Harnack Estimation for Nonlinear, Weighted, Heat-Type Equation along Geometric Flow and Applications

• DOI: 10.3390/math11112516
• Publication Date: 2023-05-31T06:27:30Z
• Authors: Yanlin Li; Sujit Bhattacharyya; Shahroud Azami; Apurba Saha; Shyamal Kumar Hui
• Keywords: Harnack inequality; parabolic equation; Harnack inequality; Harnack estimate; weighted Laplacian; parabolic equation; gradient estimate; QA1-939; Harnack estimate; Mathematics; weighted Laplacian; gradient estimate

The method of gradient estimation for the heat-type equation using the Harnack quantity is a classical approach used for understanding the nature of the solution of these heat-type equations. Most of the studies in this field involve the Laplace–Beltrami operator, but in our case, we studied the weighted heat equation that involves weighted Laplacian. This produces a number of terms involving the weight function. Thus, in this article, we derive the Harnack estimate for a positive solution of a weighted nonlinear parabolic heat equation on a weighted Riemannian manifold evolving under a geometric flow. Applying this estimation, we derive the Li–Yau-type gradient estimation and Harnack-type inequality for the positive solution. A monotonicity formula for the entropy functional regarding the estimation is derived. We specify our results for various different flows. Our results generalize some works.