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Global existence for a quasilinear wave equation outside of star-shaped domains. (English) Zbl 1016.35500
Journées “Équations aux dérivées partielles”, Plestin-les-Grèves, France, 5 au 8 juin 2001. Exposés Nos. I-XIV. Nantes: Université de Nantes. Exp. No. 12, 6 p. (2001).
Summary: This talk describes joint work with Chris Sogge and Markus Keel, in which we establish a global existence theorem for null-type quasilinear wave equations in three space dimensions, where we impose Dirichlet conditions on a smooth, compact star-shaped obstacle \({\mathcal K}\subset \mathbb{R}^3\). The key tool, following D. Christodoulou [Commun. Pure Appl. Math. 39, 267-282 (1986; Zbl 0612.35090], is to use the Penrose compactification of Minkowski space. In the case under consideration, this reduces matters to a local existence theorem for a singular obstacle problem. Full details appear in our joint paper of the same title [J. Funct. Anal. 189, 155-226 (2002; Zbl 1001.35087)].
For the entire collection see [Zbl 0990.00046].
MSC:
35L70 Second-order nonlinear hyperbolic equations
35B45 A priori estimates in context of PDEs
35L20 Initial-boundary value problems for second-order hyperbolic equations
35L55 Higher-order hyperbolic systems
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