  We consider a parabolic sine-Gordon model with periodic boundary conditions.
We prove a fundamental maximum principle which gives a priori uniform control
of the solution. In the one-dimensional case we classify all bounded steady
states and exhibit some explicit solutions. For the numerical discretization we
employ first order IMEX, and second order BDF2 discretization without any
additional stabilization term. We rigorously prove the energy stability of the
numerical schemes under nearly sharp and quite mild time step constraints. We
demonstrate the striking similarity of the parabolic sine-Gordon model with the
standard Allen-Cahn equations with double well potentials.
