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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

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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

M. Darwich, Blowup for the Damped $L^2$critical nonlinear Shrödinger equations, Advances in Differential Equations, 17 (2012), 337-367.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London, 1969. Google Scholar

[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.  Google Scholar

[11]

T. Kato, On nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Phys. Théor., 46 (1987), 113-129.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  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 Journées "Équations aux Dérivées Partielles'' (Forges-les-Eaux, 2002), pages Exp. No. XII, 5. Univ. Nantes, Nantes, 2002.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  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, J. Amer. Math. Soc., 19 (2006), 37-90 (electronic). 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, Discrete Contin. Dyn. Syst., 23 (2009), 1313-1325. 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, Physica D, 203 (2005), 167-184. 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, Ann. Henri Poincaré, 8 (2007), 1177-1219. 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, Math. Ann., 331 (2005), 577-609. 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, SIAM J. Math. Anal., 15 (1984), 357-366. 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, Comm. Partial Differential Equations, 35 (2010), 4832-4845. doi: 10.1080/03605300903540943.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

M. Darwich, Blowup for the Damped $L^2$critical nonlinear Shrödinger equations, Advances in Differential Equations, 17 (2012), 337-367.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London, 1969. Google Scholar

[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.  Google Scholar

[11]

T. Kato, On nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Phys. Théor., 46 (1987), 113-129.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  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 Journées "Équations aux Dérivées Partielles'' (Forges-les-Eaux, 2002), pages Exp. No. XII, 5. Univ. Nantes, Nantes, 2002.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  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, J. Amer. Math. Soc., 19 (2006), 37-90 (electronic). 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, Discrete Contin. Dyn. Syst., 23 (2009), 1313-1325. 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, Physica D, 203 (2005), 167-184. 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, Ann. Henri Poincaré, 8 (2007), 1177-1219. 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, Math. Ann., 331 (2005), 577-609. 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, SIAM J. Math. Anal., 15 (1984), 357-366. 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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