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

i

This article has been updated. A Correction to this article was published on .

i

The Original Article was published on

i

The Original Article was published on

i

This article has been updated

i

This article was retracted on

i

This article has been updated

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.

Keywords

differential Harnack inequality / semilinear parabolic equations / maximum principle

CHANGE HISTORY

A Correction to this paper has been published:
https://doi.org

The Retraction Note was updated to reflect reception of statements of disagreement from all authors.

This article has been retracted.Please see the Retraction Notice for more detail:
https://doi.org

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

[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

[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

[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

[5] J. Moser, Correction to: “A Harnack inequality for parabolic differential equations”, Comm. Pure Appl. Math., 20 (1967), 231-236.

[6] M. Băileşteanu, 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

[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

[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

[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

[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

[11] A. Abolarinwa, A. Osilagun, S. Azami, A Harnack inequality for a class of 1D nonlinear reactiondiffusion equations and applications to wave solutions, Int. J. Geom. Methods Mod. Phys., 21 (2024), 2450111. https://doi.org/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

[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

[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

[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

[16] R. Hamilton, Li-Yau estimates and their Harnack inequalities, Adv. Lect. Math., 17 (2011), 329-362.