A Liouville-type theorem of a weighted semilinear parabolic equation on weighted manifolds with boundary

Junsheng Gong; Jiancheng Liu; Junsheng Gong; Jiancheng Liu

doi:10.3934/era.2025102

Electronic Research Archive

2025, Volume 33, Issue 4: 2312-2324. doi: 10.3934/era.2025102

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A Liouville-type theorem of a weighted semilinear parabolic equation on weighted manifolds with boundary

Junsheng Gong ,
Jiancheng Liu ^,

College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China

Received: 18 January 2025 Revised: 30 March 2025 Accepted: 01 April 2025 Published: 21 April 2025

We derive a Liouville-type theorem for positive ancient solutions to a weighted semilinear parabolic equation with a Dirichlet boundary condition on complete noncompact weighted manifolds with a compact boundary. This result can be viewed as an extension of Dung et al.'s work on a linear heat equation.
- Liouville-type theorem,
- ancient solution,
- semilinear parabolic equation,
- weighted manifold,
- gradient estimate
Citation: Junsheng Gong, Jiancheng Liu. A Liouville-type theorem of a weighted semilinear parabolic equation on weighted manifolds with boundary[J]. Electronic Research Archive, 2025, 33(4): 2312-2324. doi: 10.3934/era.2025102

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Abstract

We derive a Liouville-type theorem for positive ancient solutions to a weighted semilinear parabolic equation with a Dirichlet boundary condition on complete noncompact weighted manifolds with a compact boundary. This result can be viewed as an extension of Dung et al.'s work on a linear heat equation.

References

[1]	G. Wei, W. Wylie, Comparison geometry for the Bakry-$\rm\acute{E}mery$ Ricci tensor, J. Differ. Geom., 83 (2009), 377–405. https://doi.org/10.4310/jdg/1261495336 doi: 10.4310/jdg/1261495336
[2]	H. T. Dung, N. T. Dung, Sharp gradient estimates for a heat equation in Riemannian manifolds, Proc. Am. Math. Soc., 147 (2019), 5329–5338. https://doi.org/10.1090/proc/14645 doi: 10.1090/proc/14645
[3]	B. Ma, F. Zeng, Hamilton-Souplet-Zhang's gradient estimates and Liouville theorems for a nonlinear parabolic equation, C. R. Math., 356 (2018), 550–557. https://doi.org/10.1016/j.crma.2018.04.003 doi: 10.1016/j.crma.2018.04.003
[4]	X. Zhu, Gradient estimates and Liouville theorems for nonlinear parabolic equations on noncompact Riemannian manifolds, Nonlinear Anal., 74 (2011), 5141–5146. https://doi.org/10.1016/j.na.2011.05.008 doi: 10.1016/j.na.2011.05.008
[5]	J. Wu, Elliptic gradient estimates for a weighted heat equation and applications, Math. Z., 280 (2015), 451–468. https://doi.org/10.1007/s00209-015-1432-9 doi: 10.1007/s00209-015-1432-9
[6]	A. Abolarinwa, Elliptic gradient estimates and Liouville theorems for a weighted nonlinear parabolic equation, J. Math. Anal. Appl., 473 (2019), 297–312. https://doi.org/10.1016/j.jmaa.2018.12.049 doi: 10.1016/j.jmaa.2018.12.049
[7]	R. Filippucci, P. Pucci, P. Souplet, A Liouville-type theorem in a half-space and its applications to the gradient blow-up behavior for superquadratic diffusive Hamilton-Jacobi equations, Commun. Partial Differ. Equations, 45 (2020), 321–349. https://doi.org/10.1080/03605302.2019.1684941 doi: 10.1080/03605302.2019.1684941
[8]	W. Liang, Z. Zhang, A priori estimates and Liouville-type theorems for the semilinear parabolic equations involving the nonlinear gradient source, Calculus Var. Partial Differ. Equations, 64 (2025), 47. https://doi.org/10.1007/s00526-024-02907-1 doi: 10.1007/s00526-024-02907-1
[9]	K. Kunikawa, Y. Sakurai, Yau and Souplet-Zhang type gradient estimates on Riemannian manifolds with boundary under Dirichlet boundary condition, Proc. Am. Math. Soc., 150 (2022), 1767–1777. https://doi.org/10.1090/proc/15768 doi: 10.1090/proc/15768
[10]	H. T. Dung, N. T. Dung, J. Wu, Sharp gradient estimates on weighted manifolds with compact boundary, Commun. Pure Appl. Anal., 20 (2021), 4127–4138. https://doi.org/10.3934/cpaa.2021148 doi: 10.3934/cpaa.2021148
[11]	P. Souplet, Q. S. Zhang, Global solutions of inhomogeneous Hamilton-Jacobi equations, J. Anal. Math., 99 (2006), 355–396. https://doi.org/10.1007/BF02789452 doi: 10.1007/BF02789452
[12]	Y. Sakurai, Rigidity of manifolds with boundary under a lower Ricci curvature bound, Osaka J. Math., 54 (2017), 85–119.
[13]	N. T. Dung, J. Wu, Gradient estimates for weighted harmonic function with Dirichlet boundary condition, Nonlinear Anal., 213 (2021), 112498. https://doi.org/10.1016/j.na.2021.112498 doi: 10.1016/j.na.2021.112498
[14]	R. Reilly, Applications of the Hessian operator in a Riemannian manifold, Indiana Univ. Math. J., 26 (1977), 459–472. https://doi.org/10.1512/iumj.1977.26.26036 doi: 10.1512/iumj.1977.26.26036
[15]	P. Li, S. T. Yau, On the parabolic kernel of the $\rm Schr\ddot{o}dinger$ operator, Acta Math., 156 (1986), 153–201. https://doi.org/10.1007/BF02399203 doi: 10.1007/BF02399203
[16]	P. Souplet, Q. S. Zhang, Sharp gradient estimate and Yau's Liouville theorem for the heat equation on noncompact manifolds, Bull. London Math. Soc., 38 (2006), 1045–1053. https://doi.org/10.1112/S0024609306018947 doi: 10.1112/S0024609306018947
[17]	E. Calabi, An extension of E. Hopf's maximum principle with an application to Riemannian geometry, Duke Math. J., 25 (1958), 45–56. https://doi.org/10.1215/s0012-7094-58-02505-5 doi: 10.1215/s0012-7094-58-02505-5

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