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Blow-up solutions and strong instability of ground states for the inhomogeneous nonlinear Schrödinger equation

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  • Using variational methods we study the stability and strong instability of ground states for the focusing inhomogeneous nonlinear Schrödinger equation (INLS)

    $ \begin{equation*} i\partial_{t}u+\Delta u+|x|^{-b}|u|^{p-1}u = 0. \end{equation*} $

    We construct two kinds of invariant sets under the evolution flow of (INLS). Then we show that the solution of (INLS) is global and bounded in $ H^{1}(\mathbb{R^{N}}) $ in the first kind of the invariant sets, while the solution blow-up in finite time in the other invariant set. Consequently, we prove that if the nonlinearity is $ L^{2} $-supercritical, then the ground states are strongly unstable by blow-up.

    Mathematics Subject Classification: Primary: 35Q55; Secondary: 35B44.

    Citation:

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  • [1] G. P. AgrawalNonlinear Fiber Optics, Academic Press, 2007. 
    [2] A. H. Ardila and V. D. Dinh, Some qualitative studies of the focusing inhomogeneous Gross-Pitaevskii equation, Z. Angew. Math. Phys., 71 (2020), 24pp. doi: 10.1007/s00033-020-01301-z.
    [3] T. Cazenave, Semilinear Schrödinger Equations, American Mathematical Society, Courant Institute of Mathematical Sciences, 2003. doi: 10.1090/cln/010.
    [4] J. Chen, On a class of nonlinear inhomogeneous Schrödinger equations, J. Appl. Math. Comput., 32 (2010), 237-253.  doi: 10.1007/s12190-009-0246-5.
    [5] J. Chen and B. Guo, Sharp global existence and blowing up results for inhomogeneous Schrödinger equations, Discrete Contin. Dyn. Syst. Ser. B, 8 (2007), 357-367.  doi: 10.3934/dcdsb.2007.8.357.
    [6] V. Combet and F. Genoud, Classification of minimal mass blow-up solutions for an ${L}^{2}$ critical inhomogeneous NLS, J. Evol. Equ., 16 (2016), 483-500.  doi: 10.1007/s00028-015-0309-z.
    [7] D. DuY. Wu and K. Zhang, On blow-up criterion for the nonlinear Schrödinger equation, Discrete Contin.Dynl. Sys, 36 (2016), 3639-3650.  doi: 10.3934/dcds.2016.36.3639.
    [8] A. de Bouard and R. Fukuizumi, Stability of standing waves for nonlinear Schrödinger equations with inhomogeneous nonlinearities, Ann. Henri Poincaré, 6 (2015), 1157-1177.  doi: 10.1007/s00023-005-0236-6.
    [9] L. G. Farah, Global well-posedness and blow-up on the energy space for the inhomogeneous nonlinear Schrödinger equation, J. Evol. Equ., 16 (2016), 193-208.  doi: 10.1007/s00028-015-0298-y.
    [10] G. Fibich and X. P. Wang, Equations with inhomogeneous nonlinearities, Physica D, 175 (2003), 96-108.  doi: 10.1016/S0167-2789(02)00626-7.
    [11] R. Fukuizumi, Equations with critical power nonlinearity and potentials., Adv. Differ. Equ., 10 (2005), 259-276. 
    [12] F. Genoud, Bifurcation and stability of travelling waves in self-focusing planar waveguides, Adv. Nonlinear Stu., 10 (2010), 357-400.  doi: 10.1515/ans-2010-0207.
    [13] F. Genoud, An inhomogeneous, ${L}^2$-critical, nonlinear Schrödinger equation, Z. Anal. Anwend., 31 (2012), 283-290.  doi: 10.4171/ZAA/1460.
    [14] F. Genoud and C. Stuart, Schrödinger equations with a spatially decaying nonlinearity: existence and stability of standing waves, Discrete Contin. Dyn. Syst., 21 (2008), 137-186.  doi: 10.3934/dcds.2008.21.137.
    [15] T. Saanouni, Remarks on the inhomogeneous fractional nonlinear Schrödinger equation, J. Math Phys., 57 (2016) 081503. doi: 10.1063/1.4960045.
    [16] J. Toland, Uniqueness of positive solutions of some semilinear Sturm-Liouville problemson the half line, Proc. Roy. Soc. Edinburgh Sect. A, 97 (1984), 259-263.  doi: 10.1017/S0308210500032042.
    [17] E. Yanagida, Uniqueness of positive radial solutions of $\delta u+g(r)u+h(r)u^{p}=0$ in $\mathbb{R}^{N}$, Arch. Rat. Mech. Anal, 115 (1991), 257-274.  doi: 10.1007/BF00380770.
    [18] S. Zhu, Blow-up solutions for the inhomogeneous Schrödinger equation with ${L}^{2}$ supercritical nonlinearity, J. Math. Anal. Appl., 409 (2014), 760-776.  doi: 10.1016/j.jmaa.2013.07.029.
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