Arrest of blowup for the 3-D semi-relativistic Schrödinger-Poisson system with pseudo-relativistic diffusion

Walid Abou Salem, Thomas Chen, Vitali Vougalter

Published 2014-03-12Version 1

We show global well-posedness in energy norm of the semi-relativistic Schr\"odinger-Poisson system of equations with attractive Coulomb interaction in ${\mathbb R}^3$ in the presence of pseudo-relativistic diffusion. We also discuss sufficient conditions to have well-posedness in ${\mathcal L}^2$. 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.