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Sharp sub-Gaussian upper bounds for subsolutions of Trudinger's equation on Riemannian manifolds

Philipp Sürig

Abstract

We consider on Riemannian manifolds the nonlinear evolution equation \begin{equation*} \partial _{t}u=Δ_{p}(u^{1/(p-1)}), \end{equation*}% where $p>1$. This equation is also known as a doubly non-linear parabolic equation or Trudinger's equation. We prove that weak subsolutions of this equation have a sub-Gaussian upper bound and prove that this upper bound is sharp for a specific class of manifolds including $\mathbb{R}^{n}$.

Sharp sub-Gaussian upper bounds for subsolutions of Trudinger's equation on Riemannian manifolds

Abstract

We consider on Riemannian manifolds the nonlinear evolution equation \begin{equation*} \partial _{t}u=Δ_{p}(u^{1/(p-1)}), \end{equation*}% where . This equation is also known as a doubly non-linear parabolic equation or Trudinger's equation. We prove that weak subsolutions of this equation have a sub-Gaussian upper bound and prove that this upper bound is sharp for a specific class of manifolds including .
Paper Structure (15 sections, 17 theorems, 177 equations, 4 figures)

This paper contains 15 sections, 17 theorems, 177 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Weak subsolutions
  3. Definition and basic properties
  4. Caccioppoli type inequality
  5. Integral estimates for subsolutions
  6. Integral maximum principle
  7. Davies-Gaffney type inequality
  8. Mean value inequality
  9. Sobolev and Faber-Krahn inequalities
  10. Comparison in two cylinders
  11. Iterations and the mean value theorem
  12. Sub-Gaussian upper bounds for subsolutions
  13. Appendix
  14. Radial solution on polynomial models
  15. An auxiliary lemma

Key Result

Theorem 1.1

Let $M$ satisfy a relative Faber-Krahn inequality (see Subsection secMoser). Let $u$ be a bounded non-negative solution to (evoeq1) in $M\times [0, \infty)$ with an initial function $u_{0}=u(\cdot ,0$) and set $A=\textnormal{supp}~u_{0}$. Then, for all $x\in M$ and all $t>0$, where $C,c$ are positive constants.

Figures (4)

  • Figure 1: Sets $A$ and $B$
  • Figure 2: Cylinders $Q$ and $Q^{\prime }$
  • Figure 3: Cylinders $Q_{k}$
  • Figure 4: Balls $B_{k}$ and $B(x, { \if@compatibility \mathchar"011A {} \mathchar"011A }_{k})$

Theorems & Definitions (27)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 1.3
  • Definition 1
  • Remark 1
  • Lemma 2.1: grigor2023finite
  • Remark 2
  • Remark 3
  • Remark 4
  • Lemma 2.2
  • ...and 17 more