## 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

### Citations:

Zbl 1061.35135; Zbl 1185.35263
Full Text:

### References:

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