American Institute of Mathematical Sciences

December  2014, 4(4): 481-500. doi: 10.3934/mcrf.2014.4.481

Asymptotic stability of Webster-Lokshin equation

 1 Université de Toulouse; ISAE, DMIA, 10, avenue E. Belin, B.P. 54032, F-31055 Toulouse cedex 4, France 2 GIPSA-lab, CNRS, 11 rue des Mathématiques, Grenoble Campus, 38402 Grenoble Cedex, France

Received  May 2013 Revised  February 2014 Published  September 2014

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.
Citation: Denis Matignon, Christophe Prieur. Asymptotic stability of Webster-Lokshin equation. Mathematical Control & Related Fields, 2014, 4 (4) : 481-500. doi: 10.3934/mcrf.2014.4.481
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