Finite-time blowup for a Schr\"odinger equation with nonlinear source term.

@inproceedings{Cazenave2018FinitetimeBF,
  title={Finite-time blowup for a Schr\"odinger equation with nonlinear source term.},
  author={Thierry Cazenave and Yvan Martel and Lifeng Zhao},
  year={2018}
}
We consider the nonlinear Schr\"odinger equation \[ u_t = i \Delta u + | u |^\alpha u \quad \mbox{on ${\mathbb R}^N $, $\alpha>0$,} \] for $H^1$-subcritical or critical nonlinearities: $(N-2) \alpha \le 4$. Under the additional technical assumptions $\alpha\geq 2$ (and thus $N\leq 4$), we construct $H^1$ solutions that blow up in finite time with explicit blow-up profiles and blow-up rates. In particular, blowup can occur at any given finite set of points of ${\mathbb R}^N$. The construction… CONTINUE READING

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