Liouville-type theorems for integral and PDE systems with critical exponents in a half space

2015 ◽  
Vol 61 (6) ◽  
pp. 731-746
Author(s):  
Caifeng Zhang ◽  
Zhao Liu
Keyword(s):  
Half Space ◽  
Critical Exponents ◽  
Liouville Type ◽  
Liouville Type Theorems ◽  
Pde Systems

scholarly journals Liouville Theorems for Fractional Parabolic Equations

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Wenxiong Chen ◽  
Leyun Wu
Keyword(s):  
Half Space ◽  
Parabolic Equations ◽  
Parabolic Problems ◽  
Maximum Principles ◽  
Qualitative Properties ◽  
Liouville Type ◽  
Liouville Theorems ◽  
Narrow Region ◽  
Liouville Type Theorems ◽  
Whole Space

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 → 0 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 R + n × R \mathbb{R}_{+}^{n}\times\mathbb{R} and obtain some new connections between the nonexistence of solutions in a half space R + n × R \mathbb{R}_{+}^{n}\times\mathbb{R} and in the whole space R n - 1 × R \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.

A Liouville Type Theorem for Poly-harmonic System with Dirichlet Boundary Conditions in a Half Space

2015 ◽  
Vol 15 (1) ◽  
Author(s):  
Zhao Liu ◽  
Wei Dai
Keyword(s):  
Boundary Conditions ◽  
Green’S Function ◽  
Half Space ◽  
Green's Function ◽  
Dirichlet Boundary ◽  
Type Theorem ◽  
Dirichlet Boundary Conditions ◽  
Liouville Type ◽  
Liouville Type Theorem ◽  
Harmonic System

AbstractIn this paper, we consider the following poly-harmonic system with Dirichlet boundary conditions in a half space ℝwherewhereis the Green’s function in ℝ

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