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On a sharp lower bound on the blow-up rate for the \(L^2\) critical nonlinear Schrödinger equation. (English) Zbl 1075.35077

Summary: We consider the \(L^2\) critical nonlinear Schrödinger equation \(iu_t=-\Delta u-| u|^{\frac{4}{N}}u\) with initial condition in the energy space \(u(0,x)=u_0\in H^1\) and study the dynamics of finite time blow up solutions. In an earlier sequence of papers, the authors established for a certain class of initial data on the basis of dispersive properties in \(L^2_{\text{loc}}\) a sharp and stable upper bound on the blow up rate: \[ |\nabla u(t)|_{L^2}\leq C\left(\frac{\log|\log(T-t)|}{T-t}\right)^{\frac{1}{2}}. \] In an earlier paper, the authors then addressed the question of a lower bound on the blow up rate and proved for this class of initial data the nonexistence of self-similar solutions, that is, \(\lim_{t\to T}\sqrt{T-t}|\nabla u(t)|_{L^2}=+\infty.\)
In this paper, we prove the sharp lower bound \[ |\nabla u(t)|_{L^2}\geq C_2 \left(\frac{\log|\log(T-t)|}{T-t}\right)^{\frac{1}{2}} \] by exhibiting the dispersive structure in the scaling invariant space \(L^2\) for this log-log regime. In addition, we extend to the pure energy space \(H^1\) a dynamical characterization of the solitons among the zero energy solutions.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35B44 Blow-up in context of PDEs
35C08 Soliton solutions
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