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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