October  2012, 32(10): 3733-3771. doi: 10.3934/dcds.2012.32.3733

Bounds on the growth of high Sobolev norms of solutions to 2D Hartree equations

1. 

Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, United States

Received  May 2010 Revised  January 2012 Published  May 2012

In this paper, we consider Hartree-type equations on the two-dimensional torus and on the plane. We prove polynomial bounds on the growth of high Sobolev norms of solutions to these equations. The proofs of our results are based on the adaptation to two dimensions of the techniques we had previously used in [49, 50] to study the analogous problem in one dimension. Since we are working in two dimensions, a more detailed analysis of the resonant frequencies is needed, as was previously used in the work of Colliander-Keel-Staffilani-Takaoka-Tao [19].
Citation: Vedran Sohinger. Bounds on the growth of high Sobolev norms of solutions to 2D Hartree equations. Discrete & Continuous Dynamical Systems - A, 2012, 32 (10) : 3733-3771. doi: 10.3934/dcds.2012.32.3733
References:
[1]

D. Benney and A. Newell, Random wave closures,, Stud. Appl. Math., 48 (1969), 29.   Google Scholar

[2]

D. Benney and P. Saffman, Nonlinear interactions of random waves in a dispersive medium,, Proc. Roy. Soc. A, 289 (1966), 301.  doi: 10.1098/rspa.1966.0013.  Google Scholar

[3]

J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations,, Geom. Funct. Anal., 3 (1993), 107.   Google Scholar

[4]

J. Bourgain, On the growth in time of higher Sobolev norms of smooth solutions of Hamiltonian PDE,, Int. Math. Research Notices, 1996 (): 277.   Google Scholar

[5]

J. Bourgain, Refinements of Strichartz's inequality and applications to 2D-NLS with critical nonlinearity,, Int. Math. Research Notices, 1998 (): 253.   Google Scholar

[6]

J. Bourgain, "Nonlinear Schrödinger Equations,", in, 5 (1999), 3.   Google Scholar

[7]

J. Bourgain, Global solutions of nonlinear Schrödinger equations,, AMS Colloquium Publications, 46 (1999).   Google Scholar

[8]

J. Bourgain, On growth of Sobolev norms in linear Schrödinger equations with smooth time dependent potential,, J. Anal. Math., 77 (1999), 315.  doi: 10.1007/BF02791265.  Google Scholar

[9]

N. Burq, P. Gérard and N. Tzvetkov, An instability property of the nonlinear Schrödinger equation on $S^d$,, Mathematical Research Letters, 9 (2002), 323.   Google Scholar

[10]

N. Burq, P. Gérard and N. Tzvetkov, Bilinear eigenfunction estimates and the nonlinear Schrödinger equation on surfaces,, Invent. Math., 159 (2005), 187.  doi: 10.1007/s00222-004-0388-x.  Google Scholar

[11]

F. Catoire and W.-M. Wang, Bounds on Sobolev norms for the nonlinear Schrödinger equation on general tori,, preprint, (2008).   Google Scholar

[12]

T. Cazenave, "Semilinear Schrödinger Equations,", Courant Lecture Notes in Mathematics, 10 (2003).   Google Scholar

[13]

J. Colliander, J.-M. Delort, C. E. Kenig and G. Staffilani, Bilinear estimates and applications to 2D NLS,, Trans. of the American Math. Soc., 353 (2001), 3307.  doi: 10.1090/S0002-9947-01-02760-X.  Google Scholar

[14]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global well-posedness for Schrödinger equations with derivative,, SIAM J. Math. Anal., 33 (2001), 649.  doi: 10.1137/S0036141001384387.  Google Scholar

[15]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Polynomial upper bounds for the orbital instability of the 1D cubic NLS below the energy norm,, Discrete Contin. Dyn. Syst., 9 (2003), 31.   Google Scholar

[16]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Sharp global well-posedness for KdV and modified KdV on $\mathbbR$ and $\mathbbT$,, J. Amer. Math. Soc., 16 (2003), 705.  doi: 10.1090/S0894-0347-03-00421-1.  Google Scholar

[17]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Multilinear estimates for periodic KdV equations, and applications,, J. Funct. Anal., 211 (2004), 173.  doi: 10.1016/S0022-1236(03)00218-0.  Google Scholar

[18]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, A refined global well-posedness for Schrödinger equations with derivative,, SIAM J. Math. Anal., 34 (2002), 64.  doi: 10.1137/S0036141001394541.  Google Scholar

[19]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Resonant decompositions and the I-method for cubic nonlinear Schrödinger equation on $\mathbbR^2$,, Disc. and Cont. Dynam. Sys., 21 (2008), 665.  doi: 10.3934/dcds.2008.21.665.  Google Scholar

[20]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in $\mathbbR^3$,, Ann. of Math. (2), 167 (2008), 767.  doi: 10.4007/annals.2008.167.767.  Google Scholar

[21]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Transfer of energy to high frequencies in the cubic nonlinear Schrödinger equation,, Invent. Math., 181 (2010), 39.  doi: 10.1007/s00222-010-0242-2.  Google Scholar

[22]

J.-M. Delort, Growth of Sobolev norms of solutions of linear Schrödinger equations on some compact manifolds,, preprint, 2010 (): 2305.   Google Scholar

[23]

B. Dodson, Global well-posedness and scattering for the defocusing, $L^2 $- critical, nonlinear Schrödinger equation when $d \geq 3$,, preprint, (2009).   Google Scholar

[24]

B. Dodson, Global well-posedness and scattering for the defocusing, $L^2 $- critical, nonlinear Schrödinger equation when $d=2$,, preprint, (2010).   Google Scholar

[25]

B. Dodson, Global well-posedness and scattering for the defocusing, $L^2 $- critical, nonlinear Schrödinger equation when $d=1$,, preprint, (2009).   Google Scholar

[26]

J. Duoandikoetxea, "Fourier Analysis,", Graduate Studies in Mathematics, 29 (2001).   Google Scholar

[27]

J. Fröhlich and E. Lenzmann, "Mean-Field Limit of Quantum Bose Gases and Nonlinear Hartree Equation,", Séminaire: É.D.P. 2003-2004, (2004), 2003.   Google Scholar

[28]

J. Ginibre and T. Ozawa, Long-range scattering for non-linear Schrödinger and Hartree equations in space dimension $n\geq 2$,, Comm. Math. Phys., 151 (1993), 619.  doi: 10.1007/BF02097031.  Google Scholar

[29]

J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equations. I. The Cauchy problem, general case,, J. Funct. Anal., 32 (1979), 1.  doi: 10.1016/0022-1236(79)90076-4.  Google Scholar

[30]

J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equations with nonlocal interaction,, Math. Z., 170 (1980), 109.  doi: 10.1007/BF01214768.  Google Scholar

[31]

J. Ginibre and G. Velo, Scattering theory in the energy space for a class of Hartree equations,, Rev. Math. Phys., 12 (2000), 361.  doi: 10.1142/S0129055X00000137.  Google Scholar

[32]

J. Ginibre and G. Velo, Long range scattering and modified wave operators for some Hartree type equations. II,, Ann. H. P., 1 (2000), 753.   Google Scholar

[33]

J. Ginibre and G. Velo, Long range scattering and modified wave operators for some Hartree type equations. III. Gevrey spaces and low dimensions,, J. Diff. Eq., 175 (2001), 415.  doi: 10.1006/jdeq.2000.3969.  Google Scholar

[34]

A. Grünrock, On the Cauchy- and periodic boundary value problem for a certain class of derivative nonlinear Schrödinger equations,, preprint, (2006).   Google Scholar

[35]

, Z. Hani,, Private communication., ().   Google Scholar

[36]

N. Hayashi, P. Naumkin and T. Ozawa, "Scattering Theory for the Hartree Equation,", Hokkaido University Preprints, (1996).   Google Scholar

[37]

C. Kenig, G. Ponce and L. Vega, The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices,, Duke Math. J., 71 (1993), 1.   Google Scholar

[38]

C. Kenig, G. Ponce and L. Vega, Quadratic forms for the 1-D semilinear Schrödinger equation,, Transactions of the AMS, 348 (1996), 3323.  doi: 10.1090/S0002-9947-96-01645-5.  Google Scholar

[39]

C. Miao, Y. Wu and G. Xu, Dynamics for the focusing, energy-critical nonlinear Hartree equation,, preprint, (2011).   Google Scholar

[40]

C. Miao, G. Xu and L. Zhao, Global well-posedness and scattering for the energy critical, defocusing Hartree equation for radial data,, J. Funct. Anal., 253 (2007), 605.  doi: 10.1016/j.jfa.2007.09.008.  Google Scholar

[41]

C. Miao, G. Xu and L. Zhao, The Cauchy problem for the Hartree equation,, J. PDEs, 21 (2008), 22.   Google Scholar

[42]

C. Miao, G. Xu and L. Zhao, Global well-posedness, scattering, and blow-up for the energy critical, focusing Hartree equation in the radial case,, Coll. Math., 114 (2009), 213.  doi: 10.4064/cm114-2-5.  Google Scholar

[43]

C. Miao, G. Xu and L. Zhao, Global well-posedness and scattering for the defocusing $H^{\frac{1}{2}}$ -subcritical Hartree equation on $\mathbbR^d$,, Ann. I. H. Poincaré, 26 (2009), 1831.   Google Scholar

[44]

C. Miao, G. Xu and L. Zhao, Global well-posedness and scattering for the energy-critical, defocusing Hartree equation in $\mathbbR^{1+n}$,, Comm. PDEs, 36 (2011), 729.  doi: 10.1080/03605302.2010.531073.  Google Scholar

[45]

C. Morawetz and W. A. Strauss, Decay and scattering of solutions of a nonlinear relativistic wave equation,, Comm. Pure. Appl. Math., 25 (1972), 1.  doi: 10.1002/cpa.3160250103.  Google Scholar

[46]

B. Schlein, "Derivation of Effective Evolution Equations from Microscopic Quantum Dynamics,", Lecture Notes, (2008).   Google Scholar

[47]

C. Sogge, Oscillatory integrals and spherical harmonics,, Duke Math. Jour., 53 (1986), 43.  doi: 10.1215/S0012-7094-86-05303-2.  Google Scholar

[48]

C. Sogge, Concerning the $\ell^p$ norm of spectral clusters for second order elliptic operators on compact manifolds,, Jour. of Funct. Anal., 77 (1988), 123.  doi: 10.1016/0022-1236(88)90081-X.  Google Scholar

[49]

V. Sohinger, Bounds on the growth of high Sobolev norms of solutions to Nonlinear Schrödinger Equations on $S^1$,, to appear in Diff. and Int. Eqs., (2010).   Google Scholar

[50]

V. Sohinger, Bounds on the growth of high Sobolev norms of solutions to nonlinear Schrödinger equations on $\mathbbR$,, to appear in Indiana Univ. Math. J., (2010).   Google Scholar

[51]

G. Staffilani, On the growth of high Sobolev norms of solutions for KdV and Schrödinger equations,, Duke Math. J., 86 (1997), 109.  doi: 10.1215/S0012-7094-97-08604-X.  Google Scholar

[52]

G. Staffilani, Quadratic forms for a 2-D semilinear Schrödinger equation,, Duke Math. J., 86 (1997), 79.  doi: 10.1215/S0012-7094-97-08603-8.  Google Scholar

[53]

T. Tao, "Nonlinear Dispersive Equations: Local and Global Analysis,", CBMS Reg. Conf. Series in Math., 106 (2006).   Google Scholar

[54]

V. E. Zakharov, Stability of periodic waves of finite amplitude on a surface of deep fluid,, J. Appl. Mech. Tech. Phys., 9 (1968), 190.   Google Scholar

[55]

S.-J. Zhong, The growth in time of higher Sobolev norms of solutions to Schrödinger equations on compact Riemannian manifolds,, J. Differential Equations, 245 (2008), 359.  doi: 10.1016/j.jde.2008.03.008.  Google Scholar

show all references

References:
[1]

D. Benney and A. Newell, Random wave closures,, Stud. Appl. Math., 48 (1969), 29.   Google Scholar

[2]

D. Benney and P. Saffman, Nonlinear interactions of random waves in a dispersive medium,, Proc. Roy. Soc. A, 289 (1966), 301.  doi: 10.1098/rspa.1966.0013.  Google Scholar

[3]

J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations,, Geom. Funct. Anal., 3 (1993), 107.   Google Scholar

[4]

J. Bourgain, On the growth in time of higher Sobolev norms of smooth solutions of Hamiltonian PDE,, Int. Math. Research Notices, 1996 (): 277.   Google Scholar

[5]

J. Bourgain, Refinements of Strichartz's inequality and applications to 2D-NLS with critical nonlinearity,, Int. Math. Research Notices, 1998 (): 253.   Google Scholar

[6]

J. Bourgain, "Nonlinear Schrödinger Equations,", in, 5 (1999), 3.   Google Scholar

[7]

J. Bourgain, Global solutions of nonlinear Schrödinger equations,, AMS Colloquium Publications, 46 (1999).   Google Scholar

[8]

J. Bourgain, On growth of Sobolev norms in linear Schrödinger equations with smooth time dependent potential,, J. Anal. Math., 77 (1999), 315.  doi: 10.1007/BF02791265.  Google Scholar

[9]

N. Burq, P. Gérard and N. Tzvetkov, An instability property of the nonlinear Schrödinger equation on $S^d$,, Mathematical Research Letters, 9 (2002), 323.   Google Scholar

[10]

N. Burq, P. Gérard and N. Tzvetkov, Bilinear eigenfunction estimates and the nonlinear Schrödinger equation on surfaces,, Invent. Math., 159 (2005), 187.  doi: 10.1007/s00222-004-0388-x.  Google Scholar

[11]

F. Catoire and W.-M. Wang, Bounds on Sobolev norms for the nonlinear Schrödinger equation on general tori,, preprint, (2008).   Google Scholar

[12]

T. Cazenave, "Semilinear Schrödinger Equations,", Courant Lecture Notes in Mathematics, 10 (2003).   Google Scholar

[13]

J. Colliander, J.-M. Delort, C. E. Kenig and G. Staffilani, Bilinear estimates and applications to 2D NLS,, Trans. of the American Math. Soc., 353 (2001), 3307.  doi: 10.1090/S0002-9947-01-02760-X.  Google Scholar

[14]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global well-posedness for Schrödinger equations with derivative,, SIAM J. Math. Anal., 33 (2001), 649.  doi: 10.1137/S0036141001384387.  Google Scholar

[15]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Polynomial upper bounds for the orbital instability of the 1D cubic NLS below the energy norm,, Discrete Contin. Dyn. Syst., 9 (2003), 31.   Google Scholar

[16]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Sharp global well-posedness for KdV and modified KdV on $\mathbbR$ and $\mathbbT$,, J. Amer. Math. Soc., 16 (2003), 705.  doi: 10.1090/S0894-0347-03-00421-1.  Google Scholar

[17]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Multilinear estimates for periodic KdV equations, and applications,, J. Funct. Anal., 211 (2004), 173.  doi: 10.1016/S0022-1236(03)00218-0.  Google Scholar

[18]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, A refined global well-posedness for Schrödinger equations with derivative,, SIAM J. Math. Anal., 34 (2002), 64.  doi: 10.1137/S0036141001394541.  Google Scholar

[19]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Resonant decompositions and the I-method for cubic nonlinear Schrödinger equation on $\mathbbR^2$,, Disc. and Cont. Dynam. Sys., 21 (2008), 665.  doi: 10.3934/dcds.2008.21.665.  Google Scholar

[20]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in $\mathbbR^3$,, Ann. of Math. (2), 167 (2008), 767.  doi: 10.4007/annals.2008.167.767.  Google Scholar

[21]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Transfer of energy to high frequencies in the cubic nonlinear Schrödinger equation,, Invent. Math., 181 (2010), 39.  doi: 10.1007/s00222-010-0242-2.  Google Scholar

[22]

J.-M. Delort, Growth of Sobolev norms of solutions of linear Schrödinger equations on some compact manifolds,, preprint, 2010 (): 2305.   Google Scholar

[23]

B. Dodson, Global well-posedness and scattering for the defocusing, $L^2 $- critical, nonlinear Schrödinger equation when $d \geq 3$,, preprint, (2009).   Google Scholar

[24]

B. Dodson, Global well-posedness and scattering for the defocusing, $L^2 $- critical, nonlinear Schrödinger equation when $d=2$,, preprint, (2010).   Google Scholar

[25]

B. Dodson, Global well-posedness and scattering for the defocusing, $L^2 $- critical, nonlinear Schrödinger equation when $d=1$,, preprint, (2009).   Google Scholar

[26]

J. Duoandikoetxea, "Fourier Analysis,", Graduate Studies in Mathematics, 29 (2001).   Google Scholar

[27]

J. Fröhlich and E. Lenzmann, "Mean-Field Limit of Quantum Bose Gases and Nonlinear Hartree Equation,", Séminaire: É.D.P. 2003-2004, (2004), 2003.   Google Scholar

[28]

J. Ginibre and T. Ozawa, Long-range scattering for non-linear Schrödinger and Hartree equations in space dimension $n\geq 2$,, Comm. Math. Phys., 151 (1993), 619.  doi: 10.1007/BF02097031.  Google Scholar

[29]

J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equations. I. The Cauchy problem, general case,, J. Funct. Anal., 32 (1979), 1.  doi: 10.1016/0022-1236(79)90076-4.  Google Scholar

[30]

J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equations with nonlocal interaction,, Math. Z., 170 (1980), 109.  doi: 10.1007/BF01214768.  Google Scholar

[31]

J. Ginibre and G. Velo, Scattering theory in the energy space for a class of Hartree equations,, Rev. Math. Phys., 12 (2000), 361.  doi: 10.1142/S0129055X00000137.  Google Scholar

[32]

J. Ginibre and G. Velo, Long range scattering and modified wave operators for some Hartree type equations. II,, Ann. H. P., 1 (2000), 753.   Google Scholar

[33]

J. Ginibre and G. Velo, Long range scattering and modified wave operators for some Hartree type equations. III. Gevrey spaces and low dimensions,, J. Diff. Eq., 175 (2001), 415.  doi: 10.1006/jdeq.2000.3969.  Google Scholar

[34]

A. Grünrock, On the Cauchy- and periodic boundary value problem for a certain class of derivative nonlinear Schrödinger equations,, preprint, (2006).   Google Scholar

[35]

, Z. Hani,, Private communication., ().   Google Scholar

[36]

N. Hayashi, P. Naumkin and T. Ozawa, "Scattering Theory for the Hartree Equation,", Hokkaido University Preprints, (1996).   Google Scholar

[37]

C. Kenig, G. Ponce and L. Vega, The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices,, Duke Math. J., 71 (1993), 1.   Google Scholar

[38]

C. Kenig, G. Ponce and L. Vega, Quadratic forms for the 1-D semilinear Schrödinger equation,, Transactions of the AMS, 348 (1996), 3323.  doi: 10.1090/S0002-9947-96-01645-5.  Google Scholar

[39]

C. Miao, Y. Wu and G. Xu, Dynamics for the focusing, energy-critical nonlinear Hartree equation,, preprint, (2011).   Google Scholar

[40]

C. Miao, G. Xu and L. Zhao, Global well-posedness and scattering for the energy critical, defocusing Hartree equation for radial data,, J. Funct. Anal., 253 (2007), 605.  doi: 10.1016/j.jfa.2007.09.008.  Google Scholar

[41]

C. Miao, G. Xu and L. Zhao, The Cauchy problem for the Hartree equation,, J. PDEs, 21 (2008), 22.   Google Scholar

[42]

C. Miao, G. Xu and L. Zhao, Global well-posedness, scattering, and blow-up for the energy critical, focusing Hartree equation in the radial case,, Coll. Math., 114 (2009), 213.  doi: 10.4064/cm114-2-5.  Google Scholar

[43]

C. Miao, G. Xu and L. Zhao, Global well-posedness and scattering for the defocusing $H^{\frac{1}{2}}$ -subcritical Hartree equation on $\mathbbR^d$,, Ann. I. H. Poincaré, 26 (2009), 1831.   Google Scholar

[44]

C. Miao, G. Xu and L. Zhao, Global well-posedness and scattering for the energy-critical, defocusing Hartree equation in $\mathbbR^{1+n}$,, Comm. PDEs, 36 (2011), 729.  doi: 10.1080/03605302.2010.531073.  Google Scholar

[45]

C. Morawetz and W. A. Strauss, Decay and scattering of solutions of a nonlinear relativistic wave equation,, Comm. Pure. Appl. Math., 25 (1972), 1.  doi: 10.1002/cpa.3160250103.  Google Scholar

[46]

B. Schlein, "Derivation of Effective Evolution Equations from Microscopic Quantum Dynamics,", Lecture Notes, (2008).   Google Scholar

[47]

C. Sogge, Oscillatory integrals and spherical harmonics,, Duke Math. Jour., 53 (1986), 43.  doi: 10.1215/S0012-7094-86-05303-2.  Google Scholar

[48]

C. Sogge, Concerning the $\ell^p$ norm of spectral clusters for second order elliptic operators on compact manifolds,, Jour. of Funct. Anal., 77 (1988), 123.  doi: 10.1016/0022-1236(88)90081-X.  Google Scholar

[49]

V. Sohinger, Bounds on the growth of high Sobolev norms of solutions to Nonlinear Schrödinger Equations on $S^1$,, to appear in Diff. and Int. Eqs., (2010).   Google Scholar

[50]

V. Sohinger, Bounds on the growth of high Sobolev norms of solutions to nonlinear Schrödinger equations on $\mathbbR$,, to appear in Indiana Univ. Math. J., (2010).   Google Scholar

[51]

G. Staffilani, On the growth of high Sobolev norms of solutions for KdV and Schrödinger equations,, Duke Math. J., 86 (1997), 109.  doi: 10.1215/S0012-7094-97-08604-X.  Google Scholar

[52]

G. Staffilani, Quadratic forms for a 2-D semilinear Schrödinger equation,, Duke Math. J., 86 (1997), 79.  doi: 10.1215/S0012-7094-97-08603-8.  Google Scholar

[53]

T. Tao, "Nonlinear Dispersive Equations: Local and Global Analysis,", CBMS Reg. Conf. Series in Math., 106 (2006).   Google Scholar

[54]

V. E. Zakharov, Stability of periodic waves of finite amplitude on a surface of deep fluid,, J. Appl. Mech. Tech. Phys., 9 (1968), 190.   Google Scholar

[55]

S.-J. Zhong, The growth in time of higher Sobolev norms of solutions to Schrödinger equations on compact Riemannian manifolds,, J. Differential Equations, 245 (2008), 359.  doi: 10.1016/j.jde.2008.03.008.  Google Scholar

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