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We show global well-posedness in energy norm of the semi-relativisticSchr\"odinger-Poisson system of equations with attractive Coulomb interactionin ${\mathbb R}^3$ in the presence of pseudo-relativistic diffusion. We alsodiscuss sufficient conditions to have well-posedness in ${\mathcal L}^2$. Inthe absence of dissipation, we show that the solution corresponding to aninitial condition with negative energy blows up in finite time, which is asexpected, since the nonlinearity is critical.

Arrest of blowup for the 3-D semi-relativistic Schr\"odinger-Poisson system with pseudo-relativistic diffusion