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

Advances in Mathematical Physics Volume 2010, Article ID 504324,35pages doi:10.1155/2010/504324

Research Article

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

Nakao Hayashi

and Pavel I. Naumkin

1Department of Mathematics, Graduate School of Science, Osaka University, Osaka, Toyonaka 560-0043, Japan

2Instituto de Matem´aticas, UNAM Campus Morelia, AP 61-3 (Xangari), Morelia CP 58089, Mexico

Correspondence should be addressed to Nakao Hayashi,nhayashi@math.sci.osaka-u.ac.jp Received 18 September 2009; Accepted 23 February 2010

Academic Editor: Dongho Chae

Copyrightq2010 N. Hayashi and P. I. Naumkin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We study the initial value problem for the quadratic nonlinear Klein-Gordon equationLu i∂x⁻¹u²,t, x∈R×R,u0, x u0x,x∈R, whereL∂tii∂xandi∂x1−∂²x. 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.

1. Introduction

Let us consider the Cauchy problem for the nonlinear Klein-Gordon equation with a quadratic nonlinearity in one dimensional case

Luλi∂x⁻¹u², t, x∈R×R,

u0, x u₀x, x∈R, 1.1

whereλ∈C,L∂_tii∂x, andi∂x 1−∂²_x.

Our purpose is to obtain the large time asymptotic profile of small solutions to the Cauchy problem1.1without the restriction of a compact support on the initial data which was assumed in the previous work1. One of the important tools of paper1was based on the transformation of the equation by virtue of the hyperbolic polar coordinates following to paper2. 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

1.1is related to the Cauchy problem

vttv−vxxμv², t, x∈R×R,

v0, x v0x, vt0, x v1x, x∈R, 1.2

wherev₀andv₁are the real-valued functions, andμ ∈R.Indeed we can putu 1/2v ii∂x⁻¹vt,thenusatisfies

Lu i

2μi∂x⁻¹uu², t, x∈R×R, u0, x u0x, x∈R,

1.3

whereu0 1/2v0ii∂x⁻¹v1.

There are a lot of works devoted to the study of the cubic nonlinear Klein-Gordon equation

v_ttv−v_xxμv³, t, x∈R×R,

v0, x v0x, vt0, x v1x, x∈R 1.4

withμ ∈R.Whenμ < 0, the global existence of solutions to1.4can be easily obtained in the energy space, which is, however, insufficient for determining the large-time asymptotic behavior of solutions. The sharpL^∞- time decay estimates of solutions and nonexistence of the usual scattering states for1.4were shown in3by 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 termv³ was excluded in these works. In paper4, 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 forms7and the other reveals an additional time decay rate via the operatorx∂_tt∂xwhich was used in2. This nonlinear transformation was refined in8and 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 in1, which allow us to prove global existence and to find sharp asymptotics of small solutions to the Cauchy problem including1.2with 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 also9,10in which asymptotic behavior of solutions to1.4was studied as in1by using hyperbolic polar coordinates. Compactness condition on the data was removed in11in the case of the cubic nonlinearityv³ and a real-valued solution. Final value problem with the cubic nonlinearity was studied in12for 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 nonlinearityv³and the complex valued initial data. When the initial data are complex-valued, global existence andL^∞-time decay estimates of small solutions to the Klein-Gordon equation with cubic nonlinearity|v|²vwere obtained in paper13under the conditions that the initial data are smooth and have a compact support.

The scattering problem and the time decay rates of small solutions to1.4with super- critical nonlinearities|v|^p−1vand|v|^pwithp > 3 were studied in papers of14,15. 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 of7 and the time decay estimate through the operatorx∂_tt∂_xof17, it was shown that every quadratic nonlinearity is nonresonant.

We denote the Lebesgue space byL^p {φ ∈ S;φ_Lp < ∞}, with the normφ_Lp

R|φx|^pdx^1/pif 1≤p <∞andφ_L∞ sup_x∈R|φx|ifp∞.The weighted Sobolev space is

H^m,sp

φ∈L^p;x^si∂x^mφ_Lp<∞

, 1.5

form, s ∈ R, 1 ≤ p ≤ ∞,wherex √

1x².For simplicity we writeH^m,s H^m,s₂ . The index 0 we usually omit if it does not cause a confusion. We denote by Fφ Fx→ξφ ≡ φ 1/√

2π

Re^−ixξφxdxthe Fourier transform of the function φ.Then the inverse Fourier transformation isF⁻¹φF⁻¹_ξ→xφ 1/√

2π

Re^ixξφξdξ.

Our main result of this paper is the following.

Theorem 1.1. Letu₀ ∈ H^3,1and the norm u0_H^3,1 ε. Then there existsε₀ > 0 such that for all 0< ε < ε0the Cauchy problem1.1has a unique global solution

ut∈C

0,∞;H^3,1 1.6

satisfying the time decay estimate

ut_H^1,0_∞ ≤Cε1t^−1/2. 1.7

Furthermore there exists a unique final stateW ∈H^0,1_∞ ∩H^0,1such that ut−e^−ii∂xtF⁻¹We^{−2i|λ|2Ω|W|2logt}

H^1,0 ≤Cε^3/2t^γ−1/4, Fe^ii∂xtut−We^{−2i|λ|2Ω|W|2logt}

H^0,1∞

≤Cε^3/2t^γ−1/4,

1.8

whereγ∈0,1/4,Ωξ≡ ξ²/2ξ2ξ2ξ.

An important tool for obtaining the time decay estimates of solutions to the nonlinear Klein-Gordon equation is implementation of the operator

Ji∂xe^−ii∂xtxe^ii∂xtF⁻¹ξe^−iξti∂ξe^iξtFi∂xxit∂x, 1.9

which is analogous to the operatorxit∂x e^−it/2∂2xxe^it/2∂2x in the case of the nonlinear Schr ¨odinger 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 αi∂x^α−2∂x; therefore the commutator L,J LJ − JL0, whereL∂tii∂xis a linear part of1.1. SinceJis not a purely differential operator, it is apparently difficult to calculate the action ofJon the nonlinearity in1.1. So, instead we use the first-order differential operator

Pt∂xx∂t 1.10

which is closely related toJby the identityPLx−iJand acts easily on the nonlinearity.

Moreover, it almost commutes withL, sinceL,P −ii∂x⁻¹∂xL.

Also we use the method of normal forms of7by which we transform the quadratic nonlinearity into a cubic one with a nonlocal operator. We multiply both sides of equation 1.1by the free Klein-Gordon evolution groupFU−t Fe^iti∂x e^itξFand putvt, ξ e^itξuto get

vtt, ξ λe^itξF

i∂x⁻¹u² λ

√2πξ

Re^itAv t, η−ξ

v t,−η

dη, 1.11

whereAξ, η ξξ−ηη.Integrating1.11with respect to time, we find

vt, ξ v0, ξ λ

√2πξ _t

0

dτ

Re^iτAv

τ, η−ξ v

τ,−η

dη. 1.12

Then we integrate by parts with respect toτ,taking into account1.11,

vt, ξ iλ

Re^itAv t, η−ξ

v

t,−η dη

√2πξA ξ, η v0, ξ iλ

Rv

0, η−ξ v

0,−η dη

√2πξA ξ, η

−2i|λ|² _t

0

dτ

R

√ dη

2πξA

ξ, ηe^iτA−ηv

τ, η−ξ Fx→η

i∂x⁻¹u² .

1.13

Returning to the functionut, x UtF⁻¹_ξ→xvF⁻¹_ξ→xe^−itξvt, ξ, we obtain the following equation:

LuiλGu, u −2i|λ|²G

u,i∂x⁻¹u² , 1.14

with the symmetric bilinear operator

G φ, ψ

F⁻¹_ξ→x

Rg

ξ−η, η φ ξ−η

ψ

η

dη, 1.15

where

g

ζ, η

1

√2π

ζη ζη

ζ

η, 1.16

andζξ−η.Our main point in this paper is to show that the right-hand side of1.14can be decomposed into two terms; one of them is a cubic nonlinearity

−2i|λ|²1

tUtF⁻¹Ωξ|FU−tut, ξ|²FU−tut, ξ, 1.17 and the other one is a remainder term with an estimate likeOt^−5/4FU−tut³_H1,3. Remark 1.2. We believe that all quadratic nonlinear terms u², u²,|u|² of problem 1.3also could be considered by this approach. In the same way as in the derivation of1.14we get from1.3

L

u i

2μGu, u G1u, u 2G2u, u

iμ²G

u,i∂x⁻¹uu² iμ²G1

u,i∂x⁻¹uu² iμ²G2

uu,i∂x⁻¹uu² ,

1.18

where

Gj

φ, ψ

F_ξ⁻¹_→x

Rg_j

ξ−η, η φ ξ−η

ψ

η dη,

g_j ζ, η

1

√2πA_j

ζη, η ζη

1.19

withA1ξ, η ξ−ξ−η−η, A2ξ, η ξ−ξ−ηη.Some more regularity conditions are necessary to treat the bilinear operatorsGj. Also we have to show that

G

u,i∂x⁻¹

u²2|u|²

, G1

u,i∂x⁻¹

u²u² , G2

u,i∂x⁻¹

u²u²

, G2

u,i∂x⁻¹

u²2|u|² 1.20

are the nonresonant termsi.e., remaindersand to remove the resonant terms G

u,i∂x⁻¹u² , G1

u,i∂x⁻¹|u|² , G2

u,i∂x⁻¹|u|² , G2

u,i∂x⁻¹u² 1.21

by an appropriate phase function. We will dedicate a separate paper to this problem.

We prove our main result inSection 3. 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

G φ, ψ

F⁻¹_ξ→x

Rg

ξ−η, η φ ξ−η

ψ

η

dη, 2.1

where

g

ζ, η

1

√2π

ζη ζη

ζ

η. 2.2

Denote the kernel as follows:

g y, z

1 4π²

R²

e^iyζizη ζη

ζ

η

ζηdηdζ. 2.3

Lemma 2.1. The representation is true G

φ, ψ

R²g y, z

φ x−y

ψx−zdydz, 2.4

where the kernelgy, zobeys the following estimate:

g

y, z≤C y₋₃

z⁻³ln²

1 1

z−y

2.5

for ally, z∈R, y /z.Moreover the following estimates are valid:

G φ, ψ

L^p ≤Cφ

L^qψ

L^r, PG

φ, ψ

L^p ≤CPφ

L^qψ

L^r CPψ

L^qφ

L^r

C∂_tφ

L^qψ

L^r C∂_tψ

L^qφ

L^r

2.6

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 representation2.4, we substitute the direct Fourier transforms

φζ 1

√2π

Re^−ix−yζφ x−y

dy,

ψ

η 1

√2π

Re^−ix−zηψx−zdz

2.7

into the definition of the operatorG. Then changingζξ−η, we find

G φ, ψ

2π^−3/2

R²φ x−y

ψx−z

R²g ζ, η

e^iyζizηdηdζ

dydz

R²g y, z

φ x−y

ψx−zdydz,

2.8

where the kernelgy, zis

g y, z

2π^−3/2

R²g ζ, η

e^iyζizηdηdζ

1 4π²

R²

e^iyζizη ζη

ζ

η

ζηdηdζ.

2.9

Changing the variables of integrationζξ/2−ηandηξ/2ηthe prime we will omit, we get

g y, z

1 4π²

R²

e^iyζizη ζη

ζ

η

ζηdηdζ 1

4π²

R

dξ

ξe^iyz/2ξ

RB ξ, η

e^iz−yηdη,

2.10

whereBξ, η 1/ξξ/2−ηξ/2η.We changesyz/2 andρ z−yand denote

g

s, ρ 1

4π²

R

dξ ξe^isξ

RB ξ, η

e^iρηdη. 2.11

For the case of|ρ| ≤ 1,|s| ≤1 we integrate by parts using the identitye^iρη Aη∂ηηe^iρη, whereAη 1/1iρη.Then we get

g

s, ρ − 1

4π²

R

dξ ξe^isξ

Re^iρηη∂η

A η

B ξ, η

dη. 2.12

Note that

η∂_ηB ξ, η

η ξ

ξ/2−η

ξ/2η₂

η−ξ/2

η−ξ/2 ηξ/2 ξ/2η

. 2.13

Then

ξ⁻¹η∂η

A η

B

ξ, η≤ C

ξ

1ρηξ

η. 2.14

Hence we can estimate the kernelgs, ρ as follows:

g

s, ρ≤C

R

dξ ξ

R

dη

1ρηξ η

≤C _∞

1

dη 1ρη

_∞

1

dξ ξ

ηξ C _∞

1

dη 1ρη

1 η

ln ξ

ηξ ^∞

1

dξ

≤C _∞

1

lnη 1ρη

ηdη≤C _1/|ρ|

1

lnηdη

η C

ρ _∞

1/|ρ| lnηdη η²

≤Cln²

1 1 ρ

2.15

for the case of|ρ| ≤1,|s| ≤1.For the case of|ρ| ≤1,|s| ≥1 we integrate three times by parts with respect toξ

g

s, ρ

C|s|⁻³

Rdξe^isξ

Re^iρη∂³_ξ

ξ⁻¹η∂η

A η

B

ξ, η dη. 2.16

Note that

∂³_ξ

ξ⁻¹η∂_η A

η B

ξ, η ≤ C

ξ

1ρηξ

η. 2.17

Hence we can estimate the kernelgy, zas follows:

g

s, ρ≤C|s|⁻³

R

dξ ξ

R

dη

1ρηξ

η ≤C|s|⁻³ln²

1 1 ρ

2.18

for all|ρ| ≤1,|s| ≥1.For the case|ρ| ≥1,|s| ≤1 we integrate by parts three times with respect toη

g

s, ρ

Cρ⁻³

R

dξ ξe^isξ

Re^iρη∂³_ηB ξ, η

dη. 2.19

Note that|∂³_ηBξ, η| ≤Cξη⁻³.Hence

g

s, ρ≤Cρ⁻³

R

dξ ξ²

R

dη

η₂ ≤Cρ⁻³. 2.20

Finally for the case of|ρ| ≥1,|s| ≥1 we integrate by parts three times with respect toξandη

g

s, ρ

Csρ⁻³

Rdξe^isξ

Re^iρη∂³_ξ∂³_η ξ⁻¹B

ξ, η dη. 2.21

Since

∂³_ξ∂³_η ξ⁻¹B

ξ, η ≤ C

ξ² ξ

η₂, 2.22

then we can estimate the kernelgs, ρ as follows:

g

s, ρ≤Csρ⁻³

R

dξ ξ²

R

dη

η₂ ≤Csρ⁻³ 2.23

for all|ρ| ≥1,|s| ≥1.Hence estimate2.5is true.

By virtue of estimate2.5applying the H ¨older inequality with 1/p1/p₁1/p₂and the Young inequality with 1/p₁11/q1/q₁and 1/p₂11/r1/r₁, we find

G φ, ψ

L^p ≤C

R

φ

x−y dy y₃

L^px¹

Rln²

1 1

z−y

ψx−z dz z³

L^px²L^∞y

≤Cφ_Lqψ_Lr,

2.24

where 1≤p≤ ∞,1< q≤p/α,1< r ≤p/1−α, α∈0,1.

We now estimate the operatorPt∂xx∂tas follows:

PG φ, ψ

G Pφ, ψ

R²yg y, z

ψx−z∂tφ x−y

dydz

G φ,Pψ

R²zg y, z

φ x−y

∂tψx−zdydz.

2.25

Then by virtue of estimate2.4applying the H ¨older and Young inequalities, we get PG

φ, ψ

L^p ≤G

Pφ, ψ

L^pG

φ,Pψ

L^p

R²yg y, z

ψx−z∂tφ x−y

dydz L^p

R²zg y, z

φ x−y

∂_tψx−zdydz L^p

≤CPφ

L^qψ

L^rCPψ

L^qφ

L^rC∂_tφ

L^qψ

L^r

C∂tψ

L^qφ

L^r.

2.26

Lemma 2.1is proved.

We now decompose the free Klein-Gordon evolution groupUt e^−ii∂xtF⁻¹EtF, whereEt e^−itξsimilarly to the factorization of the free Schr ¨odinger evolution group. We denote the dilation operator by

Dωφ 1

√iωφx

ω , Dω⁻¹iD1/ω. 2.27

Define the multiplication factorMt e^−itixθx,whereθx 1 for|x|<1 andθx 0 for

|x| ≥1.We introduce the operator

Bφ θx ix^3/2φ

x ix

. 2.28

The inverse operatorB⁻¹acts on the functionsφxdefined on−1,1as follows:

B⁻¹φ 1 ξ^3/2φ

ξ ξ

2.29

for allξ∈R,sinceξx/ix ∈R andxξ/ξ ∈−1,1.We now introduce the operators Vt B⁻¹MtD⁻¹_t F⁻¹e^−itξ,

Wt Mt1−θD⁻¹_t F⁻¹e^−itξ,

2.30

so that we have the representation for the free Klein-Gordon evolution group UtF⁻¹e^−iti∂xF⁻¹F⁻¹e^−itξDtMtBVt Wt

DtMtBDtMtBVt−1 DtMtWt. 2.31

The first term DtMtBφ 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 2.31is 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

FU−t Fe^iti∂xe^itξFV⁻¹tB⁻¹MtD⁻¹_t W⁻¹tD_t⁻¹ B⁻¹MtD_t⁻¹

V⁻¹t−1 B⁻¹MtD⁻¹_t W⁻¹tD_t⁻¹,

2.32

where the right-inverse operators are

V⁻¹t EtFDtMtB,

W⁻¹t EtFDt1−θ, 2.33

whereEt e^−itξ.

In the next lemma we state the estimates of the operatorsVt B⁻¹MtD⁻¹_t F⁻¹e^−itξ. Lemma 2.2. The estimates hold as follows:

Vtφ

L²≤Cφ

L², ξ^ρ∂_ξVtφ

L² ≤Cξ^α∂_ξφ

L²Cξ^α−1φ

L², 2.34

whereρ < α, α∈1,2,and

ξ^ρ∂_ξV⁻¹tφ

L²≤Cξ^1/4α∂_ξφ

L²Cξ^α−3/4φ

L², 2.35

whereρ < α, α∈0,1,provided the right-hand sides are finite.

Proof. Changing the variable of integrationxξξ⁻¹, we see that B⁻¹φ²

L²

R

φ ξ

ξ ² dξ

ξ³ ₁

−1

φx²dxφ²

L²−1,1. 2.36

HenceVtφ_L2 φ_L2.

Consider the estimate for the derivative∂_ξVtφ. DefineSξ, η η −ξη1/ξ.

Note that∂S/∂ηη/η −ξ/ξ 1ηξ −ηξ/ξηηξη−ξand∂S/∂ξ ξ−η/ξ³.Hence integrating by parts one time yields

∂_ξ

Re^−itSφ η

dη−itξ⁻³

Re^−itSφ η

ξ−η dη

ξ⁻³

Re^−itS φ

η g

ξ, η φ

η gη

ξ, η dη,

2.37

wheregξ, η ξ−η/ξ/ξ −η/η ξηηξ/1ηξ −ηξ. Also we have 1

η

ξ −ηξ₋₁

≤C ξ η η

ξ2. 2.38

Hence the estimate is true

g ξ, η

ξ

η

η ξ 1

η

ξ −ηξ ≤Cξ² η₂ η

ξ. 2.39

Since∂_ξξ^aaξξ^a−2, we get

ξ^ρ∂_ξVtφξ^ρ

it

2π∂_ξξ^−3/2

Re^−itSφ η

dη

−3

2ξξ^ρ−2Vtφξ^ρ−9/2

it 2π

Re^−itSφ η

g ξ, η

dη

ξ^ρ−9/2

it 2π

Re^−itSφ η

g_η ξ, η

dη.

2.40

Consider theL²-estimate of the integral

√tξ^ρ−9/2

Re^itξη/ξψ η

g ξ, η

dη. 2.41

We have

√ t

Re^itξη/ξψ η

g ξ, η

ξ^ρ−9/2dη ²

L²

t

Rdηψ η

Rdζψζ

Re^{itξ/ξη−ζ}g ξ, η

gξ, ζξ^2ρ−9dξ

Rdηψ η

RdζψζG η, ζ

,

2.42

where the kernel

G η, ζ

t

Re^{itξ/ξη−ζ}g ξ, η

gξ, ζξ^2ρ−9dξ

t η

ζ

Re^{itξ/ξη−ζ}

η ξ 1

η

ξ −ηξ ζξ

1ζξ −ζξξ^2ρ−7dξ.

2.43

After a changeξx/ix, we find

G η, ζ

t η

ζ ₁

−1e^itxη−ζ

η

1/ix 1

η

/ix −ηx/ix ζ1/ix

1ζ/ix −ζx/ixix^4−2ρdx.

2.44

We now change the contourzxiy∈Γof integration inGasysignη−ζix²; then

G η, ζ

t η

ζ

Γe^itzη−ζ

η

1/iz 1

η

/iz −ηz/iz ζ1/iz

1ζ/iz −ζz/iziz^4−2ρdz.

2.45

Since|iz| ix²y²²4x²y^21/4 ∼ ix, using2.38and the inequalityηξ ≥ η^α−1ξ^2−αforα∈1,2,we get|η1/iz/1η/iz −ηz/iz| ≤Cη^α−1ix^α−2; also we have

e^itzη−ζe^{−ix2t|η−ζ|}≤ Cix^−2δ

1tη−ζ^δ 2.46

forδ >1. Therefore we obtain the estimate

G

η, ζ≤ Ct η_α

ζ^α 1tη−ζ^δ

₁

−1ix^{2α−ρ−δ}dx≤ Ct η_α

ζ^α

1tη−ζ^δ, 2.47

if we chooseρ < α, α∈1,2and 1< δ <1α−ρ.Thus we get √

t

Re^itξη/ξψ η

g ξ, η

ξ^ρ−9/2dη ²

L²

Rdηψ η

RdζψζG η, ζ

≤Cη_α ψ

η

L²

Rdζψζζ^α t

1tη−ζ^δ

L²

≤Ct·^αψ²

L²

R

dη

1tη^δ ≤C·^αψ²

L²

2.48

sinceδ >1.

In the same manner we consider the estimate of the integral √

t

Re^itξη/ξψ η

gη

ξ, η

ξ^ρ−9/2dη ²

L²

t

Rdηψ η

Rdζψζ

Re^{itξ/ξη−ζ}gη

ξ, η

gζξ, ζξ^2ρ−9dξ

Rdηψ η

RdζψζG η, ζ

,

2.49

where the kernel G

η, ζ t

Re^{itξ/ξη−ζ}gη

ξ, η

gζξ, ζξ^2ρ−9dξ

t

Re^{itξ/ξη−ζ}

ηξ η/

η ξ 1

η

ξ −ηξ ζξ ζ/ζξ

1ζξ −ζξ ξ^2ρ−7dξ,

2.50

since by a direct calculation

∂_η η

η ξ 1

η

ξ −ηξ ηξ η/

η ξ

1 η

ξ −ηξ . 2.51

In the same way as in2.47we have G

η, ζ≤ Ct η_α−1

ζ^α−1

1tη−ζ^δ 2.52

if we chooseρ < α, α∈1,2, and 1< δ <1α−ρ.Therefore we get √

t

Re^itξη/ξψ η

g_η ξ, η

ξ^ρ−9/2dη ²

L² ≤C η_α−1

ψ η²

L²

R

tdη

1tη^δ ≤C·^α−1ψ²

L². 2.53

Henceξ^ρ∂ξVtφ_L2≤Cξ^α∂ξφ_L2Cξ^α−1φ_L2.Thus we have the second estimate of the lemma.

Consider the estimate for the derivative∂_ξV⁻¹tφ. Note that∂Sη, ξ/∂ξ ξ/ξ − η/η 1ηξ −ηξ/ξηηξξ−ηand∂Sη, ξ/∂η η−ξ/η³.Hence integrating by parts one time yields

ξ^ρ∂_ξV⁻¹tφ−

√t

√2iπξ^ρ−1

R

1 η

ξ −ηξ η

ξ η1/2

φ η

de^itSη,ξ

√t

√2iπξ^ρ−1

Re^itSη,ξ φ

η g₁

ξ, η φ

η g_1η

ξ, η dη,

2.54

whereSη, ξ ξ −ξη1/ηandg1ξ, η 1ηξ −ηξ/ηξη^1/2.The estimate is true

g₁ ξ, η

1 η

ξ −ηξ η

ξ

η1/2≤C η_3/2 ξ

η

ξ. 2.55

Consider theL²-estimate of the integral

ξ^ρ−1√ t

Re^−itξη/ηψ η

g₁ ξ, η

dη. 2.56

We have, changingη/ηyandζ/ζz, ξ^ρ−1√

t

Re^−itξη/ηψ η

g₁ ξ, η

dη ²

L²

t

Rdηψ η

Rdζψζ

Re−itξη/η−ζ/ζg₁ ξ, η

g₁ξ, ζξ^2ρ−2dξ

t ₁

−1dy iy₃

ψ y

iy ₁

−1dziz³ψ z

iz

×

Re^−itξy−zg₁

ξ, y iy

g₁

ξ, z

iz

ξ^2ρ−2dξ

₁

−1dy iy₃

ψ y

iy ₁

−1dziz³ψ z

iz

G1

y, z ,

2.57

where the kernel G₁

y, z t

Re^−itξy−zg1

ξ, y

iy

g1

ξ, z

iz

ξ^2ρ−2dξ

t iy_−1/2

iz^−1/2

Re^−itξy−z1ξ/

iy

−yξ/

iy 1/

iy ξ

1ξ/iz −zξ/iz

1/izξ ξ^2ρ−2dξ.

2.58

We now change the contourξxiy∈Γof integration inGasysigny−zx ²; then G₁

y, z t

iy_−1/2

iz^−1/2

Γe^−itξy−z1ξ/

iy

−yξ/

iy 1/

iy ξ

1ξ/iz −zξ/iz

1/izξ ξ^2ρ−2dξ.

2.59

Since|ξ| x ²−y²²4x²y^21/2∼ x and by2.55|1ξ/iy−yξ/iy/1/iy ξ| ≤Cξ^1−αiy^−αforα∈0,1, and also

e^−itξy−z≤ C

1tx y−z² ≤ C tx ^δy−z^δ

y−z 2.60

forδ <1,we obtain the estimate G1

y, z≤ C

iy_−1/2−α

iz^−1/2−α y−z^δ

y−z

Rx ^{−2α2ρ−δ}dx

≤ C

iy_−1/2−α

iz^−1/2−α y−z^δ

y−z ,

2.61

if we chooseρ < α,and 1> δ >1−2α2ρ.Therefore we get √

t

Re^itξη/ξψ η

g ξ, η

ξ^ρ−9/2dη ²

L²

₁

−1dy iy_5/2−α

ψ y

iy ₁

−1dziz^5/2−αψ z

iz

C y−z^δ

y−z

≤C ₁

−1dy

iy_5/2−2α ψ

y iy

2

≤C ₁

−1dξξ^1/22αψξ²≤C·^1/4αψ²

L².

2.62

In the same manner we consider the estimate of the integral withφηg1ηξ, η.Hence ξ^ρ∂_ξV⁻¹tφ

L²≤Cξ^1/4α∂_ξφ

L²Cξ^α−3/4φ

L², 2.63

whereα > ρ.Thus we have the second estimate of the lemma.Lemma 2.2is proved.

In the next lemma we prove an auxiliary asymptotics for the integral_∞

0e^−itz2Φξ, zdz.

Lemma 2.3. The estimate holds as follows:

ξ^α

_∞

0

e^−itz2Φξ, zdz−Φξ,0 π

4it

≤Ct^−3/4ξ^αΦzξ, z

L^∞_ξL²z 2.64

provided the right-hand side is finite, whereα∈R.

Proof. We represent the integral _∞

0

e^−itz2Φξ, zdz Φξ,0 _∞

0

e^−itz2dzR Φξ,0 π

4itR, 2.65

where the remainder term is

R _∞

0

e^−itz2Φξ, z−Φξ,0dz. 2.66

In the remainder term we integrate by parts via identitye^−itz2 1/1−2itz²∂zze^−itz2 ξ^αR

L^∞_ξ ≤C ξ^α

_∞

0

1

1tz²Φξ, z−Φξ,0dz L^∞_ξ

C ξ^α

_∞

0

z

1tz²Φzξ, zdz L^∞_ξ

≤Cξ^αΦzξ, z

L^∞_ξL²z

√ z

1tz^{2 −1} L¹z

Cξ^αΦzξ, z

L^∞_ξL²z

z

1tz^{2 −1} L²z

≤Ct^−3/4ξ^αΦzξ, z_L∞

ξL²z,

2.67

since

|Φξ, z−Φξ,0| ≤ _z

0

|Φzξ, z|dz≤C

|z|Φzξ, z_L^∞

ξL²z. 2.68

Thus we have the estimate for the remainder in the asymptotic formula.Lemma 2.3is proved.

In the next lemma we obtain the asymptotics for the operator Vt B⁻¹MtD⁻¹_t F⁻¹e^−itξand the right-inverse operatorV⁻¹t EtFDtMtB.

Lemma 2.4. The estimates hold as follows:

ξ^βVt−1φ

L^∞≤Ct^−1/4ξ^β3/4∂_ξφ

L²ξ^β−1/4φ

L² , ξ^β−3/4

V⁻¹t−1 φ

L^∞ ≤Ct^−1/4ξ^3/2β∂_ξφ

L²ξ^1/2βφ

L² ,

2.69

for allt≥1,where 0≤β≤3/2,provided the right-hand sides are finite.

Proof. We have the identities

S ξ, η

η

− ξ − ξ ξ

η−ξ

1 η

ξ −ηξ η

ξ₂ ξ

η−ξ2

,

Sη

ξ, η η

η− ξ

ξ 1 η

ξ −ηξ ξ

η

η

ξ η−ξ

,

2.70

and Sηηξ, η η⁻³.We now change z

Sξ, η.We denote The inverse functions by η_jz,so thatη₁z:0,∞ → ξ,∞andη₂z:0,∞ → −∞, ξ.

Thus the stationary point z 0 transforms into η ξ. Hence we can write the representation

Vtφξ^−3/2

it 2π

Re^−itSξ,ηφ η

dη

2ξ^−3/2

it 2π

2 j1

_∞

0

e^−itz2φ η_jz

S

ξ, η_jz Sη

ξ, ηjzdz.

2.71

By a direct calculation we find

dz dη Sη

ξ, η

S ξ, η

1

η

ξ −ηξ η

ξ^1/2 . 2.72

Therefore we obtain

Vtφ _∞

0

e^−itz2Φξ, zdz, 2.73

where

Φξ, z 2ξ⁻¹

it 2π

2 j1

η φ

η

1ξ η

−ξη

ηηjz

. 2.74

Application ofLemma 2.3yields

Vtφφξ O

t^−3/4Φzξ, z_L^∞

ξL²z . 2.75

By a direct calculation we have

Φzξ, z dz

dη

it 2π

2 j1

2 η_3/2

φ η ξ^3/4

1 η

ξ −ηξ3/4

it 2π

2 j1

η

ηξξ η

/ η

ξ ξ^3/4

η_1/2 1

η

ξ −ηξ_3/4φ η

.

2.76

By2.38we get the estimate

ξ^βΦzξ, z

L^∞_ξL²z

ξ^βΦzξ, z dz

dη L^∞_ξL²η

≤C√ t

ξ^β η_9/4 η

ξ_3/2φ η

L^∞_ξL²η

C√ t

ξ^β η_5/4 η

ξ_3/2φ η

L^∞_ξL²η

≤C√ t

η_3/4β φ

L²C√ t

η_β−1/4 φ

L²

2.77

for 0 ≤ β≤ 3/2 sinceηξ ≥ Cξ^2/3βη^1−2/3β.This yields the first estimate of the lemma.

We now prove the second estimate. We have

V⁻¹tφ

√t

√2iπ ₁

−1e^{itξ−ξy−iy} iy_−3/2

φ y

iy

dy, 2.78

then changingyηη⁻¹, we find

V⁻¹tφx

√t

√2iπ

R

e^itSη,ξφ η

η_−3/2

dη, 2.79

where

S η, ξ

ξ −ξη1

η 1 η

ξ −ηξ η

ξ₂ η

η−ξ2

. 2.80

As above we changez

Sη, ξand represent

V⁻¹tφx

√t

√2iπ

Re^itSη,ξφ η

η_−3/2 dη

√2t

√iπ 2 j1

_∞

0

e^itz2φ ηjz

ηjz_−3/2

S

ηjz, ξ S_η

η_jz, ξdz.

2.81

By a direct calculation we findS_ηη, ξ η−ξ/η³ and

dz

dη S_η η, ξ 2

S

η, ξ

η ξ 2

η5/2 1

η

ξ −ηξ

. 2.82

Therefore we obtain

V⁻¹tφx _∞

0

e^itz2Φξ, zdz, 2.83

where

Φξ, z 2√

√ 2t iπ

2 j1

η 1

η

ξ −ηξ η

ξ φ η

ηηjz

. 2.84

Application ofLemma 2.3yields

V⁻¹tφx φξ O

t^−3/4Φzξ, z_L^∞

ξL²z . 2.85

By a direct calculation we have

Φzξ, z

dz dη 2√

√2t iπ

2 j1

1 η

ξ −ηξ_3/4 η

ξ_3/2 η_9/4

φ η

2√

√ 2t iπ

2 j1

η5/4 1

η

ξ −ηξ1/4

η ξ

⎛

⎜⎝ η

1 η

ξ −ηξ η

ξ

⎞

⎟⎠

η

φ η

, 2.86