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The defocusing energy-supercritical cubic nonlinear wave equation in dimension five


Author: Aynur Bulut
Journal: Trans. Amer. Math. Soc. 367 (2015), 6017-6061
MSC (2010): Primary 35L71, 35B44, 35P25
DOI: https://doi.org/10.1090/tran/6068
Published electronically: April 20, 2015
MathSciNet review: 3356928
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Abstract: We consider the energy-supercritical nonlinear wave equation $ u_{tt}-\Delta u+\vert u\vert^2u=0$ with defocusing cubic nonlinearity in dimension $ d=5$ with no radial assumption on the initial data. We prove that a uniform-in-time
a priori bound on the critical norm implies that solutions exist globally in time and scatter at infinity in both time directions. Together with our earlier works in dimensions $ d\geq 6$ with general data and dimension $ d=5$ with radial data, the present work completes the study of global well-posedness and scattering in the energy-supercritical regime for the cubic nonlinearity under the assumption of uniform-in-time control over the critical norm.


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Additional Information

Aynur Bulut
Affiliation: School of Mathematics, Institute for Advanced Study, Princeton, New Jersey 08540
Email: abulut@math.ias.edu

DOI: https://doi.org/10.1090/tran/6068
Received by editor(s): October 14, 2012
Received by editor(s) in revised form: December 23, 2012
Published electronically: April 20, 2015
Article copyright: © Copyright 2015 American Mathematical Society