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A note on global existence of solutions to nonlinear Klein-Gordon equations in one space dimension. (English) Zbl 0945.35056

In this paper the following Cauchy problem for the nonlinear Klein-Gordon equation is considered:

$\left(\square +1\right)u=F\left(u,{u}_{t},{u}_{x},{u}_{tx},{u}_{xx}\right)\phantom{\rule{1.em}{0ex}}\text{for}\phantom{\rule{1.em}{0ex}}\left(t,x\right)\in \left(0,\infty \right)×ℝ,\phantom{\rule{2.em}{0ex}}\left(1\right)$
$u\left(0,x\right)=\epsilon f\left(x\right),\phantom{\rule{1.em}{0ex}}{u}_{t}\left(0,x\right)=\epsilon g\left(x\right)\phantom{\rule{1.em}{0ex}}\text{for}\phantom{\rule{1.em}{0ex}}x\in ℝ,\phantom{\rule{2.em}{0ex}}\left(2\right)$

where $\square =\left({\partial }^{2}/\partial {t}^{2}\right)-\left({\partial }^{2}/\partial {x}^{2}\right)$, ${u}_{t}=\left(\partial /\partial t\right)u$, ${u}_{x}=\left(\partial /\partial x\right)u$, etc., $\epsilon$ is a small positive parameter, $f$, $g\in {C}_{0}^{\infty }\left(ℝ\right)$ and the nonlinear function $F\left(\cdots \right)=F\left(\lambda \right)$ with $\lambda =\left(u,{u}_{t},{u}_{x},{u}_{tx},{u}_{xx}\right)$ can be written in the form $F\left(\lambda \right)={\sum }_{i=1}^{10}{c}_{i}{G}_{i}\left(\lambda \right)+N\left(\lambda \right)+H\left(\lambda \right)$ in a neighborhood of $\lambda =0$; here ${\left\{{c}_{i}\right\}}_{i=1}^{10}\subset ℝ$, ${\left\{{G}_{i}\right\}}_{i=1}^{10}$ are linearly independent polynomials of $\lambda$ defined by

$\begin{array}{cc}\hfill {G}_{1}\left(\lambda \right)& =u\left(-{u}^{2}+3{u}_{t}^{2}-3{u}_{x}^{2}\right),\hfill \\ \hfill {G}_{2}\left(\lambda \right)& ={u}_{t}\left(-3{u}^{2}+{u}_{t}^{2}-{u}_{x}^{2}\right)+2u\left({u}_{t}{u}_{xx}-{u}_{x}{u}_{tx}\right),\hfill \\ \hfill {G}_{3}\left(\lambda \right)& ={u}_{x}\left(-{u}^{2}+{u}_{t}^{2}-{u}_{x}^{2}\right)+2u\left({u}_{t}{u}_{tx}-{u}_{x}{u}_{tx}\right),\hfill \\ \hfill {G}_{4}\left(\lambda \right)& ={u}^{3}-2{u}^{2}{u}_{xx}-3u{u}_{t}^{2}+2{u}_{t}^{2}{u}_{xx}-2{u}_{t}{u}_{x}{u}_{tx}-u\left({u}_{tx}^{2}-{u}_{xx}^{2}\right),\hfill \\ \hfill {G}_{5}\left(\lambda \right)& =\left(-{u}^{2}+{u}_{t}^{2}-{u}_{x}^{2}\right){u}_{tx}-2u{u}_{t}{u}_{x},\hfill \\ \hfill {G}_{6}\left(\lambda \right)& =-u{u}_{x}^{2}+2{u}_{x}\left({u}_{t}{u}_{tx}-{u}_{x}{u}_{xx}\right)+u\left({u}_{tx}^{2}-{u}_{xx}^{2}\right),\hfill \\ \hfill {G}_{7}\left(\lambda \right)& =3{u}^{2}{u}_{t}-6u{u}_{t}{u}_{xx}-{u}_{t}^{3}-3{u}_{t}\left({u}_{tx}^{2}-{u}_{xx}^{2}\right),\hfill \\ \hfill {G}_{8}\left(\lambda \right)& ={u}^{2}{u}_{x}-2u{u}_{t}{u}_{tx}-2u{u}_{x}{u}_{xx}-{u}_{t}^{2}{u}_{x}-{u}_{x}\left({u}_{tx}^{2}-{u}_{xx}^{2}\right),\hfill \\ \hfill {G}_{9}\left(\lambda \right)& =-2u{u}_{x}{u}_{tx}-{u}_{t}{u}_{x}^{2}+{u}_{t}\left({u}_{tx}^{2}-{u}_{xx}^{2}\right),\hfill \\ \hfill {G}_{10}\left(\lambda \right)& =-{u}_{x}^{3}+3{u}_{x}\left({u}_{tx}^{2}-{u}_{xx}^{2}\right),\hfill \end{array}$

$N$ is of the form $N\left(\lambda \right)={P}_{1}\left(\lambda \right)\left({u}_{t}{u}_{tx}-{u}_{x}{u}_{xx}+u{u}_{x}\right)+{P}_{2}\left(\lambda \right)\left({u}_{t}{u}_{xx}-{u}_{x}{u}_{tx}\right)+{P}_{3}\left(\lambda \right)\left({u}_{tx}^{2}-{u}_{xx}^{2}+u{u}_{xx}\right)$, ${\left\{{P}_{i}\left(\lambda \right)\right\}}_{i=1}^{3}$ are homogeneous polynomials of degree 1, and $H\left(\lambda \right)$ is a smooth function of $\lambda$ of degree 4, i.e. $H\left(\lambda \right)=O\left(|\lambda {|}^{4}\right)$ near $\lambda =0$. The main result of the paper is the following:

Theorem 1.1. For any integer $k\ge 15$ there exists a positive constant ${\epsilon }_{0}$ such that for any $\epsilon \in \left(0,{\epsilon }_{0}\right]$ the Cauchy problem (1), (2) has a unique classical solution $u=u\left(t,x\right)\in {C}^{\infty }\left(\left[0,\infty \right)×ℝ\right)$. Moreover, the solution $u$ has a free profile in the sense that there exist ${u}_{0}\in {H}^{k+1}\left(ℝ\right)$ and ${u}_{1}\in {H}^{k}\left(ℝ\right)$ (${H}^{k}\left(ℝ\right)$ is the usual Sobolev space) such that

${\parallel \left(u-U\right)\left(t,·\right)\parallel }_{{H}^{k+1}\left(ℝ\right)}+{\parallel {\partial }_{t}\left(u-U\right)\left(t,·\right)\parallel }_{{H}^{k}\left(ℝ\right)}\to 0\phantom{\rule{1.em}{0ex}}\text{as}\phantom{\rule{1.em}{0ex}}t\to +\infty ,$

where $U$ is the solution to the Cauchy problem for the linear Klein-Gordon equation $\left(\square +1\right)U=0$ in $\left(0,\infty \right)×ℝ$ with initial data $U\left(0,x\right)={u}_{0}\left(x\right)$ and ${U}_{t}\left(0,x\right)={u}_{1}\left(x\right)$, $x\in ℝ$.

##### MSC:
 35L70 Nonlinear second-order hyperbolic equations 35L15 Second order hyperbolic equations, initial value problems
##### Keywords:
Cauchy problem; global classical solutions