Differential Harnack estimates for the semilinear parabolic equation with three exponents on $ \mathbb{R}^{n} $

Fanqi Zeng; Wenli Geng; Ke An Liu; Boya Wang; Fanqi Zeng; Wenli Geng; Ke An Liu; Boya Wang

doi:10.3934/era.2025008

Electronic Research Archive

2025, Volume 33, Issue 1: 142-157. doi: 10.3934/era.2025008

Previous Article Next Article

Research article

Differential Harnack estimates for the semilinear parabolic equation with three exponents on $ \mathbb{R}^{n} $

School of Mathematics and Statistics, Xinyang Normal University, Xinyang 464000, China

Received: 20 September 2024 Revised: 26 November 2024 Accepted: 09 January 2025 Published: 16 January 2025

In this paper, we thought about the positive solutions to the semilinear parabolic equation with three exponents, and obtained several differential Harnack estimates of the positive solutions to the equation. As applications of the main theorems, we found blow-up solutions for the equation and classical Harnack inequalities. Our results generalize some recent works in this direction.
- differential Harnack inequality,
- semilinear parabolic equations,
- maximum principle
Citation: Fanqi Zeng, Wenli Geng, Ke An Liu, Boya Wang. Differential Harnack estimates for the semilinear parabolic equation with three exponents on $ \mathbb{R}^{n} $[J]. Electronic Research Archive, 2025, 33(1): 142-157. doi: 10.3934/era.2025008

Related Papers:

Abstract

In this paper, we thought about the positive solutions to the semilinear parabolic equation with three exponents, and obtained several differential Harnack estimates of the positive solutions to the equation. As applications of the main theorems, we found blow-up solutions for the equation and classical Harnack inequalities. Our results generalize some recent works in this direction.

References

[1]	R. Hamilton, The Harnack estimate for the Ricci flow, J. Differential Geom., 37 (1993), 225–243. https://doi.org/10.4310/jdg/1214453430 doi: 10.4310/jdg/1214453430
[2]	J. Li, Gradient estimates and Harnack inequalities for nonlinear parabolic and nonlinear elliptic equations on Riemannian manifolds, J. Funct. Anal., 100 (1991), 233-256. https://doi.org/10.1016/0022-1236(91)90110-Q doi: 10.1016/0022-1236(91)90110-Q
[3]	P. Li, S. T. Yau, On the parabolic kernel of the Schrödinger operator, Acta Math., 156 (1986), 153–201. https://doi.org/10.1007/BF02399203 doi: 10.1007/BF02399203
[4]	J. Moser, A Harnack inequality for parabolic differential equations, Comm. Pure Appl. Math., 17 (1964), 101–134. https://doi.org/10.1002/cpa.3160170106 doi: 10.1002/cpa.3160170106
[5]	J. Moser, Correction to: "A Harnack inequality for parabolic differential equations", Comm. Pure Appl. Math., 20 (1967), 231–236.
[6]	M. Bǎileşteanu, A Harnack inequality for the parabolic Allen-Cahn equation, Ann. Global Anal. Geom., 51 (2017), 367–378. https://doi.org/10.1007/s10455-016-9540-2 doi: 10.1007/s10455-016-9540-2
[7]	D. Booth, J. Burkart, X. Cao, M. Hallgren, Z. Munro, J. Snyder, et al., A differential Harnack inequality for the Newell-Whitehead-Segel equation, Anal. Theory Appl., 35 (2019), 192–204. https://doi.org/10.4208/ata.OA-0005 doi: 10.4208/ata.OA-0005
[8]	X. Cao, M. Cerenzia, D. Kazaras, Harnack estimate for the endangered species equation, Proc. Amer. Math. Soc., 143 (2015), 4537–4545. https://doi.org/10.1090/S0002-9939-2015-12576-2 doi: 10.1090/S0002-9939-2015-12576-2
[9]	X. Cao, B. Liu, I. Pendleton, A. Ward, Differential Harnack estimates for Fisher's equation, Pacific J. Math., 290 (2017), 273–300. https://doi.org/10.2140/pjm.2017.290.273 doi: 10.2140/pjm.2017.290.273
[10]	S. Hou, L. Zou, Harnack estimate for a semilinear parabolic equation, Sci. China Math., 60 (2017), 833–840. https://doi.org/10.1007/s11425-016-0270-6 doi: 10.1007/s11425-016-0270-6
[11]	A. Abolarinwa, A. Osilagun, S. Azami, A Harnack inequality for a class of 1D nonlinear reaction-diffusion equations and applications to wave solutions, Int. J. Geom. Methods Mod. Phys., 21 (2024), 2450111. https://doi.org/10.1142/S0219887824501111 doi: 10.1142/S0219887824501111
[12]	H. Fujita, On the blowing up of solutions of the Cauchy problem for $u_{t} = \Delta u+u^{1+\alpha}$, J. Fac. Sci. Univ. Tokyo Sect. I, 13 (1966), 109–124. https://doi.org/10.1090/proc/14297 doi: 10.1090/proc/14297
[13]	G. Huang, Z. Huang, H. Li, Gradient estimates and differential Harnack inequalities for a nonlinear parabolic equation on Riemannian manifolds, Ann. Global Anal. Geom., 43 (2013), 209–232. https://doi.org/10.1007/s10455-012-9342-0 doi: 10.1007/s10455-012-9342-0
[14]	H. Wu, C. Kong, Differential Harnack estimate of solutions to a class of semilinear parabolic equation, Math. Inequal. Appl., 25 (2022), 397–405. https://doi.org/10.7153/mia-2022-25-24 doi: 10.7153/mia-2022-25-24
[15]	H. Wu, L. Min, Differential Harnack estimate for a semilinear parabolic equation on hyperbolic space, Appl. Math. Lett., 50 (2015), 69–77. https://doi.org/10.1016/j.aml.2015.06.002 doi: 10.1016/j.aml.2015.06.002
[16]	R. Hamilton, Li-Yau estimates and their Harnack inequalities, Adv. Lect. Math., 17 (2011), 329–362.

Reader Comments

Your name:*

Email:*
© 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)