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

Author

Listed:
  • Mashael Alammari

    (Florida Institute of Technology)

  • Stanley Snelson

    (Florida Institute of Technology)

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.

Suggested Citation

  • Mashael Alammari & Stanley Snelson, 2025. "Existence and stability of near-constant solutions of variable-coefficient scalar field equations," Partial Differential Equations and Applications, Springer, vol. 6(1), pages 1-28, March.
    • Handle: RePEc:spr:pardea:v:6:y:2025:i:1:d:10.1007_s42985-024-00310-1
    DOI: 10.1007/s42985-024-00310-1
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