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Characterization of large energy solutions of the equivariant wave map problem. I. (English) Zbl 1315.35130

The first important result refers to the global existence and scattering for wave maps in the zero topological class, below a sharp energy threshold. In fact one proves that for smooth data in this class, there exist a unique global evolution and it scatters to zero. The result is sharp in the sense that twice the energy of the harmonic map given by stereographic projection is a true threshold. The proof is based on the concentration compactness/rigidity method developed by C. E. Kenig and F. Merle, in two papers: [Invent. Math. 166, No. 3, 645–675 (2006; Zbl 1115.35125)] and [Acta Math. 201, No. 2, 147–212 (2008; Zbl 1183.35202)]. Further, wave maps of topological degree one are studied. For this case, a result on classification of blow-up solutions with energies below three time the energy of the harmonic map is shown. One provides a list of all known results used in the proofs of theorems. Preliminaries are presented in Section 2, a brief outline of the concentration compactness /rigidity method is given in Section 3. Next sections are devoted to the proofs.
For Part II see [the authors, ibid. 137, No. 1, 209–250 (2015; Zbl 1315.35131)].

MSC:

35L71 Second-order semilinear hyperbolic equations
35L15 Initial value problems for second-order hyperbolic equations
35P25 Scattering theory for PDEs
47A40 Scattering theory of linear operators
35B44 Blow-up in context of PDEs
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