论文标题
使用傅立叶定律的热弹性板方程的最佳大型估计和单数极限
Optimal large-time estimates and singular limits for thermoelastic plate equations with the Fourier law
论文作者
论文摘要
在本文中，我们研究了整个空间中使用热传导的经典热弹性板方程的渐近行为，$ \ mathbb {r}^n $，我们在其中引入了基于三阶（时间）微分方程和精制傅立叶分析的还原方法。当$ n \ leqslant 3 $，$ n = 4 $时，我们得出最佳的增长估计，而有限的估计以及衰减估计估计是$ n \ geqslant 5 $ 5 $ for $ l^2 $ norm的垂直位移。特别是，发现了新的关键维度$ n = 4 $，以区分板模型和热传导傅立叶定律之间的决定性作用。此外，关于温度方程中的小热参数，我们研究了单数极限问题。我们不仅显示了热弹性板和结构阻尼板之间垂直位移的全局（时间）收敛，而且还严格证明了溶液的新二阶轮廓。我们的方法可以解决热弹性方面的几个密切相关的问题。
In this paper, we study asymptotic behaviors for classical thermoelastic plate equations with the Fourier law of heat conduction in the whole space $\mathbb{R}^n$, where we introduce a reduction methodology basing on third-order (in time) differential equations and refined Fourier analysis. We derive optimal growth estimates when $n\leqslant 3$, bounded estimates when $n=4$, and decay estimates when $n\geqslant 5$ for the vertical displacement in the $L^2$ norm. Particularly, the new critical dimension $n=4$ for distinguishing the decisive role between the plate model and the Fourier law of heat conduction is discovered. Moreover, concerning the small thermal parameter in the temperature equation, we study the singular limit problem. We not only show global (in time) convergence of the vertical displacements between thermoelastic plates and structurally damped plates, but also rigorously demonstrate a new second-order profile of the solution. Our methodology can settle several closely related problems in thermoelasticity.
