On the L^2-critical nonlinear Schrödinger Equation with a nonlinear damping

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November 2014, 13(6): 2377-2394. doi: 10.3934/cpaa.2014.13.2377

On the $L^2$-critical nonlinear Schrödinger Equation with a nonlinear damping

Mohamad Darwich ^1,

1.	Laboratoire de Mathématiques et Physique Théorique, UMR 7350, Tours, France

Received October 2013 Revised May 2014 Published July 2014

Abstract
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We consider the Cauchy problem for the $L^2$-critical nonlinear Schrödinger equation with a nonlinear damping. According to the power of the damping term, we prove the global existence or the existence of finite time blowup dynamics with the log-log blow-up speed for $\|\nabla u(t)\|_{L^2}$.

Keywords: PDE's., Shrödinger equations.

Mathematics Subject Classification: Primary: 35M11; Secondary: 35A0.

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

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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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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