January  2019, 39(1): 553-583. doi: 10.3934/dcds.2019023

The radial mass-subcritical NLS in negative order Sobolev spaces

1. 

Department of Mathematics, University of California Los Angeles, Los Angeles, CA, USA

2. 

Department of Systems Innovation, Graduate School of Engineering Sciences, Toyonaka, Osaka, Japan

3. 

Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MO, USA

* Corresponding author: Jason Murphy

Received  April 2018 Published  October 2018

We consider the mass-subcritical NLS in dimensions $d≥ 3$ with radial initial data. In the defocusing case, we prove that any solution that remains bounded in the critical Sobolev space throughout its lifespan must be global and scatter. In the focusing case, we prove the existence of a threshold solution that has a compact flow.

Citation: Rowan Killip, Satoshi Masaki, Jason Murphy, Monica Visan. The radial mass-subcritical NLS in negative order Sobolev spaces. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 553-583. doi: 10.3934/dcds.2019023
References:
[1]

J. Bourgain, Global wellposedness of defocusing critical nonlinear Schrödinger equation in the radial case, J. Amer. Math. Soc., 12 (1999), 145-171. doi: 10.1090/S0894-0347-99-00283-0. Google Scholar

[2]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. doi: 10.1090/cln/010. Google Scholar

[3]

M. Christ, J. Colliander and T. Tao, Ill-posedness for nonlinear Schrödinger and wave equations, Preprint, arXiv: math/0311048.Google Scholar

[4]

J. CollianderM. KeelG. StaffilaniT. Takaoka and T. Tao, Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in ${\mathbb{R}}^3$, Ann. of Math., 167 (2008), 767-865. doi: 10.4007/annals.2008.167.767. Google Scholar

[5]

B. Dodson, Global well-posedness and scattering for the defocusing, L2-critical, nonlinear Schrödinger equation when d = 1, Amer. J. of Math., 138 (2016), 531-569. doi: 10.1353/ajm.2016.0016. Google Scholar

[6]

B. Dodson, Global well-posedness and scattering for the defocusing, L2-critical, nonlinear Schrödinger equation when d = 2, Duke Math. J., 165 (2016), 3435-3516. doi: 10.1215/00127094-3673888. Google Scholar

[7]

B. Dodson, Global well-posedness and scattering for the defocusing, L2-critical nonlinear Schrödinger equation when d ≥ 3, J. Amer. Math. Soc., 25 (2012), 429-463. doi: 10.1090/S0894-0347-2011-00727-3. Google Scholar

[8]

B. Dodson, Global well - posedness and scattering for the focusing, energy-critical nonlinear Schrödinger problem in dimension d = 4 for initial data below a ground state threshold, Preprint, arXiv: 1409.1950.Google Scholar

[9]

B. Dodson, Global well-posedness and scattering for the mass critical nonlinear Schrödinger equation with mass below the mass of the ground state, Adv. Math., 285 (2015), 1589-1618. doi: 10.1016/j.aim.2015.04.030. Google Scholar

[10]

B. DodsonC. MiaoJ. Murphy and J. Zheng, The defocusing quintic NLS in four space dimensions, Ann. Inst. H. Poincaré Anal. Non Linéaire, 34 (2017), 759-787. doi: 10.1016/j.anihpc.2016.05.004. Google Scholar

[11]

C. Gao, C. Miao and J. Yang, The intercritical defocusing nonlinear Schrödinger equation with radial initial data in dimensions four and higher, Preprint, arXiv: 1707.04686.Google Scholar

[12]

J. Ginibre and G. Velo, Smoothing properties and retarded estimates for some dispersive evolution equations, Comm. Math. Phys., 144 (1992), 163-188. doi: 10.1007/BF02099195. Google Scholar

[13]

M. Grillakis, On nonlinear Schrödinger equations, Comm. Partial Differential Equations, 25 (2000), 1827-1844. doi: 10.1080/03605300008821569. Google Scholar

[14]

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. Google Scholar

[15]

K. Hidano, Nonlinear Schrödinger equations with radially symmetric data of critical regularity, Funkcial. Ekvac., 51 (2008), 135-147. doi: 10.1619/fesi.51.135. Google Scholar

[16]

M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980. doi: 10.1353/ajm.1998.0039. Google Scholar

[17]

C. E. Kenig and F. Merle, Global well-posedness, scattering and blow-up for the energycritical, focusing, non-linear Schrödinger equation in the radial case, Invent. Math., 166 (2006), 645-675. doi: 10.1007/s00222-006-0011-4. Google Scholar

[18]

C. E. Kenig and F. Merle, Scattering for ${\dot H^{1/2}}$ bounded solutions to the cubic, defocusing NLS in 3 dimensions, Trans. Amer. Math. Soc., 362 (2010), 1937-1962. doi: 10.1090/S0002-9947-09-04722-9. Google Scholar

[19]

C. E. KenigG. Ponce and L. Vega, On the ill-posedness of some canonical dispersive equations, Duke Math. J., 106 (201), 617-633. doi: 10.1215/S0012-7094-01-10638-8. Google Scholar

[20]

R. Killip, S. Masaki, J. Murphy and M. Visan, Large data mass-subcritical NLS: critical weighted bounds imply scattering, NoDEA Nonlinear Differential Equations Appl., 24 (2017), Art. 38, 33pp. doi: 10.1007/s00030-017-0463-9. Google Scholar

[21]

R. KillipT. Tao and M. Visan, The cubic nonlinear Schrödinger equation in two dimensions with radial data, J. Eur. Math. Soc. (JEMS), 11 (2009), 1203-1258. doi: 10.4171/JEMS/180. Google Scholar

[22]

R. Killip and M. Visan, Energy-supercritical NLS: Critical ${\dot H^{s}}$-bounds imply scattering, Comm, Comm. Partial Differential Equations, 35 (2010), 945-987. doi: 10.1080/03605301003717084. Google Scholar

[23]

R. Killip and M. Visan, The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher, Amer. J. Math., 132 (2010), 361-424. doi: 10.1353/ajm.0.0107. Google Scholar

[24]

R. Killip and M. Visan, Global well-posedness and scattering for the defocusing quintic NLS in three dimensions, Anal. PDE, 5 (2012), 855-885. doi: 10.2140/apde.2012.5.855. Google Scholar

[25]

R. Killip and M. Visan, Nonlinear Schrödinger equations at critical regularity, Clay Math. Proc., 17 (2013), 325-437. Google Scholar

[26]

R. KillipM. Visan and X. Zhang, The mass-critical nonlinear Schrödinger equation with radial data in dimensions three and higher, Anal. PDE, 1 (2008), 229-266. doi: 10.2140/apde.2008.1.229. Google Scholar

[27]

H. Koch, D. Tataru and M. Visan, Dispersive Equations and Nonlinear Waves. Generalized Korteweg-de Vries, Nonlinear Schrödinger, Wave and Schrödinger Maps, Oberwolfach Seminars, 45. Birkhäuser/Springer, Basel, 2014. xii+312 pp Google Scholar

[28]

J. Lin and W. Strauss, Decay and scattering of solutions of a nonlinear Schrödinger equation, J. Funct. Anal., 30 (1978), 245-263. doi: 10.1016/0022-1236(78)90073-3. Google Scholar

[29]

C. Lu and J. Zheng, The radial defocusing energy-supercritical NLS in dimension four, J. Differential Equations, 262 (2017), 4390-4414. doi: 10.1016/j.jde.2017.01.005. Google Scholar

[30]

S. Masaki, A sharp scattering condition for focusing mass-subcritical nonlinear Schrödinger equation, Commun. Pure Appl. Anal., 14 (2015), 1481-1531. doi: 10.3934/cpaa.2015.14.1481. Google Scholar

[31]

S. Masaki, On minimal nonscattering solution for focusing mass-subcritical nonlinear Schrödinger equation, Comm. Partial Differential Equations, 42 (2017), 626-653. doi: 10.1080/03605302.2017.1286672. Google Scholar

[32]

S. Masaki, Two minimization problems on non-scattering solutions to mass-subcrticial nonlinear Schrödinger equation, Preprin, t arXiv: 1605.09234.Google Scholar

[33]

S. Masaki, On the Scattering Problem of Mass-Subcritical Hartree Equation, Adv. Stud. Pure Math., in press.Google Scholar

[34]

S. Masaki and J. Segata, Existence of a minimal non-scattering solution to the masssubcritical generalized Korteweg-de Vries equation, Ann. Inst. H. Poincaré C, Anal. Non Lineairé, 35 (2018), 283-326. doi: 10.1016/j.anihpc.2017.04.003. Google Scholar

[35]

S. Masaki and J. Segata, Refinement of Strichartz estimate for Airy equation in non-diagonal case and its application, SIAM J. Math. Anal., 50 (2018), 2839-2866. doi: 10.1137/17M1153893. Google Scholar

[36]

C. MiaoJ. Murphy and J. Zheng, The defocusing energy-supercritical NLS in four space dimensions, J. Funct. Anal., 267 (2014), 1662-1724. doi: 10.1016/j.jfa.2014.06.016. Google Scholar

[37]

J. Murphy, Intercritical NLS: Critical ${\dot H^{s}}$-bounds imply scattering, SIAM J. Math. Anal., 46 (2014), 939-997. doi: 10.1137/120898280. Google Scholar

[38]

J. Murphy, The defocusing ${\dot H^{1/2}}$-critical NLS in high dimensions, Discrete Contin. Dyn. Syst., 34 (2014), 733-748. doi: 10.3934/dcds.2014.34.733. Google Scholar

[39]

J. Murphy, The radial defocusing nonlinear Schrödinger equation in three space dimensions, Comm. Partial Differential Equations, 40 (2015), 265-308. doi: 10.1080/03605302.2014.949379. Google Scholar

[40]

E. Ryckman and M. Visan, Global well-posedness and scattering for the defocusing energycritical nonlinear Schrödinger equation in ${\mathbb{R}}^{1+4}$, Amer. J. Math., 129 (2007), 1-60. doi: 10.1353/ajm.2007.0004. Google Scholar

[41]

E. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Mathematical Series, 43. Princeton University Press, Princeton, NJ, 1993. Google Scholar

[42]

R. S. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke. Math. J., 44 (1977), 705-714. doi: 10.1215/S0012-7094-77-04430-1. Google Scholar

[43]

T. Tao, Global well-posedness and scattering for the higher-dimensional energy-critical nonlinear Schrödinger equation for radial data, New York J. Math., 11 (2005), 57-80. Google Scholar

[44]

T. TaoM. Visan and X. Zhang, Global well-posedness and scattering for the defocusing mass-critical nonlinear Schrödinger equation for radial data in high dimensions, Duke Math. J., 140 (2007), 165-202. doi: 10.1215/S0012-7094-07-14015-8. Google Scholar

[45]

M. Visan, The Defocusing Energy-Critical Nonlinear Schrödinger Equation in Dimensions Five and Higher, Ph.D Thesis, UCLA, 2006. Google Scholar

[46]

M. Visan, The defocusing energy-critical nonlinear Schrödinger equation in higher dimensions, Duke Math. J., 138 (2007), 281-374. doi: 10.1215/S0012-7094-07-13825-0. Google Scholar

[47]

M. Visan, Global well-posedness and scattering for the defocusing cubic nonlinear Schrödinger equation in four dimensions, Int. Math. Res. Not., IMRN 2012, 1037–1067. doi: 10.1093/imrn/rnr051. Google Scholar

[48]

J. Xie and D. Fang, Global well-posedness and scattering for the defocusing ${\dot H^{s}}$-critical NLS, Chin. Ann. Math. Ser. B, 34 (2013), 801-842. doi: 10.1007/s11401-013-0808-6. Google Scholar

[49]

T. Zhao, The defocusing energy-supercritical NLS in higher dimensions, Acta. Math Sin. (Eng. Ser.), 33 (2017), 911-925. doi: 10.1007/s10114-017-6499-2. Google Scholar

show all references

References:
[1]

J. Bourgain, Global wellposedness of defocusing critical nonlinear Schrödinger equation in the radial case, J. Amer. Math. Soc., 12 (1999), 145-171. doi: 10.1090/S0894-0347-99-00283-0. Google Scholar

[2]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. doi: 10.1090/cln/010. Google Scholar

[3]

M. Christ, J. Colliander and T. Tao, Ill-posedness for nonlinear Schrödinger and wave equations, Preprint, arXiv: math/0311048.Google Scholar

[4]

J. CollianderM. KeelG. StaffilaniT. Takaoka and T. Tao, Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in ${\mathbb{R}}^3$, Ann. of Math., 167 (2008), 767-865. doi: 10.4007/annals.2008.167.767. Google Scholar

[5]

B. Dodson, Global well-posedness and scattering for the defocusing, L2-critical, nonlinear Schrödinger equation when d = 1, Amer. J. of Math., 138 (2016), 531-569. doi: 10.1353/ajm.2016.0016. Google Scholar

[6]

B. Dodson, Global well-posedness and scattering for the defocusing, L2-critical, nonlinear Schrödinger equation when d = 2, Duke Math. J., 165 (2016), 3435-3516. doi: 10.1215/00127094-3673888. Google Scholar

[7]

B. Dodson, Global well-posedness and scattering for the defocusing, L2-critical nonlinear Schrödinger equation when d ≥ 3, J. Amer. Math. Soc., 25 (2012), 429-463. doi: 10.1090/S0894-0347-2011-00727-3. Google Scholar

[8]

B. Dodson, Global well - posedness and scattering for the focusing, energy-critical nonlinear Schrödinger problem in dimension d = 4 for initial data below a ground state threshold, Preprint, arXiv: 1409.1950.Google Scholar

[9]

B. Dodson, Global well-posedness and scattering for the mass critical nonlinear Schrödinger equation with mass below the mass of the ground state, Adv. Math., 285 (2015), 1589-1618. doi: 10.1016/j.aim.2015.04.030. Google Scholar

[10]

B. DodsonC. MiaoJ. Murphy and J. Zheng, The defocusing quintic NLS in four space dimensions, Ann. Inst. H. Poincaré Anal. Non Linéaire, 34 (2017), 759-787. doi: 10.1016/j.anihpc.2016.05.004. Google Scholar

[11]

C. Gao, C. Miao and J. Yang, The intercritical defocusing nonlinear Schrödinger equation with radial initial data in dimensions four and higher, Preprint, arXiv: 1707.04686.Google Scholar

[12]

J. Ginibre and G. Velo, Smoothing properties and retarded estimates for some dispersive evolution equations, Comm. Math. Phys., 144 (1992), 163-188. doi: 10.1007/BF02099195. Google Scholar

[13]

M. Grillakis, On nonlinear Schrödinger equations, Comm. Partial Differential Equations, 25 (2000), 1827-1844. doi: 10.1080/03605300008821569. Google Scholar

[14]

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. Google Scholar

[15]

K. Hidano, Nonlinear Schrödinger equations with radially symmetric data of critical regularity, Funkcial. Ekvac., 51 (2008), 135-147. doi: 10.1619/fesi.51.135. Google Scholar

[16]

M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980. doi: 10.1353/ajm.1998.0039. Google Scholar

[17]

C. E. Kenig and F. Merle, Global well-posedness, scattering and blow-up for the energycritical, focusing, non-linear Schrödinger equation in the radial case, Invent. Math., 166 (2006), 645-675. doi: 10.1007/s00222-006-0011-4. Google Scholar

[18]

C. E. Kenig and F. Merle, Scattering for ${\dot H^{1/2}}$ bounded solutions to the cubic, defocusing NLS in 3 dimensions, Trans. Amer. Math. Soc., 362 (2010), 1937-1962. doi: 10.1090/S0002-9947-09-04722-9. Google Scholar

[19]

C. E. KenigG. Ponce and L. Vega, On the ill-posedness of some canonical dispersive equations, Duke Math. J., 106 (201), 617-633. doi: 10.1215/S0012-7094-01-10638-8. Google Scholar

[20]

R. Killip, S. Masaki, J. Murphy and M. Visan, Large data mass-subcritical NLS: critical weighted bounds imply scattering, NoDEA Nonlinear Differential Equations Appl., 24 (2017), Art. 38, 33pp. doi: 10.1007/s00030-017-0463-9. Google Scholar

[21]

R. KillipT. Tao and M. Visan, The cubic nonlinear Schrödinger equation in two dimensions with radial data, J. Eur. Math. Soc. (JEMS), 11 (2009), 1203-1258. doi: 10.4171/JEMS/180. Google Scholar

[22]

R. Killip and M. Visan, Energy-supercritical NLS: Critical ${\dot H^{s}}$-bounds imply scattering, Comm, Comm. Partial Differential Equations, 35 (2010), 945-987. doi: 10.1080/03605301003717084. Google Scholar

[23]

R. Killip and M. Visan, The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher, Amer. J. Math., 132 (2010), 361-424. doi: 10.1353/ajm.0.0107. Google Scholar

[24]

R. Killip and M. Visan, Global well-posedness and scattering for the defocusing quintic NLS in three dimensions, Anal. PDE, 5 (2012), 855-885. doi: 10.2140/apde.2012.5.855. Google Scholar

[25]

R. Killip and M. Visan, Nonlinear Schrödinger equations at critical regularity, Clay Math. Proc., 17 (2013), 325-437. Google Scholar

[26]

R. KillipM. Visan and X. Zhang, The mass-critical nonlinear Schrödinger equation with radial data in dimensions three and higher, Anal. PDE, 1 (2008), 229-266. doi: 10.2140/apde.2008.1.229. Google Scholar

[27]

H. Koch, D. Tataru and M. Visan, Dispersive Equations and Nonlinear Waves. Generalized Korteweg-de Vries, Nonlinear Schrödinger, Wave and Schrödinger Maps, Oberwolfach Seminars, 45. Birkhäuser/Springer, Basel, 2014. xii+312 pp Google Scholar

[28]

J. Lin and W. Strauss, Decay and scattering of solutions of a nonlinear Schrödinger equation, J. Funct. Anal., 30 (1978), 245-263. doi: 10.1016/0022-1236(78)90073-3. Google Scholar

[29]

C. Lu and J. Zheng, The radial defocusing energy-supercritical NLS in dimension four, J. Differential Equations, 262 (2017), 4390-4414. doi: 10.1016/j.jde.2017.01.005. Google Scholar

[30]

S. Masaki, A sharp scattering condition for focusing mass-subcritical nonlinear Schrödinger equation, Commun. Pure Appl. Anal., 14 (2015), 1481-1531. doi: 10.3934/cpaa.2015.14.1481. Google Scholar

[31]

S. Masaki, On minimal nonscattering solution for focusing mass-subcritical nonlinear Schrödinger equation, Comm. Partial Differential Equations, 42 (2017), 626-653. doi: 10.1080/03605302.2017.1286672. Google Scholar

[32]

S. Masaki, Two minimization problems on non-scattering solutions to mass-subcrticial nonlinear Schrödinger equation, Preprin, t arXiv: 1605.09234.Google Scholar

[33]

S. Masaki, On the Scattering Problem of Mass-Subcritical Hartree Equation, Adv. Stud. Pure Math., in press.Google Scholar

[34]

S. Masaki and J. Segata, Existence of a minimal non-scattering solution to the masssubcritical generalized Korteweg-de Vries equation, Ann. Inst. H. Poincaré C, Anal. Non Lineairé, 35 (2018), 283-326. doi: 10.1016/j.anihpc.2017.04.003. Google Scholar

[35]

S. Masaki and J. Segata, Refinement of Strichartz estimate for Airy equation in non-diagonal case and its application, SIAM J. Math. Anal., 50 (2018), 2839-2866. doi: 10.1137/17M1153893. Google Scholar

[36]

C. MiaoJ. Murphy and J. Zheng, The defocusing energy-supercritical NLS in four space dimensions, J. Funct. Anal., 267 (2014), 1662-1724. doi: 10.1016/j.jfa.2014.06.016. Google Scholar

[37]

J. Murphy, Intercritical NLS: Critical ${\dot H^{s}}$-bounds imply scattering, SIAM J. Math. Anal., 46 (2014), 939-997. doi: 10.1137/120898280. Google Scholar

[38]

J. Murphy, The defocusing ${\dot H^{1/2}}$-critical NLS in high dimensions, Discrete Contin. Dyn. Syst., 34 (2014), 733-748. doi: 10.3934/dcds.2014.34.733. Google Scholar

[39]

J. Murphy, The radial defocusing nonlinear Schrödinger equation in three space dimensions, Comm. Partial Differential Equations, 40 (2015), 265-308. doi: 10.1080/03605302.2014.949379. Google Scholar

[40]

E. Ryckman and M. Visan, Global well-posedness and scattering for the defocusing energycritical nonlinear Schrödinger equation in ${\mathbb{R}}^{1+4}$, Amer. J. Math., 129 (2007), 1-60. doi: 10.1353/ajm.2007.0004. Google Scholar

[41]

E. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Mathematical Series, 43. Princeton University Press, Princeton, NJ, 1993. Google Scholar

[42]

R. S. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke. Math. J., 44 (1977), 705-714. doi: 10.1215/S0012-7094-77-04430-1. Google Scholar

[43]

T. Tao, Global well-posedness and scattering for the higher-dimensional energy-critical nonlinear Schrödinger equation for radial data, New York J. Math., 11 (2005), 57-80. Google Scholar

[44]

T. TaoM. Visan and X. Zhang, Global well-posedness and scattering for the defocusing mass-critical nonlinear Schrödinger equation for radial data in high dimensions, Duke Math. J., 140 (2007), 165-202. doi: 10.1215/S0012-7094-07-14015-8. Google Scholar

[45]

M. Visan, The Defocusing Energy-Critical Nonlinear Schrödinger Equation in Dimensions Five and Higher, Ph.D Thesis, UCLA, 2006. Google Scholar

[46]

M. Visan, The defocusing energy-critical nonlinear Schrödinger equation in higher dimensions, Duke Math. J., 138 (2007), 281-374. doi: 10.1215/S0012-7094-07-13825-0. Google Scholar

[47]

M. Visan, Global well-posedness and scattering for the defocusing cubic nonlinear Schrödinger equation in four dimensions, Int. Math. Res. Not., IMRN 2012, 1037–1067. doi: 10.1093/imrn/rnr051. Google Scholar

[48]

J. Xie and D. Fang, Global well-posedness and scattering for the defocusing ${\dot H^{s}}$-critical NLS, Chin. Ann. Math. Ser. B, 34 (2013), 801-842. doi: 10.1007/s11401-013-0808-6. Google Scholar

[49]

T. Zhao, The defocusing energy-supercritical NLS in higher dimensions, Acta. Math Sin. (Eng. Ser.), 33 (2017), 911-925. doi: 10.1007/s10114-017-6499-2. Google Scholar

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Mohamad Darwich. On the $L^2$-critical nonlinear Schrödinger Equation with a nonlinear damping. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2377-2394. doi: 10.3934/cpaa.2014.13.2377

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