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 $\mbi{R}^{n}_{+} = \{x = (x_{1}, \ldots, x_{n}) \mid x_{n} > 0\}$:

$u_{tt} + \Delta^{2}u + \Delta\theta = 0$   and

$\theta_{t} - \Delta \theta - \Delta u_{t} = 0$ $\mbi{R}_{+}^{n}\times(0, \infty),$

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

$\|\nabla^{j}(\nabla^{2} u(\cdot, t), u_{t}(\cdot, t), \theta(\cdot,t))\|_{L_{q}(\mbi{R}_{+}^{n})}$

$\leq C_{p,q}t^{-\frac{j}{2}-\frac{n}{2}\big(\frac{1}{p} - \frac{1}{q}\big)} \|(\nabla^{2} u_{0}, v_{0}, \theta_{0})\|_{L_{p}(\mbi{R}_{+}^{n})} \quad (t > 0)$

$(t > 0)$

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