Liouville Theorems for Fractional Parabolic Equations
Wenxiong Chen, Leyun Wu
Abstract
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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>u</m:mi> <m:mo>→</m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> u\to 0 at infinity to a polynomial growth on 𝑢 by constructing proper auxiliary functions. Then we derive monotonicity for the solutions in a half space <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msubsup> <m:mi mathvariant="double-struck">R</m:mi> <m:mo>+</m:mo> <m:mi>n</m:mi> </m:msubsup> <m:mo>×</m:mo> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> </m:math> \mathbb{R}_{+}^{n}\times\mathbb{R} and obtain some new connections between the nonexistence of solutions in a half space <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msubsup> <m:mi mathvariant="double-struck">R</m:mi> <m:mo>+</m:mo> <m:mi>n</m:mi> </m:msubsup> <m:mo>×</m:mo> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> </m:math> \mathbb{R}_{+}^{n}\times\mathbb{R} and in the whole space <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mi mathvariant="double-struck">R</m:mi> <m:mrow> <m:mi>n</m:mi> <m:mo>-</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msup> <m:mo>×</m:mo> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> </m:math> \mathbb{R}^{n-1}\times\mathbb{R} and therefore prove the corresponding Liouville type theorems. To overcome the difficulty caused by the nonlocality of the fractional Laplacian, we introduce several new ideas which will become useful tools in investigating qualitative properties of solutions for a variety of nonlocal parabolic problems.