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Global existence for high dimensional quasilinear wave equations exterior to star-shaped obstacles. (English) Zbl 1213.35314

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.

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
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