Abstract
This chapter is devoted to rigorous results concerning solutions of the elliptic NLS equation with an attracting nonlinearity
, at critical (σd = 2) and supercritical (σd > 2) dimensions that become infinite in a finite time (see Cazenave 1994 for a review). This is the phenomenon of “blowup” or “wave collapse”, which has important physical consequences in that it corresponds to a violent energy transfer from large to small scales where dissipative processes can act efficiently. In the case of rapidly decaying initial solutions, the blow up results are direct consequences of the variance identity discussed in Section 2.4.1. In some instances like that of isotropic solutions, the proof was extended to solutions with infinite variance. Other estimates concern the rate of blowup and, at critical dimension, the phenomenon of L2-norm (or mass) concentration near the singularity.
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© 1999 Springer-Verlag New York, Inc.
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(1999). Blowup Solutions. In: Sulem, C., Sulem, PL. (eds) The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse. Applied Mathematical Sciences, vol 139. Springer, New York, NY. https://doi.org/10.1007/978-0-387-22768-9_5
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DOI: https://doi.org/10.1007/978-0-387-22768-9_5
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-98611-1
Online ISBN: 978-0-387-22768-9
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