Nondetectable signals in dimension N in systems governed by equations of parabolic type

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For evolution equations of parabolic type with a singular interaction term σ in a Banach space X in the form u1 = Âu + F (t), u(0_) = f, (si u) (0) = σ, Bu (t) = Mathematical bold italic small phi sign (t) (where "(si ·) (0)" denotes "singular interaction · at t = 0", B is a boundary operator), one shows the existence even in classical context (with  = Laplacian) of σ ≠ 0 giving rise to a (unique) solution u = 0 ("nondetectable signals"), in open sets of ℝN, ∀ N. We give a result strongly limiting the possibility of appearance of such phenomena which we already know cannot happen when σ is limited to distributional values but (as seen here) appears when σ is allowed to take hyperfunction values. We use both functional analysis and classical methods. We discuss physical consequences.

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Comptes Rendus de l'Academie des Sciences - Series I: Mathematics

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