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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).
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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