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


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 \(H^1\).


Nonlinear Schrödinger equation Fujita critical exponent Low energy scattering 

Mathematics Subject Classification

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


  1. 1.
    Bergh, J., Löfström, J.: Interpolation spaces. An introduction. Grundlehren der Mathematischen Wissenschaften, vol. 223. Springer, New York. doi: 10.1007/978-3-642-66451-9 (1976)
  2. 2.
    Cazenave, T.: Semilinear Schrödinger equations, Courant Lecture Notes in Mathematics, vol. 10. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI (2003)Google Scholar
  3. 3.
    Cazenave, T., Weissler, F.B.: The Cauchy problem for the critical nonlinear Schrödinger equation in \(H^{s}\). Nonlinear Anal. 14(10), 807–836 (1990). doi: 10.1016/0362-546X(90)90023-A MATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Cazenave, T., Weissler, F.B.: Rapidly decaying solutions of the nonlinear Schrödinger equation. Comm. Math. Phys. 147, 75–100. (1992)
  5. 5.
    Deng, K., Levine, H.A.: The role of critical exponents in blow-up theorems: the sequel. J. Math. Anal. Appl. 243(1), 85–126 (2000). doi: 10.1006/jmaa.1999.6663 MATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Foschi, D.: Inhomogeneous Strichartz estimates. J. Hyperbolic Differ. Equ. 2(1), 1–24 (2005). doi: 10.1142/S0219891605000361 MATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Fujita, H.: On the blowing-up of solutions of the Cauchy problem for \(u_{t}=\triangle u+u^{\alpha +1}\). J. Fac. Sci. Univ. Tokyo Sect. IA Math. 13, 109–124. (1966)
  8. 8.
    Ginibre, J., Ozawa, T., Velo, G.: On the existence of the wave operators for a class of nonlinear Schrödinger equations. Ann. Inst. H. Poincaré Phys. Théor. 60(2), 211–239. (1994)
  9. 9.
    Ginibre, J., Velo, G.: Scattering theory in the energy space for a class of nonlinear Schrödinger equations. J. Math. Pures Appl. 64(4), 363–401 (1985)Google Scholar
  10. 10.
    Kato, T.: On nonlinear Schrödinger equations. Ann. Inst. H. Poincaré Phys. Théor. 46(1), 113–129. (1987)
  11. 11.
    Kato, T.: An \(L^{q,r}\)-theory for nonlinear Schrödinger equations. In Spectral and Scattering Theory and Applications, Advanced studies in Pure Mathematics, vol. 23, pp. 223–238. Mathematical Society of Japan, Tokyo (1994)Google Scholar
  12. 12.
    Levine, H.A.: The role of critical exponents in blow-up theorems. SIAM Rev. 32(2), 262–288 (1990). doi: 10.1137/1032046 MATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Nakanishi, K., Ozawa, T.: Remarks on scattering for nonlinear Schrödinger equations. NoDEA Nonlinear Differ. Equ. Appl. 9(1), 45–68 (2002). doi: 10.1007/s00030-002-8118-9 MATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Okazawa, N., Yokota, T.: Monotonicity method for the complex Ginzburg–Landau equation, including smoothing effect. In: Proceedings of the Third World Congress of Nonlinear Analysts, Part 1 (Catania, 2000). Nonlinear Anal. 47(1), 79–88. doi: 10.1016/S0362-546X(01)00158-4 (2001)
  15. 15.
    Strauss, W.A.: Nonlinear scattering theory. In: Scattering Theory in Mathematical Physics, NATO Advanced Study Institutes Series, vol. 9, pp. 53–78. Springer Netherlandsl. doi: 10.1007/978-94-010-2147-0_3 (1974)
  16. 16.
    Vilela, M.-C.: Inhomogeneous Strichartz estimates for the Schrödinger equation. Trans. Am. Math. Soc. 359(5), 2123–2136 (2007). doi: 10.1090/S0002-9947-06-04099-2 MATHMathSciNetCrossRefGoogle Scholar

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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