Finite time blow-up of non-radial solutions for some inhomogeneous Schrödinger equations

Abstract

This work studies the inhomogeneous Schrödinger equation $$i{\partial }_{t}u-{\mathcal{K}}_{s,\lambda }u+F(x,u)=0,u(t,x):\mathbb{R}\times {\mathbb{R}}^N\to \mathbb{C}.$$ Here, $s\in \{1,2\}$, $N>2s$ and $\lambda >-\frac{(N-2{)}^2}{4}$. The linear Schrödinger operator reads ${\mathcal{K}}_{s,\lambda }:=(-\Delta {)}^s+(2-s)\frac{\lambda }{|x{|}^2}$ and the focusing source term is local or non-local $$F(x,u)\in \{|x{|}^{-2\tau }|u{|}^{2(q-1)}u,|x{|}^{-\tau }|u{|}^{p-2}(J_{\alpha }\ast |\cdot {|}^{-\tau }|u{|}^p)u\}.$$ 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 $1+\frac{2(1-\tau )}{N}<q<1+\frac{2(1-\tau )}{N-2s}$ and $1+\frac{2-2\tau +\alpha }{N}<p<1+\frac{2-2\tau +\alpha }{N-2s}$. The purpose is to prove the finite time blow-up of solutions with datum in the energy space, non necessarily radial or with finite variance. The assumption on the data is expressed in terms of non-conserved quantities. This is weaker than the ground state threshold standard condition. The blow-up under the ground threshold or with negative energy are consequences. The proof is based on Morawetz estimates and a non-global ordinary differential inequality.

24 pages