Abstract
We consider finite time blow up solutions to the critical nonlinear Schrödinger equation For a suitable class of initial data in the energy space H1, we prove that the solution splits in two parts: the first part corresponds to the singular part and accumulates a quantized amount of L2 mass at the blow up point, the second part corresponds to the regular part and has a strong L2 limit at blow up time.
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Berestycki, H., Lions, P.-L.: Nonlinear scalar field equations. I. Existence of a ground state. Arch. Rat. Mech. Anal. 82(4), 313–345 (1983)
Bourgain, J., Wang, W.: Construction of blowup solutions for the nonlinear Schrödinger equation with critical nonlinearity. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25(1–2), 197–215 (1998)
Cazenave, Th., Weissler, F.: Some remarks on the nonlinear Schrödinger equation in the critical case. In: Nonlinear semigroups, partial differential equations and attractors (Washington, DC, 1987), Lecture Notes in Math. 1394, Berlin: Springer, 1989, pp. 18–29
Ginibre, J., Velo, G.: On a class of nonlinear Schrödinger equations. I. The Cauchy problem, general case. J. Funct. Anal. 32(1), 1–32 (1979)
Glangetas, L., Merle, F.: Existence of self-similar blow-up solutions for Zakharov equation in dimension two. I. Commun. Math. Phys. 160(1), 173–215 (1994)
Kwong, M.K.: Uniqueness of positive solutions of Δu-u+up=0 in Rn. Arch. Rati. Mech. Anal. 105(3), 243–266 (1989)
Landman, M.J., Papanicolaou, G.C., Sulem, C., Sulem, P.-L.: Rate of blowup for solutions of the nonlinear Schrödinger equation at critical dimension. Phys. Rev. A (3) 38(8), 3837–3843 (1988)
Martel, Y., Merle, F.: Stability of blow-up profile and lower bounds for blow-up rate for the critical generalized KdV equation, Ann. of Math. (2) 155(1), 235–280 (2002)
Merle, F.: Construction of solutions with exact k blow up points for the Schrödinger equation with critical power. Commun. Math.Phys. 129, 223–240 (1990)
Merle, F.: Determination of blow-up solutions with minimal mass for nonlinear Schrödinger equations with critical power. Duke Math. J. 69(2), 427–454 (1993)
Merle, F., Raphael, P.: Blow up dynamic and upper bound on the blow up rate for critical nonlinear Schrödinger equation. To appear in Annals of Math.
Merle, F., Raphael, P.: Sharp upper bound on the blow up rate for critical nonlinear Schrödinger equation, Geom. Funct. Anal. 13, 591–642 (2003)
Merle, F., Raphael, P.: On Universality of Blow up Profile for L2 critical nonlinear Schrödinger equation. Invent. Math. 156, 565–672 (2004)
Merle, F., Raphael, P.: Sharp lower bound on the blow up rate for critical nonlinear Schrödinger equation. Preprint
Merle, F., Tsutsumi, Y.: L2 concentration of blow up solutions for the nonlinear Schrödinger equation with critical power nonlinearity. J. Diff. Eq. 84, 205–214 (1990)
Nawa, H.: Asymptotic and limiting profiles of blowup solutions of the nonlinear Schrödinger equation with critical power. Commun. Pure Appl. Math. 52(2), 193–270 (1999)
Perelman, G.: On the blow up phenomenon for the critical nonlinear Schrödinger equation in 1D. Ann. Henri. Poincaré 2, 605–673 (2001)
Raphael, P.: Stability of the log-log bound for blow up solutions to the critical nonlinear Schrödinger equation. To appear in Math. Annalen
Sulem, C., Sulem, P.L.: The nonlinear Schrödinger equation. Self-focusing and wave collapse. Applied Mathematical Sciences 139, New York: Springer-Verlag, 1999
Weinstein, M.I.: Nonlinear Schrödinger equations and sharp interpolation estimates. Commun. Math. Phys. 87, 567–576 (1983)
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Communicated by P. Constantin
Part of this work has been supported by grant DMS-0111298.
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Merle, F., Raphael, P. Profiles and Quantization of the Blow Up Mass for Critical Nonlinear Schrödinger Equation. Commun. Math. Phys. 253, 675–704 (2005). https://doi.org/10.1007/s00220-004-1198-0
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DOI: https://doi.org/10.1007/s00220-004-1198-0