March 2019, 18(2): 689-708. doi: 10.3934/cpaa.2019034

On blow-up solutions to the focusing mass-critical nonlinear fractional Schrödinger equation

Institut de Mathématiques de Toulouse UMR5219, Université Toulouse CNRS, 31062 Toulouse Cedex 9, France

Received  January 2018 Revised  May 2018 Published  October 2018

In this paper we study dynamical properties of blow-up solutions to the focusing mass-critical nonlinear fractional Schrödinger equation. We establish a profile decomposition and a compactness lemma related to the equation. As a result, we obtain the $L^2$-concentration and the limiting profile with minimal mass of blow-up solutions.

Citation: Van Duong Dinh. On blow-up solutions to the focusing mass-critical nonlinear fractional Schrödinger equation. Communications on Pure & Applied Analysis, 2019, 18 (2) : 689-708. doi: 10.3934/cpaa.2019034
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]

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

[3]

D. CaiJ. MajdaD. W. McLaughlin and E. G. Tabak, Dispersive wave turbulence in one dimension, Phys. D: Nonlinear Phenomena, 152-153 (2001), 551-572. doi: 10.1016/S0167-2789(01)00193-2.

[4]

Y. Cho and S. Lee, Strichartz estimates in spherical coordinates, Indiana Univ. Math. J., 62 (2013), 991-1020.

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

[6]

Y. ChoT. Ozawa and S. Xia, Remarks on some dispersive estimates, Commun. Pure Appl. Anal., 10 (2011), 1121-1128. doi: 10.3934/cpaa.2011.10.1121.

[7]

Y. ChoH. HajaiejG. Hwang and T. Ozawa, On the Cauchy problem of fractional Schrödigner equation with Hartree type nonlinearity, Funkcial. Ekvac., 56 (2013), 193-224. doi: 10.1619/fesi.56.193.

[8]

Y. ChoH. HajaiejG. Hwang and T. Ozawa, On the orbital stability of fractional Schrödinger equations, Commun. Pure Appl. Anal., 13 (2014), 1267-1282. doi: 10.3934/cpaa.2014.13.1267.

[9]

Y. ChoG. Hwang and Y. Shim, Energy concentration of the focusing energy-critical fNLS, J. Math. Anal. Appl., 437 (2016), 310-329. doi: 10.1016/j.jmaa.2015.12.060.

[10]

M. Christ and I. Weinstein, Dispersion of small amplitude solutions of the generalized Korteweg-de Vries equation, J. Funct. Anal., 100 (1991), 87-109. doi: 10.1016/0022-1236(91)90103-C.

[11]

V. D. Dinh, Well-posedness of nonlinear fractional Schrödinger and wave equations in Sobolev spaces, Int. J. Appl. Math., 31 (2018), 483-525. doi: 10.12732/ijam.v31i4.1.

[12]

D. Fang and C. Wang, Weighted Strichartz estimates with angular regularity and their applications, Forum Math., 23 (2011), 181-205. doi: 10.1515/form.2011.009.

[13]

B. Feng, On the blow-up solutions for the fractional nonlinear Schrödinger equation with combined power-type nonlinearities, Commun. Pure Appl. Anal., 17 (2018), 1785-1804. doi: 10.3934/cpaa.2018085.

[14]

R. L. Frank and E. Lenzmann, Uniqueness of nonlinear gound states for fractional Laplacians in $\mathbb{R}$, Acta Math., 210 (2013), 261-318. doi: 10.1007/s11511-013-0095-9.

[15]

R. L. FrankE. Lenzmann and L. Silvestre, Uniqueness of radial solutions for the fractional Laplacian, Comm. Pure Appl. Math., 69 (2016), 1671-1725. doi: 10.1002/cpa.21591.

[16]

J. FröhlichG. Jonsson and E. Lenzmann, Boson stars as solitary waves, Comm. Math. Phys., 274 (2007), 1-30. doi: 10.1007/s00220-007-0272-9.

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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.

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Z. Guo, Y. Sire, Y. Wang and L. Zhao, On the energy-critical fractional Schrödinger equation in the radial case, preprint, arXiv: 1310.8616.

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Z. Guo and Y. Wang, Improved Strichartz estimates for a class of dispersive equations in the radial case and their applications to nonlinear Schrödinger and wave equations, J. Anal. Math., 124 (2014), 1-38. doi: 10.1007/s11854-014-0025-6.

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Q. Guo and S. Zhu, Sharp criteria of scattering for the fractional NLS, preprint, arXiv: 1706.02549.

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Y. Hong and Y. Sire, On fractional Schrödinger equations in Sobolev spaces, Commun. Pure Appl. Anal., 14 (2015), 2265-2282. doi: 10.3934/cpaa.2015.14.2265.

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

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C. Klein, C. Sparber and P. Markowich, Numerical study of fractional nonlinear Schrödinger equations, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 470 (2014), 20140364. doi: 10.1098/rspa.2014.0364.

[24]

J. KriegerE. Lenzmann and P. Raphaël, Non dispersive solutions for the $L^2$ critical Half-Wave equation, Arch. Rational Mech. Anal., 209 (2013), 61-129. doi: 10.1007/s00205-013-0620-1.

[25]

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[26]

C. E. KenigG. Ponce and L. Vega, Well-posedness and scattering results for the gereralized Korteveg-de Vries equation via the contraction principle, Comm. Pure Appl. Math., 46 (1993), 527-620. doi: 10.1002/cpa.3160460405.

[27]

Y. Ke, Remark on the Strichartz estimates in the radial case, J. Math. Anal. Appl., 387 (2012), 857-861. doi: 10.1016/j.jmaa.2011.09.039.

[28]

K. KirkpatrickE. Lenzmann and G. Staffilani, On the continuum limit for discrete NLS with long-range lattice interactions, Comm. Math. Phys., 317 (2013), 563-591. doi: 10.1007/s00220-012-1621-x.

[29]

N. Laskin, Fractional Schrödinger equation, Phys. Rev. E, 66 (2002), 056108. doi: 10.1103/PhysRevE.66.056108.

[30]

F. Merle and Y. Tsutsumi, $L^2$ concentration of blow up solutions for the nonlinear Schrödinger equation with critical power nonlinearity, J. Differential Equations, 84 (1990), 205-214. doi: 10.1016/0022-0396(90)90075-Z.

[31]

F. Merle, On uniqueness and continuation properties after blow-up time of self-similar solutions of nonlinear Schrödinger equation with critical exponent and critical mass, Comm. Pure Appl. Math., 45 (1992), 203-254. doi: 10.1002/cpa.3160450204.

[32]

F. Merle, Determination of blow-up solutions with minimal mass for nonlinear Schrödinger equations with critical power, Duke Math. J., 69 (1993), 427-454. doi: 10.1215/S0012-7094-93-06919-0.

[33]

F. Merle and P. Raphaël, On universality of blow-up profile for $L^2$-critical nonlinear Schrödinger equation, Invent. Math., 156 (2004), 565-672. doi: 10.1007/s00222-003-0346-z.

[34]

F. Merle and P. Raphaël, Blow-up dynamic and upper bound on the blow-up rate for critical nonlinear Schrödinger equation, Ann. Math., 161 (2005), 157-222. doi: 10.4007/annals.2005.161.157.

[35]

F. Merle and P. Raphaël, On a sharp lower bound on the blow-up rate for the $L^2$-critical nonlinear Schrödinger equation, J. Amer. Math. Soc., 19 (2006), 37-90. doi: 10.1090/S0894-0347-05-00499-6.

[36]

F. Merle and P. Raphaël, Blow up of critical norm for some radial $L^2$ super critical nonlinear Schrödinger equations, Amer. J. Math., 130 (2008), 945-978. doi: 10.1353/ajm.0.0012.

[37]

H. Nava, Asymptotic and limiting profiles of blowup solutions of the nonlinear Schrödinger equation with critical power, Comm. Pure Appl. Math., 52 (1999), 193-207. doi: 10.1002/(SICI)1097-0312(199902)52:2<193::AID-CPA2>3.0.CO;2-3.

[38]

T. Ogawa and Y. Tsutsumi, Blow-up of $H^1$ solutions for the nonlinear Schrödinger equation, J. Differential Equations, 92 (1991), 317-330. doi: 10.1016/0022-0396(91)90052-B.

[39]

T. Ogawa and Y. Tsutsumi, Blow-up of $H^1$ solutions for the one dimensional nonlinear Schrödinger equation with critical power nonlinearity, Proc. Amer. Math. Soc., 111 (1991), 487-496. doi: 10.1090/S0002-9939-1991-1045145-5.

[40]

T. Ozawa and N. Visciglia, An improvement on the Brezis-Gallouët technique for 2D NLS and 1D half-wave equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 1069-1079. doi: 10.1016/j.anihpc.2015.03.004.

[41]

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

[42]

C. SunH. WangX. Yao and J. Zheng, Scattering below ground state of focusing fractional nonlinear Schrödinger equation with radial data, Discrete Contin. Dyn. Syst., 38 (2018), 2207-2228. doi: 10.3934/dcds.2018091.

[43]

T. Tao, Nonlinear Dispersive Equations: Local and Global Analysis, CBMS Regional Conference Series in Mathematics 106, AMS, 2006.

[44]

Y. Tsutsumi, Rate of $L^2$-concentration of blow-up solutions for the nonlinear Schrödinger equation with critical power, Nonlinear Anal., 15 (1990), 719-724. doi: 10.1016/0362-546X(90)90088-X.

[45]

W. I. Weinstein, On the structure and formation of singularities in solutions to nonlinear dispersive evolution equations, Comm. Partial Differential Equations, 11 (1986), 545-565. doi: 10.1080/03605308608820435.

[46]

S. Zhu, On the blow-up solutions for the nonlinear fractional Schrödinger equation, J. Differential Equations, 261 (2016), 1506-1531. doi: 10.1016/j.jde.2016.04.007.

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]

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

[3]

D. CaiJ. MajdaD. W. McLaughlin and E. G. Tabak, Dispersive wave turbulence in one dimension, Phys. D: Nonlinear Phenomena, 152-153 (2001), 551-572. doi: 10.1016/S0167-2789(01)00193-2.

[4]

Y. Cho and S. Lee, Strichartz estimates in spherical coordinates, Indiana Univ. Math. J., 62 (2013), 991-1020.

[5]

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

[6]

Y. ChoT. Ozawa and S. Xia, Remarks on some dispersive estimates, Commun. Pure Appl. Anal., 10 (2011), 1121-1128. doi: 10.3934/cpaa.2011.10.1121.

[7]

Y. ChoH. HajaiejG. Hwang and T. Ozawa, On the Cauchy problem of fractional Schrödigner equation with Hartree type nonlinearity, Funkcial. Ekvac., 56 (2013), 193-224. doi: 10.1619/fesi.56.193.

[8]

Y. ChoH. HajaiejG. Hwang and T. Ozawa, On the orbital stability of fractional Schrödinger equations, Commun. Pure Appl. Anal., 13 (2014), 1267-1282. doi: 10.3934/cpaa.2014.13.1267.

[9]

Y. ChoG. Hwang and Y. Shim, Energy concentration of the focusing energy-critical fNLS, J. Math. Anal. Appl., 437 (2016), 310-329. doi: 10.1016/j.jmaa.2015.12.060.

[10]

M. Christ and I. Weinstein, Dispersion of small amplitude solutions of the generalized Korteweg-de Vries equation, J. Funct. Anal., 100 (1991), 87-109. doi: 10.1016/0022-1236(91)90103-C.

[11]

V. D. Dinh, Well-posedness of nonlinear fractional Schrödinger and wave equations in Sobolev spaces, Int. J. Appl. Math., 31 (2018), 483-525. doi: 10.12732/ijam.v31i4.1.

[12]

D. Fang and C. Wang, Weighted Strichartz estimates with angular regularity and their applications, Forum Math., 23 (2011), 181-205. doi: 10.1515/form.2011.009.

[13]

B. Feng, On the blow-up solutions for the fractional nonlinear Schrödinger equation with combined power-type nonlinearities, Commun. Pure Appl. Anal., 17 (2018), 1785-1804. doi: 10.3934/cpaa.2018085.

[14]

R. L. Frank and E. Lenzmann, Uniqueness of nonlinear gound states for fractional Laplacians in $\mathbb{R}$, Acta Math., 210 (2013), 261-318. doi: 10.1007/s11511-013-0095-9.

[15]

R. L. FrankE. Lenzmann and L. Silvestre, Uniqueness of radial solutions for the fractional Laplacian, Comm. Pure Appl. Math., 69 (2016), 1671-1725. doi: 10.1002/cpa.21591.

[16]

J. FröhlichG. Jonsson and E. Lenzmann, Boson stars as solitary waves, Comm. Math. Phys., 274 (2007), 1-30. doi: 10.1007/s00220-007-0272-9.

[17]

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.

[18]

Z. Guo, Y. Sire, Y. Wang and L. Zhao, On the energy-critical fractional Schrödinger equation in the radial case, preprint, arXiv: 1310.8616.

[19]

Z. Guo and Y. Wang, Improved Strichartz estimates for a class of dispersive equations in the radial case and their applications to nonlinear Schrödinger and wave equations, J. Anal. Math., 124 (2014), 1-38. doi: 10.1007/s11854-014-0025-6.

[20]

Q. Guo and S. Zhu, Sharp criteria of scattering for the fractional NLS, preprint, arXiv: 1706.02549.

[21]

Y. Hong and Y. Sire, On fractional Schrödinger equations in Sobolev spaces, Commun. Pure Appl. Anal., 14 (2015), 2265-2282. doi: 10.3934/cpaa.2015.14.2265.

[22]

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

[23]

C. Klein, C. Sparber and P. Markowich, Numerical study of fractional nonlinear Schrödinger equations, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 470 (2014), 20140364. doi: 10.1098/rspa.2014.0364.

[24]

J. KriegerE. Lenzmann and P. Raphaël, Non dispersive solutions for the $L^2$ critical Half-Wave equation, Arch. Rational Mech. Anal., 209 (2013), 61-129. doi: 10.1007/s00205-013-0620-1.

[25]

A. D. Ionescu and F. Pusateri, Nonlinear fractional Schrödinger equation in one dimension, J. Funct. Anal., 266 (2014), 139-176. doi: 10.1016/j.jfa.2013.08.027.

[26]

C. E. KenigG. Ponce and L. Vega, Well-posedness and scattering results for the gereralized Korteveg-de Vries equation via the contraction principle, Comm. Pure Appl. Math., 46 (1993), 527-620. doi: 10.1002/cpa.3160460405.

[27]

Y. Ke, Remark on the Strichartz estimates in the radial case, J. Math. Anal. Appl., 387 (2012), 857-861. doi: 10.1016/j.jmaa.2011.09.039.

[28]

K. KirkpatrickE. Lenzmann and G. Staffilani, On the continuum limit for discrete NLS with long-range lattice interactions, Comm. Math. Phys., 317 (2013), 563-591. doi: 10.1007/s00220-012-1621-x.

[29]

N. Laskin, Fractional Schrödinger equation, Phys. Rev. E, 66 (2002), 056108. doi: 10.1103/PhysRevE.66.056108.

[30]

F. Merle and Y. Tsutsumi, $L^2$ concentration of blow up solutions for the nonlinear Schrödinger equation with critical power nonlinearity, J. Differential Equations, 84 (1990), 205-214. doi: 10.1016/0022-0396(90)90075-Z.

[31]

F. Merle, On uniqueness and continuation properties after blow-up time of self-similar solutions of nonlinear Schrödinger equation with critical exponent and critical mass, Comm. Pure Appl. Math., 45 (1992), 203-254. doi: 10.1002/cpa.3160450204.

[32]

F. Merle, Determination of blow-up solutions with minimal mass for nonlinear Schrödinger equations with critical power, Duke Math. J., 69 (1993), 427-454. doi: 10.1215/S0012-7094-93-06919-0.

[33]

F. Merle and P. Raphaël, On universality of blow-up profile for $L^2$-critical nonlinear Schrödinger equation, Invent. Math., 156 (2004), 565-672. doi: 10.1007/s00222-003-0346-z.

[34]

F. Merle and P. Raphaël, Blow-up dynamic and upper bound on the blow-up rate for critical nonlinear Schrödinger equation, Ann. Math., 161 (2005), 157-222. doi: 10.4007/annals.2005.161.157.

[35]

F. Merle and P. Raphaël, On a sharp lower bound on the blow-up rate for the $L^2$-critical nonlinear Schrödinger equation, J. Amer. Math. Soc., 19 (2006), 37-90. doi: 10.1090/S0894-0347-05-00499-6.

[36]

F. Merle and P. Raphaël, Blow up of critical norm for some radial $L^2$ super critical nonlinear Schrödinger equations, Amer. J. Math., 130 (2008), 945-978. doi: 10.1353/ajm.0.0012.

[38]

T. Ogawa and Y. Tsutsumi, Blow-up of $H^1$ solutions for the nonlinear Schrödinger equation, J. Differential Equations, 92 (1991), 317-330. doi: 10.1016/0022-0396(91)90052-B.

[39]

T. Ogawa and Y. Tsutsumi, Blow-up of $H^1$ solutions for the one dimensional nonlinear Schrödinger equation with critical power nonlinearity, Proc. Amer. Math. Soc., 111 (1991), 487-496. doi: 10.1090/S0002-9939-1991-1045145-5.

[40]

T. Ozawa and N. Visciglia, An improvement on the Brezis-Gallouët technique for 2D NLS and 1D half-wave equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 1069-1079. doi: 10.1016/j.anihpc.2015.03.004.

[41]

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

[42]

C. SunH. WangX. Yao and J. Zheng, Scattering below ground state of focusing fractional nonlinear Schrödinger equation with radial data, Discrete Contin. Dyn. Syst., 38 (2018), 2207-2228. doi: 10.3934/dcds.2018091.

[43]

T. Tao, Nonlinear Dispersive Equations: Local and Global Analysis, CBMS Regional Conference Series in Mathematics 106, AMS, 2006.

[44]

Y. Tsutsumi, Rate of $L^2$-concentration of blow-up solutions for the nonlinear Schrödinger equation with critical power, Nonlinear Anal., 15 (1990), 719-724. doi: 10.1016/0362-546X(90)90088-X.

[45]

W. I. Weinstein, On the structure and formation of singularities in solutions to nonlinear dispersive evolution equations, Comm. Partial Differential Equations, 11 (1986), 545-565. doi: 10.1080/03605308608820435.

[46]

S. Zhu, On the blow-up solutions for the nonlinear fractional Schrödinger equation, J. Differential Equations, 261 (2016), 1506-1531. doi: 10.1016/j.jde.2016.04.007.

Table 1.  Local well-posedness (LWP) in $H^s$ for NLFS
$s$$\alpha$LWP
$d=1$$\frac{1}{3}<s<\frac{1}{2}$$0<\alpha<\frac{4s}{1-2s}$$u_0$ non-radial
$d=1$$\frac{1}{2}<s<1$$0<\alpha<\infty$$u_0$ non-radial
$d=2$$\frac{1}{2}<s<1$$0<\alpha<\frac{4s}{2-2s}$$u_0$ non-radial
$d=3$$\frac{3}{5}\leq s\leq \frac{3}{4}$$0<\alpha<\frac{4s}{3-2s}$$u_0$ radial
$d=3$$\frac{3}{4}<s< 1$$0<\alpha<\frac{4s}{3-2s}$$u_0$ non-radial
$d\geq 4$$\frac{d}{2d-1}\leq s<1$$0<\alpha<\frac{4s}{d-2s}$$u_0$ radial
$s$$\alpha$LWP
$d=1$$\frac{1}{3}<s<\frac{1}{2}$$0<\alpha<\frac{4s}{1-2s}$$u_0$ non-radial
$d=1$$\frac{1}{2}<s<1$$0<\alpha<\infty$$u_0$ non-radial
$d=2$$\frac{1}{2}<s<1$$0<\alpha<\frac{4s}{2-2s}$$u_0$ non-radial
$d=3$$\frac{3}{5}\leq s\leq \frac{3}{4}$$0<\alpha<\frac{4s}{3-2s}$$u_0$ radial
$d=3$$\frac{3}{4}<s< 1$$0<\alpha<\frac{4s}{3-2s}$$u_0$ non-radial
$d\geq 4$$\frac{d}{2d-1}\leq s<1$$0<\alpha<\frac{4s}{d-2s}$$u_0$ radial
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