\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

A unified approach for energy scattering for focusing nonlinear Schrödinger equations

Abstract / Introduction Full Text(HTML) Related Papers Cited by
  • We consider the Cauchy problem for focusing nonlinear Schrödinger equation

    $ i\partial_t u + \Delta u = - |u|^\alpha u, \quad (t, x) \in \mathbb R \times \mathbb R^N, $

    where $ N\geq 1 $, $ \alpha>\frac{4}{N} $ and $ \alpha <\frac{4}{N-2} $ if $ N\geq 3 $. We give a criterion for energy scattering for the equation that covers well-known scattering results below, at and above the mass and energy ground state threshold. The proof is based on a recent argument of Dodson-Murphy [Math. Res. Lett. 25(6):1805-1825, 2018] using the interaction Morawetz estimate.

    Mathematics Subject Classification: Primary: 35A01; Secondary: 35Q55.

    Citation:

    \begin{equation} \\ \end{equation}
  • 加载中
  • [1] T. Akahori and H. Nawa, Blowup and scattering problems for the nonlinear Schrödinger equations, Kyoto J. Math., 53 (2013), 629-672.  doi: 10.1215/21562261-2265914.
    [2] A. K. AroraB. Dodson and J. Murphy, Scattering below the ground state for the 2D radial nonlinear Schrödinger equation, Proc. Amer. Math. Soc., 148 (2020), 1653-1663.  doi: 10.1090/proc/14824.
    [3] T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. doi: 10.1090/cln/010.
    [4] T. Cazenave and F. B. Weissler, Rapidly decaying solutions of the nonlinear Schrödinger equation, Commun. Math. Phys., 147 (1992), 75-100.  doi: 10.1007/BF02099529.
    [5] B. Dodson and J. Murphy, A new proof of scattering below the ground state for the non-radial focusing NLS, Math. Res. Lett., 25 (2018), 1805-1825.  doi: 10.4310/MRL.2018.v25.n6.a5.
    [6] T. DuyckaertsJ. Holmer and S. Roudenko, Scattering for the non-radial 3D cubic nonlinear Schrödinger equation, Math. Res. Lett., 15 (2008), 1233-1250.  doi: 10.4310/MRL.2008.v15.n6.a13.
    [7] T. Duyckaerts and S. Roudenko, Threshold solutions for the focusing 3D cubic Schrödinger equation, Rev. Mat. Iberoam., 26 (2010), 1-56.  doi: 10.4171/RMI/592.
    [8] T. Duyckaerts and S. Roudenko, Going beyond the threshold: Scattering and blpw-up in the focusing NLS equation, Commun. Math. Phys., 334 (2015), 1573-1615.  doi: 10.1007/s00220-014-2202-y.
    [9] D. FangJ. Xie and T. Cazenave, Scattering for the focusing energy-subcritical nonlinear Schrödinger equation, Sci. China Math., 54 (2011), 2037-2062.  doi: 10.1007/s11425-011-4283-9.
    [10] Y. Gao and Z. Wang, Below and beyond the mass-energy threshold: Scattering for the Hartree equation with radial data in $d\geq 5$, Z. Angew. Math. Phys., 71 (2020), 23pp. doi: 10.1007/s00033-020-1274-0.
    [11] C. D. Guevara, Global behavior of finite energy solutions to the $d$-dimensional focusing nonlinear Schrödinger equation, Appl. Math. Res. Express. AMRX, 2014, 177–243. doi: 10.1002/cta.2381.
    [12] J. Holmer and S. Roudenko, A sharp condition for scattering of the radial 3D cubic nonlinear Schrödinger equation, Comm. Math. Phys., 282 (2008), 435-467.  doi: 10.1007/s00220-008-0529-y.
    [13] M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980.  doi: 10.1353/ajm.1998.0039.
    [14] 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.  doi: 10.1007/s00222-006-0011-4.
    [15] P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. I, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109-145.  doi: 10.1016/S0294-1449(16)30428-0.
    [16] S. Xia and C. Xu, On dynamics of the system of two coupled nonlinear Schrödinger in $ \mathbb R^3$, Math. Meth. Appl. Sci., 42 (2019), 7096-7112.  doi: 10.1002/mma.5814.
  • 加载中
SHARE

Article Metrics

HTML views(1838) PDF downloads(217) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return