Non global solutions for non-radial inhomogeneous nonlinear Schrodinger equations

This work concerns the inhomogeneous Schr"odinger equation $$ \mathrm{i}\partial_t u-\mathcal{K}_{s,\lambda}u +F(x,u)=0 , \quad u(t,x):\mathbb{R}\times\mathbb{R}^N\to\mathbb{C}. $$ Here, \(s\in\{1,2\}\), \(N>2s\) and \(\lambda>-(N-2)^2/4\). The linear Schr"odinger operator is \(\mathcal{K}_{s,\lambda}:= (-\Delta)^s +(2-s)\frac{\lambda}{|x|^2}\), and the focusing source term can be local or non-local $$ F(x,u)\in\{|x|^{-2\tau}|u|^{2(q-1)}u,|x|^{-\tau}|u|^{p-2} \big(J_\alpha *|\cdot|^{-\tau}|u|^p\big)u\}. $$ The Riesz potential is \(J_\alpha(x)=C_{N,\alpha}|x|^{-(N-\alpha)}\), for certain \(0<\alpha<N\). The singular decaying term \(|x|^{-2\tau}\), for some \(\tau>0\) gives a inhomogeneous non-linearity. One considers the inter-critical regime, namely \(1+\frac{2(s-\tau)}N<q<1+\frac{2(s-\tau)}{N-2s}\) and \(1+\frac{2(s-\tau)+\alpha}{N}<p<1+\frac{2(s-\tau)+\alpha}{N-2s}\). The purpose is to prove the finite time blow-up of solutions with datum in the energy space, not necessarily radial or with finite variance. The assumption on the data is expressed in two different ways. The first one is in the spirit of the potential well method due to Payne-Sattinger. The second one is the ground state threshold standard condition. The proof is based on Morawetz estimates and a non-global ordinary differential inequality. This work complements the recent paper by Bai and Li [4] in many directions. For more information see https://ejde.math.txstate.edu/Volumes/2025/55/abstr.html