Blow-up criteria and instability of standing waves for the inhomogeneous fractional Schrödinger equation. (English) Zbl 1466.35058
Summary: In this article, we study the blow-up and instability of standing waves for the inhomogeneous fractional Schrödinger equation \[ i\partial_tu-(-\Delta)^s u+ |x|^{-b}|u|^p u=0, \] where \(s\in(\frac{1}{2}, 1)\), \(0<b<\min\{2s, N\}\) and \(0<p<\frac{4s-2b}{N-2s}\). In the \(L^2\)-critical and \(L^2\)-supercritical cases, i.e., \(\frac{4s-2b}{N}\leq p<\frac{4s-2b}{N-2s}\), we establish general blow-up criteria for non-radial solutions by using localized virial estimates. Based on these blow-up criteria, we prove the strong instability of standing waves.
MSC:
| 35B44 | Blow-up in context of PDEs |
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 35R11 | Fractional partial differential equations |
References:
| [1] | T. Boulenger, D. Himmelsbach, E. Lenzmann; Blowup for fractional Schr¨odinger equation, J. Funct. Anal.,271(2016), 2569-2603. · Zbl 1348.35038 |
| [2] | T. Cazenave;Semilinear Schr¨odinger 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 |
| [3] | Y. Cho, H. Hajaiej, G. Hwang, T. Ozawa; On the Cauchy problem of fractional Schr¨odinger equations with Hartree type nonlimearity,Funkcial. Ekvac.,56(2013), 193-224. · Zbl 1341.35138 |
| [4] | Y. Cho, G. Hwang, S. Kwon, S. Lee; On finite time blow-up for the mass-critical Hartree equations,Proc. Roy. Soc. Edinburgh Sect. A,145(2015), 467-479. · Zbl 1326.35298 |
| [5] | V. D. Dinh; Well-posedness of nonlinear fractional Schr¨odinger and wave equations in Sobolev spaces,Int. J. Appl. Math.,31(2018), 483-525. |
| [6] | V. D. Dinh; On instability of standing waves for the mass-supercritical fractional nonlinear Schr¨odinger equation,Z. Angew. Math. Phys.,70(2019), 17 pp. · Zbl 1420.35348 |
| [7] | V. D. Dinh; Blowup ofH1solutions for a class of the focusing inhomogeneous nonlinear Schr¨odinger equation,Nonlinear Anal.,174(2018), 169-188. · Zbl 1388.35177 |
| [8] | V. D. Dinh; Blow-up criteria for fractional nonlinear Schr¨odinger equations,Nonlinear Anal. Real World Appl.,48(2019), 117-140. · Zbl 1428.35501 |
| [9] | D. Du, Y. Wu, K. Zhang; On blow-up criterion for the nonlinear Schr¨odinger equation, Discrete Contin. Dyn. Syst.,36(2016), 3639-3650. · Zbl 1333.35250 |
| [10] | L. G. Farah; Global well-posedness and blow-up on the energy space for the inhomogeneous nonlinear Schr¨odinger equation,J. Evol. Equ.,16(2016), 193-208. · Zbl 1339.35287 |
| [11] | B. Feng; On the blow-up solutions for the fractional nonlinear Schr¨odinger equation with combined power-type nonlinearities,Comm. Pure Appl. Anal.,17(2018), 1785-1804. · Zbl 1397.35269 |
| [12] | B. Feng, R. Chen, J. Liu; Blow-up criteria and instability of normalized standing waves for the fractional Schr¨odinger-Choquard equation,Adv. Nonlinear Anal.,10(2021), 311-330. · Zbl 1447.35291 |
| [13] | B. Feng, R. Chen, Q. Wang; Instability of standing waves for the nonlinear Schr¨odingerPoisson equation in theL2-critical case,J. Dynam. Differential Equations,32(2020), 13571370. · Zbl 1445.35133 |
| [14] | B. Feng, J. Liu, H. Niu, B. Zhang; Strong instability of standing waves for a fourth-order nonlinear Schr¨odinger equation with the mixed dispersions,Nonlinear Anal.,196(2020), 111791, 16 pp. · Zbl 1437.35624 |
| [15] | B. Feng, Q Wang; Strong Instability of Standing Waves for the Nonlinear Schr¨odinger Equation in Trapped Dipolar Quantum Gases,J. Dynam. Differential Equations, (2020), https://doi.org/10.1007/s10884-020-09881-0. · Zbl 1504.35475 |
| [16] | C. Peng, D. Zhao; Global existence and blowup on the energy space for the inhomogeneous fractional nonlinear Schr¨odinger equation,Discrete Continuous Dynamical Systems-Series B,24(2019), 3335-3356. · Zbl 1420.35373 |
| [17] | T. Saanouni; Strong instability of standing waves for the fractional Choquard equation,J. Math. Phys.,59(2018), 081509. · Zbl 1395.35175 |
| [18] | Y. Su, H. Chen, S. Liu, X. Fang; Fractional Schr¨odinger-Poisson systems with weighted Hardy potential and critical exponent,Electron. J. Differential Equations, (2020), 17 pp. · Zbl 1429.35204 |
| [19] | J. Sun, T. Wu, Z. Feng; Two positive solutions to non-autonomous Schr¨odinger-Poisson systems,Nonlinearity,32(2019), 4002-4032. · Zbl 1425.35027 |
| [20] | J. Zhang, S.H. Zhu; Sharp blow-up criteria for the Davey-Stewartson system inR3,Dyn. Partial Differ. Equ.,8(2011), 239-260. · Zbl 1256.35144 |
| [21] | J. Zhang, S. H. Zhu; Stability of standing waves for the nonlinear fractional Schr¨odinger equation,J. Dynam. Differential Equations,29(2017), 1017-1030. · Zbl 1384.35123 |
| [22] | S. H. Zhu; On the blow-up solutions for the nonlinear fractional Schr¨odinger equation,J. Differential Equations,261(2016), 1506-1531. · Zbl 1339.35260 |
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