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The nonlinear porous medium equation for the f-Laplacian: Hamilton-Souplet-Zhang type gradient estimates and implications

Ali Taheri, Vahideh Vahidifar

TL;DR

The paper develops Hamilton-Souplet-Zhang type gradient estimates for positive solutions of the nonlinear porous medium equation $\partial_t u - \Delta_f u^p = \mathscr N(t,x,u)$ on smooth metric measure spaces with time-dependent geometry. By applying the pressure transform $v = \tfrac{p}{p-1}u^{p-1}$ and analyzing the evolution of $w = |\nabla v|^2/v^{\beta}$ under the operator $\mathscr L_v^p = \partial_t -(p-1)v\Delta_f$, the authors derive local and global gradient bounds in two exponent regimes, using Bochner-type formulas, curvature-dimension conditions, and space-time cut-offs. They also establish refined evolution inequalities (including an II version) and develop cylindrical localisation techniques to prove results for equations with forcing and to study ancient solutions, including Liouville-type rigidity statements. A Ricci-Perelman super-flow inequality is used to obtain global gradient bounds on closed SMMS, linking geometric evolution with nonlinear diffusion. These contributions extend and sharpen prior results in the nonlinear diffusion and SMMS setting, with implications for ancient solutions and parabolic Liouville-type theorems.

Abstract

This article presents new gradient estimates for positive solutions to the nonlinear porous medium equation (NPME) in the context of smooth metric measure spaces. The diffusion operator here is the f-Laplacian and the gradient estimates of interest are mainly of Hamilton-Souplet-Zhang types. These estimates are established using a variety of methods and techniques and several implications, most notably, to parabolic Liouville-type results and characterisation of ancient solutions are given. The problem is posed in the general framework where the metric and potential evolve with time and the proofs make use of natural lower bounds on the time derivative of the metric and the Bakry-Émery m-Ricci curvature tensors. Our results extend and improve various existing ones in the literature.

The nonlinear porous medium equation for the f-Laplacian: Hamilton-Souplet-Zhang type gradient estimates and implications

TL;DR

The paper develops Hamilton-Souplet-Zhang type gradient estimates for positive solutions of the nonlinear porous medium equation on smooth metric measure spaces with time-dependent geometry. By applying the pressure transform and analyzing the evolution of under the operator , the authors derive local and global gradient bounds in two exponent regimes, using Bochner-type formulas, curvature-dimension conditions, and space-time cut-offs. They also establish refined evolution inequalities (including an II version) and develop cylindrical localisation techniques to prove results for equations with forcing and to study ancient solutions, including Liouville-type rigidity statements. A Ricci-Perelman super-flow inequality is used to obtain global gradient bounds on closed SMMS, linking geometric evolution with nonlinear diffusion. These contributions extend and sharpen prior results in the nonlinear diffusion and SMMS setting, with implications for ancient solutions and parabolic Liouville-type theorems.

Abstract

This article presents new gradient estimates for positive solutions to the nonlinear porous medium equation (NPME) in the context of smooth metric measure spaces. The diffusion operator here is the f-Laplacian and the gradient estimates of interest are mainly of Hamilton-Souplet-Zhang types. These estimates are established using a variety of methods and techniques and several implications, most notably, to parabolic Liouville-type results and characterisation of ancient solutions are given. The problem is posed in the general framework where the metric and potential evolve with time and the proofs make use of natural lower bounds on the time derivative of the metric and the Bakry-Émery m-Ricci curvature tensors. Our results extend and improve various existing ones in the literature.

Paper Structure

This paper contains 10 sections, 21 theorems, 142 equations.

Table of Contents

  1. Introduction
  2. A Hamilton-Souplet-Zhang estimate for \ref{['eq11']}: $1< p < 1+1/(\sqrt{2m} +1)$
  3. Evolution inequalities ${\bf I}$: $\mathscr L_v^p=\partial_t - (p-1)v\Delta_f$ and $w = |\nabla v|^2/v^{\beta}$
  4. Space-time cut-offs, cylindrical localisation and the proof of Theorem \ref{['thm1']}
  5. Ancient solutions to $\partial_t u - \Delta_f u^p = {\mathscr N}(u)$ ${\bf I}$: Proof of Theorem \ref{['ancient-1']}
  6. Another Hamilton-Souplet-Zhang estimate for \ref{['eq11']}: $1< p < 1+1/\sqrt{m-1}$
  7. Evolution inequalities ${\bf II}$: $\mathscr L_v^p=\partial_t - (p-1)v \Delta_f$ and $w = |\nabla v|^2/v^{\beta}$
  8. Cylindrical localisation and the proof of Theorem \ref{['thm1-EQ-PME']}
  9. Ancient solutions to $\partial_t u - \Delta_f u^p = {\mathscr N}(u)$ ${\bf II}$: Proof of Theorem \ref{['ancient-2']}
  10. A Ricci-Perelman super flow inequality $\partial_t g + 2(p-1) v {\mathscr Ric}^m_f(g) \ge - 2\mathsf{k}g$

Key Result

Theorem 2.1

Let $(\mathscr M, g, d\mu)$ be a complete smooth metric measure space with $d\mu=e^{-f} dv_g$. Assume the metric and potential are time dependent, of class $\mathscr{C}^2$ and that for suitable constants $k, h \ge 0$ and $m \ge n$ satisfy ${\mathscr Ric}_f^m (g) \ge -(m-1)k g$, $\partial_t g \ge -2 Here $\beta \in (\beta_1, \beta_2)$ where $\beta_1<\beta_2< 0$ are the roots of $\beta^2 + (2-p)/(p

Theorems & Definitions (35)

  • Theorem 2.1
  • Remark 2.2
  • Remark 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 25 more