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

1. 

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

Received  October 2013 Revised  May 2014 Published  July 2014

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 $\|\nabla u(t)\|_{L^2}$.
Citation: 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
References:
[1]

P. Antonelli and C. Sparber, Global well-posedness for cubic NLS with nonlinear damping,, \emph{Comm. Partial Differential Equations}, 35 (2010), 4832.  doi: 10.1080/03605300903540943.  Google Scholar

[2]

H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. II. Existence of infinitely many solutions,, \emph{Arch. Rational Mech. Anal.}, 82 (1983), 347.  doi: 10.1007/BF00250556.  Google Scholar

[3]

T. Cazenave, Semilinear Schrödinger Equations, volume 10 of Courant Lecture Notes in Mathematics,, New York University Courant Institute of Mathematical Sciences, (2003).   Google Scholar

[4]

T. Cazenave and F. Weissler, Some remarks on the nonlinear Schrödinger equation in the subcritical case,, in \emph{New Methods and Results in Nonlinear Field Equations} (Bielefeld, 347 (1987), 59.  doi: 10.1007/BFb0025761.  Google Scholar

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J. Colliander and P. Raphael, Rough blowup solutions to the $L^2$ critical NLS,, \emph{Math. Ann.}, 345 (2009), 307.  doi: 10.1007/s00208-009-0355-3.  Google Scholar

[6]

M. Darwich, Blowup for the Damped $L^2$critical nonlinear Shrödinger equations,, \emph{Advances in Differential Equations}, 17 (2012), 337.   Google Scholar

[7]

G. Fibich and F. Merle, Self-focusing on bounded domains,, \emph{Phys. D}, 155 (2001), 132.  doi: 10.1016/S0167-2789(01)00249-4.  Google Scholar

[8]

G. Fibich and M. Klein, Nonlinear-damping continuation of the nonlinear Schrödinger equation-a numerical study,, \emph{Physica D}, 241 (2012), 519.  doi: 10.1016/j.physd.2011.11.008.  Google Scholar

[9]

A. Friedman, Partial Differential Equations,, Holt, (1969).   Google Scholar

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T. Hmidi and S. Keraani, Blowup theory for the critical nonlinear Schrödinger equations revisited,, \emph{Int. Math. Res. Not.}, 46 (2005), 2815.  doi: 10.1155/IMRN.2005.2815.  Google Scholar

[11]

T. Kato, On nonlinear Schrödinger equations,, \emph{Ann. Inst. H. Poincar\'e Phys. Th\'eor.}, 46 (1987), 113.   Google Scholar

[12]

M. K. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p=0$ in $R^n$,, \emph{Arch. Rational Mech. Anal.}, 105 (1989), 243.  doi: 10.1007/BF00251502.  Google Scholar

[13]

P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. II,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, 1 (1984), 223.   Google Scholar

[14]

F. Merle, Determination of blow-up solutions with minimal mass for nonlinear Schröinger equations with critical power,, \emph{Duke Math. J.}, 69 (1993), 427.  doi: 10.1215/S0012-7094-93-06919-0.  Google Scholar

[15]

F. Merle and P. Raphael, Blow up dynamic and upper bound on the blow up rate for critical nonlinear Schrödinger equation,, In \emph{Journ\'ees, (2002).   Google Scholar

[16]

F. Merle and P. Raphael, Sharp upper bound on the blow-up rate for the critical nonlinear Schrödinger equation,, \emph{Geom. Funct. Anal.}, 13 (2003), 591.  doi: 10.1007/s00039-003-0424-9.  Google Scholar

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

[18]

F. Merle and P. Raphael, Profiles and quantization of the blow up mass for critical nonlinear Schrödinger equation,, \emph{Comm. Math. Phys.}, 253 (2005), 675.  doi: 10.1007/s00220-004-1198-0.  Google Scholar

[19]

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

[20]

M. Ohta and G. Todorova, Remarks on global existence and blowup for damped nonlinear Schrödinger equations,, \emph{Discrete Contin. Dyn. Syst.}, 23 (2009), 1313.  doi: 10.3934/dcds.2009.23.1313.  Google Scholar

[21]

T. Passota, C. Sulemb and P. L. Sulem, Linear versus nonlinear dissipation for critical NLS equation,, \emph{Physica D}, 203 (2005), 167.  doi: 10.1016/j.physd.2005.03.011.  Google Scholar

[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,, \emph{Ann. Henri Poincar\'e}, 8 (2007), 1177.  doi: 10.1007/s00023-007-0332-x.  Google Scholar

[23]

P. Raphael, Stability of the log-log bound for blow up solutions to the critical non linear Schrödinger equation,, \emph{Math. Ann.}, 331 (2005), 577.  doi: 10.1007/s00208-004-0596-0.  Google Scholar

[24]

M. Tsutsumi, Nonexistence of global solutions to the Cauchy problem for the damped nonlinear Schrödinger equations,, \emph{SIAM J. Math. Anal.}, 15 (1984), 357.  doi: 10.1137/0515028.  Google Scholar

[25]

M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates,, \emph{Comm. Math. Phys.}, 87 (): 567.   Google Scholar

show all references

References:
[1]

P. Antonelli and C. Sparber, Global well-posedness for cubic NLS with nonlinear damping,, \emph{Comm. Partial Differential Equations}, 35 (2010), 4832.  doi: 10.1080/03605300903540943.  Google Scholar

[2]

H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. II. Existence of infinitely many solutions,, \emph{Arch. Rational Mech. Anal.}, 82 (1983), 347.  doi: 10.1007/BF00250556.  Google Scholar

[3]

T. Cazenave, Semilinear Schrödinger Equations, volume 10 of Courant Lecture Notes in Mathematics,, New York University Courant Institute of Mathematical Sciences, (2003).   Google Scholar

[4]

T. Cazenave and F. Weissler, Some remarks on the nonlinear Schrödinger equation in the subcritical case,, in \emph{New Methods and Results in Nonlinear Field Equations} (Bielefeld, 347 (1987), 59.  doi: 10.1007/BFb0025761.  Google Scholar

[5]

J. Colliander and P. Raphael, Rough blowup solutions to the $L^2$ critical NLS,, \emph{Math. Ann.}, 345 (2009), 307.  doi: 10.1007/s00208-009-0355-3.  Google Scholar

[6]

M. Darwich, Blowup for the Damped $L^2$critical nonlinear Shrödinger equations,, \emph{Advances in Differential Equations}, 17 (2012), 337.   Google Scholar

[7]

G. Fibich and F. Merle, Self-focusing on bounded domains,, \emph{Phys. D}, 155 (2001), 132.  doi: 10.1016/S0167-2789(01)00249-4.  Google Scholar

[8]

G. Fibich and M. Klein, Nonlinear-damping continuation of the nonlinear Schrödinger equation-a numerical study,, \emph{Physica D}, 241 (2012), 519.  doi: 10.1016/j.physd.2011.11.008.  Google Scholar

[9]

A. Friedman, Partial Differential Equations,, Holt, (1969).   Google Scholar

[10]

T. Hmidi and S. Keraani, Blowup theory for the critical nonlinear Schrödinger equations revisited,, \emph{Int. Math. Res. Not.}, 46 (2005), 2815.  doi: 10.1155/IMRN.2005.2815.  Google Scholar

[11]

T. Kato, On nonlinear Schrödinger equations,, \emph{Ann. Inst. H. Poincar\'e Phys. Th\'eor.}, 46 (1987), 113.   Google Scholar

[12]

M. K. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p=0$ in $R^n$,, \emph{Arch. Rational Mech. Anal.}, 105 (1989), 243.  doi: 10.1007/BF00251502.  Google Scholar

[13]

P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. II,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, 1 (1984), 223.   Google Scholar

[14]

F. Merle, Determination of blow-up solutions with minimal mass for nonlinear Schröinger equations with critical power,, \emph{Duke Math. J.}, 69 (1993), 427.  doi: 10.1215/S0012-7094-93-06919-0.  Google Scholar

[15]

F. Merle and P. Raphael, Blow up dynamic and upper bound on the blow up rate for critical nonlinear Schrödinger equation,, In \emph{Journ\'ees, (2002).   Google Scholar

[16]

F. Merle and P. Raphael, Sharp upper bound on the blow-up rate for the critical nonlinear Schrödinger equation,, \emph{Geom. Funct. Anal.}, 13 (2003), 591.  doi: 10.1007/s00039-003-0424-9.  Google Scholar

[17]

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

[18]

F. Merle and P. Raphael, Profiles and quantization of the blow up mass for critical nonlinear Schrödinger equation,, \emph{Comm. Math. Phys.}, 253 (2005), 675.  doi: 10.1007/s00220-004-1198-0.  Google Scholar

[19]

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

[20]

M. Ohta and G. Todorova, Remarks on global existence and blowup for damped nonlinear Schrödinger equations,, \emph{Discrete Contin. Dyn. Syst.}, 23 (2009), 1313.  doi: 10.3934/dcds.2009.23.1313.  Google Scholar

[21]

T. Passota, C. Sulemb and P. L. Sulem, Linear versus nonlinear dissipation for critical NLS equation,, \emph{Physica D}, 203 (2005), 167.  doi: 10.1016/j.physd.2005.03.011.  Google Scholar

[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,, \emph{Ann. Henri Poincar\'e}, 8 (2007), 1177.  doi: 10.1007/s00023-007-0332-x.  Google Scholar

[23]

P. Raphael, Stability of the log-log bound for blow up solutions to the critical non linear Schrödinger equation,, \emph{Math. Ann.}, 331 (2005), 577.  doi: 10.1007/s00208-004-0596-0.  Google Scholar

[24]

M. Tsutsumi, Nonexistence of global solutions to the Cauchy problem for the damped nonlinear Schrödinger equations,, \emph{SIAM J. Math. Anal.}, 15 (1984), 357.  doi: 10.1137/0515028.  Google Scholar

[25]

M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates,, \emph{Comm. Math. Phys.}, 87 (): 567.   Google Scholar

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