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Gradient estimates for a weighted parabolic equation under geometric flow

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Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

Abstract

Let \((M^{n},g,e^{-\phi }dv)\) be a weighted Riemannian manifold evolving by geometric flow \(\frac{\partial g}{\partial t}=2\,h(t),\,\,\,\frac{\partial \phi }{\partial t}=\Delta \phi \). In this paper, we obtain a series of space-time gradient estimates for positive solutions of a parabolic partial equation

$$\begin{aligned} (\Delta _{\phi }-\partial _{t})u(x,t)=q(x,t)u^{a+1}(x,t)+p(x,t)A(u(x,t))),\,\,\,\,(x,t)\in M\times [0,T], \end{aligned}$$

on a weighted Riemannian manifold under geometric flow. By integrating the gradient estimates, we find the corresponding Harnack inequalities.

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  1. Department of Pure Mathematics, Faculty of Science, Imam Khomeini International University, Qazvin, Iran

    Shahroud Azami

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Azami, S. Gradient estimates for a weighted parabolic equation under geometric flow. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 117, 74 (2023). https://doi.org/10.1007/s13398-023-01408-8

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