Communications in Mathematical Physics

, Volume 353, Issue 2, pp 549–596 | Cite as

Hyperboloidal Evolution and Global Dynamics for the Focusing Cubic Wave Equation

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Abstract

The focusing cubic wave equation in three spatial dimensions has the explicit solution \({\sqrt{2}/t}\). We study the stability of the blowup described by this solution as \({t \to 0}\) without symmetry restrictions on the data. Via the conformal invariance of the equation we obtain a companion result for the stability of slow decay in the framework of a hyperboloidal initial value formulation. More precisely, we identify a codimension-1 Lipschitz manifold of initial data leading to solutions that converge to Lorentz boosts of \({\sqrt{2}/t}\) as \({t\to\infty}\). These global solutions thus exhibit a slow nondispersive decay, in contrast to small data evolutions.

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© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Mathematical Institute and Hausdorff Center for MathematicsUniversity of BonnBonnGermany
  2. 2.Faculty of MathematicsUniversity of ViennaViennaAustria

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