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The global harnack estimates for a nonlinear heat equation with potential under finsler-geometric flow

  • Shahroud Azami EMAIL logo
Published/Copyright: December 4, 2022
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Mathematica Slovaca
From the journal Mathematica Slovaca Volume 72 Issue 6

Abstract

Let (Mn, F(t), m), t ∈ [0, T], be a compact Finsler manifold with F(t) evolving by the Finsler-geometric flow ∂g(x,t)∂t=2h(x,t), where g(t) is the symmetric metric tensor associated with F, and h(t) is a symmetric (0, 2)-tensor. In this paper, we consider local Li-Yau type gradient estimates for positive solutions of the following nonlinear heat equation with potential

∂tu(x,t)=Δmu(x,t)−R(x,t)u(x,t)−au(x,t)log⁡u(x,t),(x,t)∈M×[0,T],

along the Finsler-geometric flow, where 𝓡 is a smooth function, and a is a real nonpositive constant. As an application we obtain a global estimate and a Harnack estimate. Our results are also natural extension of similar results on Riemannian-geometric flow.

Keywords: Gradient estimate; nonlinear heat equation; Finsler-geometric flow
MSC 2010: Primary 53C44; 58J35; Secondary 53B40

  1. ( Communicated by Alberto Lastra )

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Received: 2021-06-04
Accepted: 2021-11-15
Published Online: 2022-12-04
Published in Print: 2022-12-16

© 2022 Mathematical Institute Slovak Academy of Sciences

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https://doi.org/10.1515/ms-2022-0109
Keywords for this article
Gradient estimate; nonlinear heat equation; Finsler-geometric flow

Articles in the same Issue

  1. Prof. RNDr. Július korbaš, CSC. passed away
  2. Kalmbach measurability In d0-algebras
  3. Quantifiers on L-algebras
  4. Ideals of functions with compact support in the integer-valued case
  5. An algebraic study of the logic S5’(BL)
  6. Triangular numbers and generalized fibonacci polynomial
  7. A general matrix series inversion pair and associated polynomials
  8. Quantum ostrowski type inequalities for pre-invex functions
  9. Hermite-Hadamard type inequalities for interval-valued fractional integrals with respect to another function
  10. Approximating families for lattice outer measures on unsharp quantum logics
  11. On a generalized Lamé-Navier system in ℝ3
  12. Coercive and noncoercive elliptic problems with variable exponent Laplacian under Robin boundary conditions
  13. On some classical properties of normed spaces via generalized vector valued almost convergence
  14. Fan-Hemicontinuity for the gradient of the norm in Hilbert space
  15. Solvability of mixed problems for heat equations with two nonlocal conditions
  16. The global harnack estimates for a nonlinear heat equation with potential under finsler-geometric flow
  17. A note on set-star-K-Menger spaces
  18. A bivariate extension of the Omega distribution for two-dimensional proportional data
  19. Bernstein polynomials based iterative method for solving fractional integral equations
  20. The symmetric 4-Player gambler’s problem with unequal initial stakes
  21. Enveloping action: Convergence spaces
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