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- W2094400963 abstract "In this paper, we focus on the one-dimensional wave equations with variable damping, either in the interior or on the boundary of an interval. We show that, generically with respect to the non-linearity, they satisfy the Kupka–Smale property, that is that all their equilibrium points are hyperbolic and that their stable and unstable manifolds intersect transversally. More generally, we show that the generic Kupka–Smale property holds for an abstract equation: (0.1) u t t + B ( u + Γ u t ) = f ( x , u ) , for which we assume, besides other hypotheses, that the rootvectors of the associated linear operator form a Riesz basis. We notice that other one-dimensional damped equations fit into the frame of Eq. ( 0.1 ). Additional qualitative properties of ( 0.1 ) are also obtained. Dans cet article, nous nous intéressons aux équations des ondes avec dissipation variable à l'intérieur ou au bord d'un intervalle. Nous montrons que, génériquement par rapport à la non-linéarité, elles possèdent la propriété de Kupka–Smale, c'est-à-dire que leurs points d'équilibre sont hyperboliques et leurs variétés stables et instables sont transverses. Plus généralement, nous montrons que cela est vrai pour une équation abstraite du type : (0.2) u t t + B ( u + Γ u t ) = f ( x , u ) , pour laquelle on suppose, entre autres, que les vecteurs propres généralisés de l'opérateur linéarisé associé forment une base de Riesz. L'Éq. ( 0.2 ) modélise d'autres équations des ondes dissipatives en dimension un. Nous montrons, au passage, différentes propriétés qualitatives de l'Éq. ( 0.2 )." @default.
- W2094400963 created "2016-06-24" @default.
- W2094400963 creator A5061759154 @default.
- W2094400963 date "2005-08-01" @default.
- W2094400963 modified "2023-10-18" @default.
- W2094400963 title "Generic transversality property for a class of wave equations with variable damping" @default.
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- W2094400963 doi "https://doi.org/10.1016/j.matpur.2005.01.002" @default.
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