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Evolution problems with perturbed $1$-Laplacian type operators on random walk spaces

W. Górny, J. M. Mazón, J. Toledo

TL;DR

This work develops a unified variational framework for nonlocal evolution with inhomogeneous growth on random walk spaces, encompassing both graph-based and continuum kernels. By formulating two-structure gradient flows for energies with $(1,p)$ or $(q,p)$ growth and a partitioned growth functional, it proves existence, uniqueness, contraction, and mass-conservation results via completely accretive subdifferentials and nonlinear semigroup theory. The authors derive extinction-time estimates under Poincaré-type inequalities and provide explicit graph and nonlocal examples to reveal diverse diffusion regimes. The results offer a robust toolkit for modeling inhomogeneous nonlocal PDEs on general random-walk spaces, with potential applications in heterogeneous media and network-driven processes.

Abstract

Random walk spaces are a general framework for the study of PDEs. They include as particular cases locally finite weighted connected graphs and nonlocal settings involving symmetric integrable kernels on $\mathbb{R}^N$. We are interested in the study of evolution problems involving two random walk structures so that the associated functionals have different growth on each structure. We also deal with the case of a functional with different growth on a partition of the random walk.

Evolution problems with perturbed $1$-Laplacian type operators on random walk spaces

TL;DR

This work develops a unified variational framework for nonlocal evolution with inhomogeneous growth on random walk spaces, encompassing both graph-based and continuum kernels. By formulating two-structure gradient flows for energies with or growth and a partitioned growth functional, it proves existence, uniqueness, contraction, and mass-conservation results via completely accretive subdifferentials and nonlinear semigroup theory. The authors derive extinction-time estimates under Poincaré-type inequalities and provide explicit graph and nonlocal examples to reveal diverse diffusion regimes. The results offer a robust toolkit for modeling inhomogeneous nonlocal PDEs on general random-walk spaces, with potential applications in heterogeneous media and network-driven processes.

Abstract

Random walk spaces are a general framework for the study of PDEs. They include as particular cases locally finite weighted connected graphs and nonlocal settings involving symmetric integrable kernels on . We are interested in the study of evolution problems involving two random walk structures so that the associated functionals have different growth on each structure. We also deal with the case of a functional with different growth on a partition of the random walk.

Paper Structure

This paper contains 17 sections, 29 theorems, 374 equations, 15 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Convex analysis and nonlinear semigroup theory
  4. Random walk spaces
  5. Nonlocal differential operators
  6. Mixing random walks with different growth functionals
  7. The $(1,p)$-Laplace operator
  8. The gradient flow
  9. Asymptotic behaviour
  10. The $(1,1)$-Laplace operator
  11. The gradient flow
  12. Asymptotic behaviour
  13. The $(q,p)$-Laplace operator.
  14. Different growth functionals on a partition of the random walk domain
  15. Examples
  16. ...and 2 more sections

Key Result

Theorem 2.2

Let $\mathcal{F} : H \to (-\infty, \infty]$ be a proper, convex, and lower semicontinuous functional. Given $u_0 \in \overline{D(\partial \mathcal{F})}$ and $f \in L^2(0,T; H)$, there exists a unique strong solution $u(t)$ of the abstract Cauchy problem ACP1.

Figures (15)

  • Figure 1: Graph in Example \ref{['ex3points']}
  • Figure 2: Example \ref{['ex3points']}, case A. $a=b=1$, $c=1$. $x(t)$ continuous line; $y(t)=z(t)$ dashed line. After $t\approx 0.51986$, $x(t)=y(t)=z(t)$.
  • Figure 3: Example \ref{['ex3points']} case B, $a=1$, $b=1$, $c=10$. $x(t)$ continuous line; $y(t)$ dashed line; $z(t)$ dotted line; $0\le t\lesssim 1.609438$.
  • Figure 4: Example \ref{['ex3points']}, case B. $a=1$, $b=1$, $c=10$. $x(t)$ continuous line; $y(t)=z(t)$ dashed line; $t\gtrsim 1.609438$. After $t\approx 1.861792$, $x(t)=y(t)=z(t).$
  • Figure 5: Example \ref{['ex3points']}, case C. $a=10$, $b=1$, $c=1$. $x(t)$ continuous line; $y(t)$ dashed line; $z(t)$ dotted line; $0\le t\lesssim 0.376844$.
  • ...and 10 more figures

Theorems & Definitions (65)

  • Example 1.1
  • Definition 2.1
  • Theorem 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Definition 2.5
  • Proposition 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • ...and 55 more