Abstract
This paper is concerned with the dynamical system generated by certain semilinear damped wave equations. In §1 we reproduce a result obtained in a previous paper (Mora [1986]), which shows that, when the damping is sufficiently large this dynamical system has the property that its global attractor is contained in a finite-dimensional local invariant manifold of classC 1. In the present paper, we will show that, on the other hand, when the damping is small, it is a fairly generic fact that there is no finite-dimensional local invariant manifold of class C 1containing the global attractor. The exact result obtained in this connection is stated in Theorem 4.1. In the way towards this result, we have developed some auxiliary results which have some interest by themselves, namely, a result giving optimal inner products for linear wave equations (Theorem 2.1), and a C 1 linearization theorem (Theorem 3.1).
Work partially supported by the CAICYT.
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Mora, X., Solà-Morales, J. (1987). Existence and Non-Existence of Finite-Dimensional Globally Attracting Invariant Manifolds in Semilinear Damped Wave Equations. In: Chow, SN., Hale, J.K. (eds) Dynamics of Infinite Dimensional Systems. NATO ASI Series, vol 37. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-86458-2_21
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DOI: https://doi.org/10.1007/978-3-642-86458-2_21
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