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Global existence for a Leibenson type equation with reaction on Riemannian manifolds

Giulia Meglioli, Francescantonio Oliva, Francesco Petitta

TL;DR

This work analyzes global-in-time existence for a reaction-diffusion equation of Leibenson type on complete non-compact Riemannian manifolds with infinite volume: u_t = Δ_p u^m + u^q, under 1<p<N and m>1, with q>m(p−1). By assuming Sobolev inequality (and, when available, Poincaré inequality), the authors prove global existence for small initial data in appropriate L^s spaces, obtaining explicit L^∞ estimates that decay in time. The key methods combine Caccioppoli-type energy estimates with a Moser iteration, applied to suitably truncated problems and followed by a careful passage to the limit on expanding domains. The results reveal a delicate interaction between geometry (noncompactness, infinite measure) and nonlinear diffusion that yields global behavior without Euclidean analogues, notably permitting global solutions in a regime where the diffusion-dominated dynamics interacts with reaction terms differently than in bounded Euclidean domains.

Abstract

We show a global existence result for a doubly nonlinear porous medium type equation of the form $$u_t = Δ_p u^m +\, u^q$$ on a complete and non-compact Riemannian manifold $M$ of infinite volume. Here, for $1<p<N$, we assume $m(p-1)\ge1$, $m>1$ and $q>m(p-1)$. In particular, under the assumptions that $M$ supports the Sobolev inequality, we prove that a solution for such a problem exists globally in time provided $q>m(p-1)+\frac pN$ and the initial datum is small enough; namely, we establish an explicit bound on the $L^\infty$ norm of the solution at all positive times, in terms of the $L^1$ norm of the data. Under the additional assumption that a Poincaré-type inequality also holds in $M$, we can establish the same result in the larger interval, i.e. $q>m(p-1)$. This result has no Euclidean counterpart, as it differs entirely from the case of a bounded Euclidean domain due to the fact that $M$ is non-compact and has infinite measure.

Global existence for a Leibenson type equation with reaction on Riemannian manifolds

TL;DR

This work analyzes global-in-time existence for a reaction-diffusion equation of Leibenson type on complete non-compact Riemannian manifolds with infinite volume: u_t = Δ_p u^m + u^q, under 1<p<N and m>1, with q>m(p−1). By assuming Sobolev inequality (and, when available, Poincaré inequality), the authors prove global existence for small initial data in appropriate L^s spaces, obtaining explicit L^∞ estimates that decay in time. The key methods combine Caccioppoli-type energy estimates with a Moser iteration, applied to suitably truncated problems and followed by a careful passage to the limit on expanding domains. The results reveal a delicate interaction between geometry (noncompactness, infinite measure) and nonlinear diffusion that yields global behavior without Euclidean analogues, notably permitting global solutions in a regime where the diffusion-dominated dynamics interacts with reaction terms differently than in bounded Euclidean domains.

Abstract

We show a global existence result for a doubly nonlinear porous medium type equation of the form on a complete and non-compact Riemannian manifold of infinite volume. Here, for , we assume , and . In particular, under the assumptions that supports the Sobolev inequality, we prove that a solution for such a problem exists globally in time provided and the initial datum is small enough; namely, we establish an explicit bound on the norm of the solution at all positive times, in terms of the norm of the data. Under the additional assumption that a Poincaré-type inequality also holds in , we can establish the same result in the larger interval, i.e. . This result has no Euclidean counterpart, as it differs entirely from the case of a bounded Euclidean domain due to the fact that is non-compact and has infinite measure.
Paper Structure (10 sections, 13 theorems, 213 equations, 2 figures)

This paper contains 10 sections, 13 theorems, 213 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Related results in the literature
  3. Outline of our results
  4. A short overview of the applied techniques
  5. Statement of main results
  6. Caccioppoli-type estimates and the iterative scheme
  7. An extra $L^s-L^s$ estimate
  8. Auxiliary results when Poincaré inequality holds true
  9. Proof of Theorem \ref{['teo1']}
  10. Proof of Theorem \ref{['teo2']}

Key Result

Theorem 2.2

Let $M$ be a complete, non-compact Riemannian manifold of infinite volume such that the Sobolev inequality S holds. Let also $m,p$ be as in hp, and assume that and $s>q_0$, where $q_0$ is defined in q0. Moreover, assume that the initial datum $u_0\ge0$ satisfies where $\varepsilon_0=\varepsilon_0(q,m,p,N, \mathcal{S}_{p})>0$ is sufficiently small. Then problem problema admits a solution for any

Figures (2)

  • Figure 1: The cylinders $U_i = B_R \times (\theta_i, t)$
  • Figure 2: Constants approximating $C_0$

Theorems & Definitions (31)

  • Definition 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Definition 3.1
  • Remark 3.2
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • Remark 3.5
  • proof
  • ...and 21 more