## Global well-posedness, scattering and blow-up for the energy-critical, focusing, nonlinear Schrödinger equation in the radial case.(English)Zbl 1115.35125

Initial value problem for nonlinear Schrödinger equation $i\partial _t u + \Delta u\pm | u | ^{\frac{4}{N - 2}}u = 0, \quad (x,t) \in \mathbb R^N\times \mathbb R,$
$u| _{t = 0} = u_0 \in \dot H^1(\mathbb R^N),$ is considered where the $$-$$ sign corresponds to the defocusing problem, while the $$+$$ sign corresponds to the focusing problem. The authors show: if $$E(u_0 ) < E(W),\| {u_0 } \| _{\dot H^1}< \| W \| _{\dot H^1}$$, $$N = 3,4,5$$ and $$u_0$$ is radial, then the solution $$u$$ with data $$u_0$$ at $$t = 0$$ is defined for all time and there exists $$u_{0, + }$$, $$u_{0, - }$$ in $$\dot H^1$$ such that $\lim _{t \to + \infty} \| u( t) - e^{it\Delta}u_{0, + }\| _{\dot H^1}= 0, \quad \lim_{t \to - \infty} \| u( t) - e^{it\Delta}u_{0, - }\| _{\dot H^1}= 0.$ Here $$W(x) = 1 / (1 + | x | ^2 / N(N - 2))^{\frac{N - 2}{2}}$$ is in $$\dot H^1(\mathbb R^N)$$ and solves the elliptic equation $\Delta W + | W| ^{\frac{4}{N - 2}}W = 0.$ The authors also show that for $$u_0$$ radial, $$| x| u_0 \in L^2(\mathbb R^N)$$, $$E(u_0 ) < E(W)$$, but $$\| {u_0 }\| _{\dot H^1}>\| W\| _{\dot H^1}$$, the solution must break down in finite time.

### MSC:

 35Q55 NLS equations (nonlinear Schrödinger equations) 35B40 Asymptotic behavior of solutions to PDEs 35P25 Scattering theory for PDEs
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### References:

  Aubin, T.: Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire. J. Math. Pures Appl., IX. Sér. 55, 269–296 (1976) · Zbl 0336.53033  Bahouri, H., Gérard, P.: High frequency approximation of solutions to critical nonlinear wave equations. Am. J. Math. 121, 131–175 (1999) · Zbl 0919.35089  Berestycki, H., Cazenave, T.: Instabilité des états stationnaires dans les équations de Schrödinger et de Klein–Gordon non linéaires. C. R. Acad. Sci., Paris, Sér. I, Math. 293, 489–492 (1981) · Zbl 0492.35010  Bergh, J., Lofstrom, J.: Interpolation Spaces. An Introduction. Grundlehren der Mathematischen Wissenschaften, No. 223. Berlin, New York: Springer 1976  Bourgain, J.: Global well-posedness of defocusing critical nonlinear Schrödinger equation in the radial case. J. Am. Math. Soc. 12, 145–171 (1999) · Zbl 0958.35126  Bourgain, J.: New global well-posedness results for nonlinear Schrödinger equations. AMS Colloquium Publications, 46, 1999 · Zbl 0933.35178  Cazenave, T.: Semilinear Schrödinger Equations. Courant Lecture Notes in Mathematics, vol. 10. New York: New York University Courant Institute of Mathematical Sciences 2003 · Zbl 1055.35003  Cazenave, T., Weissler, F.B.: The Cauchy problem for the critical nonlinear Schrödinger equation in H s . Nonlinear Anal., Theory Methods Appl. 14, 807–836 (1990) · Zbl 0706.35127  Colliander, J., Keel, M., Staffilani, G., Takaoke, H., Tao, T.: Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in $$\mathbb{R}$$3. To appear in Ann. Math. · Zbl 1178.35345  Gérard, P.: Description du défaut de compacité de l’injection de Sobolev. ESAIM Control Optim. Calc. Var. 3, 213–233 (1998) · Zbl 0907.46027  Gerard, P., Meyer, Y., Oru, F.: Inégalités de Sobolev précisées, Séminaire sur les Équations aux Dérivées Partielles, 1996–1997, Exp. No. IV, 11. École Polytech. 1997  Glassey, R.T.: On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations. J. Math. Phys. 18, 1794–1797 (1977) · Zbl 0372.35009  Grillakis, M.G.: On nonlinear Schrödinger equations. Commun. Partial Differ. Equations 25, 1827–1844 (2000) · Zbl 0970.35134  Keel, M., Tao, T.: Endpoint Strichartz estimates. Am. J. Math. 120, 955–980 (1998) · Zbl 0922.35028  Keraani, S.: On the defect of compactness for the Strichartz estimates of the Schrödinger equations. J. Differ. Equations 175, 353–392 (2001) · Zbl 1038.35119  Keraani, S.: On the blow up phenomenon of the critical Schrödinger equation. J. Funct. Anal. 235, 171–192 (2006) · Zbl 1099.35132  Merle, F.: Determination of blow-up solutions with minimal mass for nonlinear Schrödinger equations with critical power. Duke Math. J. 69, 427–454 (1993) · Zbl 0808.35141  Merle, F.: On uniqueness and continuation properties after blow-up time of self-similar solutions of nonlinear Schrödinger equation with critical exponent and critical mass. Comm. Pure Appl. Math. 45, 203–254 (1992) · Zbl 0767.35084  Merle, F.: Existence of blow-up solutions in the energy space for the critical generalized KdV equation. J. Am. Math. Soc. 14, 555–578 (2001) · Zbl 0970.35128  Merle, F., Tsutsumi, Y.: L 2 concentration of blow-up solutions for the nonlinear Schrödinger equation with critical power nonlinearity. J. Differ. Equations 84, 205–214 (1990) · Zbl 0722.35047  Merle, F., Vega, L.: Compactness at blow-up time for L 2 solutions of the critical nonlinear Schrödinger equation in 2D. Int. Math. Res. Not. 8, 399–425 (1998) · Zbl 0913.35126  Ogawa, T., Tsutsumi, Y.: Blow-up of H 1 solution for the nonlinear Schrödinger equation. J. Differ. Equations 92, 317–330 (1991) · Zbl 0739.35093  Raphael, P.: Existence and stability of a solution blowing-up on a sphere for a L 2 supercritical nonlinear Schrödinger equation. Duke Math. J. 134(2), 199–258 (2006) · Zbl 1117.35077  Ryckman, E., Visan, M.: Global well-posedness and scattering for the defocusing energy-critical nonlinear Schrödinger equation in $$\mathbb{R}$$1+4. To appear in Amer. J. Math. Preprint, http://arkiv.org/abs/math.AP/0501462 · Zbl 1160.35067  Talenti, G.: Best constant in Sobolev inequality. Ann. Mat. Pura Appl. 110, 353–372 (1976) · Zbl 0353.46018  Tao, T.: Global well-posedness and scattering for the higher-dimensional energy-critical nonlinear Schrödinger equation for radial data. New York J. Math. 11, 57–80 (2005) · Zbl 1119.35092  Tao, T., Visan, M.: Stability of energy-critical nonlinear Schrödinger equations in high dimensions. Electron. J. Differ. Equ. 118, 28 (2005) · Zbl 1245.35122  Visan, M.: The defocusing energy-critical nonlinear Schrödinger equation in higher dimensions. Preprint, http://arkiv.org/abs/math.AP/0508298 · Zbl 1131.35081  Weinstein, M.: Nonlinear Schrödinger equations and sharp interpolation estimates. Commun. Math. Phys. 87, 567–576 (1982/83) · Zbl 0527.35023  Zhang, J.: Sharp conditions of global existence for nonlinear Schrödinger and Klein–Gordon equations. Nonlinear Anal., Theory Methods Appl. 48, 191–207 (2002) · Zbl 1038.35131
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