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On the profile of energy concentration at blow-up points for subconformal focusing nonlinear waves


Authors: Spyros Alexakis and Arick Shao
Journal: Trans. Amer. Math. Soc. 369 (2017), 5525-5542
MSC (2010): Primary 35L70
DOI: https://doi.org/10.1090/tran/6820
Published electronically: February 13, 2017
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Abstract: We consider singularities of the focusing subconformal nonlinear wave equation and some generalizations of it. At noncharacteristic points on the singularity surface, Merle and Zaag have identified the rate of blow-up of the $ H^1$-norm of the solution inside cones that terminate at the singularity. We derive bounds that restrict how this $ H^1$-energy can be distributed inside such cones. Our proof relies on new localized estimates--obtained using Carleman-type inequalities--for such nonlinear waves. These bound the $ L^{p+1}$-norm in the interior of timelike cones by their $ H^1$-norm near the boundary of the cones. Such estimates can also be applied to obtain certain integrated decay estimates for globally regular solutions to such equations in the interior of time cones.


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Additional Information

Spyros Alexakis
Affiliation: Department of Mathematics, University of Toronto, 40 St George Street, Room 6290, Toronto, Ontario M5S 2E4, Canada
Email: alexakis@math.utoronto.ca

Arick Shao
Affiliation: Department of Mathematics, South Kensington Campus, Imperial College, London SW7 2AZ, United Kingdom
Email: c.shao@imperial.ac.uk

DOI: https://doi.org/10.1090/tran/6820
Received by editor(s): April 24, 2015
Received by editor(s) in revised form: September 1, 2015
Published electronically: February 13, 2017
Article copyright: © Copyright 2017 American Mathematical Society