Blowup on an Arbitrary Compact Set for a Schrödinger Equation with Nonlinear Source Term

Abstract

We consider the nonlinear Schrödinger equation on ${\mathbb R}^N$, $N\ge 1$,

$$\begin{aligned} \partial _t u = i \varDelta u + \lambda | u |^\alpha u \end{aligned}$$

with $\lambda \in {\mathbb C}$ and $\mathfrak {R}\lambda >0$, 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 $U_0(t,x) = ( \mathfrak {R}\lambda )^{- \frac{1}{\alpha }} (-\alpha t + A(x) )^{ -\frac{1}{\alpha } - i \frac{\mathfrak {I}\lambda }{\alpha \mathfrak {R}\lambda } }$, 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 Cazenave et al. (Discrete Contin Dyn Syst 39(2):1171–1183, 2019. https://doi.org/10.3934/dcds.2019050; Solutions blowing up on any given compact set for the energy subcritical wave equation. 2018. arXiv:1812.03949).