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

Abstract

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 ${\mathbb{R}}^N$, $\alpha >0$,} \end{equation*} with $\lambda \in \mathbb{C}$ and $\Re \lambda >0$, for $H^1$-subcritical nonlinearities, i.e. $\alpha >0$ and $(N-2)\alpha <4$. 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 $U_0(t,x)=(\Re \lambda {)}^{-\frac{1}{\alpha }}(-\alpha t+A(x){)}^{-\frac{1}{\alpha }-i\frac{\Im \lambda }{\alpha \Re \lambda }}$, where $A\ge 0$ vanishes exactly on $K$, which is a solution of the ODE $u^{\prime }=\lambda |u{|}^{\alpha }u$. 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].