Stability of the log-log bound for blow up solutions to the critical nonlinear Schrödinger equation
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 \textit{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 \(H^1(\mathbb R^N)\).
scientific article; zbMATH DE number 2156235
scientific article