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Gradient estimates for a weighted nonlinear parabolic equation

Gradient estimates for a weighted nonlinear parabolic equation Abstract Let \((M^{n},g,e^{-f}dv)\) be a complete smooth metric measure space. We prove elliptic gradient estimates for positive solutions of a weighted nonlinear parabolic equation $$\begin{aligned} \left( \varDelta _{f}-\frac{\partial }{\partial t}\right) u(x,t)+q(x,t)u(x,t)+au(x,t)(\ln u(x,t))^{\alpha }=0, \end{aligned}$$ where \((x,t)\in M\times (-\infty ,\infty )\) and a, \(\alpha \) are arbitrary constants. Under the assumption that the \(\infty \)-Bakry-Émery Ricci curvature is bounded from below, we obtain a local elliptic (Hamilton’s type and Souplet–Zhang’s type) gradient estimates to positive smooth solutions of this equation. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Annals of Functional Analysis Springer Journals

Gradient estimates for a weighted nonlinear parabolic equation

Annals of Functional Analysis , Volume 11 (2): 16 – Apr 1, 2020

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References (26)

Publisher
Springer Journals
Copyright
2019 Tusi Mathematical Research Group (TMRG)
ISSN
2639-7390
eISSN
2008-8752
DOI
10.1007/s43034-019-00006-3
Publisher site
See Article on Publisher Site

Abstract

Abstract Let \((M^{n},g,e^{-f}dv)\) be a complete smooth metric measure space. We prove elliptic gradient estimates for positive solutions of a weighted nonlinear parabolic equation $$\begin{aligned} \left( \varDelta _{f}-\frac{\partial }{\partial t}\right) u(x,t)+q(x,t)u(x,t)+au(x,t)(\ln u(x,t))^{\alpha }=0, \end{aligned}$$ where \((x,t)\in M\times (-\infty ,\infty )\) and a, \(\alpha \) are arbitrary constants. Under the assumption that the \(\infty \)-Bakry-Émery Ricci curvature is bounded from below, we obtain a local elliptic (Hamilton’s type and Souplet–Zhang’s type) gradient estimates to positive smooth solutions of this equation.

Journal

Annals of Functional AnalysisSpringer Journals

Published: Apr 1, 2020

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