  This article studies a class of semilinear scalar field equations on the real
line with variable coefficients in the linear terms. These coefficients are not
necessarily small perturbations of a constant. We prove that under suitable
conditions, the non-translation-invariant linear operator leads to steady
states that are ``almost constant'' in the spatial variable. The main challenge
of the proof is due to a spectral obstruction that cannot be treated
perturbatively. Next, we consider stability of constant and near-constant
steady states. We establish asymptotic stability for the vacuum state with
respect to perturbations in $H^1\times L^2$, without placing any parity
assumptions on the coefficients, potential, or initial data. Finally, under a
parity assumption, we show asymptotic stability for near-constant steady
states.
