Doubly nonlinear equation with perturbation governed by hypergraph Laplacian

Masahiro Ikeda; Shun Uchida; Masahiro Ikeda; Shun Uchida

doi:10.3934/cam.2026026

Communications in Analysis and Mechanics

2026, Volume 18, Issue 3: 626-648. doi: 10.3934/cam.2026026

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Research article

Doubly nonlinear equation with perturbation governed by hypergraph Laplacian

Masahiro Ikeda ^1,2,
Shun Uchida ^{3
,
,}

1.
Graduate School of Information Science and Technology, The University of Osaka, 1-5 Yamadaoka Suita-shi, Osaka 565-0871, Japan
2.
Center for Advanced Intelligence Project, RIKEN, 1-4-1 Nihonbashi, Chuo-ku, Tokyo 103-0027, Japan
3.
Faculty of Science and Technology, Oita University, 700 Dannoharu, Oita 870-1192, Japan

Received: 01 November 2025 Revised: 05 March 2026 Accepted: 06 May 2026 Published: 06 August 2026
Primary 34G25; Secondary 05C65, 34A12, 34A34, 47H04

The hypergraph Laplacian, which was introduced to study the structure of complicated networks in the information science, is defined as a set-valued operator on a finite-dimensional Euclidean space. In previous works, we have shown that a nonlinear ordinary differential equation governed by the hypergraph Laplacian describes the diffusion process on a hypergraph, and its behavior closely resembles the heat equation. In this paper, we consider some generalized doubly nonlinear equation governed by the hypergraph Laplacian in order to deal with various diffusion phenomena on discrete domains.
- hypergraph Laplacian,
- nonlinear evolution inclusion,
- doubly nonlinear equation,
- set-valued differential equation,
- subdifferential,
- solvability
Citation: Masahiro Ikeda, Shun Uchida. Doubly nonlinear equation with perturbation governed by hypergraph Laplacian[J]. Communications in Analysis and Mechanics, 2026, 18(3): 626-648. doi: 10.3934/cam.2026026

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Abstract

The hypergraph Laplacian, which was introduced to study the structure of complicated networks in the information science, is defined as a set-valued operator on a finite-dimensional Euclidean space. In previous works, we have shown that a nonlinear ordinary differential equation governed by the hypergraph Laplacian describes the diffusion process on a hypergraph, and its behavior closely resembles the heat equation. In this paper, we consider some generalized doubly nonlinear equation governed by the hypergraph Laplacian in order to deal with various diffusion phenomena on discrete domains.

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