# zbMATH — the first resource for mathematics

Global existence and asymptotic behavior of solutions for the Klein-Gordon equations with quadratic nonlinearity in two space dimensions. (English) Zbl 0877.35030
This 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].
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
##### Keywords:
normal forms; Lorentz invariant Sobolev norms
Full Text:
##### References:
 [1] Bachelot, R.: Problème de Cauchy globale pour des systèmes de Dirac-Klein-Gordon, Ann. Inst. Henri Poincaré, Physique Théorique48, 387–422 (1988) · Zbl 0672.35071 [2] Bergh, J., Löfström, J.: ”Interpolation Spaces,” Springer-Verlag, Berlin-Heidelberg-New York, 1976 · Zbl 0344.46071 [3] Friedman, A.: ”Partial Differential Equations,” Holt Rinehart and Winston, New York, 1969 [4] Georgiev, V.: Decay estimates for the Klein-Gordon equations. Commun. Part. Diff. Eqs.17, 1111–1139 (1992) · Zbl 0767.35068 [5] Georgiev, V., Popivanov, P.: Global solution to the two-dimensional Klein-Gordon equation. Commun. Part. Diff. Eqs.16, 941–995 (1991) · Zbl 0741.35039 [6] Ginibre, J., Velo, G.: Time decay of finite energy solutions of the non linear Klein–Gordon and Schrödinger equations. Ann. Inst. Henri Poincaré, Physique Théorique43, 339–442 (1985) · Zbl 0595.35089 [7] Hörmander, L.: ”Non-linear Hyperbolic Differential Equations,” Lectures 1986–1987, Lund, 1988:2 [8] Klainerman, S.: Global existence of small amplitude solutions to nonlinear Klein-Gordon equations in four space time dimensions. Comm. Pure Appl. Math.38, 631–641 (1985) · Zbl 0597.35100 [9] Klainerman, S., Ponce, G.: Global small amplitude solutions to nonlinear evolution equations. Comm. Pure Appl. Math.36, 133–141 (1983) · Zbl 0509.35009 [10] Kosecki, R.: The unit condition and global existence for a class of nonlinear Klein–Gordon equations. J. Diff. Eqs.100, 257–268 (1992) · Zbl 0781.35062 [11] Shatah, J.: Global existence of small solutions to nonlinear evolution equations. J. Diff. Eqs.46, 409–425 (1982) · Zbl 0518.35046 [12] Shatah, J.: Normal forms and quadratic nonlinear Klein-Gordon equations. Comm. Pure Appl. Math.38, 685–696 (1985) · Zbl 0597.35101 [13] Sideris, T.: Decay estimates for the three-dimensional inhomogeneous Klein-Gordon equation and applications. Commun. Part. Diff. Eqs.14, 1421–1455 (1989) · Zbl 0696.35015 [14] Simon, J.C.H., Taflin, E.: The Cauchy problem for non-linear Klein-Gordon equations. Commun. Math. Phys.152, 433–478 (1993) · Zbl 0783.35066 [15] Strauss, W.A.: ”Nonlinear Wave Equations,” CBMS Regional Conference Series in Mathematics, no. 73, Amer. Math. Soc., Providence, RI, 1989 · Zbl 0714.35003 [16] Simon, J.C.H.: A wave operator for a non-linear Klein-Gordon equation. Lett. Math. Phys.7, 387–398 (1983) · Zbl 0539.35007 [17] Simon, J.C.H., Taflin, E.: Wave operators and analytic solutions for systems of non-linear Klein-Gordon equations and non-linear Schrödinger equations. Commun. Math. Phys.99, 541–562 (1985) · Zbl 0615.47034
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.