We consider the nonlinear Schrödinger equation
${u_t} = i\Delta u + {\left| u \right|^\alpha }u\;\;\;\;{\rm{on}}\;\;\;{\mathbb{R}^N},\;\;\alpha >0$
for $H^1$-subcritical or critical nonlinearities: $(N-2) α ≤ 4$. Under the additional technical assumptions $α≥ 2$ (and thus $N≤4$), 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 $U_t=|U|^α U$. In the simplest case, $U(t,x)=(|x|^k-α t)^{-\frac 1α}$ for $t<0$, $x∈ \mathbb{R}^N$. For $k$ sufficiently large, $U$ satisfies $|Δ U|\ll U_t$ 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.
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