# American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2022111
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## Global existence and blow-up for the focusing inhomogeneous nonlinear Schrödinger equation with inverse-square potential

 Faculty of Mathematics, Kim Il Sung University, Pyongyang, Democratic People's Republic of Korea

*Corresponding author: Jinmyong Kim (jm.kim0211@ryongnamsan.edu.kp)

Received  September 2021 Revised  April 2022 Early access June 2022

In this paper, we study the Cauchy problem for the focusing inhomogeneous nonlinear Schrödinger equation with inverse-square potential
 $iu_{t} +\Delta u-c|x|^{-2}u+|x|^{-b} |u|^{\sigma } u=0,\; u(0)=u_{0} \in H_{c}^{1},\;(t, x)\in \mathbb R\times\mathbb R^{d},$
where
 $d\ge3$
,
 $0 , $ \frac{4-2b}{d}<\sigma<\frac{4-2b}{d-2} $and $ c>-c(d):=-\left(\frac{d-2}{2}\right)^{2} $. We first establish the criteria for global existence and blow-up of general (not necessarily radial or finite variance) solutions to the equation. Using these criteria, we study the global existence and blow-up of solutions to the equation with general data lying below, at, and above the ground state threshold. Our results extend the global existence and blow-up results of Campos-Guzmán (Z. Angew. Math. Phys., 2021) and Dinh-Keraani (SIAM J. Math. Anal., 2021). Citation: Jinmyong An, Roesong Jang, Jinmyong Kim. Global existence and blow-up for the focusing inhomogeneous nonlinear Schrödinger equation with inverse-square potential. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022111 ##### References:  [1] J. An and J. Kim, Local well-posedness for the inhomogeneous nonlinear Schrödinger equation in$H^{s}(\mathbb R^{n})$, Nonlinear Anal. Real World Appl., 59 (2021), 103268. doi: 10.1016/j.nonrwa.2020.103268. [2] J. An, J. Kim and K. Chae, Continuous dependence of the Cauchy problem for the inhomogeneous nonlinear Schrödinger equation in$H^{s}(\mathbb R^{n})$, Discrete Contin. Dyn. Syst. Ser. B, (2021). doi: 10.3934/dcdsb. 2021221. [3] A. H. Ardila and M. Cardoso, Blow-up solutions and strong instability of ground states for the inhomogeneous nonlinear Schrödinger equation, Commun. Pure Appl. Anal., 20 (2021), 101-119. doi: 10.3934/cpaa.2020259. [4] J. Belmonte-Beitia, V. M. Pérez-García, V. Vekslerchik and P. J. Torres, Lie symmetries and solitons in nonlinear systems with spatially inhomogeneous nonlinearities, Phys. Rev. Lett., 98 (2007), 064102. doi: 10.1103/PhysRevLett.98.064102. [5] N. Burq, F. Planchon, J. G. Stalker and A. S. Tahvildar-Zadeh, Strichartz estimates for the wave and Schrödinger equations with the inverse-square potential, J. Funct. Anal., 203 (2003), 519-549. doi: 10.1016/S0022-1236(03)00238-6. [6] L. Campos, Scattering of radial solutions to the inhomogeneous nonlinear Schrödinger equation, Nonlinear Anal., 202 (2021), 112118. doi: 10.1016/j.na.2020.112118. [7] L. Campos and C. M. Guzmán, On the inhomogeneous NLS with inverse-square potential, Z. Angew. Math. Phys., 72 (2021), Paper No. 143, 29 pp. doi: 10.1007/s00033-021-01560-4. [8] T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. doi: 10.1090/cln/010. [9] V. D. Dinh, Global exsitence and blowup for a class of focusing nonlinear Schrödinger equation with inverse-square potential, J. Math. Anal. Appl., 468 (2018), 270-303. doi: 10.1016/j.jmaa.2018.08.006. [10] V. D. Dinh and S. Keraani, Long time dynamics of non-radial solutions to inhomogeneous nonlinear Schrödinger equations, SIAM J. Math. Anal., 53 (2021), 4765-4811. doi: 10.1137/20M1383434. [11] R. T. Glassey, On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations, J. Math. Phys., 18 (1977), 1794-1797. doi: 10.1063/1.523491. [12] R. Jang, J. An and J. Kim, The Cauchy problem for the energy-critical inhomogeneous nonlinear Schrödinger equation with inverse–square potential, preprint, arXiv: 2107.09826. [13] H. Kalf, U. W. Schmincke, J. Walter and R. Wust, On the spectral theory of Schrödinger and Dirac operators with strongly singular potentials, in: Spectral Theory and Differential Equations, in: Lect. Notes in Math., vol. 448, Springer, Berlin, 1975, pp. 182–226. [14] Y. V. Kartashov, B. A. Malomed, V. A. Vysloukh, M. R. Belic and L. Torner, Rotating vortex clusters in media with inhomogeneous defocusing nonlinearity, Opt. Lett., 42 (2017), 446-449. doi: 10.1364/OL.42.000446. [15] R. Killip, C. Miao, M. Visan, J. Zhang and J. Zheng, The energy-critical NLS with inverse-square potential, Discrete Contin. Dyn. Syst., 37 (2017), 3831-3866. doi: 10.3934/dcds.2017162. [16] R. Killip, J. Murphy, M. Visan and J. Zheng, The focusing cubic NLS with inverse-square potential in three space dimensions, Differ. Integral Equ., 30 (2017), 161-206. [17] F. Linares and G. Ponce, Introduction to Nonlinear Dispersive Equations,$2^nd$edition, Universitext. Springer, New York, 2015. doi: 10.1007/978-1-4939-2181-2. [18] J. Lu, C. Miao and J. Murphy, Scattering in$H^{1}$for the intercritical NLS with an inverse-square potential, J. Differential Equations, 264 (2018), 3174-3211. doi: 10.1016/j.jde.2017.11.015. [19] T. Suzuki, Solvability of nonlinear Schrödinger equations with some critical singular potential via generalized Hardy-Rellich inequalities, Funkcial. Ekvac., 59 (2016), 1-34. doi: 10.1619/fesi.59.1. [20] B. Wang, Z. Huo, C. Hao and Z. Guo, Harmonic Analysis Method for Nonlinear Evolution Equations, I, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2011. doi: 10.1142/9789814360746. [21] M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Commun. Math. Phys., 87 (1983), 567-576. [22] K. Yang, Scattering of the focusing energy-critical NLS with inverse-square potential in the radial case, Comm. Pure Appl. Anal., 20 (2021), 77-99. doi: 10.3934/cpaa.2020258. show all references ##### References:  [1] J. An and J. Kim, Local well-posedness for the inhomogeneous nonlinear Schrödinger equation in$H^{s}(\mathbb R^{n})$, Nonlinear Anal. Real World Appl., 59 (2021), 103268. doi: 10.1016/j.nonrwa.2020.103268. [2] J. An, J. Kim and K. Chae, Continuous dependence of the Cauchy problem for the inhomogeneous nonlinear Schrödinger equation in$H^{s}(\mathbb R^{n})$, Discrete Contin. Dyn. Syst. Ser. B, (2021). doi: 10.3934/dcdsb. 2021221. [3] A. H. Ardila and M. Cardoso, Blow-up solutions and strong instability of ground states for the inhomogeneous nonlinear Schrödinger equation, Commun. Pure Appl. Anal., 20 (2021), 101-119. doi: 10.3934/cpaa.2020259. [4] J. Belmonte-Beitia, V. M. Pérez-García, V. Vekslerchik and P. J. Torres, Lie symmetries and solitons in nonlinear systems with spatially inhomogeneous nonlinearities, Phys. Rev. Lett., 98 (2007), 064102. doi: 10.1103/PhysRevLett.98.064102. [5] N. Burq, F. Planchon, J. G. Stalker and A. S. Tahvildar-Zadeh, Strichartz estimates for the wave and Schrödinger equations with the inverse-square potential, J. Funct. Anal., 203 (2003), 519-549. doi: 10.1016/S0022-1236(03)00238-6. [6] L. Campos, Scattering of radial solutions to the inhomogeneous nonlinear Schrödinger equation, Nonlinear Anal., 202 (2021), 112118. doi: 10.1016/j.na.2020.112118. [7] L. Campos and C. M. Guzmán, On the inhomogeneous NLS with inverse-square potential, Z. Angew. Math. Phys., 72 (2021), Paper No. 143, 29 pp. doi: 10.1007/s00033-021-01560-4. [8] T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. doi: 10.1090/cln/010. [9] V. D. Dinh, Global exsitence and blowup for a class of focusing nonlinear Schrödinger equation with inverse-square potential, J. Math. Anal. Appl., 468 (2018), 270-303. doi: 10.1016/j.jmaa.2018.08.006. [10] V. D. Dinh and S. Keraani, Long time dynamics of non-radial solutions to inhomogeneous nonlinear Schrödinger equations, SIAM J. Math. Anal., 53 (2021), 4765-4811. doi: 10.1137/20M1383434. [11] R. T. Glassey, On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations, J. Math. Phys., 18 (1977), 1794-1797. doi: 10.1063/1.523491. [12] R. Jang, J. An and J. Kim, The Cauchy problem for the energy-critical inhomogeneous nonlinear Schrödinger equation with inverse–square potential, preprint, arXiv: 2107.09826. [13] H. Kalf, U. W. Schmincke, J. Walter and R. Wust, On the spectral theory of Schrödinger and Dirac operators with strongly singular potentials, in: Spectral Theory and Differential Equations, in: Lect. Notes in Math., vol. 448, Springer, Berlin, 1975, pp. 182–226. [14] Y. V. Kartashov, B. A. Malomed, V. A. Vysloukh, M. R. Belic and L. Torner, Rotating vortex clusters in media with inhomogeneous defocusing nonlinearity, Opt. Lett., 42 (2017), 446-449. doi: 10.1364/OL.42.000446. [15] R. Killip, C. Miao, M. Visan, J. Zhang and J. Zheng, The energy-critical NLS with inverse-square potential, Discrete Contin. Dyn. Syst., 37 (2017), 3831-3866. doi: 10.3934/dcds.2017162. [16] R. Killip, J. Murphy, M. Visan and J. Zheng, The focusing cubic NLS with inverse-square potential in three space dimensions, Differ. Integral Equ., 30 (2017), 161-206. [17] F. Linares and G. Ponce, Introduction to Nonlinear Dispersive Equations,$2^nd$edition, Universitext. Springer, New York, 2015. doi: 10.1007/978-1-4939-2181-2. [18] J. Lu, C. Miao and J. Murphy, Scattering in$H^{1}$for the intercritical NLS with an inverse-square potential, J. Differential Equations, 264 (2018), 3174-3211. doi: 10.1016/j.jde.2017.11.015. [19] T. Suzuki, Solvability of nonlinear Schrödinger equations with some critical singular potential via generalized Hardy-Rellich inequalities, Funkcial. Ekvac., 59 (2016), 1-34. doi: 10.1619/fesi.59.1. [20] B. Wang, Z. Huo, C. Hao and Z. Guo, Harmonic Analysis Method for Nonlinear Evolution Equations, I, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2011. doi: 10.1142/9789814360746. [21] M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Commun. Math. Phys., 87 (1983), 567-576. [22] K. Yang, Scattering of the focusing energy-critical NLS with inverse-square potential in the radial case, Comm. Pure Appl. Anal., 20 (2021), 77-99. doi: 10.3934/cpaa.2020258.  [1] Alex H. Ardila, Mykael Cardoso. 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