Phragmen-Liouville-type theorems and Liouville theorems for a linear parabolic equation

1985 ◽  
Vol 37 (1) ◽  
pp. 67-70
Author(s):  
R. Ya. Glagoleva
Keyword(s):  
Parabolic Equation ◽  
Linear Parabolic Equation ◽  
Liouville Type ◽  
Liouville Theorems ◽  
Liouville Type Theorems

Liouville theorems for a class of fourth order elliptic equations

1980 ◽  
Vol 86 (1-2) ◽  
pp. 129-137 ◽  
Author(s):  
Vinod B. Goyal ◽  
Philip W. Schaefer
Keyword(s):  
Fourth Order ◽  
Elliptic Equations ◽  
Entire Solutions ◽  
Liouville Type ◽  
Liouville Theorems ◽  
Liouville Type Theorems ◽  
Solutions Of Equations ◽  
Fourth Order Elliptic Equations

SynopsisLiouville type theorems are obtained for bounded entire solutions of equations of the form Δ2u − q(x)Δu + p(x)u = 0 by means of subharmonic functionals and Green type inequalities.

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.

scholarly journals A Note on Parabolic Liouville Theorems and Blow-Up Rates for a Higher-Order Semilinear Parabolic System

2011 ◽  
Vol 2011 ◽  
pp. 1-9 ◽  
Author(s):  
Guocai Cai ◽  
Hongjing Pan ◽  
Ruixiang Xing
Keyword(s):  
Parabolic System ◽  
Blow Up ◽  
Parabolic Systems ◽  
Time Variable ◽  
Liouville Type ◽  
Liouville Theorems ◽  
Semilinear Parabolic System ◽  
Liouville Type Theorems ◽  
Systems Of Inequalities ◽  
Semilinear Parabolic

We improve some results of Pan and Xing (2008) and extend the exponent range in Liouville-type theorems for some parabolic systems of inequalities with the time variable onR. As an immediate application of the parabolic Liouville-type theorems, the range of the exponent in blow-up rates for the corresponding systems is also improved.

Liouville theorems on some indefinite equations

1999 ◽  
Vol 129 (3) ◽  
pp. 649-661 ◽  
Author(s):  
Meijun Zhu
Keyword(s):  
Elliptic Equations ◽  
Liouville Type ◽  
Liouville Theorems ◽  
Liouville Type Theorems

In this note, we present some Liouville type theorems about the non-negative solutions to some indefinite elliptic equations.

scholarly journals Some Liouville Theorems on Finsler Manifolds

Mathematics ◽  
2019 ◽  
Vol 7 (4) ◽  
pp. 351 ◽  
Author(s):  
Minqiu Wang ◽  
Songting Yin
Keyword(s):  
Recent Literature ◽  
Finsler Manifold ◽  
Finsler Metric ◽  
Liouville Type ◽  
Finsler Manifolds ◽  
Liouville Theorems ◽  
Superharmonic Functions ◽  
Liouville Type Theorems ◽  
Geometric Relationship

We give some Liouville type theorems of L p harmonic (resp. subharmonic, superharmonic) functions on a complete noncompact Finsler manifold. Using the geometric relationship between a Finsler metric and its reverse metric, we remove some restrictions on the reversibility. These improve the recent literature (Zhang and Xia, 2014).

ENERGY ESTIMATES AND LIOUVILLE THEOREMS FOR HARMONIC MAPS

2000 ◽  
Vol 11 (03) ◽  
pp. 413-448 ◽  
Author(s):  
MARCO RIGOLI ◽  
ALBERTO G. SETTI
Keyword(s):  
Vector Bundle ◽  
Riemannian Manifolds ◽  
Harmonic Maps ◽  
Energy Estimates ◽  
Liouville Type ◽  
Harmonic Forms ◽  
Liouville Theorems ◽  
Liouville Type Theorems

We obtain lower and upper energy estimates for harmonic maps between Riemannian manifolds under natural curvature conditions leading to various Liouville-type theorems. Some of the methods described may also be applied to vanishing-type problems for vector bundle-valued harmonic forms.

Time and space meshes adaptivity for the resolution of a semi-linear heat equation

2021 ◽  
pp. 1-10
Author(s):  
Nejmeddine Chorfi
Keyword(s):  
Parabolic Equation ◽  
Heat Equation ◽  
Order Of Convergence ◽  
Linear Parabolic Equation ◽  
Spectral Elements ◽  
Euler Scheme ◽  
Time Step ◽  
Implicit Euler Scheme ◽  
Linear Heat ◽  
Error Indicators

The aim of this work is to highlight that the adaptivity of the time step when combined with the adaptivity of the spectral mesh is optimal for a semi-linear parabolic equation discretized by an implicit Euler scheme in time and spectral elements method in space. The numerical results confirm the optimality of the order of convergence. The later is similar to the order of the error indicators.

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