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Sharp long distance upper bounds for solutions of Leibenson's equation on Riemannian manifolds

Alexander Grigor'yan, Jin Sun, Philipp Sürig

Abstract

We consider on Riemannian manifolds the Leibenson equation $\partial _{t}u=Δ_{p}u^{q}$ that is also known as a doubly nonlinear evolution equation. We prove sharp upper estimates of weak subsolutions to this equation on Riemannian manifolds with non-negative Ricci curvature in the whole range of $p>1$ and $q>0$ satisfying $q(p-1)<1$. In this way, we improve the result of \cite{Grigoryan2024a} and prove Conjecture 1.2 from \cite{Grigoryan2024a}.

Sharp long distance upper bounds for solutions of Leibenson's equation on Riemannian manifolds

Abstract

We consider on Riemannian manifolds the Leibenson equation that is also known as a doubly nonlinear evolution equation. We prove sharp upper estimates of weak subsolutions to this equation on Riemannian manifolds with non-negative Ricci curvature in the whole range of and satisfying . In this way, we improve the result of \cite{Grigoryan2024a} and prove Conjecture 1.2 from \cite{Grigoryan2024a}.

Paper Structure

This paper contains 9 sections, 9 theorems, 118 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Weak subsolutions
  3. Faber-Krahn inequality
  4. Long distance decay
  5. Long time decay
  6. Comparison in two cylinders
  7. Iterations and the mean value theorem
  8. Optimal long time decay
  9. Combined estimate

Key Result

Theorem 1.1

Let $M$ satisfy a relative Faber-Krahn inequality (see Section appFk for definition) and assume that, for all $x\in M$ and all $R\geq 1,$ for some $c,{ \if@compatibility \mathchar"010B {} \mathchar"010B } >0.$ Assume that (orising) holds and that Let $u$ be a bounded non-negative solution of(evoeq) in $M\times [0, \infty)$ with initial function $u_{0}=u\left( \cdot ,0\right)\in L^{1}(M)\cap

Figures (4)

  • Figure 1: Cylinders $Q$ and $Q^{\prime }$
  • Figure 2: Balls $B_{k}$ and $B(x, { \if@compatibility \mathchar"011A {} \mathchar"011A }_{k})$
  • Figure 3: Cylinders $Q_{0}$ and $Q_{1 }$
  • Figure 4: Cylinders $Q^{\prime}$ and $Q$

Theorems & Definitions (17)

  • Theorem 1.1
  • Definition 1
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 4.1
  • Remark 1
  • Remark 2
  • Lemma 4.2
  • Remark 3
  • Lemma 5.1
  • ...and 7 more