Wave operators to a quadratic nonlinear Klein–Gordon equation in two space dimensions

Abstract

We study asymptotics around the final states of solutions to the nonlinear Klein–Gordon equations with quadratic nonlinearities in two space dimensions $({\partial }_t^2-\Delta +m^2)u=\lambda u^2,(t,x)\in R\times R^2$, where $\lambda \in C$. We prove that if the final states

$$u_1^{+}\in H_{\frac{q}{q-1}}^{4-\frac{4}{q}}\left(R^2\right)\cap H^{\frac{5}{2},1}\left(R^2\right)\cap H_1^2\left(R^2\right)\text{,}$$$$u_2^{+}\in H_{\frac{q}{q-1}}^{3-\frac{4}{q}}\left(R^2\right)\cap H^{\frac{3}{2},1}\left(R^2\right)\cap H_1^1\left(R^2\right)\text{,}$$

and

$${\Vert u_1^{+}\Vert }_{H_1^2}+{\Vert u_2^{+}\Vert }_{H_1^1}$$

is sufficiently small, where $4<q\le \infty$, then there exists a unique global solution $u\in C([T,\infty );L^2\left(R^2\right))$ to the nonlinear Klein–Gordon equations such that $u\left(t\right)$ tends as $t\to \infty$ in the $L^2$ sense to the solution $u_0\left(t\right)=u_1^{+}cos\left({〈i\nabla 〉}_{m}t\right)+\left({〈i\nabla 〉}_m^{-1}{u}_2^{+}\right)sin\left({〈i\nabla 〉}_{m}t\right)$ of the free Klein–Gordon equation.

Introduction

In this paper, we show the existence of wave operators for the nonlinear Klein–Gordon equation

$$({\partial }_t^2-\Delta +m^2)u=\lambda u^2\text{,}(t,x)\in R\times R^2,\lambda \in C,m>0$$

in two space dimensions in a lower order Sobolev space than the previous result in [1]. If we define a new variable

$$w=\left(\begin{array}{c}{w}_1\hfill \\ w_2\hfill \end{array}\right)=\frac{1}{2}\left(\begin{array}{cc}1\hfill & i{〈i\nabla 〉}_m^{-1}{\partial }_t\hfill \\ 1\hfill & -i{〈i\nabla 〉}_m^{-1}{\partial }_t\hfill \end{array}\right)\left(\begin{array}{c}u\hfill \\ u\hfill \end{array}\right)\text{,}$$

then (1.1) is written as

$$(E{\partial }_t+iA{〈i\nabla 〉}_m)w=\frac{i\lambda }{2}b{〈i\nabla 〉}_m^{-1}{(w_1+w_2)}^2\text{,}$$

where ${〈i\nabla 〉}_m={(m^2-\Delta )}^{\frac{1}{2}}$ and

$$E=\left(\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill \end{array}\right)\text{,}A=\left(\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -1\hfill \end{array}\right)\text{,}b=\left(\begin{array}{c}\hfill 1\hfill \\ \hfill -1\hfill \end{array}\right)\text{.}$$

If we replace the nonlinear term $\lambda u^2$ by $-\left|u\right|u$ in (1.1), it was shown in [2], [3] that there are no nontrivial asymptotically free solutions. We note that if the data are real, then $w_1=\overline{w_2}$ and

$${(w_1+w_2)}^2={(w_1+\overline{w_1})}^2=w_1^2+2{\left|w_1\right|}^2+{\overline{w_1}}^2\text{.}$$

Hence (1.2) becomes

$$({\partial }_t+i{〈i\nabla 〉}_m)w_1=\frac{i\lambda }{2}{〈i\nabla 〉}_m^{-1}(w_1^2+2{\left|w_1\right|}^2+{\overline{w_1}}^2)\text{.}$$

The Klein–Gordon equation is a relativistic version of the Schrödinger equation, so that the quadratic nonlinear equation (1.2) can be considered as a relativistic version of the quadratic nonlinear Schrödinger equation

$$(i{\partial }_t+\frac{1}{2}\Delta )u={\lambda }_1{u}^2+{\lambda }_2{\left|u\right|}^2+{\lambda }_3{\overline{u}}^2+{\lambda }_4\left|u\right|u\text{,}(t,x)\in R\times R^2$$

if ${\lambda }_4=0$. Asymptotic behavior of solutions to (1.4) with ${\lambda }_2=0,{\lambda }_4\in R$ was studied in papers [4], [5], [6], where the wave operators and the modified wave operator were constructed when ${\lambda }_4=0$ and ${\lambda }_4\in R$, respectively. In [7], it was shown that the wave operator exists under the condition for the coefficients ${\lambda }_4=0,{\lambda }_1=\frac{\overline{{\lambda }_3}}{2}$ and ${\lambda }_2=\frac{{\lambda }_3^2}{2\overline{{\lambda }_3}}$. On the other hand, the non-existence of the wave operators was proved in [8], [9] for the case of ${\lambda }_1={\lambda }_2=0,{\lambda }_4\in R$. In the case of the nonlinear Schrödinger equation

$$(i{\partial }_t+\frac{1}{2}\Delta )u={\lambda }_4{\left|u\right|}^{p-1}u\text{,}(t,x)\in R\times R^n,{\lambda }_4\in R$$

in general space dimensions, we have a lot of results in the critical case $p=1+\frac{2}{n}$ on the asymptotic behavior of solutions for $n\le 3$ (see [10], [11] for instance) and the non-existence of the wave operators (see [12], [13] for instance). On the other hand, there are no results on the asymptotic behavior of solutions to the nonlinear Klein–Gordon equation

$$({\partial }_t+i{〈i\nabla 〉}_m)w_1=\frac{i\lambda }{2}{〈i\nabla 〉}_m^{-1}\left|w_1\right|w_1\text{,}(t,x)\in R\times R^2,\lambda \in R\text{.}$$

The initial value problem for the nonlinear Klein–Gordon equations with quadratic nonlinearities including that of (1.1) was studied in papers [14], [15], [1], [16], [17], when the initial data are smooth and decay quickly at infinity. In particular, in paper [15], the Cauchy problem was considered.

$$({\partial }_t^2-\Delta +m^2)u=\mathcal{N}(u,{\partial }_{t}u,\nabla u)\text{,}(t,x)\in R\times R^2$$

with the initial data $u\left(0\right)=u_1,{\partial }_{t}u\left(0\right)=u_2$, where $\partial =({\partial }_0,\nabla )=(i{\partial }_t,{\partial }_1,{\partial }_2),{\partial }_j={\partial }_{x_j}$. Under the assumption that the initial data $u_1\in H_2^{k+1,k}\left(R^2\right),u_2\in H_2^{k,k}\left(R^2\right)$ are small and therefore obtain a global existence result, where $k\ge 36$, and the weighted Sobolev space $H_p^{k,s}\left(R^n\right)$ is defined by

$$H_p^{k,s}\left(R^n\right)=\{\varphi \in L^p\left(R^n\right);{\Vert \varphi \Vert }_{H_p^{k,s}}={\Vert {〈i\nabla 〉}^k{〈x〉}^s\varphi \Vert }_{L^p}<\infty \}\text{,}$$

with $〈x〉=\sqrt{1+{\left|x\right|}^2}$. In [1], they extended the result of [15] to the quasi-linear case with the condition $k\ge 40$. Their proof depends on the method of the normal form in [18] and the time decay estimate in [19]. On the other hand there are a few results as far as we know on the existence of wave operators in a quadratic case except in papers [1], [20]. In [1], they considered nonlinearities such that

$$\mathcal{N}(u,\partial u,\partial \nabla u)=a{u}^2+b\sum _{j=0}^2{\left({\partial }_{j}u\right)}^2\text{,}{a}^2+b^2\ne 0,a,b\in R$$

and constructed a wave operator for given small final data $u_1^{+}\in H_2^{k+1,k}\left(R^2\right),u_2^{+}\in H_2^{k,k}\left(R^2\right)$, where $k\ge 12$ (see also [20]). The above nonlinearity has a remarkable property that using the method of the normal form of [18] does not cause a derivative loss.

Denote by $u_0$ the solution of the free Klein–Gordon equation

$$({\partial }_t^2-\Delta +m^2)u_0=0\text{,}(t,x)\in R\times R^2\text{,}$$

then $u_0$ satisfies

$$\left(\begin{array}{c}\hfill u_0\left(t\right)\hfill \\ \hfill {〈i\nabla 〉}_m^{-1}{\partial }_t{u}_0\left(t\right)\hfill \end{array}\right)=\left(\begin{array}{cc}\hfill cos{〈i\nabla 〉}_{m}t\hfill & \hfill sin{〈i\nabla 〉}_{m}t\hfill \\ \hfill -sin{〈i\nabla 〉}_{m}t\hfill & \hfill cos{〈i\nabla 〉}_{m}t\hfill \end{array}\right)\left(\begin{array}{c}\hfill u_1^{+}\hfill \\ \hfill {〈i\nabla 〉}_m^{-1}{u}_2^{+}\hfill \end{array}\right)$$

with the initial conditions $u_0\left(0\right)=u_1^{+},{\partial }_t{u}_0\left(0\right)=u_2^{+}$. We say that the wave operator is constructed if a unique solution $u$ of (1.1) can be found in the neighborhood of the solution $u_0$ of the free Klein–Gordon equation (1.5). For simplicity, we denote $H^{k,s}\left(R^n\right)=H_2^{k,s}\left(R^n\right),H^k\left(R^n\right)=H^{k,0}\left(R^n\right),H_p^k\left(R^n\right)=H_p^{k,0}\left(R^n\right)$.

Our main result in the present paper is the following.

Theorem 1

We assume that

$$u_1^{+}\in H_{\frac{q}{q-1}}^{4-\frac{4}{q}}\left(R^2\right)\cap H^{\frac{5}{2},1}\left(R^2\right)\cap H_1^2\left(R^2\right)\text{,}$$$$u_2^{+}\in H_{\frac{q}{q-1}}^{3-\frac{4}{q}}\left(R^2\right)\cap H^{\frac{3}{2},1}\left(R^2\right)\cap H_1^1\left(R^2\right)\text{,}$$

and

$${\Vert u_1^{+}\Vert }_{H_1^2}+{\Vert u_2^{+}\Vert }_{H_1^1}\le \rho \text{,}$$

where $4<q\le \infty$ . Then there exist $\rho >0$ and $T>1$ such that(1.1)has a unique global solution $u\in C([T,\infty );L^2\left(R^2\right))$ satisfying the asymptotics

$${\Vert u\left(t\right)-u_0\left(t\right)\Vert }_{L^2}\le C{t}^{-\delta }$$

for all $t>T$ , where $\frac{1}{2}<\delta <1-\frac{2}{q}$.

Remark 1

If we assume that ${\Vert u_1^{+}\Vert }_{H^{\frac{5}{2}}}+{\Vert u_2^{+}\Vert }_{H^{\frac{3}{2}}}$ is sufficiently small, then we can take $T=0$.

Remark 2

We note that every quadratic term $\mathcal{N}(u_0,{\partial }_t{u}_0,\nabla u_0)$ can be represented in the form

$$\mathcal{N}(u_0,{\partial }_t{u}_0,\nabla u_0)=({\partial }_t^2-\Delta +m^2)f\left(u_0\right)+\Gamma \text{,}$$

where $u_0$ is a solution of the free Klein–Gordon equation (1.5) and the remainder term $\Gamma$ has some additional time decay. However we encounter the derivative loss difficulty, when applying the Strichartz type estimates to the differences $\mathcal{N}(u,{\partial }_{t}u,\nabla u)-\mathcal{N}(u_0,{\partial }_t{u}_0,\nabla u_0)$, so the case of general quadratic nonlinearity $\mathcal{N}(u,{\partial }_{t}u,\nabla u)$ could not be treated by the same method as in the proof of Theorem 1.

Remark 3

Our result works for the same nonlinearities as given in (1.4) without any restriction on coefficients.

We compare the result above with that for (1.4). Denote the homogeneous Sobolev space

$${\dot{H}}^m\left(R^n\right)=\{u\in {\mathcal{S}}^{\prime };{\Vert u\Vert }_{{\dot{H}}^m}={\Vert {(-\Delta )}^{\frac{m}{2}}u\Vert }_{L^2}<\infty \}\text{.}$$

In [6] we have

Proposition 1

Let ${\lambda }_2={\lambda }_4=0,u^{+}\in H^{0,2}\left(R^2\right)\cap {\dot{H}}^{-2\delta }\left(R^2\right)$ and ${\Vert u^{+}\Vert }_{H^{0,2}}+{\Vert u^{+}\Vert }_{{\dot{H}}^{-2\delta }}$ be sufficiently small, where $\frac{1}{2}<\delta <1$ . Then there exists a unique global solution $u$ of(1.4)such that $u\in C(R^{+};L^2\left(R^2\right))$,

$${\Vert u\left(t\right)-u_s\left(t\right)\Vert }_{L^2}\le C{t}^{-\delta }\text{,}$$

where $u_s\left(t\right)$ is the solution of $i{\partial }_{t}u+\frac{1}{2}\Delta u=0,u\left(0\right)=u^{+}$.

The rest of the paper is organized as follows. In Section 2 we give some preliminary calculations following the method of papers [16], [17], [14]. Then Section 3 is devoted to the proof of Theorem 1.