Blowup on an arbitrary compact set for a Sch\"odinger equation with nonlinear source term

Abstract

We consider the nonlinear Schr\"odinger equation on ${\mathbb R}^N $, $N\ge 1$, \begin{equation*} \partial _t u = i \Delta u + \lambda | u |^\alpha u \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 $u'= \lambda | u |^\alpha u$. 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].