略超线性heat equations的全局零能控性和非负能控性
Global null-controllability and nonnegative-controllability of slightly superlinear heat equations
摘要：
考虑半线性热方程在$\mathbb{R}^{N}$的光滑有界区域$Ω$上的Dirichlet或Neumann边界条件.控制输入是局部化在$Ω$的某个任意非空开子集$ω$中的源项。本文的目的是证明半线性系统$f(s) = \pm |s| \log^α(2+|s|)$的一致大时间全局零能控性，其中$α\in [3/2,2)$是Enrique Fernando-Cara和Enrique Zuazua在2000年提出的未解决的问题.值得一提的是，自由解（没有控制）可以爆破。首先，我们建立了系统的小时间全局非负能控性（分别为非正能控性），即，可以在任意时间将任何初始数据引导到非负（相应地非正）状态。特别地，由于控制项，人们可以局部地行动，以防止爆破发生。证明依赖于具有有界势$a(t,x)$的线性热方程的精确可观测性估计。更准确地说，我们表明，可观测性举行的顺序$\exp\left(C |a|\_{\infty}^{1/2}\right)$的非负初始数据的尖锐常数。这个不等式来自一个新的$L^1$ Carleman估计。角谷不动点论证使得可以回到半线性热方程。其次，一致的大时间零能控性的结果来自三个成分：全局非负能控性，自由解和基本常微分方程的解之间的比较原理，该原理提供了自由解在$L^{\infty}(Ω)$-范数下向$0$的收敛性，以及半线性热方程的局部零能控性。
英文摘要：
We consider the semilinear heat equation posed on a smooth bounded domain $Ω$ of $\mathbb{R}^{N}$ with Dirichlet or Neumann boundary conditions. The control input is a source term localized in some arbitrary nonempty open subset $ω$ of $Ω$. The goal of this paper is to prove the uniform large time global null-controllability for semilinearities $f(s) = \pm |s| \log^α(2+|s|)$ where $α\in [3/2,2)$ which is the case left open by Enrique Fernandez-Cara and Enrique Zuazua in 2000. It is worth mentioning that the free solution (without control) can blow-up. First, we establish the small-time global nonnegative-controllability (respectively nonpositive-controllability) of the system, i.e., one can steer any initial data to a nonnegative (respectively nonpositive) state in arbitrary time. In particular, one can act locally thanks to the control term in order to prevent the blow-up from happening. The proof relies on precise observability estimates for the linear heat equation with a bounded potential $a(t,x)$. More precisely, we show that observability holds with a sharp constant of the order $\exp\left(C |a|\_{\infty}^{1/2}\right)$ for nonnegative initial data. This inequality comes from a new $L^1$ Carleman estimate. A Kakutani's fixed point argument enables to go back to the semilinear heat equation. Secondly, the uniform large time null-controllability result comes from three ingredients: the global nonnegative-controllability, a comparison principle between the free solution and the solution to the underlying ordinary differential equation which provides the convergence of the free solution toward $0$ in $L^{\infty}(Ω)$-norm, and the local null-controllability of the semilinear heat equation.
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。
