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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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