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

Abstract

The paper is concerned with linear thermoelastic plate equations in the half-space ${\mathit{R}}_{+}^n=\{x=(x_1,\dots ,x_n)❘x_n>0\}$:

$u_{\mathit{tt}}+{\mathrm{\Delta }}^{2}u+\mathrm{\Delta }\theta =0$ 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 $W_{p,D}^2=\{u\in W_p^2❘u{|}_{x_n=0}=D_{n}u{|}_{x_n=0}=0\}$. 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:

$\parallel {\nabla }^j({\nabla }^{2}u(·,t),u_t(·,t),\theta (·,t)){\parallel }_{L_q({\mathit{R}}_{+}^n)}$

$\le C_{p,q}{t}^{-\frac{j}{2}-\frac{n}{2}(\frac{1}{p}-\frac{1}{q})}\parallel ({\nabla }^2{u}_0,v_0,{\theta }_0){\parallel }_{L_p\left({\mathit{R}}_{+}^n\right)}$

$(t>0)$

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 $j$.