## Abstract

The paper is concerned with linear thermoelastic plate equations in the half-space ${\mathit{R}}_{+}^{n}=\{x=({x}_{1},\dots ,{x}_{n})\u2758{x}_{n}>0\}$:

${u}_{\mathit{tt}}+{\mathrm{\Delta}}^{2}u+\mathrm{\Delta}\theta =0$ and

${\theta}_{t}-\mathrm{\Delta}\theta -\mathrm{\Delta}{u}_{t}=0$ ${\mathit{R}}_{+}^{n}\times (0,\infty ),$

subject to the boundary condition: $u{|}_{{x}_{n}=0}={D}_{n}u{|}_{{x}_{n}=0}=\theta {|}_{{x}_{n}=0}=0$ and initial condition: $(u,{D}_{t}u,\theta ){|}_{t=0}=({u}_{0},{v}_{0},{\theta}_{0})\in {\mathcal{H}}_{p}={W}_{p,D}^{2}\times {L}_{p}\times {L}_{p}$, where ${W}_{p,D}^{2}=\{u\in {W}_{p}^{2}\u2758u{|}_{{x}_{n}=0}={D}_{n}u{|}_{{x}_{n}=0}=0\}$. We show that for any $p\in (1,\infty )$, the associated semigroup $\{T(t){\}}_{t\ge 0}$ is analytic in the underlying space ${\mathcal{H}}_{p}$. Moreover, a solution $(u,\theta )$ satisfies the estimates:

$\parallel {\nabla}^{j}({\nabla}^{2}u(\xb7,t),{u}_{t}(\xb7,t),\theta (\xb7,t)){\parallel}_{{L}_{q}({\mathit{R}}_{+}^{n})}$

$\le {C}_{p,q}{t}^{-\frac{j}{2}-\frac{n}{2}(\frac{1}{p}-\frac{1}{q})}\parallel ({\nabla}^{2}{u}_{0},{v}_{0},{\theta}_{0}){\parallel}_{{L}_{p}\left({\mathit{R}}_{+}^{n}\right)}$

$(t>0)$

for $j=0,1,2$ provided that $$ when $j=0$, 1 and that $$ when $j=2$, where ${\nabla}^{j}$ stands for space gradient of order $j$.

## Citation

Yuka NAITO. Yoshihiro SHIBATA. "On the ${L}_{p}$ analytic semigroup associated with the linear thermoelastic plate equations in the half-space." J. Math. Soc. Japan 61 (4) 971 - 1011, October, 2009. https://doi.org/10.2969/jmsj/06140971

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