Harnack inequalities for nonlinear parabolic equations under integral ricci curvature bounds

Document Type : Research Paper

Authors

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

2 Department of Mathematics, Payame Noor University, PO BOX 19395-4697, Tehran, Iran

Abstract

This work is devoted to the study of gradient estimates and Harnack inequalities for positive solutions to a class of nonlinear parabolic equations on compact Riemannian manifolds under integral Ricci curvature bounds. More precisely, we consider equations of the form:
\[
\partial_t v = \Delta v + a v (\ln v)^b + q(x,t) A(v),
\]
where $a,b \in \mathbb{R}$ and $A(v)$ is a smooth function. These equations naturally generalize the classical heat equation and arise in various geometric and physical contexts. We establish Li-Yau, Hamilton, and Souplet-Zhang type gradient estimates, and deduce corresponding Harnack inequalities. Our results extend previous works by relaxing the curvature conditions to integral lower bounds on the Ricci tensor, thereby allowing for more general geometric settings.
 

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References

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History

  • Receive Date: 18 October 2024
  • Revise Date: 24 February 2025
  • Accept Date: 29 April 2025
  • Publish Date: 28 October 2025

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