Global existence and asymptotic behavior of solutions for the Klein-Gordon equations with quadratic nonlinearity in two space dimensions.

*(English)*Zbl 0877.35030This paper is devoted to the proof of existence of global solutions of the Cauchy problem for the nonlinear Klein-Gordon (NLKG) equation
\[
\square u+ u= F(u,\partial_tu,\nabla u)\tag{1}
\]
in space-time dimension \(2+1\), for a smooth nonlinearity \(F\) which is of order two in its arguments near zero, and for small initial data. In addition, the solutions thereby obtained are proved to be asymptotically free, namely to be asymptotic in suitable norms to solutions of the linear Klein-Gordon equation as \(t\to\pm\infty\).

The proof combines three ingredients: (1) The method of normal forms introduced by Shatah: one performs a change of unknown functions \(v= u-B(u\partial_tu)\), where \(B(u,\partial_tu)\) is defined by an integral operator and is quadratic in \(u\) and \(\partial_tu\). The equation (1) is then transformed into an NLKG equation for \(v\) with a nonlinearity which is at least cubic near zero. (2) The use of Lorentz invariant Sobolev norms defined by means of the generators of the Lorentz group, introduced and previously used by Klainerman in higher space dimensions. (3) A pointwise decay estimate of solutions of the linear Klein-Gordon equation in terms of norms of this type, due to Georgiev. The results of this paper have been subsequently extended by the authors to the quasilinear case [Proc. of the 4th MSJ Intern. Res. Inst. on Nonlinear Waves, Sapporo 1995].

The proof combines three ingredients: (1) The method of normal forms introduced by Shatah: one performs a change of unknown functions \(v= u-B(u\partial_tu)\), where \(B(u,\partial_tu)\) is defined by an integral operator and is quadratic in \(u\) and \(\partial_tu\). The equation (1) is then transformed into an NLKG equation for \(v\) with a nonlinearity which is at least cubic near zero. (2) The use of Lorentz invariant Sobolev norms defined by means of the generators of the Lorentz group, introduced and previously used by Klainerman in higher space dimensions. (3) A pointwise decay estimate of solutions of the linear Klein-Gordon equation in terms of norms of this type, due to Georgiev. The results of this paper have been subsequently extended by the authors to the quasilinear case [Proc. of the 4th MSJ Intern. Res. Inst. on Nonlinear Waves, Sapporo 1995].

Reviewer: J.Ginibre (Orsay)

##### MSC:

35G25 | Initial value problems for nonlinear higher-order PDEs |

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

35L70 | Second-order nonlinear hyperbolic equations |

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