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Kyungpook Mathematical Journal 2021; 61(3): 631-644

Published online September 30, 2021

Copyright © Kyungpook Mathematical Journal.

Harnack Estimate for Positive Solutions to a Nonlinear Equation Under Geometric Flow

Ghodratallah Fasihi-Ramandi, Shahroud Azami*

Department of Pure Mathematics, Faculty of Science Imam Khomeini International University, Qazvin, Iran
e-mail : fasihi@sci.ikiu.ac.ir and azami@sci.ikiu.ac.ir

Received: August 28, 2019; Revised: June 16, 2020; Accepted: August 18, 2020

Abstract

In the present paper, we obtain gradient estimates for positive solutions to the following nonlinear parabolic equation under general geometric flow on complete noncompact manifolds ut=u+a(x,t)up+b(x,t)uq where, 0 < p, q < 1 are real constants and a(x, t) and b(x, t) are functions which are C2 in the x-variable and C1 in the t-variable. We shall get an interesting Harnack inequality as an application.

Keywords: Geometric Flow, Harnack Estimate, Nonlinear Parabolic Equations.