Liouville theorems for fractional parabolic equations

TL;DR

This paper proves Liouville theorems for solutions of fractional parabolic equations, introducing new techniques to handle non-local operators and weakening decay conditions to polynomial growth.

Contribution

It develops new tools like narrow region and maximum principles for antisymmetric functions, and establishes connections between solutions in half spaces and whole spaces for fractional parabolic equations.

Findings

Established Liouville theorems for fractional parabolic equations.

Weakened decay conditions to polynomial growth for solutions.

Introduced new methods to analyze non-local fractional Laplacian problems.

Abstract

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.