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On the $L^2$-critical nonlinear Schrödinger Equation with a nonlinear damping
1. | Laboratoire de Mathématiques et Physique Théorique, UMR 7350, Tours, France |
References:
[1] |
P. Antonelli and C. Sparber, Global well-posedness for cubic NLS with nonlinear damping, Comm. Partial Differential Equations, 35 (2010), 4832-4845.
doi: 10.1080/03605300903540943. |
[2] |
H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. II. Existence of infinitely many solutions, Arch. Rational Mech. Anal., 82 (1983), 347-375.
doi: 10.1007/BF00250556. |
[3] |
T. Cazenave, Semilinear Schrödinger Equations, volume 10 of Courant Lecture Notes in Mathematics, New York University Courant Institute of Mathematical Sciences, New York, 2003. |
[4] |
T. Cazenave and F. Weissler, Some remarks on the nonlinear Schrödinger equation in the subcritical case, in New Methods and Results in Nonlinear Field Equations (Bielefeld, 1987), 59-69, Lecture Notes in Phys., 347, Springer, Berlin, 1989.
doi: 10.1007/BFb0025761. |
[5] |
J. Colliander and P. Raphael, Rough blowup solutions to the $L^2$ critical NLS, Math. Ann., 345 (2009), 307-366.
doi: 10.1007/s00208-009-0355-3. |
[6] |
M. Darwich, Blowup for the Damped $L^2$critical nonlinear Shrödinger equations, Advances in Differential Equations, 17 (2012), 337-367. |
[7] |
G. Fibich and F. Merle, Self-focusing on bounded domains, Phys. D, 155 (2001), 132-158.
doi: 10.1016/S0167-2789(01)00249-4. |
[8] |
G. Fibich and M. Klein, Nonlinear-damping continuation of the nonlinear Schrödinger equation-a numerical study, Physica D, 241 (2012), 519-527.
doi: 10.1016/j.physd.2011.11.008. |
[9] |
A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London, 1969. |
[10] |
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. |
[11] |
T. Kato, On nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Phys. Théor., 46 (1987), 113-129. |
[12] |
M. K. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p=0$ in $R^n$, Arch. Rational Mech. Anal., 105 (1989), 243-266.
doi: 10.1007/BF00251502. |
[13] |
P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. II, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 223-283. |
[14] |
F. Merle, Determination of blow-up solutions with minimal mass for nonlinear Schröinger equations with critical power, Duke Math. J., 69 (1993), 427-454.
doi: 10.1215/S0012-7094-93-06919-0. |
[15] |
F. Merle and P. Raphael, Blow up dynamic and upper bound on the blow up rate for critical nonlinear Schrödinger equation, In Journées "Équations aux Dérivées Partielles'' (Forges-les-Eaux, 2002), pages Exp. No. XII, 5. Univ. Nantes, Nantes, 2002. |
[16] |
F. Merle and P. Raphael, Sharp upper bound on the blow-up rate for the critical nonlinear Schrödinger equation, Geom. Funct. Anal., 13 (2003), 591-642.
doi: 10.1007/s00039-003-0424-9. |
[17] |
F. Merle and P. Raphael, 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. |
[18] |
F. Merle and P. Raphael, Profiles and quantization of the blow up mass for critical nonlinear Schrödinger equation, Comm. Math. Phys., 253 (2005), 675-704.
doi: 10.1007/s00220-004-1198-0. |
[19] |
F. Merle and P. Raphael, 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 (electronic).
doi: 10.1090/S0894-0347-05-00499-6. |
[20] |
M. Ohta and G. Todorova, Remarks on global existence and blowup for damped nonlinear Schrödinger equations, Discrete Contin. Dyn. Syst., 23 (2009), 1313-1325.
doi: 10.3934/dcds.2009.23.1313. |
[21] |
T. Passota, C. Sulemb and P. L. Sulem, Linear versus nonlinear dissipation for critical NLS equation, Physica D, 203 (2005), 167-184.
doi: 10.1016/j.physd.2005.03.011. |
[22] |
F. Planchon and P. Raphaël, Existence and stability of the log-log blow-up dynamics for the $L^2$-critical nonlinear Schrödinger equation in a domain, Ann. Henri Poincaré, 8 (2007), 1177-1219.
doi: 10.1007/s00023-007-0332-x. |
[23] |
P. Raphael, Stability of the log-log bound for blow up solutions to the critical non linear Schrödinger equation, Math. Ann., 331 (2005), 577-609.
doi: 10.1007/s00208-004-0596-0. |
[24] |
M. Tsutsumi, Nonexistence of global solutions to the Cauchy problem for the damped nonlinear Schrödinger equations, SIAM J. Math. Anal., 15 (1984), 357-366.
doi: 10.1137/0515028. |
[25] |
M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys., 87 (1982/83), 567-576. |
show all references
References:
[1] |
P. Antonelli and C. Sparber, Global well-posedness for cubic NLS with nonlinear damping, Comm. Partial Differential Equations, 35 (2010), 4832-4845.
doi: 10.1080/03605300903540943. |
[2] |
H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. II. Existence of infinitely many solutions, Arch. Rational Mech. Anal., 82 (1983), 347-375.
doi: 10.1007/BF00250556. |
[3] |
T. Cazenave, Semilinear Schrödinger Equations, volume 10 of Courant Lecture Notes in Mathematics, New York University Courant Institute of Mathematical Sciences, New York, 2003. |
[4] |
T. Cazenave and F. Weissler, Some remarks on the nonlinear Schrödinger equation in the subcritical case, in New Methods and Results in Nonlinear Field Equations (Bielefeld, 1987), 59-69, Lecture Notes in Phys., 347, Springer, Berlin, 1989.
doi: 10.1007/BFb0025761. |
[5] |
J. Colliander and P. Raphael, Rough blowup solutions to the $L^2$ critical NLS, Math. Ann., 345 (2009), 307-366.
doi: 10.1007/s00208-009-0355-3. |
[6] |
M. Darwich, Blowup for the Damped $L^2$critical nonlinear Shrödinger equations, Advances in Differential Equations, 17 (2012), 337-367. |
[7] |
G. Fibich and F. Merle, Self-focusing on bounded domains, Phys. D, 155 (2001), 132-158.
doi: 10.1016/S0167-2789(01)00249-4. |
[8] |
G. Fibich and M. Klein, Nonlinear-damping continuation of the nonlinear Schrödinger equation-a numerical study, Physica D, 241 (2012), 519-527.
doi: 10.1016/j.physd.2011.11.008. |
[9] |
A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London, 1969. |
[10] |
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. |
[11] |
T. Kato, On nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Phys. Théor., 46 (1987), 113-129. |
[12] |
M. K. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p=0$ in $R^n$, Arch. Rational Mech. Anal., 105 (1989), 243-266.
doi: 10.1007/BF00251502. |
[13] |
P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. II, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 223-283. |
[14] |
F. Merle, Determination of blow-up solutions with minimal mass for nonlinear Schröinger equations with critical power, Duke Math. J., 69 (1993), 427-454.
doi: 10.1215/S0012-7094-93-06919-0. |
[15] |
F. Merle and P. Raphael, Blow up dynamic and upper bound on the blow up rate for critical nonlinear Schrödinger equation, In Journées "Équations aux Dérivées Partielles'' (Forges-les-Eaux, 2002), pages Exp. No. XII, 5. Univ. Nantes, Nantes, 2002. |
[16] |
F. Merle and P. Raphael, Sharp upper bound on the blow-up rate for the critical nonlinear Schrödinger equation, Geom. Funct. Anal., 13 (2003), 591-642.
doi: 10.1007/s00039-003-0424-9. |
[17] |
F. Merle and P. Raphael, 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. |
[18] |
F. Merle and P. Raphael, Profiles and quantization of the blow up mass for critical nonlinear Schrödinger equation, Comm. Math. Phys., 253 (2005), 675-704.
doi: 10.1007/s00220-004-1198-0. |
[19] |
F. Merle and P. Raphael, 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 (electronic).
doi: 10.1090/S0894-0347-05-00499-6. |
[20] |
M. Ohta and G. Todorova, Remarks on global existence and blowup for damped nonlinear Schrödinger equations, Discrete Contin. Dyn. Syst., 23 (2009), 1313-1325.
doi: 10.3934/dcds.2009.23.1313. |
[21] |
T. Passota, C. Sulemb and P. L. Sulem, Linear versus nonlinear dissipation for critical NLS equation, Physica D, 203 (2005), 167-184.
doi: 10.1016/j.physd.2005.03.011. |
[22] |
F. Planchon and P. Raphaël, Existence and stability of the log-log blow-up dynamics for the $L^2$-critical nonlinear Schrödinger equation in a domain, Ann. Henri Poincaré, 8 (2007), 1177-1219.
doi: 10.1007/s00023-007-0332-x. |
[23] |
P. Raphael, Stability of the log-log bound for blow up solutions to the critical non linear Schrödinger equation, Math. Ann., 331 (2005), 577-609.
doi: 10.1007/s00208-004-0596-0. |
[24] |
M. Tsutsumi, Nonexistence of global solutions to the Cauchy problem for the damped nonlinear Schrödinger equations, SIAM J. Math. Anal., 15 (1984), 357-366.
doi: 10.1137/0515028. |
[25] |
M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys., 87 (1982/83), 567-576. |
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