• Previous Article
    Uniqueness and nondegeneracy of ground states for nonlinear Schrödinger equations with attractive inverse-power potential
  • CPAA Home
  • This Issue
  • Next Article
    Scattering of the focusing energy-critical NLS with inverse square potential in the radial case
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

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.
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:
[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. 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.  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

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. 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.  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

[1]

Justin Holmer, Chang Liu. Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity II: Supercritical blow-up profiles. Communications on Pure & Applied Analysis, 2021, 20 (1) : 215-242. doi: 10.3934/cpaa.2020264

[2]

Youshan Tao, Michael Winkler. Critical mass for infinite-time blow-up in a haptotaxis system with nonlinear zero-order interaction. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 439-454. doi: 10.3934/dcds.2020216

[3]

Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020450

[4]

Zedong Yang, Guotao Wang, Ravi P. Agarwal, Haiyong Xu. Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020436

[5]

Kihoon Seong. Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5437-5473. doi: 10.3934/cpaa.2020247

[6]

José Luis López. A quantum approach to Keller-Segel dynamics via a dissipative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020376

[7]

Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, 2021, 20 (1) : 449-465. doi: 10.3934/cpaa.2020276

[8]

Oussama Landoulsi. Construction of a solitary wave solution of the nonlinear focusing schrödinger equation outside a strictly convex obstacle in the $ L^2 $-supercritical case. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 701-746. doi: 10.3934/dcds.2020298

[9]

Karoline Disser. Global existence and uniqueness for a volume-surface reaction-nonlinear-diffusion system. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 321-330. doi: 10.3934/dcdss.2020326

[10]

Ahmad Z. Fino, Wenhui Chen. A global existence result for two-dimensional semilinear strongly damped wave equation with mixed nonlinearity in an exterior domain. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5387-5411. doi: 10.3934/cpaa.2020243

[11]

Serge Dumont, Olivier Goubet, Youcef Mammeri. Decay of solutions to one dimensional nonlinear Schrödinger equations with white noise dispersion. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020456

[12]

Noriyoshi Fukaya. Uniqueness and nondegeneracy of ground states for nonlinear Schrödinger equations with attractive inverse-power potential. Communications on Pure & Applied Analysis, 2021, 20 (1) : 121-143. doi: 10.3934/cpaa.2020260

[13]

Xiyou Cheng, Zhitao Zhang. Structure of positive solutions to a class of Schrödinger systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020461

[14]

Eduard Feireisl, Elisabetta Rocca, Giulio Schimperna, Arghir Zarnescu. Weak sequential stability for a nonlinear model of nematic electrolytes. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 219-241. doi: 10.3934/dcdss.2020366

[15]

Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020345

[16]

Shiqi Ma. On recent progress of single-realization recoveries of random Schrödinger systems. Electronic Research Archive, , () : -. doi: 10.3934/era.2020121

[17]

Cheng He, Changzheng Qu. Global weak solutions for the two-component Novikov equation. Electronic Research Archive, 2020, 28 (4) : 1545-1562. doi: 10.3934/era.2020081

[18]

Reza Chaharpashlou, Abdon Atangana, Reza Saadati. On the fuzzy stability results for fractional stochastic Volterra integral equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020432

[19]

Denis Bonheure, Silvia Cingolani, Simone Secchi. Concentration phenomena for the Schrödinger-Poisson system in $ \mathbb{R}^2 $. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020447

[20]

Xin-Guang Yang, Lu Li, Xingjie Yan, Ling Ding. The structure and stability of pullback attractors for 3D Brinkman-Forchheimer equation with delay. Electronic Research Archive, 2020, 28 (4) : 1395-1418. doi: 10.3934/era.2020074

2019 Impact Factor: 1.105

Metrics

  • PDF downloads (36)
  • HTML views (79)
  • Cited by (0)

Other articles
by authors

[Back to Top]