  In this article, we study a class of semilinear scalar field equations on the
real line with variable coefficients in the linear terms. 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. Next, under a parity assumption, we show asymptotic
stability for steady states that are "almost constant" in the spatial variable,
which can be constructed because the of the non-translation-invariant linear
part of our equation.
