλ𝒢(u¯,u¯)∥H2≤Cɛ〈t〉γ+C∥x𝒢(u¯,〈i∂x〉-1u2)∥H2≤Cɛ〈t〉γ+C∥u∥H∞12∥xu∥H2+C∥u∥H∞12∥u∥H2≤Cɛ〈t〉γ.
We use Lemma 2.8 to obtain for the new variable v(t)=eit〈ξ〉û(t,ξ)(3.13)〈ξ〉ℱ𝒰(-t)𝒢(u¯,〈i∂x〉-1u2)=1t〈ξ〉Ω(ξ)|v(t,ξ)|2v(t,ξ)+O(t-5/4∥v∥H1,33)
for t≥1 uniformly with respect to ξ∈R, where Ω(ξ)≡〈ξ〉2/〈2ξ〉(2〈ξ〉+〈2ξ〉). Hence for the new function ϕ=eit〈ξ〉ℱ(u+iλ𝒢(u¯,u¯)) we get(3.14)∂t〈ξ〉ϕ=-2i|λ|21t〈ξ〉Ω(ξ)|ϕ(t,ξ)|2ϕ(t,ξ)+O(t-5/4ɛ3〈t〉3γ)
in Lr, 2≤r≤∞. Then integrating, we obtain supt≥1∥〈ξ〉ϕ(t)∥L∞≤Cɛ. Hence(3.15)supt≥1∥v(t)∥H∞0,1≤Cɛ.
By the decomposition of the free evolutions group we have the identity(3.16)〈i∂x〉u(t)=(𝒟tM(t)ℬ𝒱(t)+𝒟t𝒲(t))〈ξ〉ℱei〈i∂x〉tu(t)=𝒟tM(t)ℬ〈ξ〉ℱei〈i∂x〉tu(t)+(𝒟tM(t)ℬ〈ξ〉-3/2〈ξ〉3/2(𝒱(t)-1)+𝒟t𝒲(t))〈ξ〉ℱei〈i∂x〉tu(t).
By Lemmas 2.4 and 2.5 we find that the last term of the right-hand side of (3.16) is a remainder. Indeed we have the estimate(3.17)∥u(t)∥H∞1≤C∥𝒟tM(t)ℬ〈ξ〉ℱei〈i∂x〉tu(t)∥L∞+Cɛ〈t〉-5/4∥〈ξ〉ℱei〈i∂x〉tu(t)∥H1,2≤C〈t〉-1/2supt≥1∥v(t)∥H∞0,1+C〈t〉-5/4∥𝒥u(t)∥H2,
which yields(3.18)supt≥1〈t〉1/2∥u(t)∥H∞1≤Cɛ.
From (3.15) and (3.18) it follows that(3.19)∥u∥XT≤Cɛ<ɛ,
which implies the desired contradiction. Thus there exists a unique global solution u∈C([0,∞);H3,1) to (1.1) with the time decay estimate.We now prove the asymptotics of solutions. By (3.14) as in the proof of Lemma 2.7 we have(3.20)∥ψ(t)-ψ(s)∥H∞0,1+∥ψ(t)-ψ(s)∥H0,1≤Cɛ3/2∫stτγ-5/4dτ≤Cɛ3/2sγ-1/4
with γ∈(0,1/4), where(3.21)ψ(t)=ve-2i|λ|2Ω|v|2logt=ℱei〈i∂x〉tu(t)e-2i|λ|2Ω|ℱei〈i∂x〉tu(t)|2logt.
Thus we see that there exists a unique final state ψ+∈H∞0,1∩H0,1 such that(3.22)∥ψ(t)-ψ+∥H∞0,1+∥ψ(t)-ψ+∥H0,1≤Cɛ3/2tγ-1/4.
We consider the asymptotics of the phase function(3.23)Φ(t)=iΩ∫1t(|v(τ)|2-|v(t)|2)dττ=iΩ∫1t(|ψ(τ)|2-|ψ(t)|2)dττ.
By a direct calculation we have(3.24)Φ(t)-Φ(s)=iΩ(∫st(|ψ(τ)|2-|ψ(t)|2)dττ+(|ψ(t)|2-|ψ(s)|2)logs),
where 1<s<τ<t. Hence(3.25)∥Φ(t)-Φ(s)∥H∞0,1≤C∫st∥ψ(τ)-ψ(t)∥H∞0,1(∥ψ(τ)∥L∞+∥ψ(t)∥L∞)dττ+C∥ψ(s)-ψ(t)∥H∞0,1(∥ψ(s)∥L∞+∥ψ(t)∥L∞)logs≤Cɛ5/2∫stτγ-5/4dτ+Cɛ5/2sγ-1/4logs,
from which it follows that there exists a unique real valued function Φ+ such that(3.26)∥Φ(t)-iΦ+∥H∞0,1≤Cɛ3/2tγ-1/4.
Similarly, we find ∥Φ(t)-iΦ+∥H0,1≤Cɛ3/2tγ-1/4. Therefore we have the asymptotics of the phase function(3.27)iΩ∫1t|ψ(τ)|2dττ=iΩ∫1t|ℱei〈i∂x〉tu(τ)|2dττ=iΦ++iΩ|ψ+|2logt+(Φ(t)-iΦ+)+iΩ(|ℱei〈i∂x〉tu(t)|2-|ψ+|2)logt.
We also find(3.28)ℱei〈i∂x〉tu(t)-ψ+e2|λ|2(iΦ++iΩ|ψ+|2logt)=ℱei〈i∂x〉tu(t)-ψ+e2|λ|2iΩ∫1t|ψ(τ)|2(dτ/τ)+ψ+(e2|λ|2iΩ∫1t|ψ(τ)|2(dτ/τ)-e2|λ|2iΦ++2|λ|2iΩ|ψ+|2logt)=ψ(t)e2|λ|2iΩ∫1t|ψ(t)|2(dτ/τ)-ψ+e2|λ|2iΩ∫1t|ψ(τ)|2(dτ/τ)+ψ+(e2|λ|2iΩ∫1t|ψ(τ)|2(dτ/τ)-e(2|λ|2iΦ++2|λ|2iΩ|ψ+|2logt)).
Collecting these estimates, we obtain(3.29)∥ℱei〈i∂x〉tu(t)-ψ+e2|λ|2iΦ++2|λ|2iΩ|ψ+|2logt∥H0,1≤C∥ψ(t)-ψ+∥H0,1+C∥ψ+∥H0,1∥Φ(t)-iΦ+∥L2+∥ψ+∥H0,1∥ψ(t)-ψ+∥L∞(∥ψ+∥L2+∥ψ(t)∥L2)logt≤Cɛ3/2tγ-1/4,
and similarly(3.30)∥ℱei〈i∂x〉tu(t)-ψ+e2|λ|2iΦ++2|λ|2iΩ|ψ+|2logt∥H∞0,1≤Cɛ3/2tγ-1/4.
Therefore we have(3.31)∥ℱei〈i∂x〉tu(t)-Ŵ+eiΩ|Ŵ+|2logt∥H0,1≤Cɛ3/2tγ-1/4,∥ℱei〈i∂x〉tu(t)-Ŵ+eiΩ|Ŵ+|2logt∥H∞0,1≤Cɛ3/2tγ-1/4,
where Ŵ+=ψ+exp(2|λ|2iΦ+). Estimate (3.31) means that(3.32)∥u(t)-e-i〈i∂x〉tℱ-1Ŵ+e2|λ|2iΩ|Ŵ+|2logt∥H1,0≤Cɛ3/2tγ-1/4.
Theorem 1.1 is proved.



---

*Source: 504324-2010-03-29.xml*