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Stability of the log-log bound for blow up solutions to the critical nonlinear Schrödinger equation. (English) Zbl 1082.35143

The paper continues the investigation of the blow up solutions of the critical nonlinear Schrödinger equation in \(\mathbb R^N\) \[ iu_t=-\Delta u-u| u| ^{4/N} \] started by F. Merle and the author in their previous papers [ibid. 101, No. 1, 157–222 (2005; Zbl 1185.35263) and Geom. Funct. Anal. 13, No. 3, 591–642 (2003; Zbl 1061.35135)]. In those papers, the universal sharp upper bound for the blow up solutions near the blow up time \(T_u\): the so called log-log law which has the form \[ \| \nabla u(t)\| _{L^2}\leq C\left(\frac{\log| \log(T_u-t)| }{T_u-t}\right)^{1/2} \tag{1} \] has been established for the case of initial data with negative energy, assuming some spectral property for linear Schrödinger operators involving the ground state \(Q\) (which is rigorously verified for \(N=1\)).
In the present paper, the author studies the case where the initial data has positive energy, assuming the same spectral property. In particular, it is proven that if the solution \(u(t)\) for initial data with positive energy blows up in finite time \(T_u\), then \(u(t)\) satisfies either the log-log upper bound {1} or the lower bound \[ \| \nabla u(t)\| _{L^2}\geq C(T_u-t)^{-1} \tag{2} \] for \(t\) close to \(T_u\) (the last lower bound is sharp since the known explicit blow up solution \(S(t)\) constructed by the ground state \(Q\) and the pseudo-conformal transformation has exactly the same asymptotic behaviour near the blow up). Moreover, it is proved that the log-log type blow up is stable in the sense that the set of initial data \(u_0\) such that the corresponding solution \(u(t)\) blows up in finite time with the log-log upper bound is open in \(H^1(\mathbb R^N)\).

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35B44 Blow-up in context of PDEs
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[1] Berestycki, H., Lions, P.-L.: Nonlinear scalar field equations. I. Existence of a ground state. Arch. Rational Mech. Anal. 82, 313-345 (1983) · Zbl 0533.35029
[2] Bourgain, J.: Global solutions of nonlinear Schrödinger equations. American Mathematical Society Colloquium Publications, 46. American Mathematical Society, Providence, RI, 1999 · Zbl 0933.35178
[3] 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 (1997) (1998) · Zbl 1043.35137
[4] Cazenave, Th., Weissler, F.: Some remarks on the nonlinear Schrödinger equation in the critical case. Nonlinear semigroups, partial differential equations and attractors (Washington, DC, 1987), 18-29, Lecture Notes in Math., 1394, Springer, Berlin, 1989
[5] Ginibre, J., Velo, G.: On a class of nonlinear Schrödinger equations. I. The Cauchy problem, general case. J. Funct. Anal. 32, 1-32 (1979) · Zbl 0396.35028
[6] Kato, T.: On nonlinear Schrödinger equations. Ann. Inst. H. Poincaré Phys. Theor. 46, 113-129 (1987) · Zbl 0632.35038
[7] Kwong, M. K.: Uniqueness of positive solutions of ?u-u+up=0 in Rn. Arch. Rational Mech. Anal. 105, 243-266 (1989) · Zbl 0676.35032
[8] 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, 3837-3843 (1988)
[9] 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
[10] Merle, F., Raphael, P.: Blow up dynamic and upper bound on the blow up rate for critical nonlinear Schrödinger equation. To appear in Ann. of Math. · Zbl 1117.35075
[11] 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
[12] Merle, F., Raphael, P.: On universility of blow-up profile for L2 critical non linear Schrödinger equation. Invent. Math. 156, 565-572 (2004) · Zbl 1067.35110
[13] Perelman, G.: On the blow up phenomenon for the critical nonlinear Schrödinger equation in 1D. Ann. H. Poincaré 2, 605-673 (2001) · Zbl 1007.35087
[14] Weinstein, M.I.: Modulational stability of ground states of nonlinear Schrödinger equations. SIAM J. Math. Anal. 16, 472-491 (1985) · Zbl 0583.35028
[15] Weinstein, M.I.: Nonlinear Schrödinger equations and sharp interpolation estimates. Commun. Math. Phys. 87 , 567-576 (1983) · Zbl 0527.35023
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