Blowup on an arbitrary compact set for a Schödinger equation with nonlinear source term

We consider the nonlinear Schrödinger equation on ${\mathbb R}^N$, $N\ge 1$, \begin{equation*} \partial _t u = i Δu + λ| u |^αu \quad \mbox{on ${\mathbb R}^N$, $α>0$,} \end{equation*} with $λ\in {\mathbb C}$ and $\Re λ>0$, for $H^1$-subcritical nonlinearities, i.e. $α>0$ and $(N-2) α< 4$. Given a compact set $K \subset {\mathbb R}^N$, we construct $H^1$ solutions that are defined on $(-T,0)$ for some $T>0$, and blow up on $K$ at $t=0$. The construction is based on an appropriate ansatz. The initial ansatz is simply $U_0(t,x) = ( \Re λ)^{- \frac {1} {α}} (-αt + A(x) )^{ -\frac {1} {α} - i \frac {\Im λ} {α\Re λ} }$, where $A\ge 0$ vanishes exactly on $K$, which is a solution of the ODE $u'= λ| u |^αu$. We refine this ansatz inductively, using ODE techniques. We complete the proof by energy estimates and a compactness argument. This strategy is reminiscent of~[3, 4].

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