## Sharp upper bound on the blow-up rate for the critical nonlinear Schrödinger equation.(English)Zbl 1061.35135

The authors consider the Cauchy problem for the critical nonlinear Schrödinger equation $\begin{cases} iu_t = - \Delta u -| u | ^{4 / N}u,&(t,x) \in [0,\,T)\times \mathbb R^N, \\ u(0,x)= u_0(x), &u_0 :\mathbb R^N \to \mathbb C,\end{cases}$ with $$u_0\in H^1(\mathbb R^N)$$, $$N\geq 1$$. It is proved that the following estimation is true $| \nabla u(t) | _{L_2 } \leq C\left( \frac{\ln | \ln (T - t)| }{T-t} \right)^{1 / 2}.$

### MSC:

 35Q55 NLS equations (nonlinear Schrödinger equations) 35B44 Blow-up in context of PDEs
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