Arrest of blow up for the 3D semi-relativistic Schrödinger–Poisson system with pseudo-relativistic diffusion

Abstract

We show global well-posedness in energy norm of the semi-relativistic Schrödinger–Poisson system of equations with attractive Coulomb interaction in [Formula: see text] in the presence of pseudo-relativistic diffusion. We also discuss sufficient conditions to have well-posedness in [Formula: see text]. In the absence of dissipation, we show that the solution corresponding to an initial condition with negative energy blows up in finite time, which is as expected, since the nonlinearity is critical.