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

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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 $\|\nabla u(t)\|_{L^2}$.
    Mathematics Subject Classification: Primary: 35M11; Secondary: 35A01.


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  • [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.


    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.


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


    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.


    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.


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


    G. Fibich and F. Merle, Self-focusing on bounded domains, Phys. D, 155 (2001), 132-158.doi: 10.1016/S0167-2789(01)00249-4.


    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.


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


    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.


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


    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.


    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.


    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.


    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.


    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.


    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.


    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.


    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.


    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.


    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.


    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.


    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.


    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.


    M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys., 87 (1982/83), 567-576.

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