Existence and stability of near-constant solutions of variable-coefficient scalar field equations

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 H1xL2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H<^>1\times L<^>2$$\end{document}, 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.