Metcalfe, Jason; Sogge, Christopher D. Global existence for high dimensional quasilinear wave equations exterior to star-shaped obstacles. (English) Zbl 1213.35314 Discrete Contin. Dyn. Syst. 28, No. 4, 1589-1601 (2010). This paper gives a global existence result for the initial-boundary value problem for \(Lu=Q(u,Du,D^2u)\), where \(L\) is the wave operator in \(4+1\) dimensions, on the exterior of a star-shaped, smooth bounded obstacle \(K\), with Dirichlet conditions on the boundary of \(K\); here \(D\) and \(D^2\) stand for space-time derivatives of first and second order respectively. The nonlinearity is quadratic near the origin (its Taylor expansion contains only terms of order two or higher) and has the form \(Q=A(u,Du)+B(u,Du)D^2u\); in addition, one requires \(\partial^2_uA(0)=0\). Results in \(n\) space dimensions with \(n\geq 5\) that do not require the last condition are also mentioned. The proofs rely on weighted estimates and commutator arguments. Reviewer: Satyanad Kichenassamy (Reims) Cited in 13 Documents MSC: 35L72 Second-order quasilinear hyperbolic equations 35A01 Existence problems for PDEs: global existence, local existence, non-existence 35L20 Initial-boundary value problems for second-order hyperbolic equations Keywords:wave equation; global existence; exterior domain; Dirichlet conditions; weighted estimates; commutator arguments PDFBibTeX XMLCite \textit{J. Metcalfe} and \textit{C. D. Sogge}, Discrete Contin. Dyn. Syst. 28, No. 4, 1589--1601 (2010; Zbl 1213.35314) Full Text: DOI arXiv