Finite-time blowup for a Schrödinger equation with nonlinear source term

We consider the nonlinear Schrodinger equation \begin{document}${u_t} = i\Delta u + {\left| u \right|^\alpha }u\;\;\;\;{\rm{on}}\;\;\;{\mathbb{R}^N},\;\;\alpha >0$ \end{document} for \begin{document}$H^1$\end{document} -subcritical or critical nonlinearities: \begin{document}$(N-2) α ≤ 4$\end{document} . Under the additional technical assumptions \begin{document}$α≥ 2$\end{document} (and thus \begin{document}$N≤4$\end{document} ), we construct \begin{document}$H^1$\end{document} 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 \begin{document}$\mathbb{R}^N$\end{document} . The construction involves explicit functions \begin{document}$U$\end{document} , solutions of the ordinary differential equation \begin{document}$U_t=|U|^α U$\end{document} . In the simplest case, \begin{document}$U(t,x)=(|x|^k-α t)^{-\frac 1α}$\end{document} for \begin{document}$t , \begin{document}$x∈ \mathbb{R}^N$\end{document} . For \begin{document}$k$\end{document} sufficiently large, \begin{document}$U$\end{document} satisfies \begin{document}$|Δ U|\ll U_t$\end{document} close to the blow-up point \begin{document}$(t,x)=(0,0)$\end{document} , so that it is a suitable approximate solution of the problem. To construct an actual solution u close to U , we use energy estimates and a compactness argument.