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

We consider the nonlinear Schr"odinger equation u_t = i \Delta u + | u |^\alpha u \quad on ${\mathbb R^N $, $\alpha>0$,} for $H^1$-subcritical or critical nonlinearities: $(N-2)\alpha \le 4$. Under the additional technical assumptions $\alpha \ge 2$ (and thus $N\le 4$), 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 involves explicit functions $U$, solutions of the ordinary differential equation $U_t=\mathrm{\mid }U{\mathrm{\mid }}^{\alpha }U$. In the simplest case, $U(t,x)=(\mathrm{\mid }x{\mathrm{\mid }}^k-\alpha t{)}^{-\frac{1}{\alpha }}$ for $t<0$, $x\in {\mathbb{R}}^N$. For $k$ sufficiently large, $U$ satisfies $\mathrm{\mid }\mathrm{\Delta }U\mathrm{\mid }\ll U_t$ close to the blow-up point $(t,x)=(0,0)$, 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.