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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Stable blow up dynamics for energy supercritical wave equations


Authors: Roland Donninger and Birgit Schörkhuber
Journal: Trans. Amer. Math. Soc. 366 (2014), 2167-2189
MSC (2010): Primary 35L05, 35B44, 35C06
Published electronically: November 14, 2013
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Abstract: We study the semilinear wave equation

$\displaystyle \partial _t^2 \psi -\Delta \psi =\vert\psi \vert^{p-1}\psi $

for $ p > 3$ with radial data in three spatial dimensions. There exists an explicit solution which blows up at $ t=T>0$ given by

$\displaystyle \psi ^T(t,x)=c_p (T-t)^{-\frac {2}{p-1}}, $

where $ c_p$ is a suitable constant. We prove that the blow up described by $ \psi ^T$ is stable in the sense that there exists an open set (in a topology strictly stronger than the energy) of radial initial data that leads to a solution which converges to $ \psi ^T$ as $ t\to T-$ in the backward lightcone of the blow up point $ (t,r)=(T,0)$.

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Additional Information

Roland Donninger
Affiliation: Department of Mathematics, École Polytechnique Fédérale de Lausanne, Station 8, CH-1015 Lausanne, Switzerland
Email: roland.donninger@epfl.ch

Birgit Schörkhuber
Affiliation: Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstraße 8-10, A-1040 Vienna, Austria
Email: birgit.schoerkhuber@tuwien.ac.at

DOI: http://dx.doi.org/10.1090/S0002-9947-2013-06038-2
PII: S 0002-9947(2013)06038-2
Received by editor(s): August 20, 2012
Published electronically: November 14, 2013
Additional Notes: The second author acknowledges partial support from the Austrian Science Fund (FWF), grants P22108, P23598, P24304, and I395; the Austrian-French Project of the Austrian Exchange Service (ÖAD); and the Innovative Ideas Program of Vienna University of Technology.
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.