# American Institute of Mathematical Sciences

• 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  January 2021 Early access  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. 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

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
 [1] Jianbo Cui, Jialin Hong, Liying Sun. On global existence and blow-up for damped stochastic nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6837-6854. doi: 10.3934/dcdsb.2019169 [2] Zaihui Gan, Jian Zhang. Blow-up, global existence and standing waves for the magnetic nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems, 2012, 32 (3) : 827-846. doi: 10.3934/dcds.2012.32.827 [3] Dapeng Du, Yifei Wu, Kaijun Zhang. On blow-up criterion for the nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems, 2016, 36 (7) : 3639-3650. doi: 10.3934/dcds.2016.36.3639 [4] 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 [5] Jian Zhang, Shihui Zhu, Xiaoguang Li. Rate of $L^2$-concentration of the blow-up solution for critical nonlinear Schrödinger equation with potential. Mathematical Control & Related Fields, 2011, 1 (1) : 119-127. doi: 10.3934/mcrf.2011.1.119 [6] Van Duong Dinh. On blow-up solutions to the focusing mass-critical nonlinear fractional Schrödinger equation. Communications on Pure & Applied Analysis, 2019, 18 (2) : 689-708. doi: 10.3934/cpaa.2019034 [7] Binhua Feng. On the blow-up solutions for the fractional nonlinear Schrödinger equation with combined power-type nonlinearities. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1785-1804. doi: 10.3934/cpaa.2018085 [8] Türker Özsarı. Blow-up of solutions of nonlinear Schrödinger equations with oscillating nonlinearities. Communications on Pure & Applied Analysis, 2019, 18 (1) : 539-558. doi: 10.3934/cpaa.2019027 [9] Van Duong Dinh. Blow-up criteria for linearly damped nonlinear Schrödinger equations. Evolution Equations & Control Theory, 2021, 10 (3) : 599-617. doi: 10.3934/eect.2020082 [10] Long Wei, Zhijun Qiao, Yang Wang, Shouming Zhou. Conserved quantities, global existence and blow-up for a generalized CH equation. Discrete & Continuous Dynamical Systems, 2017, 37 (3) : 1733-1748. doi: 10.3934/dcds.2017072 [11] Cristophe Besse, Rémi Carles, Norbert J. Mauser, Hans Peter Stimming. Monotonicity properties of the blow-up time for nonlinear Schrödinger equations: Numerical evidence. Discrete & Continuous Dynamical Systems - B, 2008, 9 (1) : 11-36. doi: 10.3934/dcdsb.2008.9.11 [12] Hristo Genev, George Venkov. Soliton and blow-up solutions to the time-dependent Schrödinger-Hartree equation. Discrete & Continuous Dynamical Systems - S, 2012, 5 (5) : 903-923. doi: 10.3934/dcdss.2012.5.903 [13] Mingyou Zhang, Qingsong Zhao, Yu Liu, Wenke Li. Finite time blow-up and global existence of solutions for semilinear parabolic equations with nonlinear dynamical boundary condition. Electronic Research Archive, 2020, 28 (1) : 369-381. doi: 10.3934/era.2020021 [14] Bin Li. On the blow-up criterion and global existence of a nonlinear PDE system in biological transport networks. Kinetic & Related Models, 2019, 12 (5) : 1131-1162. doi: 10.3934/krm.2019043 [15] Lili Du, Chunlai Mu, Zhaoyin Xiang. Global existence and blow-up to a reaction-diffusion system with nonlinear memory. Communications on Pure & Applied Analysis, 2005, 4 (4) : 721-733. doi: 10.3934/cpaa.2005.4.721 [16] Monica Marras, Stella Vernier Piro. On global existence and bounds for blow-up time in nonlinear parabolic problems with time dependent coefficients. Conference Publications, 2013, 2013 (special) : 535-544. doi: 10.3934/proc.2013.2013.535 [17] Ronghua Jiang, Jun Zhou. Blow-up and global existence of solutions to a parabolic equation associated with the fraction p-Laplacian. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1205-1226. doi: 10.3934/cpaa.2019058 [18] Shuyin Wu, Joachim Escher, Zhaoyang Yin. Global existence and blow-up phenomena for a weakly dissipative Degasperis-Procesi equation. Discrete & Continuous Dynamical Systems - B, 2009, 12 (3) : 633-645. doi: 10.3934/dcdsb.2009.12.633 [19] Xiumei Deng, Jun Zhou. Global existence and blow-up of solutions to a semilinear heat equation with singular potential and logarithmic nonlinearity. Communications on Pure & Applied Analysis, 2020, 19 (2) : 923-939. doi: 10.3934/cpaa.2020042 [20] Gongwei Liu. The existence, general decay and blow-up for a plate equation with nonlinear damping and a logarithmic source term. Electronic Research Archive, 2020, 28 (1) : 263-289. doi: 10.3934/era.2020016

2020 Impact Factor: 1.916