\(\log\)-\(\log\) blow up solutions blow up at exactly \(m\) points
The author considers the Cauchy Problem for the mass-critical focusing nonlinear Schrödinger equation \[ i u_t = -\Delta u- | u|^{4/d} u \] for initial conditions in \( H^1(\mathbb{R}^d) \) with \( d =1,2 \). He proves that for arbitrary points \( x_1, \ldots x_m \) there exists a solution \( u \) that blows up in finite time \( T \) at exactly these points. There exists an exact asymptotic formula based on a pseudo-conformal transformation of the global soliton solution showing that this blow-up follows a log-log rule.
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