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

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).

我们认为在非线性薛定谔方程\（{\ mathbb R} ^ N \） ，\（N \ GE 1 \） ，

与\（\ lambda \ in {\ mathbb C} \）和\（\ mathfrak {R} \ lambda> 0 \）中，对于\（H ^ 1 \）-次临界非线性，即\（\ alpha> 0 \）和\（（N-2）\ alpha <4 \）。给定一个紧凑集\（K \ subset {\ mathbb {R}} ^ N \），我们构造\（H ^ 1 \）在\（（-T，0）\）上为一些\（T > 0 \） ，和炸毁上ķ在\（T = 0 \） 。该结构基于适当的ansatz。最初的ansatz很简单\（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}} \），其中\（A \ ge 0 \）完全在K上消失，这是一个解决方案ODE的\（u'= \ lambda | u | ^ \ alpha u \）。我们使用ODE技术感应地完善了ansatz。我们通过能量估计和紧凑性论证来完成证明。这种策略使人联想到Cazenave等人。（离散Contin Dyn Syst 39（2）：1171–1183，2019. https://doi.org/10.3934/dcds.2019050;解决方案在能量亚临界波动方程的任何给定紧缩集上爆炸.2018.arXiv：1812.03949 ）。