1. Applying Proposition 2.2 again, there exists a T 1 = T (b 1 ) ? r 0 such that there exists a unique solution u 1 ? C([T 1 , ?) ; L 2 ) with ?u 1 ? X T 1 ,b 1 ? M . Without loss of generality, we assume T 1 ? T 0 . One sees from b 1 > b 0 that for any t ? T 1 ? 1, t b 0 ?? ?u 1 ? u p ? ?,2,?,t + t b 0 ?2? ?u 1 ? u p ? q,r,?,t ? T 1 b 0 ?b 1 ?u 1 ? X T 1 ,b ? M, which implies ?u 1 ? X T 1 ,b 0 ? M . Hence u, u 1 ? X T 1 ,b 0 ,M . By the uniqueness property of X T 1 ,b 0 ,M , it holds that u = u 1 on [T 1 , ?);Since ?u? X T 0 ,b < ? for any b ? (2?, b 0 ] is trivial, we will prove ?u? X T 0 ,b < ? for all b ? (b 0 , ? + ?
2. Nonexistence of asymptotically free solutions for a nonlinear Schr�dinger equation;Jacqueline E Barab;J. Math. Phys,1984
3. Nonlinear Schr�dinger equations with repulsive harmonic potential and applications;R�mi Carles;SIAM J. Math. Anal,2003
4. Nonlinear Schr�dinger equations with repulsive harmonic potential and applications;SIAM J. Math. Anal,2003
5. Hitchhiker's guide to the fractional Sobolev spaces;Eleonora Di Nezza;Bull. Sci. Math,2012