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Equivariant wave maps exterior to a ball. (English) Zbl 1243.35117

Summary: We consider the exterior Cauchy-Dirichlet problem for equivariant wave maps from 3 + 1 dimensional Minkowski spacetime into the three sphere. Using mixed analytical and numerical methods we show that, for a given topological degree of the map, all solutions starting from smooth finite energy initial data converge to the unique static solution (harmonic map). The asymptotics of this relaxation process is described in detail. We hope that our model will provide an attractive mathematical setting for gaining insight into dissipation-by-dispersion phenomena, in particular the soliton resolution conjecture.

MSC:

35L53 Initial-boundary value problems for second-order hyperbolic systems
35L71 Second-order semilinear hyperbolic equations
35P20 Asymptotic distributions of eigenvalues in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
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