Kyoto Journal of Mathematics

Blowup and scattering problems for the nonlinear Schrödinger equations

Takafumi Akahori and Hayato Nawa

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We consider L2-supercritical and H1-subcritical focusing nonlinear[4] Schrödinger equations. We introduce a subset PW of H1(Rd) for d1, and investigate behavior of the solutions with initial data in this set. To this end, we divide PW into two disjoint components PW+ and PW. Then, it turns out that any solution starting from a datum in PW+ behaves asymptotically free, and solution starting from a datum in PW blows up or grows up, from which we find that the ground state has two unstable directions. Our result is an extension of the one by Duyckaerts, Holmer, and Roudenko to the general powers and dimensions, and our argument mostly follows the idea of Kenig and Merle.

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Kyoto J. Math., Volume 53, Number 3 (2013), 629-672.

First available in Project Euclid: 19 August 2013

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Zentralblatt MATH identifier

Primary: 35Q55: NLS-like equations (nonlinear Schrödinger) [See also 37K10] 35B35: Stability 35B40: Asymptotic behavior of solutions 35B44: Blow-up


Akahori, Takafumi; Nawa, Hayato. Blowup and scattering problems for the nonlinear Schrödinger equations. Kyoto J. Math. 53 (2013), no. 3, 629--672. doi:10.1215/21562261-2265914.

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