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 and 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.

[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.

[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.

[4]

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

[5]

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

[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. 

[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. 

[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.

[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.

[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.

[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. 

[12]

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

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[24]

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

[25]

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

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.

[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.

[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.

[4]

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

[5]

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

[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. 

[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. 

[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.

[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.

[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.

[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. 

[12]

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

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[24]

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

[25]

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

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