Journal of the Mathematical Society of Japan

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

Yuka NAITO and Yoshihiro SHIBATA

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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 + Δ θ = 0   and

θ t - Δ θ - Δ u t = 0 R + n × ( 0 , ) ,

subject to the boundary condition: u | x n = 0 = D n u | x n = 0 = θ | x n = 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 | x n = 0 = D n u | x n = 0 = 0 } . We show that for any p ( 1 , ) , 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 - j2 - n2 ( 1p - 1q ) ( 2 u0 , v0 , θ0 ) Lp ( 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 .

Article information

Source
J. Math. Soc. Japan, Volume 61, Number 4 (2009), 971-1011.

Dates
First available in Project Euclid: 6 November 2009

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1257520498

Digital Object Identifier
doi:10.2969/jmsj/06140971

Mathematical Reviews number (MathSciNet)
MR2588502

Zentralblatt MATH identifier
1184.35062

Subjects
Primary: 35K50
Secondary: 74F05: Thermal effects

Keywords
thermoelastic plate equations whole space half space resolvent estimate $L_{p}$ analytic semigroup $L_{p}$-$L_{q}$ decay estimate

Citation

NAITO, Yuka; SHIBATA, Yoshihiro. On the $L_{p}$ analytic semigroup associated with the linear thermoelastic plate equations in the half-space. J. Math. Soc. Japan 61 (2009), no. 4, 971--1011. doi:10.2969/jmsj/06140971. https://projecteuclid.org/euclid.jmsj/1257520498


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