scholarly journals Singularity and blow-up estimates via Liouville-type theorems for Hardy–Hénon parabolic equations

2013 ◽  
Vol 13 (2) ◽  
pp. 411-442 ◽  
Author(s):  
Quoc Hung Phan
Keyword(s):  
Parabolic Equations ◽  
Blow Up ◽  
Liouville Type ◽  
Liouville Type Theorems

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.

Liouville-type Theorems for Fully Nonlinear Elliptic Equations and Systems in Half Spaces

2013 ◽  
Vol 13 (4) ◽  
Author(s):  
Guozhen Lu ◽  
Jiuyi Zhu
Keyword(s):  
Elliptic Equations ◽  
Blow Up ◽  
Nonlinear Elliptic Equations ◽  
Decay Estimates ◽  
Fully Nonlinear Elliptic Equations ◽  
Liouville Type ◽  
Fully Nonlinear Equations ◽  
Fully Nonlinear ◽  
Nonlinear Elliptic ◽  
Liouville Type Theorems

AbstractThe main purpose of this paper is to establish Liouville-type theorems and decay estimates for viscosity solutions to a class of fully nonlinear elliptic equations or systems in half spaces without the boundedness assumptions on the solutions. Using the blow-up method and doubling lemma of [18], we remove the boundedness assumption on solutions which was often required in the proof of Liouville-type theorems in the literature. We also prove the Liouville-type theorems for supersolutions of a system of fully nonlinear equations with Pucci extremal operators in half spaces. Liouville theorems and decay estimates for high order elliptic equations and systems have also been established by the authors in an earlier work [15] when no boundedness assumption was given on the solutions.

scholarly journals Liouville-type theorems for certain degenerate and singular parabolic equations

2010 ◽  
Vol 348 (15-16) ◽  
pp. 873-877 ◽  
Author(s):  
Emmanuele DiBenedetto ◽  
Ugo Gianazza ◽  
Vincenzo Vespri
Keyword(s):  
Parabolic Equations ◽  
Liouville Type ◽  
Liouville Type Theorems ◽  
Singular Parabolic Equations

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.

scholarly journals Hamilton type gradient estimates for a general type of nonlinear parabolic equations on Riemannian manifolds

2021 ◽  
Vol 6 (10) ◽  
pp. 10506-10522
Author(s):  
Fanqi Zeng ◽  
Keyword(s):  
Riemannian Manifolds ◽  
Parabolic Equations ◽  
General Type ◽  
Nonlinear Parabolic Equations ◽  
Complete Riemannian Manifold ◽  
Gradient Estimates ◽  
Geometric Flow ◽  
Liouville Type ◽  
Nonlinear Parabolic ◽  
Liouville Type Theorems

<abstract><p>In this paper, we prove Hamilton type gradient estimates for positive solutions to a general type of nonlinear parabolic equation concerning $ V $-Laplacian:</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ (\Delta_{V}-q(x, t)-\partial_{t})u(x, t) = A(u(x, t)) $\end{document} </tex-math></disp-formula></p> <p>on complete Riemannian manifold (with fixed metric). When $ V = 0 $ and the metric evolves under the geometric flow, we also derive some Hamilton type gradient estimates. Finally, as applications, we obtain some Liouville type theorems of some specific parabolic equations.</p></abstract>

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