##
**Normal forms and global existence of solutions to a class of cubic nonlinear Klein-Gordon equations in one space dimension.**
*(English)*
Zbl 0891.35096

The author investigates the initial value problem in one spatial variable for the Klein-Gordon equation
\[
xu_{tt} - u_{xx} +u = F(u,\partial u, \partial u_x), \quad t>0, x\in\mathbb{R}, \tag{1}
\]

\[ u(0,x)=u_0(x), u_t(0,x)=u_1(x) , \quad x\in\mathbb{R}, \tag{2} \] where \(\partial =(\partial _t, \partial _x)\). The function \(F=F(u,v,p)\) is a linear combination of homogeneous polynomials which are all of degree 3 and linear in the components of \(p\). As a work by B. Yordanov [Blow-up for the one dimensional Klein-Gordon equation with a cubic nonlinearity, (preprint, 1995)] shows, for \(F= u_t^2u_x\), there may be a blow-up even for small initial data of some kind. On the other hand, according to a result from the thesis of K. Yagi [Normal forms and nonlinear Klein-Gordon equations in one space dimension, Master thesis, Waseda University, March 1994] it is known that when \[ F=F_1=3uu_t^2 -3uu_x^2 -u^3, \] then there is a unique global solution to (1), (2) for any small initial data. Moreover, as \(t\to +\infty \), this solution approaches a solution to (1) with \(F\equiv 0\). In this paper the same result is proved for a larger class of functions \(F\). This class is formed by linear combinations of the above mentioned function \(F_1\) with another six functions \(F_2, \ldots , F_7\), which are respectively given by the expressions \(3u^2_tu_x-u_x^3-3u^2u_x+6uu_tu_{tx}\), \(uu_xu_{xx}- u^2u_x+u_t^2u_x +2uu_tu_{tx}\), \((u_t^2-u^2_x-u^2)u_{xx} -2uu_x^2 \), \((u_t^2-u^2_x-u^2)u_{tx} -2uu_tu_x \), \(u_t^3-3u^2_xu_t-3u^2u_t-6uu_xu_{tx} \), \(u_tu_x^2+uu_tu_{xx}+2uu_xu_{tx} \).

The result is obtained by applying the method of normal forms due to J. Shatah [Commun. Pure Appl. Math. 38, 685-696 (1985; Zbl 0597.35101)] along the lines of the above mentioned work by K. Yagi.

\[ u(0,x)=u_0(x), u_t(0,x)=u_1(x) , \quad x\in\mathbb{R}, \tag{2} \] where \(\partial =(\partial _t, \partial _x)\). The function \(F=F(u,v,p)\) is a linear combination of homogeneous polynomials which are all of degree 3 and linear in the components of \(p\). As a work by B. Yordanov [Blow-up for the one dimensional Klein-Gordon equation with a cubic nonlinearity, (preprint, 1995)] shows, for \(F= u_t^2u_x\), there may be a blow-up even for small initial data of some kind. On the other hand, according to a result from the thesis of K. Yagi [Normal forms and nonlinear Klein-Gordon equations in one space dimension, Master thesis, Waseda University, March 1994] it is known that when \[ F=F_1=3uu_t^2 -3uu_x^2 -u^3, \] then there is a unique global solution to (1), (2) for any small initial data. Moreover, as \(t\to +\infty \), this solution approaches a solution to (1) with \(F\equiv 0\). In this paper the same result is proved for a larger class of functions \(F\). This class is formed by linear combinations of the above mentioned function \(F_1\) with another six functions \(F_2, \ldots , F_7\), which are respectively given by the expressions \(3u^2_tu_x-u_x^3-3u^2u_x+6uu_tu_{tx}\), \(uu_xu_{xx}- u^2u_x+u_t^2u_x +2uu_tu_{tx}\), \((u_t^2-u^2_x-u^2)u_{xx} -2uu_x^2 \), \((u_t^2-u^2_x-u^2)u_{tx} -2uu_tu_x \), \(u_t^3-3u^2_xu_t-3u^2u_t-6uu_xu_{tx} \), \(u_tu_x^2+uu_tu_{xx}+2uu_xu_{tx} \).

The result is obtained by applying the method of normal forms due to J. Shatah [Commun. Pure Appl. Math. 38, 685-696 (1985; Zbl 0597.35101)] along the lines of the above mentioned work by K. Yagi.

Reviewer: Milan Štědrý (Praha)

### MSC:

35L70 | Second-order nonlinear hyperbolic equations |

35B40 | Asymptotic behavior of solutions to PDEs |

35L15 | Initial value problems for second-order hyperbolic equations |

35Q40 | PDEs in connection with quantum mechanics |