Remarks on blow-up of the smooth solution to quantum hydrodynamic models in $$\mathbf {R}^{d}$$

Abstract

We establish two results about the blow-up of the smooth solution to the Cauchy problem of quantum hydrodynamic models in \(\mathbf {R}^{d}\). First, we obtain a blow-up result under the assumption that the radial component of initial momentum is large enough, but without the condition non negative total fluctuation around a constant state, which is an essential condition in Guo and Wang (J Differ Equ 261:3815–3842, 2016). Second, we get a similar blow-up result for the quantum hydrodynamic models including the relaxation term.