Research Article | Open Access
The Initial Value Problem for the Quadratic Nonlinear Klein-Gordon Equation
We study the initial value problem for the quadratic nonlinear Klein-Gordon equation , , , , where and . 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.
Let us consider the Cauchy problem for the nonlinear Klein-Gordon equation with a quadratic nonlinearity in one dimensional case
where , and
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 . One of the important tools of paper  was based on the transformation of the equation by virtue of the hyperbolic polar coordinates following to paper . 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
where and are the real-valued functions, and Indeed we can put then satisfies
There are a lot of works devoted to the study of the cubic nonlinear Klein-Gordon equation
with When , 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 - time decay estimates of solutions and nonexistence of the usual scattering states for (1.4) were shown in  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 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 was excluded in these works. In paper , 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  and the other reveals an additional time decay rate via the operator which was used in . This nonlinear transformation was refined in  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 , 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  by using hyperbolic polar coordinates. Compactness condition on the data was removed in  in the case of the cubic nonlinearity and a real-valued solution. Final value problem with the cubic nonlinearity was studied in  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 and the complex valued initial data. When the initial data are complex-valued, global existence and -time decay estimates of small solutions to the Klein-Gordon equation with cubic nonlinearity were obtained in paper  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 and with 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 , where combining the method of the normal forms of  and the time decay estimate through the operator of , it was shown that every quadratic nonlinearity is nonresonant.
We denote the Lebesgue space by , with the norm if and if The weighted Sobolev space is
for , where For simplicity we write . The index we usually omit if it does not cause a confusion. We denote by the Fourier transform of the function Then the inverse Fourier transformation is .
Our main result of this paper is the following.
Theorem 1.1. Let and the norm . Then there exists such that for all the Cauchy problem (1.1) has a unique global solution satisfying the time decay estimate Furthermore there exists a unique final state such that where ,
An important tool for obtaining the time decay estimates of solutions to the nonlinear Klein-Gordon equation is implementation of the operator
which is analogous to the operator in the case of the nonlinear Schrödinger equations used in . The operator was used previously in paper of  for constructing the scattering operator for nonlinear Klein-Gordon equations with a supercritical nonlinearity. We have ; therefore the commutator , where is a linear part of (1.1). Since is not a purely differential operator, it is apparently difficult to calculate the action of on the nonlinearity in (1.1). So, instead we use the first-order differential operator
which is closely related to by the identity and acts easily on the nonlinearity. Moreover, it almost commutes with , since .
Also we use the method of normal forms of  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 and put to get
where Integrating (1.11) with respect to time, we find
Then we integrate by parts with respect to taking into account (1.11),
Returning to the function , we obtain the following equation:
with the symmetric bilinear operator
and 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
and the other one is a remainder term with an estimate like
Remark 1.2. We believe that all quadratic nonlinear terms 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 with Some more regularity conditions are necessary to treat the bilinear operators . Also we have to show that are the nonresonant terms (i.e., remainders) and to remove the resonant terms by an appropriate phase function. We will dedicate a separate paper to this problem.
We prove our main result in Section 3. In the next section we prove several lemmas used in the proof of the main result.
First we give some estimates for the symmetric bilinear operator
Denote the kernel as follows:
Lemma 2.1. The representation is true where the kernel obeys the following estimate: for all Moreover the following estimates are valid: for , 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 . Then changing , we find
where the kernel is
Changing the variables of integration and (the prime we will omit), we get
where We change and ) and denote
For the case of we integrate by parts using the identity where Then we get
Hence we can estimate the kernel as follows:
for the case of For the case of we integrate three times by parts with respect to
Hence we can estimate the kernel as follows:
for all For the case we integrate by parts three times with respect to
Note that Hence
Finally for the case of we integrate by parts three times with respect to and
then we can estimate the kernel as follows:
for all Hence estimate (2.5) is true.
By virtue of estimate (2.5) applying the Hölder inequality with and the Young inequality with and , we find where .
We now estimate the operator as follows: Then by virtue of estimate (2.4) applying the Hölder and Young inequalities, we get Lemma 2.1 is proved.
We now decompose the free Klein-Gordon evolution group where similarly to the factorization of the free Schrödinger evolution group. We denote the dilation operator by
Define the multiplication factor where for and for We introduce the operator
The inverse operator acts on the functions defined on as follows:
for all since and We now introduce the operators
so that we have the representation for the free Klein-Gordon evolution group
The first term 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 with initial data . 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
where the right-inverse operators are
In the next lemma we state the estimates of the operators .
Lemma 2.2. The estimates hold as follows: where and where provided the right-hand sides are finite.
Proof. Changing the variable of integration , we see that
Consider the estimate for the derivative . Define Note that and Hence integrating by parts one time yields where . Also we have Hence the estimate is true Since , we get Consider the -estimate of the integral We have where the kernel After a change , we find We now change the contour of integration in as then Since , using (2.38) and the inequality for we get ; also we have for . Therefore we obtain the estimate if we choose and Thus we get since
In the same manner we consider the estimate of the integral where the kernel since by a direct calculation In the same way as in (2.47) we have if we choose , and Therefore we get Hence Thus we have the second estimate of the lemma.
Consider the estimate for the derivative . Note that and Hence integrating by parts one time yields where and The estimate is true Consider the -estimate of the integral We have, changing and , where the kernel We now change the contour of integration in as ; then Since