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Gradient estimate for Fisher-KPP equation on Finsler metric measure spaces

Bin Shen, Dingli Xia

Abstract

In this manuscript, we study the positive solutions of the Finslerian Fisher-KPP equation $$ u_t=Δ^{\nabla u} u+cu(1-u). $$ The Fisher-KPP equation is widely applied and connected to many mathematical branches. We establish the global gradient estimates on compact Finsler metric measure manifold with the traditional $CD(K,N)$ condition, which is developed by S. Ohta and K.-T. Sturm in Finsler geometry. Furthermore, With the assistance of a new comparison theorem developed by the first author, we also give the gradient estimate on forward complete noncompact locally finite misalignment Finsler metric measure spaces with the mixed weighted curvature bounded below and some non-Riemannian curvatures norm-bounded.

Gradient estimate for Fisher-KPP equation on Finsler metric measure spaces

Abstract

In this manuscript, we study the positive solutions of the Finslerian Fisher-KPP equation The Fisher-KPP equation is widely applied and connected to many mathematical branches. We establish the global gradient estimates on compact Finsler metric measure manifold with the traditional condition, which is developed by S. Ohta and K.-T. Sturm in Finsler geometry. Furthermore, With the assistance of a new comparison theorem developed by the first author, we also give the gradient estimate on forward complete noncompact locally finite misalignment Finsler metric measure spaces with the mixed weighted curvature bounded below and some non-Riemannian curvatures norm-bounded.
Paper Structure (4 sections, 7 theorems, 105 equations)

This paper contains 4 sections, 7 theorems, 105 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Global gradient estimates on compact Finsler manifolds
  4. Local gradient estimates on forward complete Finsler spaces

Key Result

Theorem 1.1

Let $(M, F, \mu)$ be a compact Finsler metric measure space with dimensional $n\ge2$, whose weighted Ricci curvature satisfies $Ric^N\ge-K$, for some positive constant $K$. Assume the reversibility of $M$ has upper bound $\rho_{0}$. Suppose $u$ is a bounded positive smooth solution of the Fisher-KPP where, $M_{1}=\sup u(x,t)$.

Theorems & Definitions (20)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Remark 2.1
  • ...and 10 more