On the Lp analytic semigroup associated with the linear thermoelastic plate equations in the half-space

Yuka Naito, Yoshihiro Shibata

    Research output: Contribution to journalArticle

    7 Citations (Scopus)

    Abstract

    The paper is concerned with linear thermoelastic plate equations in the half-space R+ n = {x = (x1,..., xn) | xn > 0}: utt + Δ2u + Δθ = 0 and θt - Δθ - Δθut = 0 in R+ n × (0, ∞), subject to the boundary condition: u|xn=0 = Dn u|x n=0 = θ|xn=0 = 0 and initial condition: (u,Dtu,θ)|t=0 = (u0,v0, θ0) ∈ Hp = Wp,D 2 × LP × LP, where Wp,D 2 = {u ∈ Wp 2 | u|Xn=0 = D nu|xn=0 = 0}. We show that for any p ∈ (1, infin;), the associated semigroup {T(t)}t≥0 is analytic in the underlying space Hp. Moreover, a solution (u, θ) satisfies the estimates: ||∇j(∇2u(·, t), u t(·, t), θ(·, t))||L q(R+ n) ≤ cp,qt-1\2-n\2(1\p-1\q) ||(∇2u0,V0, θ0)|| Lq(R+ n) (t > 0) for j = 0, 1, 2 provided that 1 < p ≤ q ≤ ∞ when j = 0, 1 and that 1 < p ≤ q < ∞ when j = 2, where ∇j stands for space gradient of order j.

    Original languageEnglish
    Pages (from-to)971-1011
    Number of pages41
    JournalJournal of the Mathematical Society of Japan
    Volume61
    Issue number4
    DOIs
    Publication statusPublished - 2009 Oct

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    Plate Equation
    Analytic Semigroup
    Thermoelastic
    Half-space
    Initial conditions
    Semigroup
    Gradient
    Boundary conditions
    Estimate

    Keywords

    • Half space
    • L analytic semigroup
    • L-l decay estimate
    • Resolvent estimate
    • Thermoelastic plate equations
    • Whole space

    ASJC Scopus subject areas

    • Mathematics(all)

    Cite this

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    abstract = "The paper is concerned with linear thermoelastic plate equations in the half-space R+ n = {x = (x1,..., xn) | xn > 0}: utt + Δ2u + Δθ = 0 and θt - Δθ - Δθut = 0 in R+ n × (0, ∞), subject to the boundary condition: u|xn=0 = Dn u|x n=0 = θ|xn=0 = 0 and initial condition: (u,Dtu,θ)|t=0 = (u0,v0, θ0) ∈ Hp = Wp,D 2 × LP × LP, where Wp,D 2 = {u ∈ Wp 2 | u|Xn=0 = D nu|xn=0 = 0}. We show that for any p ∈ (1, infin;), the associated semigroup {T(t)}t≥0 is analytic in the underlying space Hp. Moreover, a solution (u, θ) satisfies the estimates: ||∇j(∇2u(·, t), u t(·, t), θ(·, t))||L q(R+ n) ≤ cp,qt-1\2-n\2(1\p-1\q) ||(∇2u0,V0, θ0)|| Lq(R+ n) (t > 0) for j = 0, 1, 2 provided that 1 < p ≤ q ≤ ∞ when j = 0, 1 and that 1 < p ≤ q < ∞ when j = 2, where ∇j stands for space gradient of order j.",
    keywords = "Half space, L analytic semigroup, L-l decay estimate, Resolvent estimate, Thermoelastic plate equations, Whole space",
    author = "Yuka Naito and Yoshihiro Shibata",
    year = "2009",
    month = "10",
    doi = "10.2969/jmsj/06140971",
    language = "English",
    volume = "61",
    pages = "971--1011",
    journal = "Journal of the Mathematical Society of Japan",
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    publisher = "Mathematical Society of Japan - Kobe University",
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    TY - JOUR

    T1 - On the Lp analytic semigroup associated with the linear thermoelastic plate equations in the half-space

    AU - Naito, Yuka

    AU - Shibata, Yoshihiro

    PY - 2009/10

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    N2 - The paper is concerned with linear thermoelastic plate equations in the half-space R+ n = {x = (x1,..., xn) | xn > 0}: utt + Δ2u + Δθ = 0 and θt - Δθ - Δθut = 0 in R+ n × (0, ∞), subject to the boundary condition: u|xn=0 = Dn u|x n=0 = θ|xn=0 = 0 and initial condition: (u,Dtu,θ)|t=0 = (u0,v0, θ0) ∈ Hp = Wp,D 2 × LP × LP, where Wp,D 2 = {u ∈ Wp 2 | u|Xn=0 = D nu|xn=0 = 0}. We show that for any p ∈ (1, infin;), the associated semigroup {T(t)}t≥0 is analytic in the underlying space Hp. Moreover, a solution (u, θ) satisfies the estimates: ||∇j(∇2u(·, t), u t(·, t), θ(·, t))||L q(R+ n) ≤ cp,qt-1\2-n\2(1\p-1\q) ||(∇2u0,V0, θ0)|| Lq(R+ n) (t > 0) for j = 0, 1, 2 provided that 1 < p ≤ q ≤ ∞ when j = 0, 1 and that 1 < p ≤ q < ∞ when j = 2, where ∇j stands for space gradient of order j.

    AB - The paper is concerned with linear thermoelastic plate equations in the half-space R+ n = {x = (x1,..., xn) | xn > 0}: utt + Δ2u + Δθ = 0 and θt - Δθ - Δθut = 0 in R+ n × (0, ∞), subject to the boundary condition: u|xn=0 = Dn u|x n=0 = θ|xn=0 = 0 and initial condition: (u,Dtu,θ)|t=0 = (u0,v0, θ0) ∈ Hp = Wp,D 2 × LP × LP, where Wp,D 2 = {u ∈ Wp 2 | u|Xn=0 = D nu|xn=0 = 0}. We show that for any p ∈ (1, infin;), the associated semigroup {T(t)}t≥0 is analytic in the underlying space Hp. Moreover, a solution (u, θ) satisfies the estimates: ||∇j(∇2u(·, t), u t(·, t), θ(·, t))||L q(R+ n) ≤ cp,qt-1\2-n\2(1\p-1\q) ||(∇2u0,V0, θ0)|| Lq(R+ n) (t > 0) for j = 0, 1, 2 provided that 1 < p ≤ q ≤ ∞ when j = 0, 1 and that 1 < p ≤ q < ∞ when j = 2, where ∇j stands for space gradient of order j.

    KW - Half space

    KW - L analytic semigroup

    KW - L-l decay estimate

    KW - Resolvent estimate

    KW - Thermoelastic plate equations

    KW - Whole space

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    U2 - 10.2969/jmsj/06140971

    DO - 10.2969/jmsj/06140971

    M3 - Article

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    VL - 61

    SP - 971

    EP - 1011

    JO - Journal of the Mathematical Society of Japan

    JF - Journal of the Mathematical Society of Japan

    SN - 0025-5645

    IS - 4

    ER -