×

zbMATH — the first resource for mathematics

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

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
PDF BibTeX XML Cite
Full Text: DOI
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).
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.