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Profiles and quantization of the blow up mass for critical nonlinear Schrödinger equation. (English) Zbl 1062.35137
Summary: We consider finite time blow up solutions to the critical nonlinear Schrödinger equation $$iu_t =- \Delta u - | u|^{\frac4N}u$$ for which $$\lim_{t\uparrow T < +\infty}|\nabla u(t)|_{L^2} = +\infty$$. For a suitable class of initial data in the energy space $$H^1$$, we prove that the solution splits in two parts: the first part corresponds to the singular part and accumulates a quantized amount of $$L^2$$ mass at the blow up point, the second part corresponds to the regular part and has a strong $$L^2$$ limit at blow up time.

##### MSC:
 35Q55 NLS equations (nonlinear Schrödinger equations) 35B44 Blow-up in context of PDEs
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##### References:
 [1] Berestycki, H., Lions, P.-L.: Nonlinear scalar field equations. I. Existence of a ground state. Arch. Rat. Mech. Anal. 82(4), 313-345 (1983) · Zbl 0533.35029 [2] 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) · Zbl 1043.35137 [3] 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 [4] 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) · Zbl 0396.35028 [5] 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) · Zbl 0808.35137 [6] Kwong, M.K.: Uniqueness of positive solutions of ?u-u+up=0 in Rn. Arch. Rati. Mech. Anal. 105(3), 243-266 (1989) · Zbl 0676.35032 [7] 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) [8] 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) · Zbl 1005.35081 [9] 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) · Zbl 0707.35021 [10] 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) · Zbl 0808.35141 [11] 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. · Zbl 1117.35075 [12] Merle, F., Raphael, P.: Sharp upper bound on the blow up rate for critical nonlinear Schrödinger equation, Geom. Funct. Anal. 13, 591-642 (2003) · Zbl 1061.35135 [13] Merle, F., Raphael, P.: On Universality of Blow up Profile for L2 critical nonlinear Schrödinger equation. Invent. Math. 156, 565-672 (2004) · Zbl 1067.35110 [14] Merle, F., Raphael, P.: Sharp lower bound on the blow up rate for critical nonlinear Schrödinger equation. Preprint · Zbl 1117.35075 [15] 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) · Zbl 0722.35047 [16] 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) · Zbl 0964.37014 [17] Perelman, G.: On the blow up phenomenon for the critical nonlinear Schrödinger equation in 1D. Ann. Henri. Poincaré 2, 605-673 (2001) · Zbl 1007.35087 [18] Raphael, P.: Stability of the log-log bound for blow up solutions to the critical nonlinear Schrödinger equation. To appear in Math. Annalen [19] Sulem, C., Sulem, P.L.: The nonlinear Schrödinger equation. Self-focusing and wave collapse. Applied Mathematical Sciences 139, New York: Springer-Verlag, 1999 · Zbl 0928.35157 [20] Weinstein, M.I.: Nonlinear Schrödinger equations and sharp interpolation estimates. Commun. Math. Phys. 87, 567-576 (1983) · Zbl 0527.35023
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