Note on stability of an abstract coupled hyperbolic-parabolic system: singular case
Abstract
In this paper we try to complete the stability analysis for an abstract system of coupled hyperbolic and parabolic equations \ arraylll utt + Au - Aα w = 0, \\ wt + Aα ut + Aβ w = 0,\\ u(0) = u0, ut(0) = u1, w(0) = w0, array . where A is a self-adjoint, positive definite operator on a complex Hilbert space H, and (α, β) ∈ [0,1] × [0,1], which is considered in Amk, and after, in liu1. Our contribution is to identify a fine scale of polynomial stability of the solution in the region S3: = \(α,β) ∈ [0,1] × [0,1]; \, β < 2α -1 \ taking into account the presence of a singularity at zero.
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