Hopf's lemmas for parabolic fractional Laplacians and parabolic fractional $p$-Laplacians

Wang, Pengyan; Chen, Wenxiong

doi:10.48550/arxiv.2010.01212

Cited by 3 publications

(2 citation statements)

References 37 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…To proceed with the proof, we need the following Hopf's lemma for antisymmetric functions, whose proof is similar to that for Theorem 3.1 in [38]. However, for readers' convenience, we attach it in the Appendix.…”

Section: Liouville Type Theorem In a Half Spacementioning

confidence: 99%

Liouville theorems for fractional parabolic equations

Chen

2021

Preprint

Self Cite

View full text Add to dashboard Cite

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 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 R n + × R and obtain some new connections between the nonexistence of solutions in a half space R n + × R and in the whole space R n−1 × 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.

show abstract

Section: Liouville Type Theorem In a Half Spacementioning

confidence: 99%

Liouville theorems for fractional parabolic equations

Chen

2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…Recently, for parabolic equations involving the fractional Laplacian, Chen etc. [16,43] developed a systematical scheme to carry out the asymptotic method of moving planes to investigate qualitative properties of solutions, either on bounded or on unbounded domains.…”

Section: Introductionmentioning

confidence: 99%

Nonexistence of solutions for indefinite fractional parabolic equations

Chen¹,

Wu²,

Wang³

2021

Preprint

Self Cite

View full text Add to dashboard Cite

We study fractional parabolic equations with indefinite nonlinearitieswhere 0 < s < 1 and 1 < p < ∞. We first prove that all positive bounded solutions are monotone increasing along the x 1 direction. Based on this we derive a contradiction and hence obtain non-existence of solutions. These monotonicity and nonexistence results are crucial tools in a priori estimates and complete blow-up for fractional parabolic equations in bounded domains.To this end, we introduce several new ideas and developed a systematic approach which may also be applied to investigate qualitative properties of solutions for many other fractional parabolic problems.

show abstract

Hopf’s lemma and radial symmetry for the Logarithmic Laplacian problem

Zhang,

Nie

2024

Fract Calc Appl Anal

View full text Add to dashboard Cite

Hopf's lemmas for parabolic fractional Laplacians and parabolic fractional $p$-Laplacians

Cited by 3 publications

References 37 publications

Liouville theorems for fractional parabolic equations

Liouville theorems for fractional parabolic equations

Nonexistence of solutions for indefinite fractional parabolic equations

Hopf’s lemma and radial symmetry for the Logarithmic Laplacian problem

Contact Info

Product

Resources

About