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

TL;DR

This paper investigates the existence and stability of near-constant solutions in variable-coefficient scalar field equations, demonstrating conditions for steady states and their asymptotic stability without perturbative assumptions.

Contribution

It establishes the existence of almost constant steady states and proves their asymptotic stability under broad conditions, including non-perturbative variable coefficients.

Findings

Existence of near-constant steady states under suitable conditions.

Asymptotic stability of the vacuum state without parity assumptions.

Stability of near-constant states under parity conditions.

Abstract

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.