On the focusing critical semi-linear wave equation

• Autores: J. Krieger, Wilhelm Schlag
• Localización: American journal of mathematics, ISSN 0002-9327, Vol. 129, Nº 3, 2007, págs. 843-912
• Idioma: inglés
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• Resumen

  • The wave equation ?tt? - ?? - ?5 = 0 in R3 is known to exhibit finite time blowup for data of negative energy. Furthermore, it admits the special static solutions F(x, a) = (3a) ¿ (1 + a|x|2)-¿ for all a > 0 which are linearly unstable. We view these functions as a curve in the energy space ?1 ×L2. We prove the existence of a family of perturbations of this curve that lead to global solutions possessing a well-defined long time asymptotic behavior as the sum of a bulk term plus a scattering term. Moreover, this family forms a co-dimension one manifold of small diameter in a suitable topology. Loosely speaking, acts as a center-stable manifold with the curve F(·, a) as an attractor in .