# American Institute of Mathematical Sciences

March  2003, 2(1): 33-50. doi: 10.3934/cpaa.2003.2.33

## Polynomial upper bounds for the instability of the nonlinear Schrödinger equation below the energy norm

 1 University of Toronto, Toronto, Ontario, M5S 2E4, Canada 2 University of Minnesota, United States 3 Massachusetts Institute of Technology 4 Kobe University, Japan 5 University of California, Los Angeles, United States

Received  December 2002 Published  December 2002

We continue the study (initiated in [18]) of the orbital stability of the ground state cylinder for focussing non-linear Schrödinger equations in the $H^s(\R^n)$ norm for $1-\varepsilon < s < 1$, for small $\varepsilon$. In the $L^2$-subcritical case we obtain a polynomial bound for the time required to move away from the ground state cylinder. If one is only in the $H^1$-subcritical case then we cannot show this, but for defocussing equations we obtain global well-posedness and polynomial growth of $H^s$ norms for $s$ sufficiently close to 1.
Citation: J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao. Polynomial upper bounds for the instability of the nonlinear Schrödinger equation below the energy norm. Communications on Pure & Applied Analysis, 2003, 2 (1) : 33-50. doi: 10.3934/cpaa.2003.2.33
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