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São Paulo Journal of Mathematical Sciences

, Volume 9, Issue 2, pp 146–161 | Cite as

A Fujita-type blowup result and low energy scattering for a nonlinear Schrödinger equation

  • Thierry Cazenave
  • Simão Correia
  • Flávio Dickstein
  • Fred B. Weissler
Article

Abstract

In this paper we consider the nonlinear Schrödinger equation \(i u_t +\Delta u +\kappa |u|^\alpha u=0\). We prove that if \(\alpha <\frac{2}{N}\) and \(\mathfrak {I}\kappa <0\), then every nontrivial \(H^1\)-solution blows up in finite or infinite time. In the case \(\alpha >\frac{2}{N}\) and \(\kappa \in \mathbb {C}\), we improve the existing low energy scattering results in dimensions \(N\ge 7\). More precisely, we prove that if \( \frac{8}{N + \sqrt{ N^2 +16N }} < \alpha \le \frac{4}{N} \), then small data give rise to global, scattering solutions in \(H^1\).

Keywords

Nonlinear Schrödinger equation Fujita critical exponent Low energy scattering 

Mathematics Subject Classification

Primary: 35Q55 Secondary: 35Q56 35B33 35B40 35B44 

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Copyright information

© Instituto de Matemática e Estatística da Universidade de São Paulo 2015

Authors and Affiliations

  • Thierry Cazenave
    • 1
  • Simão Correia
    • 2
  • Flávio Dickstein
    • 3
  • Fred B. Weissler
    • 4
  1. 1.CNRS, Laboratoire Jacques-Louis LonsUniversité Pierre et Marie CurieParis Cedex 05France
  2. 2.Centro de Matemática e Aplicações FundamentaisUniversidade de LisboaLisboaPortugal
  3. 3.Instituto de MatemáticaUniversidade Federal do Rio de JaneiroRio de JaneiroBrazil
  4. 4.CNRS UMR 7539 LAGAUniversité Paris 13 - Sorbonne Paris CitéVilletaneuseFrance

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