Changing blow-up time in nonlinear Schrödinger equations

Rémi Carles

Journées équations aux dérivées partielles (2003)

  • page 1-12
  • ISSN: 0752-0360

Abstract

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Solutions to nonlinear Schrödinger equations may blow up in finite time. We study the influence of the introduction of a potential on this phenomenon. For a linear potential (Stark effect), the blow-up time remains unchanged, but the location of the collapse is altered. The main part of our study concerns isotropic quadratic potentials. We show that the usual (confining) harmonic potential may anticipate the blow-up time, and always does when the power of the nonlinearity is L 2 -critical. On the other hand, introducing a “repulsive” harmonic potential prevents finite time blow-up, provided that this potential is sufficiently “strong”. For the L 2 -critical nonlinearity, this mechanism is explicit : according to the strength of the potential, blow-up is first delayed, then prevented.

How to cite

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Carles, Rémi. "Changing blow-up time in nonlinear Schrödinger equations." Journées équations aux dérivées partielles (2003): 1-12. <http://eudml.org/doc/93445>.

@article{Carles2003,
abstract = {Solutions to nonlinear Schrödinger equations may blow up in finite time. We study the influence of the introduction of a potential on this phenomenon. For a linear potential (Stark effect), the blow-up time remains unchanged, but the location of the collapse is altered. The main part of our study concerns isotropic quadratic potentials. We show that the usual (confining) harmonic potential may anticipate the blow-up time, and always does when the power of the nonlinearity is $L^2$-critical. On the other hand, introducing a “repulsive” harmonic potential prevents finite time blow-up, provided that this potential is sufficiently “strong”. For the $L^2$-critical nonlinearity, this mechanism is explicit : according to the strength of the potential, blow-up is first delayed, then prevented.},
author = {Carles, Rémi},
journal = {Journées équations aux dérivées partielles},
keywords = {finite time blow-up; Stark effect; harmonic potential; nonlinear Schrödinger equations; isotropic quadratic potentials; critical nonlinearity},
language = {eng},
pages = {1-12},
publisher = {Université de Nantes},
title = {Changing blow-up time in nonlinear Schrödinger equations},
url = {http://eudml.org/doc/93445},
year = {2003},
}

TY - JOUR
AU - Carles, Rémi
TI - Changing blow-up time in nonlinear Schrödinger equations
JO - Journées équations aux dérivées partielles
PY - 2003
PB - Université de Nantes
SP - 1
EP - 12
AB - Solutions to nonlinear Schrödinger equations may blow up in finite time. We study the influence of the introduction of a potential on this phenomenon. For a linear potential (Stark effect), the blow-up time remains unchanged, but the location of the collapse is altered. The main part of our study concerns isotropic quadratic potentials. We show that the usual (confining) harmonic potential may anticipate the blow-up time, and always does when the power of the nonlinearity is $L^2$-critical. On the other hand, introducing a “repulsive” harmonic potential prevents finite time blow-up, provided that this potential is sufficiently “strong”. For the $L^2$-critical nonlinearity, this mechanism is explicit : according to the strength of the potential, blow-up is first delayed, then prevented.
LA - eng
KW - finite time blow-up; Stark effect; harmonic potential; nonlinear Schrödinger equations; isotropic quadratic potentials; critical nonlinearity
UR - http://eudml.org/doc/93445
ER -

References

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