Existence for large times of strict solutions of nonlinear wave equations in three space dimensions for small initial data.

*(English)*Zbl 0662.35070The author studies the asymptotic behaviour of solutions to the quasilinear wave equation
\[
(\partial^ 2_ t-\Delta)u=a_{\alpha \beta}(u')D_{\alpha}D_{\beta}u
\]
in three space dimensions to initial data
\[
u(x,0)=\epsilon f(x),\quad u_ t(x,0)=\epsilon g(x).
\]
He proves that the time of existence T(\(\epsilon)\) of the solution satisfies \(\lim_{\epsilon \to 0}\inf \epsilon \log T(\epsilon)\geq 1/A,\) where A is a constant determined from the behaviour of the derivatives of the coefficients \(a_{\alpha \beta}(u')\) at \(u'=0\). The asymptotic behaviour of solutions of the linear wave equation enters the calculation of A in an interesting way. The result was proved independently by L. Hörmander [Institut Mittag-Leffler, Report No.5 (revised version), 1-67 (1985); see also Lect. Notes Math. 1256, 214-280 (1988; Zbl 0632.35045)], but the author gives a new proof in the paper under consideration. He also gives a new proof of the fact that if Klainerman’s null condition is satisfied (which implies \(A=0)\), then for sufficiently small \(\epsilon\) the solution exists globally. This result has been proved independently by D. Christodoulou [Commun. Pure Appl. Math. 39, 267-282 (1986; Zbl 0612.35090)] and S. Klainerman [Lect. Appl. Math. 23, Pt. 1, 293-326 (1986; Zbl 0599.35105)]. A proof has also been given by L. Hörmander [On global existence of solutions of nonlinear hyperbolic equations in \({\mathbb{R}}^{1+3}\), Institut Mittag Leffler, Report No.9, 1-22 (1985)].

Reviewer: H.D.Alber

##### MSC:

35L70 | Second-order nonlinear hyperbolic equations |

35B40 | Asymptotic behavior of solutions to PDEs |

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

35B30 | Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs |

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\textit{F. John}, Commun. Pure Appl. Math. 40, No. 1, 79--109 (1987; Zbl 0662.35070)

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##### References:

[1] | John, Comm. Pure Appl. Math. 34 pp 20– (1981) |

[2] | John, Comm. Pure Appl. Math. 29 pp 649– (1976) |

[3] | The Cauchy problem for quasi-linear symmetric hyperbolic systems, Archive for Rat. Mech. Analysis 57–58, 1974 –75, pp. 181–205. |

[4] | Klainerman, Comm. Pure Appl. Math. 38 pp 321– (1985) |

[5] | John, Comm. Pure Appl. Math. 37 pp 443– (1984) |

[6] | John, Comm. Pure Appl. Math. 36 pp 1– (1983) |

[7] | The lifespan of classical solutions of nonlinear hyperbolic equations, Report No. 5 (Revised version). Institut Mittag-Leffler, 1985, pp. 1–67. |

[8] | Christodoulou, Comm. Pure Appl. Math. 39 pp 267– (1986) |

[9] | Klainerman, Lectures in Applied Mathematics 23 pp 293– (1986) |

[10] | On global existence of solutions of nonliear hyperbolic equations in R1+3, Report No. 9. Institut Mittag Leffler, 1985, pp. 1–22. |

[11] | Blow-up of radial solutions of = /partialF(ut)/partialt in three space dimensions, MRC Technical Summary Report #2493. University of Wisconsin, 1982. |

[12] | John, Mathematica Aplicada e Computational 4 pp 3– (1985) |

[13] | On Sobolev spaces associated with some Lie algebras, Report No. 4, Institut Mittag-Leffler, 1985. |

[14] | Klainerman, Comm. Pure Appl. Math. 40 pp 111– (1987) |

[15] | Long time effects of nonlinearity in second-order hyperbolic equations, Proceedings on the Symposium on Frontiers of the Mathematical Sciences, (1985). Wiley-Interscience (to appear). |

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