Global existence for nonlinear wave equations with multiple speeds. (English) Zbl 1052.35133
Beckner, William (ed.) et al., Harmonic analysis at Mount Holyoke. Proceedings of an AMS-IMS-SIAM joint summer research conference, Mount Holyoke College, South Hadley, MA, USA, June 25–July 5, 2001. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-2903-3/pbk). Contemp. Math. 320, 353-366 (2003).
This paper considers systems of the form \((\partial_t-c_I^2\Delta)u^I = F^I[\partial u,\partial^2u]\) in \({\mathbb R}^3\), where \(u=(u^I)_{I=1,\dots D}\), and \(F^I\) satisfies a generalization of the null condition and a symmetry condition. The main result concerns global existence for small data in \({\mathcal H}^m\times {\mathcal H}^{m-1}\) if \(m\geq 10\), where \({\mathcal H}^m=\{f\in L^2 : (\sqrt{1+| x| ^2}\;\nabla_x)^\alpha f\in L^2\) for \(| \alpha| \leq m\}\). This result generalizes work of T. Sideris and S.-Y. Tu [SIAM J. Math. Anal. 33, No. 2, 477–488 (2001; Zbl 1002.35091)].
For the entire collection see [Zbl 1013.00026].
For the entire collection see [Zbl 1013.00026].
Reviewer: Satyanad Kichenassamy (Reims)
MSC:
| 35L70 | Second-order nonlinear hyperbolic equations |
| 35L15 | Initial value problems for second-order hyperbolic equations |
