### Abstract

The paper is concerned with linear thermoelastic plate equations in the half-space R_{+}
^{n} = {x = (x_{1},..., x_{n}) | x_{n} > 0}: u_{tt} + Δ^{2}u + Δθ = 0 and θ_{t} - Δθ - Δθu_{t} = 0 in R_{+}
^{n} × (0, ∞), subject to the boundary condition: u|_{xn}=0 = D_{n} u|_{x} _{n}=0 = θ|_{xn}=0 = 0 and initial condition: (u,D_{t}u,θ)|_{t=0} = (u_{0},v_{0}, θ_{0}) ∈ H_{p} = W_{p,D}
^{2} × L_{P} × L_{P}, where W_{p,D}
^{2} = {u ∈ W_{p}
^{2} | u|_{Xn}=0 = D _{n}u|_{xn}=0 = 0}. We show that for any p ∈ (1, infin;), the associated semigroup {T(t)}_{t≥0} is analytic in the underlying space H_{p}. Moreover, a solution (u, θ) satisfies the estimates: ||∇^{j}(∇^{2}u(·, t), u _{t}(·, t), θ(·, t))||_{L} _{q}(R_{+}
^{n}) ≤ c_{p,q}t-1\2-n\2(1\p-1\q) ||(∇^{2}u_{0},V_{0}, θ_{0})|| _{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 language | English |
---|---|

Pages (from-to) | 971-1011 |

Number of pages | 41 |

Journal | Journal of the Mathematical Society of Japan |

Volume | 61 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2009 Oct |

### Fingerprint

### 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

**On the L _{p} analytic semigroup associated with the linear thermoelastic plate equations in the half-space.** / Naito, Yuka; Shibata, Yoshihiro.

Research output: Contribution to journal › Article

_{p}analytic semigroup associated with the linear thermoelastic plate equations in the half-space',

*Journal of the Mathematical Society of Japan*, vol. 61, no. 4, pp. 971-1011. https://doi.org/10.2969/jmsj/06140971

}

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

Y1 - 2009/10

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

UR - http://www.scopus.com/inward/record.url?scp=74349085038&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=74349085038&partnerID=8YFLogxK

U2 - 10.2969/jmsj/06140971

DO - 10.2969/jmsj/06140971

M3 - Article

AN - SCOPUS:74349085038

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 -