论文标题
$ \ mathbb {r}^{n-k} \ times（0，\ infty）^k $ harty不平等和热量方程的大时间行为
The Hardy inequality and large time behaviour of the heat equation on $\mathbb{R}^{N-k}\times (0,\infty)^k$
论文作者
论文摘要
在本文中，我们研究了较大的时间渐进式行为，在角落空间上具有强壮的反平方电位$ \ mathbb {r}^{n-k} {n-k} \ times（0，\ infty）^k $，$ k \ geq 0 $。我们首先展示了具有强大潜力的量子谐波振荡器的新的改进的Hardy-Poincaré不平等。考虑到这一点，我们构建了适当的功能设置，以便构成库奇问题。然后，我们获得了最佳多项式大衰减率，然后在$ l^2（\ Mathbb {r}^{n-k} \ times（0，\ infty）^k）$中获得了解决方案的渐近扩展中的第一项。特别是，我们扩展并改善了Vázquez和Zuazua获得的结果（J.funct。Anal。2000），该结果与$ k = 0 $相对应的任何$ k \ geq 0 $。我们强调的是，$ k $的价值越高，时间衰减率越好。我们采用了与Vázquez和Zuazua不同的简化方法，在我们的分析中设法消除了球形谐波分解的用法。
In this paper we study the large time asymptotic behaviour of the heat equation with Hardy inverse-square potential on corner spaces $\mathbb{R}^{N-k}\times (0,\infty)^k$, $k\geq 0$. We first show a new improved Hardy-Poincaré inequality for the quantum harmonic oscillator with Hardy potential. In view of that, we construct the appropriate functional setting in order to pose the Cauchy problem. Then we obtain optimal polynomial large time decay rates and subsequently the first term in the asymptotic expansion of the solutions in $L^2(\mathbb{R}^{N-k}\times (0,\infty)^k)$. Particularly, we extend and improve the results obtained by Vázquez and Zuazua (J. Funct. Anal. 2000), which correspond to the case $k=0$, to any $k\geq 0$. We emphasize that the higher the value of $k$ the better time decay rates are. We employ a different and simplified approach than Vázquez and Zuazua, managing to remove the usage of spherical harmonics decomposition in our analysis.
