doi: 10.3934/dcdsb.2021304
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Blowup results for the fractional Schrödinger equation without gauge invariance

1. 

Department of Mathematics, Lanzhou University of Technology, Lanzhou, 730050, China

2. 

School of Mathematics and Statistics, Tianshui Normal University, Tianshui, 741001, China

3. 

Department of Mathematics, Zhejiang Normal University, Jinhua, 321004, China

*Corresponding author: Qihong Shi

Received  August 2021 Early access December 2021

Fund Project: The authors are supported by NNSF of China (Nos.12061040, 11701244, 11901266), NSF of Gansu Province of China(Nos. 20JR5RA460, 20JR5RA498) and the Innovation Ability Promotion Foundation of Universities in Gansu Province, China(No. 2020B-185)

This paper is concerned with the nonexistence of global solutions to the fractional Schrödinger equations with order $ \alpha $ and nongauge power type nonlinearity $ |u|^p $ for any space dimensions, where $ \alpha\in (0, 2] $ is assumed to be any fractional numbers. A modified test function is employed to overcome some difficulties caused by the fractional operator and to establish blowup results. Some restrictions with respect to $ \alpha, p $ and initial data in the previous literature are removed.

Citation: Qihong Shi, Congming Peng, Qingxuan Wang. Blowup results for the fractional Schrödinger equation without gauge invariance. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021304
References:
[1]

J. BellazziniV. Georgiev and N. Visciglia, Long time dynamics for semi-relativistic NLS and half wave in arbitrary dimension, Math. Ann., 371 (2018), 707-740.  doi: 10.1007/s00208-018-1666-z.  Google Scholar

[2]

J. P. Borgna and D. F. Rial, Existence of ground states for a one-dimensional relativistic Schrödinger equation, J. Math. Phys., 53 (2012), 062301.  doi: 10.1063/1.4726198.  Google Scholar

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T. BoulengerD. Himmelsbach and E. Lenzmann, An Blowup for Fractional NLS, J. Funct. Anal., 271 (2016), 2569-2603.  doi: 10.1016/j.jfa.2016.08.011.  Google Scholar

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T. A. Dao and M. Reissig, A Blow-up result for semi-linear structurally damped $\sigma$-evolution equations, Anomalies in Partial Differential Equations, 43 (2021), 213-245.   Google Scholar

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H. Fujita, On the blowing up of solutions of the Cauchy problem for $u_t = \triangle u+ u^{ 1+\alpha}$, J. Fac. Sci. Univ. Tokyo Sect. Ⅰ, 13 (1966), 109-124.   Google Scholar

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K. Fujiwara, A note for the global non-existence of semirelativistic equations with non-gauge invariant power type nonlinearity., Math. Method. Appl. Sci., 41 (2018), 4955-4966.  doi: 10.1002/mma.4944.  Google Scholar

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K. Fujiwara, S. Machihara and T. Ozawa, Remark on a semirelativistic equation in the energy space, Discrete Contin. Dyn. Syst., Dynamical systems, differential equations and applications. 10th AIMS Conference. Suppl., 2015 (2015), 473–478. doi: 10.3934/proc.2015.0473.  Google Scholar

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K. FujiwaraS. Machihara and T. Ozawa, Well-posedness for the Cauchy problem of a system of semirelativistic equations, Comm. Math. Phys., 338 (2015), 367-391.  doi: 10.1007/s00220-015-2347-3.  Google Scholar

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K. Fujiwara and T. Ozawa, Remarks on global solutions to the Cauchy problem for semirel- ativistic equations with power type nonlinearrity, International Journal of Mathematical Analysis, 9 (2015), 2599-2610.   Google Scholar

[12]

K. Hayakawa, On nonexistence of global solutions of some semilinear parabolic differential equations, Proc. Japan Acad., 49 (1973), 503-505.   Google Scholar

[13]

T. Hmidi and S. Keraani, Blowup theory for the critical nonlinear Schrödinger equations revisited, Int. Math. Res. Not., 46 (2005), 2815-2828.  doi: 10.1155/IMRN.2005.2815.  Google Scholar

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M. Ikeda and T. Inui, Some non-existence results for the semilinear Schrödinger equation without gauge invariance, J. Math. Anal. Appl., 425 (2015), 758-773.  doi: 10.1016/j.jmaa.2015.01.003.  Google Scholar

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M. Ikeda and T. Inui, Small data blow up of $L^2$ or $H^1$-solution for the semilinear Schrödinger equation without gauge invariance, J. Evol. Equ., 15 (2015), 571-581.  doi: 10.1007/s00028-015-0273-7.  Google Scholar

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M. Ikeda and Y. Wakasugi, Small data blow-up of $L^2$-solution for the nonlinear Schrödinger equation without gauge invariance, Differential Integral Equations, 26 (2013), 1275-1285.   Google Scholar

[17]

T. Inui, Some nonexistence results for a semirelativistic Schrödinger equation with nongauge power type nonlinearity, Proc. Amer. Math. Soc., 144 (2016), 2901-2909.  doi: 10.1090/proc/12938.  Google Scholar

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S. Klainerman and S. Selberg, Bilinear estimates and applications to nonlinear wave equations, Comm. Contemp. Math., 4 (2002), 223-295.  doi: 10.1142/S0219199702000634.  Google Scholar

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J. KriegerE. Lenzmann and P. Raphael, Nondispersive solutions to the $L^2$-critical half-wave equation, Arch. Ration. Mech. Anal., 209 (2013), 61-129.  doi: 10.1007/s00205-013-0620-1.  Google Scholar

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M. Kwaśnicki, Ten equivalent definitions of the fractional laplace operator, Fract. Calc. Appl. Anal., 20 (2017), 7-51.  doi: 10.1515/fca-2017-0002.  Google Scholar

[21]

N.-A. Lai and Y. Zhou, The sharp lifespan estimate for semilinear damped wave equation with Fujita critical power in high dimensions, J. Math. Pures Appl., 123 (2019), 229-243.  doi: 10.1016/j.matpur.2018.04.009.  Google Scholar

[22]

T.-Y. Lee and W.-M. Ni, Global existence, large time behavior and life span of solutions of a semilinear parabolic Cauchy problem, Trans. Amer. Math. Soc., 333 (1992), 365-378.  doi: 10.1090/S0002-9947-1992-1057781-6.  Google Scholar

[23]

C. Peng and Q. Shi, Stability of standing wave for the fractional nonlinear Schrödinger equation, J. Math. Phys., 59 (2018), 011508.  doi: 10.1063/1.5021689.  Google Scholar

[24]

T. Saanouni, Remark on the inhomogeneous fractional nonlinear Schrödinger equations, J. Math. Phys., 57 (2016), 081503.  doi: 10.1063/1.4960045.  Google Scholar

[25]

S. Sugitani, On nonexistence of global solutions for some nonlinear integral equations, Osaka J. Math., 12 (1975), 45-51.   Google Scholar

show all references

References:
[1]

J. BellazziniV. Georgiev and N. Visciglia, Long time dynamics for semi-relativistic NLS and half wave in arbitrary dimension, Math. Ann., 371 (2018), 707-740.  doi: 10.1007/s00208-018-1666-z.  Google Scholar

[2]

J. P. Borgna and D. F. Rial, Existence of ground states for a one-dimensional relativistic Schrödinger equation, J. Math. Phys., 53 (2012), 062301.  doi: 10.1063/1.4726198.  Google Scholar

[3]

T. BoulengerD. Himmelsbach and E. Lenzmann, An Blowup for Fractional NLS, J. Funct. Anal., 271 (2016), 2569-2603.  doi: 10.1016/j.jfa.2016.08.011.  Google Scholar

[4]

Y. Cho and T. Ozawa, On the semirelativistic Hartree-type equation, SIAM J. Math. Anal., 38 (2006), 1060-1074.  doi: 10.1137/060653688.  Google Scholar

[5]

M. D'Abbicco and M. Reissig, Semilinear structural damped waves, Math. Meth. Appl. Sci., 37 (2014), 1570-1592.  doi: 10.1002/mma.2913.  Google Scholar

[6]

T. A. Dao and M. Reissig, A Blow-up result for semi-linear structurally damped $\sigma$-evolution equations, Anomalies in Partial Differential Equations, 43 (2021), 213-245.   Google Scholar

[7]

H. Fujita, On the blowing up of solutions of the Cauchy problem for $u_t = \triangle u+ u^{ 1+\alpha}$, J. Fac. Sci. Univ. Tokyo Sect. Ⅰ, 13 (1966), 109-124.   Google Scholar

[8]

K. Fujiwara, A note for the global non-existence of semirelativistic equations with non-gauge invariant power type nonlinearity., Math. Method. Appl. Sci., 41 (2018), 4955-4966.  doi: 10.1002/mma.4944.  Google Scholar

[9]

K. Fujiwara, S. Machihara and T. Ozawa, Remark on a semirelativistic equation in the energy space, Discrete Contin. Dyn. Syst., Dynamical systems, differential equations and applications. 10th AIMS Conference. Suppl., 2015 (2015), 473–478. doi: 10.3934/proc.2015.0473.  Google Scholar

[10]

K. FujiwaraS. Machihara and T. Ozawa, Well-posedness for the Cauchy problem of a system of semirelativistic equations, Comm. Math. Phys., 338 (2015), 367-391.  doi: 10.1007/s00220-015-2347-3.  Google Scholar

[11]

K. Fujiwara and T. Ozawa, Remarks on global solutions to the Cauchy problem for semirel- ativistic equations with power type nonlinearrity, International Journal of Mathematical Analysis, 9 (2015), 2599-2610.   Google Scholar

[12]

K. Hayakawa, On nonexistence of global solutions of some semilinear parabolic differential equations, Proc. Japan Acad., 49 (1973), 503-505.   Google Scholar

[13]

T. Hmidi and S. Keraani, Blowup theory for the critical nonlinear Schrödinger equations revisited, Int. Math. Res. Not., 46 (2005), 2815-2828.  doi: 10.1155/IMRN.2005.2815.  Google Scholar

[14]

M. Ikeda and T. Inui, Some non-existence results for the semilinear Schrödinger equation without gauge invariance, J. Math. Anal. Appl., 425 (2015), 758-773.  doi: 10.1016/j.jmaa.2015.01.003.  Google Scholar

[15]

M. Ikeda and T. Inui, Small data blow up of $L^2$ or $H^1$-solution for the semilinear Schrödinger equation without gauge invariance, J. Evol. Equ., 15 (2015), 571-581.  doi: 10.1007/s00028-015-0273-7.  Google Scholar

[16]

M. Ikeda and Y. Wakasugi, Small data blow-up of $L^2$-solution for the nonlinear Schrödinger equation without gauge invariance, Differential Integral Equations, 26 (2013), 1275-1285.   Google Scholar

[17]

T. Inui, Some nonexistence results for a semirelativistic Schrödinger equation with nongauge power type nonlinearity, Proc. Amer. Math. Soc., 144 (2016), 2901-2909.  doi: 10.1090/proc/12938.  Google Scholar

[18]

S. Klainerman and S. Selberg, Bilinear estimates and applications to nonlinear wave equations, Comm. Contemp. Math., 4 (2002), 223-295.  doi: 10.1142/S0219199702000634.  Google Scholar

[19]

J. KriegerE. Lenzmann and P. Raphael, Nondispersive solutions to the $L^2$-critical half-wave equation, Arch. Ration. Mech. Anal., 209 (2013), 61-129.  doi: 10.1007/s00205-013-0620-1.  Google Scholar

[20]

M. Kwaśnicki, Ten equivalent definitions of the fractional laplace operator, Fract. Calc. Appl. Anal., 20 (2017), 7-51.  doi: 10.1515/fca-2017-0002.  Google Scholar

[21]

N.-A. Lai and Y. Zhou, The sharp lifespan estimate for semilinear damped wave equation with Fujita critical power in high dimensions, J. Math. Pures Appl., 123 (2019), 229-243.  doi: 10.1016/j.matpur.2018.04.009.  Google Scholar

[22]

T.-Y. Lee and W.-M. Ni, Global existence, large time behavior and life span of solutions of a semilinear parabolic Cauchy problem, Trans. Amer. Math. Soc., 333 (1992), 365-378.  doi: 10.1090/S0002-9947-1992-1057781-6.  Google Scholar

[23]

C. Peng and Q. Shi, Stability of standing wave for the fractional nonlinear Schrödinger equation, J. Math. Phys., 59 (2018), 011508.  doi: 10.1063/1.5021689.  Google Scholar

[24]

T. Saanouni, Remark on the inhomogeneous fractional nonlinear Schrödinger equations, J. Math. Phys., 57 (2016), 081503.  doi: 10.1063/1.4960045.  Google Scholar

[25]

S. Sugitani, On nonexistence of global solutions for some nonlinear integral equations, Osaka J. Math., 12 (1975), 45-51.   Google Scholar

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