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January  2021, 20(1): 101-119. doi: 10.3934/cpaa.2020259

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

Alex H. Ardila 1,, and  Mykael Cardoso 2,

1. 

Department of Mathematics, ICEx-UFMG, Universidade Federal de Minas Gerais-ICEx, Caixa Postal 702, CEP 30123-970, Belo Horizonte-MG, Brazil
2. 

Department of Mathematics, CCN, Universidade Federal do Piauí, Ininga - CEP: 64049-550, Teresina - PI, Brasil

* Corresponding author

Received  April 2020 Revised  August 2020 Published  October 2020

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.
Keywords: Nonlinear Schrödinger equation, global existence, blow-up, stability.
Mathematics Subject Classification: Primary: 35Q55; Secondary: 35B44.
Citation: Alex H. Ardila, Mykael Cardoso. Blow-up solutions and strong instability of ground states for the inhomogeneous nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2021, 20 (1) : 101-119. doi: 10.3934/cpaa.2020259
References:
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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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D. Du, Y. 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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[13]

F. Genoud, An inhomogeneous, ${L}^2$-critical, nonlinear Schrödinger equation, Z. Anal. Anwend., 31 (2012), 283-290.  doi: 10.4171/ZAA/1460.  Google Scholar

[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.  Google Scholar

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T. Saanouni, Remarks on the inhomogeneous fractional nonlinear Schrödinger equation, J. Math Phys., 57 (2016) 081503. doi: 10.1063/1.4960045.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

show all references

References:
[1] G. P. Agrawal, Nonlinear Fiber Optics, Academic Press, 2007.   Google Scholar
[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.  Google Scholar

[3]

T. Cazenave, Semilinear Schrödinger Equations, American Mathematical Society, Courant Institute of Mathematical Sciences, 2003. doi: 10.1090/cln/010.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

D. Du, Y. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

R. Fukuizumi, Equations with critical power nonlinearity and potentials., Adv. Differ. Equ., 10 (2005), 259-276.   Google Scholar

[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.  Google Scholar

[13]

F. Genoud, An inhomogeneous, ${L}^2$-critical, nonlinear Schrödinger equation, Z. Anal. Anwend., 31 (2012), 283-290.  doi: 10.4171/ZAA/1460.  Google Scholar

[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.  Google Scholar

[15]

T. Saanouni, Remarks on the inhomogeneous fractional nonlinear Schrödinger equation, J. Math Phys., 57 (2016) 081503. doi: 10.1063/1.4960045.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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