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

Mashael Alammari () and Stanley Snelson ()
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Mashael Alammari: Florida Institute of Technology
Stanley Snelson: Florida Institute of Technology

Partial Differential Equations and Applications, 2025, vol. 6, issue 1, 1-28

Abstract: 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 $$H^1\times L^2$$ H 1 × 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.

Keywords: Scalar-field equations; Variable coefficients; Stationary solutions; Asymptotic stability; 35L71; 35C07; 35B35 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s42985-024-00310-1

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