Analyticity and regularity for a class of second order evolution equations

Alain Haraux, Mitsuharu Otani

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    8 Citations (Scopus)

    Abstract

    The regularity conservation as well as the smoothing effect are studied for the equation u″+Au+cAαu′ = 0, where A is a positive selfadjoint operator on a real Hilbert space H and α ∈ (0, 1]; c > 0. When α ≥ 1/2 the equation generates an analytic semigroup on D(A1/2) × H, and if α ∈ (0, 1/2) a weaker optimal smoothing property is established. Some conservation prop- erties in other norms are also established, as a typical example, the strongly dissipative wave equation utt – Δu – cΔut = 0 with Dirichlet boundary conditions in a bounded domain is given, for which the space C0(Ω) × C0(Ω) is conserved for t > 0, which presents a sharp contrast with the conservative case utt – Δu = 0 for which C0()-regularity can be lost even starting from an initial state (u0, 0) with u0 2 C0(Ω) ⋂ C1(Ω).

    Original languageEnglish
    Pages (from-to)101-117
    Number of pages17
    JournalEvolution Equations and Control Theory
    Volume2
    Issue number1
    DOIs
    Publication statusPublished - 2013

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    Keywords

    • Analytic semi-group
    • Regularity
    • Smoothing effect

    ASJC Scopus subject areas

    • Applied Mathematics
    • Control and Optimization
    • Modelling and Simulation

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