Sharp condition for inhomogeneous nonlinear Schrödinger equations by cross-invariant manifolds. (English) Zbl 1492.35313
Summary: In this paper, we study a class of cross constrained variational problem for the inhomogeneous nonlinear Schrödinger equation in \({\mathbb{R}}^N\). By constructing cross-invariant manifolds, we derive a sharp condition for blow-up phenomenon and global well-posedness of solutions.
MSC:
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35B44 | Blow-up in context of PDEs |
Keywords:
inhomogeneous nonlinear Schrödinger equation; sharp condition; cross-invariant manifolds; global existence; finite time blowupReferences:
| [1] | Berestycki, H.; Cazenave, T., Instability of stationary states in nonlinear Schrödinger and Klein-Gordon equations, C. R. Acad. Sci. Paris, 293, 489-492 (1981) · Zbl 0492.35010 |
| [2] | Cardoso, M.; Farah, LG, Blow-up of radial solutions for the intercritical inhomogeneous NLS equation, J. Funct. Anal., 281 (2021) · Zbl 1473.35498 · doi:10.1016/j.jfa.2021.109134 |
| [3] | Cazenave, T.: Semilinear Schrödinger Equations. Courant Lecture Notes in Mathematics, vol.10, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI (2003) · Zbl 1055.35003 |
| [4] | Chen, JQ, On the inhomogeneous nonlinear Schrödinger equation with harmonic potential and unbounded coefficient, Czechoslovak Math. J., 60, 715-736 (2010) · Zbl 1224.35083 · doi:10.1007/s10587-010-0046-y |
| [5] | Chen, JQ; Guo, BL, Sharp global existence and blowing up results for inhomogeneous Schrödinger equations, Discret. Contin. Dyn. Syst. Ser. B, 8, 357-367 (2007) · Zbl 1151.35089 |
| [6] | Combet, V.; Genoud, F., Classification of minimal mass blow-up solutions for an \(L^2\) critical inhomogeneous NLS, J. Evol. Equ., 16, 483-500 (2016) · Zbl 1358.35165 · doi:10.1007/s00028-015-0309-z |
| [7] | Dinh, VD, Blowup of \(H^1\) solutions for a class of the focusing inhomogeneous nonlinear Schrödinger equation, Nonlinear Anal., 174, 169-188 (2018) · Zbl 1388.35177 · doi:10.1016/j.na.2018.04.024 |
| [8] | Dinh, VD; Keraani, S., Long time dynamics of nonradial solutions to inhomogeneous nonlinear Schrödinger equations, SIAM J. Math. Anal., 53, 4765-4811 (2021) · Zbl 1479.35776 · doi:10.1137/20M1383434 |
| [9] | Farah, LG, Global well-posedness and blow-up on the energy space for the inhomogeneous nonlinear Schrödinger equation, J. Evol. Equ., 16, 193-208 (2016) · Zbl 1339.35287 · doi:10.1007/s00028-015-0298-y |
| [10] | Feng, BH; Chen, RP; Liu, JY, Blow-up criteria and instability of normalized standing waves for the fractional Schrödinger-Choquard equation, Adv. Nonlinear Anal., 10, 311-330 (2021) · Zbl 1447.35291 |
| [11] | Gao, FS; Rădulescu, VD; Yang, MB; Zheng, Y., Standing waves for the pseudo-relativistic Hartree equation with Berestycki-Lions nonlinearity, J. Differ. Equ., 295, 70-112 (2021) · Zbl 1479.35386 · doi:10.1016/j.jde.2021.05.047 |
| [12] | Genoud, F.; Stuart, CA, Schrödinger equations with a spatially decaying nonlinearity: existence and stability of standing waves, Discret. Contin. Dyn. Syst., 21, 137-186 (2008) · Zbl 1154.35082 · doi:10.3934/dcds.2008.21.137 |
| [13] | Genoud, F., An inhomogeneous, \(L^2\)-critical, nonlinear Schrödinger equation, Z. Anal. Anwend., 31, 283-290 (2012) · Zbl 1251.35146 · doi:10.4171/ZAA/1460 |
| [14] | Gill, TS, Optical guiding of laser beam in nonuniform plasma, Pramana J. Phys., 55, 835-842 (2000) · doi:10.1007/s12043-000-0051-z |
| [15] | Guzmán, CM, On well posedness for the inhomogeneous nonlinear Schrödinger equation, Nonlinear Anal. Real World Appl., 37, 249-286 (2017) · Zbl 1375.35486 · doi:10.1016/j.nonrwa.2017.02.018 |
| [16] | He, FL; Qin, DD; Tang, XH, Existence of ground states for Kirchhoff-type problems with general potentials, J. Geom. Anal., 31, 7709-7725 (2021) · Zbl 1473.35265 · doi:10.1007/s12220-020-00546-4 |
| [17] | Lian, W.; Rădulescu, VD; Xu, RZ; Yang, YB; Zhao, N., Global well-posedness for a class of fourth-order nonlinear strongly damped wave equations, Adv. Calc. Var., 14, 589-611 (2021) · Zbl 1476.35048 · doi:10.1515/acv-2019-0039 |
| [18] | Lian, W.; Shen, JH; Xu, RZ; Yang, YB, Infinite sharp conditions by Nehari manifolds for nonlinear Schrödinger equations, J. Geom. Anal., 30, 1865-1886 (2020) · Zbl 1439.35441 · doi:10.1007/s12220-019-00281-5 |
| [19] | Liu, CS; Tripathi, VK, Laser guiding in an axially nonuniform plasma channel, Phys. Plasmas, 1, 3100-3103 (1994) · doi:10.1063/1.870501 |
| [20] | Liu, Y.; Wang, XP; Wang, K., Instability of standing waves of the Schrödinger equations with inhomogeneous nonlinearity, Trans. Am. Math. Soc., 358, 2105-2122 (2006) · Zbl 1092.35105 · doi:10.1090/S0002-9947-05-03763-3 |
| [21] | Merle, F.: Nonexistence of minimal blow-up solutions of equations \(iu_t=-\Delta u-k(x)|u|^{4/N }u\) in \({\mathbb{R}}^N\). Ann. Inst. H. Poincaré Phys. Théor. 64, 33-85 (1996) · Zbl 0846.35129 |
| [22] | Miao, C., Murphy, J., Zheng, J.: Scattering for the non-radial inhomogeneous NLS. Math. Res. Lett. arXiv:1912.01318 (2019) |
| [23] | Papageorgiou, N.S., Rădulescu, V.D., Repovs̆, D.D.: Nonlinear Analysis-Theory and Methods. Springer Monoger. Math., Springer, Cham (2019) · Zbl 1414.46003 |
| [24] | Raphaël, P., Szeftel, J.: Existence and uniqueness of minimal blow-up solutions to an inhomogeneous mass critical NLS. J. Am. Math. Soc. 24, 471-546 (2011) · Zbl 1218.35226 |
| [25] | Ruiz, D., The Schrödinger-Poisson equation under the effect of a nonlinear local term, J. Funct. Anal., 237, 655-674 (2006) · Zbl 1136.35037 · doi:10.1016/j.jfa.2006.04.005 |
| [26] | Shatah, J.; Strauss, W., Instability of nonlinear bound states, Commun. Math. Phys., 100, 173-190 (1985) · Zbl 0603.35007 · doi:10.1007/BF01212446 |
| [27] | Wang, XC; Xu, RZ, Global existence and finite time blowup for a nonlocal semilinear pseudo-parabolic equation, Adv. Nonlinear Anal., 10, 261-288 (2021) · Zbl 1447.35078 · doi:10.1515/anona-2020-0141 |
| [28] | Yanagida, E., Uniqueness of positive radial solutions of \(\Delta u+g(r)u+h(r)u^p=0\) in \({\mathbb{R}}^n\), Arch. Ratl. Mech. Anal., 115, 257-274 (1991) · Zbl 0737.35026 · doi:10.1007/BF00380770 |
| [29] | Zhang, J., Sharp threshold for global existence and blowup in nonlinear Schrödinger equation with harmonic potential, Commun. Part. Differ. Equ., 30, 1429-1443 (2005) · Zbl 1081.35109 · doi:10.1080/03605300500299539 |
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