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Existence for large times of strict solutions of nonlinear wave equations in three space dimensions for small initial data. (English) Zbl 0662.35070
The 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

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
Full Text: DOI
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