Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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FINITE TIME BLOWUP FOR THE FOURTH-ORDER NLS

  • Cho, Yonggeun (Department of Mathematics and Institute of Pure and Applied Mathematics, Chonbuk National University) ;
  • Ozawa, Tohru (Department of Applied Physics, Waseda University) ;
  • Wang, Chengbo (Department of Mathematics, Zhejiang University)
  • Received : 2015.04.11
  • Published : 2016.03.31
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Abstract

We consider the fourth-order $Schr{\ddot{o}}dinger$ equation with focusing inhomogeneous nonlinearity ($-{\mid}x{\mid}^{-2}{\mid}u{\mid}^{\frac{4}{n}}u$) in high space dimensions. Extending Glassey's virial argument, we show the finite time blowup of solutions when the energy is negative.

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References

  1. M. Ben-Artzi, H. Koch, and J.-C. Saut, Dispersion estimates for fourth order Schrodinger equations, C. R. Acad. Sci. Paris Ser. I Math. 330 (2000), no. 2, 87-92. https://doi.org/10.1016/S0764-4442(00)00120-8
  2. G. Baruch and G. Fibich, Singular solutions of the $L^2$-supercritical biharmonic noninear Schrodinger equation, Nonlinearity 24 (2011), no. 6, 1843-1859. https://doi.org/10.1088/0951-7715/24/6/009
  3. G. Baruch, G. Fibich, and E. Mandelbaum, Singular solutions of the biharmonic nonlinear Schrodinger equation, SIAM J. Appl. Math. 70 (2010), no. 8, 3319-3341. https://doi.org/10.1137/100784199
  4. L. Berge, Soliton stability versus collapse, Phys. Rev. E (3) 62 (2000), no. 3, 3071-3074. https://doi.org/10.1103/PhysRevB.62.3071
  5. J. L. Bona, G. Ponce, J.-C. Saut, and C. Sparber, Dispersive blow-up for nonlinear Schrodinger equations revisited, J. Math. Pures Appl. 102 (2014), no. 4, 782-811. https://doi.org/10.1016/j.matpur.2014.02.006
  6. R. Carles and E. Moulay, Higher order Schrodinger equations, J. Phys. A 45 (2012), no. 39, 395304, 11 pp. https://doi.org/10.1088/1751-8113/45/39/395304
  7. R. Carles and E. Moulay, Higher oerder Schrodinger and Hartee-Fock equations, in preprint (arXiv: 1305.4880). https://doi.org/10.1063/1.4936646
  8. T. Cazenave, Semilinear Schrodinger equations, Courant Lecture Notes in Mathematics, 10. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003.
  9. M. Chae, S. Hong, and S. Lee, Mass concentration for the $L^2$-critical nonlinear Schrodinger equations of higher orders, Discrete Contin. Dyn. Syst. 29 (2011), no. 3, 909-928. https://doi.org/10.3934/dcds.2011.29.909
  10. Y. Cho, G. Hwang, S. Kwon, and S. Lee, On the finite time blowup for mass-critical Hartree equations, to appear in Proc. Roy. Soc. Edinburgh Sect. A (arXiv:1208.2302).
  11. Y. Cho, T. Ozawa, and S. Xia, Remarks on some dispersive estimates, Commun. Pure Appl. Anal., 10 (2011), no. 4, 1121-1128. https://doi.org/10.3934/cpaa.2011.10.1121
  12. G. Fibich, B. Ilan, and G. Papanicolaou, Self-focusing with fourth-order dispersion, SIAM J. Appl. Math. 62 (2002), no. 4, 1437-1462. https://doi.org/10.1137/S0036139901387241
  13. J. Frohlich and E. Lenzmann, Blow-up for nonlinear wave equations describing Boson stars, Comm. Pure Appl. Math. 60 (2007), no. 11, 1691-1705. https://doi.org/10.1002/cpa.20186
  14. R. T. Glassey, On the blowing up of solutions to the Cauchy problem for nonlinear Schrodinger equations, J. Math. Phys. 18 (1977), no. 9, 1794-1797. https://doi.org/10.1063/1.523491
  15. V. I. Karpman, Stabilization of soliton instability by high-order dispersion: fourth order nonlinear Schrodinger-type equations, Phys. Rev. E 53 (1996), 1336-1339. https://doi.org/10.1103/PhysRevB.53.1336
  16. V. I. Karpman and S. G. Shagalov, Stability of soliton described by nonlinear Schrodinger type equations with higher-order dispersion, Phys. D 144 (2000), no. 1-2, 194-210. https://doi.org/10.1016/S0167-2789(00)00078-6
  17. C. Miao, G. Xu, and L. Zhao, Global well-posedness and scattering for the focusing energy-critical nonlinear Schrdinger equations of fourth order in the radial case, J. Differential Equations 246 (2009), no. 9, 3715-3749. https://doi.org/10.1016/j.jde.2008.11.011
  18. T. Ogawa and Y. Tsutsumi, Blow-up of $H^1$ solution for the nonlinear Schrodinger equation, J. Differential Equations 92 (1991), no. 2, 317-330. https://doi.org/10.1016/0022-0396(91)90052-B
  19. B. Pausader, Global well-posedness for energy critical fourth-order Schrodinger equations in the radial case, Dyn. Partial Differ. Equ. 4 (2007), no. 3, 197-225. https://doi.org/10.4310/DPDE.2007.v4.n3.a1
  20. B. Pausader, The cubic fourth-order Schrodinger equation, J. Funct. Anal. 256 (2009), no. 8, 2473-2517. https://doi.org/10.1016/j.jfa.2008.11.009
  21. B. Pausader, The focusing energy-critical fourth-order Schrodinger equation with radial data, Discrete Contin. Dyn. Syst. 24 (2009), no. 4, 1275-1292. https://doi.org/10.3934/dcds.2009.24.1275
  22. J. Segata, Modified wave operators for the fourth-order nonlinear Schrodinger-type equation with cubic nonlinearity, Math. Methods Appl. Sci. 26 (2006), no. 15, 1785-1800.
  23. S. H. Zhu, J. Zhang, and H. Yang, Limiting profile of the blow-up solutions for the fourth-order nonlinear Schrodinger equation, Dyn. Partial Differ. Equ. 7 (2010), no. 2, 187-205. https://doi.org/10.4310/DPDE.2010.v7.n2.a4
  24. S. H. Zhu, J. Zhang, and H. Yang, Blow-up of rough solutions to the fourth-order nonlinear Schrodinger equation, Nonlinear Anal. 74 (2011), no. 17, 6186-6201. https://doi.org/10.1016/j.na.2011.05.096

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