Extinction for a Fast Diffusion Equation Involving a Nonlinear Gradient Source and a Singular Potential

LIU Dengming; DING Yige; YANG Jichen

doi:10.1051/wujns/2026315437

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Extinction for a Fast Diffusion Equation Involving a Nonlinear Gradient Source and a Singular Potential

Mathematics | 更新时间：2026-10-29

- Extinction for a Fast Diffusion Equation Involving a Nonlinear Gradient Source and a Singular Potential
- Wuhan University Journal of Natural Sciences Vol. 31, Issue 5, Pages: 437-442(2026)
- 作者机构：
  
  湖南科技大学数学与统计学院，湖南湘潭 411201
- 作者简介：
  
  LIU Dengming, male, Ph. D., Professor, research direction: nonlinear partial differential equations. E-mail: liudengming@hnust.edu.cn
- 基金信息：
- DOI：10.1051/wujns/2026315437
  CLC： O175.29
- Received：27 March 2026，
  
  Published：24 October 2026
- 稿件说明：
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刘灯明,丁一格,杨吉晨.具非线性梯度源和奇异势的快扩散方程解的熄灭[J].武汉大学学报自然科学英文版,2026,31(05):437-442.

LIU Dengming, DING Yige, YANG Jichen. Extinction for a Fast Diffusion Equation Involving a Nonlinear Gradient Source and a Singular Potential[J]. Wuhan Univ J of Nat Sci, 2026, 31(5): 437-442.
刘灯明,丁一格,杨吉晨.具非线性梯度源和奇异势的快扩散方程解的熄灭[J].武汉大学学报自然科学英文版,2026,31(05):437-442. DOI： 10.1051/wujns/2026315437.

LIU Dengming, DING Yige, YANG Jichen. Extinction for a Fast Diffusion Equation Involving a Nonlinear Gradient Source and a Singular Potential[J]. Wuhan Univ J of Nat Sci, 2026, 31(5): 437-442. DOI： 10.1051/wujns/2026315437.

摘要

本文研究了含非线性梯度源与奇异势项的快扩散方程解的熄灭行为。利用能量估计方法、Hardy-Littlewood-Sobolev不等式和微分不等式技巧，给出了一类具非线性梯度源和奇异势的快扩散方程解在有限时刻发生熄灭现象的充分条件。同时，利用上、下解方法，证明了非熄灭解的存在性。

Abstract

This article deals with the extinction behavior of the solution for a fast diffusion equation involving a nonlinear gradient source and a singular potential. Based on the energy estimate method

the Hardy-Littlewood-Sobolev inequality

and the ordinary differential inequality approach

we obtain the sufficient conditions on the occurrence of the finite time extinction phenomenon of the weak solution. Meanwhile

we establish the existence of a non-extinction weak solution via constructing suitable non-extinction sub- and super-solutions and applying the standard iteration process.

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references

Kardar M , Parisi G , Zhang Y C . Dynamic scaling of growing interfaces [J ] . Physical Review Letters , 1986 , 56 ( 9 ): 889 - 892 .

Shang H F . On the Cauchy problem for the singular parabolic equations with gradient term [J ] . Journal of Mathematical Analysis and Applications , 2011 , 378 ( 2 ): 578 - 591 .

Shang H F . Cauchy problem for fast diffusion equations with a gradient term [J ] . Journal of Mathematical Analysis and Applications , 2012 , 396 ( 1 ): 133 - 144 .

Ohnuma M , Sato K . Singular degenerate parabolic equations with applications to the p -Laplace diffusion equation [J ] . Communications in Partial Differential Equations , 1997 , 22 ( 3/4 ): 381 - 411 .

Iagar R G , Laurençot P . Positivity, decay, and extinction for a singular diffusion equation with gradient absorption [J ] . Journal of Functional Analysis , 2012 , 262 ( 7 ): 3186 - 3239 .

Constantin L , Cuesta Romero C M . Doubly nonlinear parabolic equation involving a mixed local-nonlocal operator and a convection term [J ] . Journal of Mathematical Analysis and Applications , 2026 , 556 ( 1 ): 130219 .

Chaouai Z , El Hachimi A . Qualitative properties of weak solutions for p -Laplacian equations with nonlocal source and gradient absorption [J ] . Bulletin of the Korean Mathematical Society , 2020 , 57 ( 4 ): 1003 - 1031 .

Chen Y J , Wang J , Zhang H X . Extinction for a couple of fast diffusion systems with nonlinear sources [J ] . Nonlinear Analysis: Real World Applications , 2013 , 14 ( 4 ): 1931 - 1937 .

DiBenedetto E . Degenerate Parabolic Equations [M ] . New York : Springer-Verlag , 1993 .

Sugiyama Y , Yahagi Y . Extinction, decay and blow-up for Keller-Segel systems of fast diffusion type [J ] . Journal of Differential Equations , 2011 , 250 ( 7 ): 3047 - 3087 .

Gu Y G . Necessary and sufficient conditions for extinction of solution to parabolic equations [J ] . Acta Mathematica Sinica , Chinese Series, 1994 , 37 ( 1 ): 73 - 79 (Ch) .

Tian Y , Mu C L . Extinction and non-extinction for a p -Laplacian equation with nonlinear source [J ] . Nonlinear Analysis: Theory , Methods & Applications, 2008 , 69 ( 8 ): 2422 - 2431 .

Jin C H , Yin J X , Ke Y Y . Critical extinction and blow-up exponents for fast diffusive polytropic filtration equation with sources [J ] . Proceedings of the Edinburgh Mathematical Society , 2009 , 52 ( 2 ): 419 - 444 .

Zhou J , Mu C L . Critical blow-up and extinction exponents for non-Newton polytropic filtration equation with source [J ] . Bulletin of the Korean Mathematical Society , 2009 , 46 ( 6 ): 1159 - 1173 .

Mu C L , Yan L , Xiao Y B . Extinction and nonextinction for the fast diffusion equation [J ] . Abstract and Applied Analysis , 2013 , 2013 : 747613 .

Liu D M , Mu C L . Extinction for a quasilinear parabolic equation with a nonlinear gradient source [J ] . Taiwanese Journal of Mathematics , 2014 , 18 ( 5 ): 1329 - 1343 .

Liu D M , Yang L . Extinction behavior for a parabolic p -Laplacian equation with gradient source and singular potential [J ] . AIMS Mathematics , 2022 , 7 ( 1 ): 915 - 924 .

Badiale M , Tarantello G . A Sobolev-hardy inequality with applications to a nonlinear elliptic equation arising in astrophysics [J ] . Archive for Rational Mechanics and Analysis , 2002 , 163 ( 4 ): 259 - 293 .

Deng X M , Zhou J . Global existence, extinction, and non-extinction of solutions to a fast diffusion p -Laplace evolution equation with singular potential [J ] . Journal of Dynamical and Control Systems , 2020 , 26 ( 3 ): 509 - 523 .

Liu W J , Wu B . A note on extinction for fast diffusive p -Laplacian with sources [J ] . Mathematical Methods in the Applied Sciences , 2008 , 31 ( 12 ): 1383 - 1386 .

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