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

We study the initial value problem for the quadratic nonlinear Klein-Gordon equation ℒ𝑢=⟨𝑖𝜕𝑥⟩−1𝑢2, (𝑡,𝑥)∈𝐑×𝐑, 𝑢(0,𝑥)=𝑢0(𝑥), 𝑥∈𝐑, where ℒ=𝜕𝑡


Introduction
Let us consider the Cauchy problem for the nonlinear Klein-Gordon equation with a quadratic nonlinearity in one dimensional case where λ ∈ C, L ∂ t i i∂ x , and i∂ x 1 − ∂ 2 x .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 Advances in Mathematical Physics 1.1 is related to the Cauchy problem where v 0 and v 1 are the real-valued functions, and μ ∈ R. Indeed we can put u 1/2 v i i∂ x −1 v t , then u satisfies where There are a lot of works devoted to the study of the cubic nonlinear Klein-Gordon equation 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 on v, v t , v x , v xx , v tx 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 v 3 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 v 3 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 v 3 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| 2 v 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 supercritical nonlinearities |v| p−1 v 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 by L p {φ ∈ S ; φ L p < ∞}, with the norm For simplicity we write H m,s H m,s 2 .The index 0 we usually omit if it does not cause a confusion.We denote by Fφ Our main result of this paper is the following.Theorem 1.1.Let u 0 ∈ H 3,1 and the norm u 0 H 3,1 ε.Then there exists ε 0 > 0 such that for all 0 < ε < ε 0 the Cauchy problem 1.1 has a unique global solution satisfying the time decay estimate Furthermore there exists a unique final state An important tool for obtaining the time decay estimates of solutions to the nonlinear Klein-Gordon equation is implementation of the operator

J
i∂ x e −i i∂ x t xe i i∂ x t F −1 ξ e −i ξ t i∂ ξ e i ξ t F i∂ x x it∂ x , 1.9 which is analogous to the operator x it∂ x e −it/2 ∂ 2 x xe it/2 ∂ 2 x in the case of the nonlinear Schr ödinger 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 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 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 Fe it i∂ x e it ξ F and put v t, ξ e it ξ u to get where A ξ, η ξ ξ − η η .Integrating 1.11 with respect to time, we find Then we integrate by parts with respect to τ, taking into account 1.11 ,

1.13
Returning to the function u t, x , we obtain the following equation: with the symmetric bilinear operator Remark 1.2.We believe that all quadratic nonlinear terms u 2 , 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 where Some more regularity conditions are necessary to treat the bilinear operators G j .Also we have to show that 1.20 are the nonresonant terms i.e., remainders and to remove the resonant terms 21 by an appropriate phase function.We will dedicate a separate paper to this problem.

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We prove our main result in Section 3. In the next section we prove several lemmas used in the proof of the main result.

Preliminaries
First we give some estimates for the symmetric bilinear operator where Denote the kernel as follows: where the kernel g y, z obeys the following estimate: for all y, z ∈ R, y / z.Moreover the following estimates are valid: , provided that the right-hand sides are bounded.
Proof.To prove representation 2.4 , we substitute the direct Fourier transforms into the definition of the operator G. Then changing ζ ξ − η, we find where the kernel g y, z is

2.9
Changing the variables of integration ζ ξ/2 − η and η ξ/2 η the prime we will omit , we get where We change s y z/2 and ρ z − y and denote

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Note that Then Hence we can estimate the kernel g s, ρ as follows: 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 g s, ρ

2.16
Note that 2.17 Hence we can estimate the kernel g y, z as follows: for all |ρ| ≤ 1, |s| ≥ 1.For the case |ρ| ≥ 1, |s| ≤ 1 we integrate by parts three times with respect to η g s, ρ C ρ

2.20
Finally for the case of |ρ| ≥ 1, |s| ≥ 1 we integrate by parts three times with respect to ξ and η g s, ρ C sρ

2.21
Since then we can estimate the kernel g s, ρ as follows:

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Then by virtue of estimate 2.4 applying the H ölder and Young inequalities, we get
We now decompose the free Klein-Gordon evolution group U t e −i i∂ x t F −1 E t F, where E t e −it ξ similarly to the factorization of the free Schr ödinger evolution group.We denote the dilation operator by

2.27
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
The inverse operator B −1 acts on the functions φ x defined on −1, 1 as follows: for all ξ ∈ R, since ξ x/ ix ∈ R and x ξ/ ξ ∈ −1, 1 .We now introduce the operators 2.30 so that we have the representation for the free Klein-Gordon evolution group

2.31
The first term D t M 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
where the right-inverse operators are where E t e −it ξ .In the next lemma we state the estimates of the operators Lemma 2.2.The estimates hold as follows:
Proof.Changing the variable of integration x ξ ξ −1 , we see that Hence integrating by parts one time yields

2.38
Hence the estimate is true

2.40
Consider the L 2 -estimate of the integral

2.41
We have where the kernel

2.43
Advances in Mathematical Physics 13 After a change ξ x/ ix , we find

2.44
We now change the contour z x iy ∈ Γ of integration in G as y sign η − ζ ix 2 ; then

2.45
Since ∼ ix , using 2.38 and the inequality η for δ > 1. Therefore we obtain the estimate

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In the same manner we consider the estimate of the integral

2.49
where the kernel since by a direct calculation In the same way as in 2.47 we have if we choose ρ < α, α ∈ 1, 2 , and 1 < δ < 1 α − ρ.Therefore we get Thus we have the second estimate of the lemma.
Consider the estimate for the derivative Hence integrating by parts one time yields

2.55
Consider the L 2 -estimate of the integral We have, changing η/ η y and ζ/ ζ z,

2.57
where the kernel

2.58
We now change the contour ξ x i y ∈ Γ of integration in G as y sign y − z x 2 ; then

2.62
In the same manner we consider the estimate of the integral with φ η g 1η ξ, η .Hence 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 −itz 2 Φ ξ, z dz.
Lemma 2.3.The estimate holds as follows: provided the right-hand side is finite, where α ∈ R.
Proof.We represent the integral where the remainder term is

2.66
In the remainder term we integrate by parts via identity

2.68
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 operator Lemma 2.4.The estimates hold as follows:

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

2.72
Therefore we obtain where

2.74
Application of Lemma 2.3 yields

2.75
By a direct calculation we have

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As above we change z S η, ξ and represent S η j z , ξ S η η j z , ξ dz.

2.82
Therefore we obtain where

2.84
Application of Lemma 2.3 yields

2.85
By a direct calculation we have 2.87 therefore we get the estimate . This yields the second estimate of the lemma.Lemma 2.4 is proved.
In the next lemma we find the estimates for the operator Lemma 2.5.The estimates hold as follows: for all t ≥ 1, where 1 < r ≤ ∞, δ ∈ 0, 1 , provided the right-hand side is finite.
Proof.Note that W t φ 0 for |x| ≤ 1.To prove the estimate, we integrate by parts via the identity

2.91
For the case of 1 < |x| ≤ 2 we integrate by parts via the identity 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 operator J.
Lemma 2.6.The estimate is valid 2.95 for all t ≥ 0, provided that the right-hand side is finite.
, by applying the L ∞ -L 1 time decay estimate of the free evolution group e −i i∂ x t see paper of 19, Lemma 1 , we get

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Lemma 2.7.The following asymptotics is true: R e itA ξ,y φ ξ, y dy for t ≥ 1 uniformly with respect to ξ ∈ R, provided the right-hand side is finite.

2.100
By a direct calculation we find where

2.102
Therefore we obtain R e itA ξ,y φ ξ, y dy where

2.105
By a direct calculation we have

2.107
Therefore we get the estimate

2.108
This yields the estimate of the lemma.Lemma 2.7 is proved.
In the next lemma we obtain the asymptotics for the nonlinear term FU −t G u, i∂ x −1 u 2 in 1.14 .
Proof.We first find the representation for FU −t i∂ x −1 u 2 t .We introduce the operator , so that the representation for the free evolution group is true ω iD 1/ω , and FD 1/ω D ω F, we find with ω > 0. Applying this formula with ω 2 and putting u t

2.111
Since FU −t Fe i i∂ x t e it ξ F, we can write Hence we obtain

2.113
Since u t, η − ξ e it η−ξ v t, η − ξ , we get for all x ∈ R. In the same manner applying the identity

2.116
After a change η 2ξ 2y, we find

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Thus ξ R e itA ξ,y φ 2 ξ, y dy

2.122
We apply Lemma 2.7 to find

2.125
We now write the representation for Ψ t, ξ as

2.126
Advances in Mathematical Physics 29 where the remainder is

2.127
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 for all t ≥ 1.Also

2.129
Thus we find the estimate for the remainder R t as

2.131
Therefore the asymptotics of the lemma is true.Lemma 2.8 is proved.

Proof of Theorem 1.1
We introduce the function space where and γ > 0 is small.The local existence in the function space X T can be proved by a standard contraction mapping principle.We state it without a proof.Theorem 3.1.Let u 0 ∈ H 3,1 and the norm u 0 H 3,1 ε.Then there exist ε 0 > 0 and T > 1 such that for all 0 < ε < ε 0 the initial value problem 1.1 has a unique local solution u ∈ C 0, T ; H 3,1 with the estimate u X T < √ ε.
Let us prove that the existence time T 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 X T < √ ε does not hold; namely, we have u X T ≤ √ ε.We apply the energy method to 1.14 L u iλG u, u −2i|λ| 2 G u, i∂ x −1 u 2 , 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

3.5
Next we use the commutator relations Then by the energy method i.e., multiplying both sides of the above equation by i∂ x 4 P u − G u, u , taking the real part, and integrating over the space , by Lemma 2.1,

3.8
Therefore by Theorem 3.1 it follows that Pu H 2 ≤ Cε t γ .

3.9
The energy estimate and the identity Lx xL 3.11

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Then by the identity Then by the Sobolev inequality we have u L ∞ ≤ C u H 1 .Thus the desired estimate follows.Lemma 2.6 is proved.