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<p>In this article, we consider the nonlinear stability of the non-zero equilibrium state for the viscous scalar conservation laws with a delay effect in the one-dimensional whole space. The necessary and sufficient conditions for the linear stability are obtained in Ueda (Ueda, Y in Jpn. J. Ind. Appl. Math 42 2057–2072, 2025), and the situation strongly depends on equilibrium states. Based on the result of the linear stability, the purpose of this article is to derive the condition for the nonlinear stability. The method for deriving the linear stability is the analysis of the corresponding eigenvalue problem; however, it does not apply to the derivation of nonlinear stability. Thus, we apply the energy method to construct the a priori estimate and prove the existence of a global-in-time solution.</p>

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Nonlinear stability of equilibrium states for the viscous conservation laws with time delay

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  • 发表日期：

    2026年4月30日

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  • Yoshihiro Ueda

摘要

In this article, we consider the nonlinear stability of the non-zero equilibrium state for the viscous scalar conservation laws with a delay effect in the one-dimensional whole space. The necessary and sufficient conditions for the linear stability are obtained in Ueda (Ueda, Y in Jpn. J. Ind. Appl. Math 42 2057–2072, 2025), and the situation strongly depends on equilibrium states. Based on the result of the linear stability, the purpose of this article is to derive the condition for the nonlinear stability. The method for deriving the linear stability is the analysis of the corresponding eigenvalue problem; however, it does not apply to the derivation of nonlinear stability. Thus, we apply the energy method to construct the a priori estimate and prove the existence of a global-in-time solution.