Non global solutions for non-radial coupled inhomogeneous non-linear Schrödinger systems

We establish finite-time blow-up of energy solutions to an inhomogeneous nonlinear coupled Schrödinger system under suitable conditions. The novelty of our approach lies in allowing non-radial initial data and removing the assumption of finite variance. This result complements the work (J. Math. Phys. 62, 101508 (2021)) by eliminating the finite variance condition, and further extends the findings of (Potential Anal 60 (2024), 197-218) by ruling out the possibility of infinite-time blow-up in the considered setting. This is obtained by using a localized variance identity coupled with the decreasing property of the inhomogeneous term \(|x|^{-b}\) , which enables to handle the terms of large frequency. In this work, we discuss two cases: the inter-critical regime under the ground state threshold and the mass-critical one with negative energy. Herein, we deal with a coupled source term, so that we complement the scalar case considered in (Nonlinear Anal. 232 (2023), 113266) and (Nonlinearity 35 (2022), 4426). We end this note, with some numerical simulations which show the influence of the singular inhomogeneity in the blow-up rate.