×

Difficulties in obtaining finite time blowup for fourth-order semilinear Schrödinger equations in the variational method frame. (English) Zbl 1421.35345

Summary: This article concerns the Cauchy problem for fourth-order semilinear Schrödinger equations. By constructing a variational problem and some invariant manifolds, we prove the existence of a global solution. Then we analyze the difficulties in proving the finite time blowup of the solution for the corresponding problem in the frame of the variational method. Understanding the finite time blowup of solutions, without radial initial data, still remains an open problem.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35B44 Blow-up in context of PDEs
35G25 Initial value problems for nonlinear higher-order PDEs
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35A15 Variational methods applied to PDEs
PDFBibTeX XMLCite
Full Text: Link

References:

[1] G. Baruch, G. Fibich, E. Mandelbaum; Singular solutions of the biharmonic nonlinear Schr¨odinger equation, SIAM J. Appl. Math., 70 (2010), 3319-3341. · Zbl 1210.35224
[2] G. Baruch, G. Fibich, N. Gavish; Singular standing ring solutions of nonlinear partial differential equations, Physica D, 239 (2010), 1968-1983. · Zbl 1203.35259
[3] G. Baruch, G. Fibic; Singular solutions of the L2-supercritical biharmonic nonlinear Schr¨odinger equation, Nonlinearity, 24 (2011), 1843-1859.
[4] T. Boulenger, E. Lenzmann; Blowup for Biharmonic NLS, Ann. Sci. Ecole Norm. Sup., 50 (2017), 503-544. · Zbl 1375.35476
[5] T. Cazenave; Semilinear Schr¨odinger Equation. Courant Lecture Notes in Mathematics, Proc. Amer. Math. Soc., Providence, Rhode Island, 2003.
[6] Y. Cho, T. Ozawa, C. Wang; Finite time blowup for the fourth-order NLS, Bull. Korean Math. Soc., 53 (2016), 615-640. · Zbl 1342.35324
[7] T. A. Davydova, Y. A. Zaliznyak; Schr¨odinger ordinary solitons and chirped solitons: fourthorder dispersive effects and cubic-quintic nonlinearity, Physica D, 156 (2001), 260-282. · Zbl 0973.35171
[8] V. D. Dinh; On blowup solutions to the focusing intercritical nonlinear fourth-order Schr¨odinger equation, J. Dynam. Differential Equations, 30 (2018), 1-30.
[9] G. Fibich, B. Ilan, G. Papanicolaou; Self-focusing with fourth-order dispersion, SIAM J. · Zbl 1003.35112
[10] R. Z. Xu, C. Xu; Cross-constrained problems for nonlinear Schr¨odinger equation with harmonic potential, Electron. J. Differential Equations, 211 (2012), 1-12.
[11] J. Zhang; Sharp conditions of global existence for nonlinear Schr¨odinger and Klein-Gordon equations, Nonlinear Anal., 48 (2002), 191-207.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.