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Time-insensitive nonlocal parabolic Harnack estimates

Naian Liao, Marvin Weidner

Abstract

We establish new Harnack estimates that defy the waiting-time phenomenon for global solutions to nonlocal parabolic equations. Our technique allows us to consider general nonlocal operators with bounded measurable coefficients. Moreover, we show that a waiting-time is required for the nonlocal parabolic Harnack inequality when local solutions are considered.

Time-insensitive nonlocal parabolic Harnack estimates

Abstract

We establish new Harnack estimates that defy the waiting-time phenomenon for global solutions to nonlocal parabolic equations. Our technique allows us to consider general nonlocal operators with bounded measurable coefficients. Moreover, we show that a waiting-time is required for the nonlocal parabolic Harnack inequality when local solutions are considered.
Paper Structure (34 sections, 28 theorems, 256 equations)

This paper contains 34 sections, 28 theorems, 256 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Extension of the main results
  4. Strategy of the proof
  5. An improved weak Harnack inequality
  6. Further literature
  7. Outline
  8. Notation and notion of weak solutions
  9. Collecting tools
  10. Algebraic estimates
  11. An integral estimate
  12. Boundedness estimates
  13. Weak Harnack estimates
  14. Continuity of local solutions
  15. Some local estimates for supersolutions
  16. ...and 19 more sections

Key Result

theorem 1

Assume that the kernel $K$ satisfies eq:Kcomp. Let $u \ge 0$ be a global weak solution to Eq:1:1 in $(0,T] \times \mathbb{R}^d$ in the sense of Def:global-sol. Then, there exists a constant $c>1$ depending on the data $\{d,s,\lambda,\Lambda\}$, such that for any $\tau \in (0,T]$ and any $x_o \in \ma and Consequently, we have for some constant $c=c(d,s,\lambda,\Lambda)$.

Theorems & Definitions (58)

  • theorem 1: time-insensitive Harnack inequality
  • theorem 2: elliptic-type Harnack inequality
  • theorem 3
  • theorem 4
  • theorem 5: improved weak Harnack inequality
  • definition 6: local solution
  • definition 7: global solution
  • remark 8
  • remark 9
  • definition 10: Cauchy problem
  • ...and 48 more