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Asymptotic stability of Webster-Lokshin equation

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  • The Webster-Lokshin equation is a partial differential equation considered in this paper. It models the sound velocity in an acoustic domain. The dynamics contains linear fractional derivatives which can admit an infinite dimensional representation of diffusive type. The boundary conditions are described by impedance condition, which can be represented by two finite dimensional systems. Under the physical assumptions, there is a natural energy inequality. However, due to a lack of precompactness of the solutions, the LaSalle invariance principle can not be applied. The asymptotic stability of the system is proved by studying the resolvent equation, and by using the Arendt-Batty stability condition.
    Mathematics Subject Classification: 35B35, 93C20, 93D20.


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