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

Bizoń, Piotr; Chmaj, Tadeusz; Maliborski, Maciej

Nonlinearity 25, No. 5, 1299-1309 (2012).

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

Keywords:

exterior Cauchy-Dirichlet problem; mixed analytical and numerical methods; unique static solution; dissipation-by-dispersion phenomena; soliton resolution conjecture