arXivDaily arXiv每日学术速递 周一至周五更新
arXiv 2403.07643math.APmath.CAmath.OC
收藏

具有幂增长势的1D热方程的小性定量二维传播与控制

Quantitative 2D propagation of smallness and control for 1D heat equations with power growth potentials

Yunlei Wang

备注：
43 pages
阅读 PDF
arXiv 页面 ↗HTML 原文 ↗

AI总结：

本文研究平面小性传播与热方程控制的关系，建立了参数显式依赖的定量小性传播版本，并以此为关键要素，证明了具有任意非负幂增长势的1D热方程在厚集上的精确零能控性，扩展了已有结果。

摘要：

我们研究平面中小性传播与热方程控制之间的关系。前者已由Zhu证明，他展示了解在某些小集合中的值如何传播到更大的区域。通过回顾其证明，我们建立了一个具有参数显式依赖关系的定量版本。利用这一显式版本，我们建立了具有任意非负幂增长势 $V\in\mathcal{C}(\mathbb{R})$ 的1D热方程的新的精确零能控性结果。作为关键要素，我们建立了新的谱不等式。我们考虑的控制集 $Ω$ 满足 \begin{equation*}\left|Ω\cap [x-L\langle x\rangle ^{-s},x+L\langle x\rangle ^{-s}]\right|\ge γ^{\langle x\rangle^τ}2L\langle x\rangle^{-s} \end{equation*}，其中 $γ\in(0,1)$，$L>0$，$τ,s\ge 0$，且 $\langle x\rangle:=(1+|x|^2)^{1 /2} $。特别地，包含了允许密度衰减（即 $s=0$ 且 $τ\ge 0$）的厚集情况下的零能控性结果。这些结果将Zhu-Zhuge的结果在1维情况下从 $Ω$ 为等分布开集的并集扩展到厚集，并将Su-Sun-Yuan的结果从有界势扩展到某些无界势。

英文摘要：

We study the relation between propagation of smallness in the plane and control for heat equations. The former has been proved by Zhu who showed how the value of solutions in some small set propagates to a larger domain. By reviewing his proof, we establish a quantitative version with the explicit dependence of parameters. Using this explicit version, we establish new exact null-controllability results of 1D heat equations with any nonnegative power growth potentials $V\in\mathcal{C}(\mathbb{R})$. As a key ingredient, new spectral inequalities are established. The control set $Ω$ that we consider satisfy \begin{equation*} \left|Ω\cap [x-L\langle x\rangle ^{-s},x+L\langle x\rangle ^{-s}]\right|\ge γ^{\langle x\rangle^τ}2L\langle x\rangle^{-s} \end{equation*} for some $γ\in(0,1)$, $L>0$, $τ,s\ge 0$, and $\langle x\rangle:=(1+|x|^2)^{1 /2} $. In particular, the null-controllability result for the case of thick sets that allow the decay of the density (\textit{i.e.}, $s=0$ and $τ\ge 0$) is included. These extend the Zhu-Zhuge's results from $Ω$ being the union of equidistributive open sets to thick sets in the 1-dimensional case, and Su-Sun-Yuan's result from bounded potentials to certain unbounded ones.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

↑