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Gradient estimates of positive solutions for the weighted nonlinear parabolic equation

Gradient estimates of positive solutions for the weighted nonlinear parabolic equation In this paper, we prove a Li–Yau type gradient estimate for a positive solution to the weighted nonlinear parabolic type equation (Δϕ-∂t)u(x,t)+a(x,t)u(x,t)lnu(x,t)+b(x,t)u(x,t)=0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\begin{aligned} (\Delta _{\phi }-\partial _{t})u(x,t) +a(x,t)u(x,t)\ln u(x,t)+b(x,t)u(x,t)=0 \end{aligned}$$\end{document}on the complete smooth metric measure space under integral Bakry–Émery Ricci curvature bounds. This estimates optimize the obtained conclusions by Zhang and Zhu (J Funct Anal 275:478–515, 2018). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Annals of Functional Analysis Springer Journals

Gradient estimates of positive solutions for the weighted nonlinear parabolic equation

Mi, Rong
Annals of Functional Analysis , Volume 14 (2) – Apr 1, 2023
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Springer Journals
Copyright
Copyright © Tusi Mathematical Research Group (TMRG) 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
ISSN
2639-7390
eISSN
2008-8752
DOI
10.1007/s43034-023-00253-5
Publisher site
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Abstract

In this paper, we prove a Li–Yau type gradient estimate for a positive solution to the weighted nonlinear parabolic type equation (Δϕ-∂t)u(x,t)+a(x,t)u(x,t)lnu(x,t)+b(x,t)u(x,t)=0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\begin{aligned} (\Delta _{\phi }-\partial _{t})u(x,t) +a(x,t)u(x,t)\ln u(x,t)+b(x,t)u(x,t)=0 \end{aligned}$$\end{document}on the complete smooth metric measure space under integral Bakry–Émery Ricci curvature bounds. This estimates optimize the obtained conclusions by Zhang and Zhu (J Funct Anal 275:478–515, 2018).

Journal

Annals of Functional Analysis – Springer Journals

Published: Apr 1, 2023

Keywords: Smooth metric measure space; Gradient estimate; Integral Bakry–Émery Ricci curvature; 58J35; 53C21; 35K08

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