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arXiv · 1511.08600

Hyperboloidal evolution and global dynamics for the focusing cubic wave equation

Annegret Y. Burtscher·Roland Donninger
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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 which 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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Annegret Y. Burtscher, Roland Donninger. 2017-03-15. Hyperboloidal evolution and global dynamics for the focusing cubic wave equation. https://doi.org/10.1007/s00220-017-2887-9

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