On a Repulsion-Diffusion Equation with Immigration

We study a repulsion-diffusion equation with immigration, whose asymptotic behaviour is related to stability of long-term dynamics in spatial population models and other branching particle systems. We prove well-posedness and find sharp conditions on the repulsion under which a form of the maximum principle and a strong notion of global boundedness of solutions hold. The critical asymptotic strength of the repulsion is $\mathrm{\mid }x{\mathrm{\mid }}^{1-d}$, that of the Newtonian potential.