Kyoto Journal of Mathematics

Blowup and scattering problems for the nonlinear Schrödinger equations

Takafumi Akahori and Hayato Nawa

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Abstract

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.

Article information

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

Dates
First available in Project Euclid: 19 August 2013

Permanent link to this document
https://projecteuclid.org/euclid.kjm/1376917628

Digital Object Identifier
doi:10.1215/21562261-2265914

Mathematical Reviews number (MathSciNet)
MR3102564

Zentralblatt MATH identifier
1295.35365

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

Citation

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. https://projecteuclid.org/euclid.kjm/1376917628


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References

  • [1] H. Bahouri and P. Gérard, High frequency approximation of solutions to critical nonlinear wave equations, Amer. J. Math. 121 (1999), 131–175.
  • [2] H. Berestycki and P. L. Lions, Nonlinear scalar field equations, I, Arch. Ration. Mech. Anal. 82 (1983), 313–345; II, 347–375.
  • [3] J. Bourgain, Scattering in the energy space and below for $3$D NLS, J. Anal. Math. 75 (1998), 267–297.
  • [4] T. Cazenave, Semilinear Schrödinger Equations, Courant Lect. Notes Math. 10, Courant Inst. Math. Sci., New York; Amer. Math. Soc., Providence, 2003.
  • [5] T. Duyckaerts, J. Holmer, and S. Roudenko, Scattering for the non-radial $3$D cubic nonlinear Schrödinger equation, Math. Res. Lett. 15 (2008), 1233–1250.
  • [6] D. Foschi, Inhomogeneous Strichartz estimates, J. Hyperbolic Differ. Equ. 2 (2005), 1–24.
  • [7] B. Gidas, W. M. Ni, and L. Nirenberg, “Symmetry of positive solutions of nonlinear elliptic equations in $\mathbf{R}^{n}$” in Mathematical Analysis and Applications, Part A, Adv. in Math. Suppl. Stud. 7a, Academic Press, New York, 1981, 369–402.
  • [8] R. T. Glassey, On the blowing up solution to the Cauchy problem for nonlinear Schrödinger equations, J. Math. Phys. 18 (1977), 1794–1797.
  • [9] J. Ginibre and G. Velo, On a class of Schrödinger equations. I: The Cauchy problem, general case, J. Funct. Anal. 32 (1979), 1–32; II: Scattering theory, general case, 33–71.
  • [10] J. Holmer and S. Roudenko, A sharp condition for scattering of the radial $3$D cubic nonlinear Schrödinger equation, Comm. Math. Phys. 282 (2008), 435–467.
  • [11] Holmer, J. and Roudenko, S., On blow-up solutions to the $3$D cubic nonlinear Schrödinger equation, Appl. Math. Res. Express AMRX 2007, no. 1, art. ID abm004.
  • [12] S. Ibrahim, N. Masmoudi, and K. Nakanishi, Scattering threshold for the focusing nonlinear Klein–Gordon equation, Anal. PDE 4 (2011), 405–460.
  • [13] T. Kato, On nonlinear Schrödinger equations, Ann. Inst. Henri Poincaré, Phys. Theor. 46 (1987), 113–129.
  • [14] Kato, T., “Nonlinear Schrödinger equations” in Schrödinger operators (Sonderborg, 1988), Lecture Notes in Phys. 345, Springer, Berlin, 1989, 218–263.
  • [15] Kato, T., On nonlinear Schrödinger equations, II: $H^{s}$-solutions and unconditional well-posedness, J. Anal. Math. 67 (1995), 281–306.
  • [16] C. E. Kenig and F. Merle, Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case, Invent. Math. 166 (2006), 645–675.
  • [17] S. Keraani, On the defect of compactness for the Strichartz estimates of the Schrödinger equations, J. Differential Equations 175 (2001), 353–392.
  • [18] M. K. Kwong, Uniqueness of positive solutions of $\triangle u-u+u^{p}=0$ in $\mathbf{R}^{N}$, Arch. Rational Mech. Anal. 105 (1989), 243–266.
  • [19] F. Merle, Limit of the solution of a nonlinear Schrödinger equation at blow-up time, J. Funct. Anal. 84 (1989), 201–214.
  • [20] H. Nawa, “Asymptotic profiles of blow-up solutions of the nonlinear Schrödinger equation” in Singularities in Fluids, Plasmas and Optics (Heraklion, 1992), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 404, Kluwer, Dordrecht, 1993, 221–253.
  • [21] Nawa, H., Asymptotic and limiting profiles of blowup solutions of the nonlinear Schrödinger equations with critical power, Comm. Pure Appl. Math. 52 (1999), 193–270.
  • [22] T. Ogawa and Y. Tsutsumi, Blow-up of $H^{1}$-solution for the nonlinear Schrödinger equation, J. Differential Equations 92 (1991), 317–330.
  • [23] D. H. Sattinger, On global solution of nonlinear hyperbolic equations, Arch. Rational Mech. Anal. 30 (1968), 148–172.
  • [24] W. A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys. 55 (1977), 149–162.
  • [25] C. Sulem and P.-L. Sulem, The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse, Appl. Math. Sci. 139, Springer, New York, 1999.
  • [26] M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys. 87 (1982/83), 567–576.