Analyticity and regularity for a class of second order evolution equations

Alain Haraux, Mitsuharu Otani

    研究成果: Article

    7 引用 (Scopus)

    抄録

    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).

    元の言語English
    ページ(範囲)101-117
    ページ数17
    ジャーナルEvolution Equations and Control Theory
    2
    発行部数1
    DOI
    出版物ステータスPublished - 2013

    Fingerprint

    Analyticity
    Second Order Equations
    Evolution Equation
    Conservation
    Regularity
    Smoothing Effect
    Analytic Semigroup
    Dissipative Equations
    Hilbert spaces
    Positive Operator
    Wave equations
    Self-adjoint Operator
    Dirichlet Boundary Conditions
    Mathematical operators
    Smoothing
    Wave equation
    Bounded Domain
    Hilbert space
    Boundary conditions
    Norm

    ASJC Scopus subject areas

    • Applied Mathematics
    • Control and Optimization
    • Modelling and Simulation

    これを引用

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    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).",
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    N2 - 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).

    AB - 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).

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