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Diffusion bounds for non-autonomous degenerate parabolic equations

Marius Lemm, Israel Michael Sigal, Jingxuan Zhang

Abstract

We prove the Davies-Gaffney (i.e., integrated Nash-Aronson) type diffusive upper bounds on the propagators of parabolic equations in $L^p$-sense for all $1\le p\le\infty$. Our approach is based on a simple exponential deformation argument that does not require hypoellipticity. It provides a unified approach to diffusive upper bounds that covers a wide class of problems including degenerate, non-autonomous, and non-linear equations.

Diffusion bounds for non-autonomous degenerate parabolic equations

Abstract

We prove the Davies-Gaffney (i.e., integrated Nash-Aronson) type diffusive upper bounds on the propagators of parabolic equations in -sense for all . Our approach is based on a simple exponential deformation argument that does not require hypoellipticity. It provides a unified approach to diffusive upper bounds that covers a wide class of problems including degenerate, non-autonomous, and non-linear equations.
Paper Structure (26 sections, 15 theorems, 121 equations, 3 figures)

This paper contains 26 sections, 15 theorems, 121 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Setup and results
  3. Preliminaries
  4. Euclidean space
  5. Riemannian manifolds
  6. Nonlinear parabolic equations
  7. Comparison with the sharp heat kernel bounds
  8. Discussion
  9. Proof of Theorem \ref{['thm1']}
  10. Key relation
  11. Bound on deformed propagator
  12. Proof of Lemma \ref{['lem23']}
  13. Construction of $\phi$ and bounds on $\chi_XT^{-1}$ and $T\chi_Y$
  14. Construction of $\xi$
  15. Proof of Theorem \ref{['thm1']} for $p=1$
  16. ...and 11 more sections

Key Result

Theorem 2.1

Let $\Omega=\mathbb{R}^n$. Assume the coefficients of $L$ satisfy cCond--bCond. Then there exists $k=k(n)>0$ such that for any $t>s$ and $X,\,Y\subset \mathbb{R}^n$, for all $1\le p\le \infty$, provided

Figures (3)

  • Figure 1: Schematic diagram for the geometric setup of Proposition \ref{['prop24']}. Note that no regularity on the boundary of $X$ or $Y$ is assumed.
  • Figure 2: Schematic diagram for the geometric decomposition used to prove Corollary \ref{['cor:intbound']}.
  • Figure 3: Profile of the signed cutoff function $\phi(x)$.

Theorems & Definitions (26)

  • Theorem 2.1: Diffusion estimate in $\mathbb{R}^n$
  • Remark 2.2
  • Definition 2.3: $\gamma$-cutoff property
  • Theorem 2.4: Diffusion estimate on Riemannian manifolds
  • Proposition 2.5
  • Corollary 2.6
  • proof
  • Theorem 2.7: Nonlinear diffusion bound in $\Omega\subset\mathbb{R}^n$
  • Remark 2.8
  • Theorem 2.9: Sharp diffusion bound
  • ...and 16 more