标题： 分数抛物方程的Liouville定理 显示英文标题

标题： Liouville theorems for fractional parabolic equations

摘要： 在本文中，我们建立了关于分数抛物方程整体解的若干Liouville型定理。 我们首先获得了证明Liouville定理所需的关键要素，例如在无界区域中反对称函数的窄区域原理和最大值原理，其中我们通过构造辅助函数显著地将通常在空间变量上的衰减条件$u \to 0$弱化为在$u$上的多项式增长。然后我们推导出半空间$\mathbb{R}_+^n \times \mathbb{R}$中解的单调性，并得到了半空间$\mathbb{R}_+^n \times \mathbb{R}$中解的不存在性与整个空间$\mathbb{R}^{n-1} \times \mathbb{R}$中解的不存在性之间的一些新联系，从而证明了相应的Liouville型定理。 为了克服分数Laplacian的非局部性带来的困难，我们引入了几种新的思想，这些思想将在研究各种非局部抛物问题解的定性性质时成为有用的工具。 显示英文摘要

摘要： In this paper, we establish several Liouville type theorems for entire solutions to fractional parabolic equations. We first obtain the key ingredients needed in the proof of Liouville theorems, such as narrow region principles and maximum principles for antisymmetric functions in unbounded domains, in which we remarkably weaken the usual decay condition $u \to 0$ at infinity with respect to the spacial variables to a polynomial growth on $u$ by constructing auxiliary functions.Then we derive monotonicity for the solutions in a half space $\mathbb{R}_+^n \times \mathbb{R}$ and obtain some new connections between the nonexistence of solutions in a half space $\mathbb{R}_+^n \times \mathbb{R}$ and in the whole space $\mathbb{R}^{n-1} \times \mathbb{R}$ and therefore prove the corresponding Liouville type theorems. To overcome the difficulty caused by the non-locality of the fractional Laplacian, we introduce several new ideas which will become useful tools in investigating qualitative properties of solutions for a variety of non-local parabolic problems.