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Scattering for wave maps exterior to a ball. (English) Zbl 1264.35045

This paper deals with equivariant wave maps \(U : \mathbb R\times(\mathbb R^3\setminus B)\rightarrow S^3\) with the Dirichlet condition \(U(\partial B) = {N}\), where \(N\) is a fixed point on \(S^3\) and \(B\subset \mathbb{R}^3\) is the unit ball centered at 0 in \(\mathbb{R}^3\). The main result states that 1-equivariant maps of degree zero scatter to zero irrespective of their energy. For positive degrees, asymptotic stability of the unique harmonic map in the energy class determined by the degree is proved.

MSC:

35B40 Asymptotic behavior of solutions to PDEs
35P25 Scattering theory for PDEs
53C43 Differential geometric aspects of harmonic maps
58E20 Harmonic maps, etc.
35L71 Second-order semilinear hyperbolic equations
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