Stable blow up dynamics for energy supercritical wave equations
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- by Roland Donninger and Birgit Schörkhuber PDF
- Trans. Amer. Math. Soc. 366 (2014), 2167-2189 Request permission
Abstract:
We study the semilinear wave equation \[ \partial _t^2 \psi -\Delta \psi =|\psi |^{p-1}\psi \] for $p > 3$ with radial data in three spatial dimensions. There exists an explicit solution which blows up at $t=T>0$ given by \[ \psi ^T(t,x)=c_p (T-t)^{-\frac {2}{p-1}}, \] 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 $(t,r)=(T,0)$.References
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Additional Information
- Roland Donninger
- Affiliation: Department of Mathematics, École Polytechnique Fédérale de Lausanne, Station 8, CH-1015 Lausanne, Switzerland
- MR Author ID: 839104
- 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
- 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.
- © Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 366 (2014), 2167-2189
- MSC (2010): Primary 35L05, 35B44, 35C06
- DOI: https://doi.org/10.1090/S0002-9947-2013-06038-2
- MathSciNet review: 3152726