Nonlinear Diffusion Equations on Graphs: Global Well-Posedness, Blow-Up Analysis and Applications

TL;DR

The paper addresses nonlinear diffusion on graphs with non-Lipschitz nonlinearities, focusing on short-time existence, global well-posedness, and finite-time blow-up under nontrivial potentials. It develops a graph-based potential-well framework using the energy $J$, Nehari functional $N$, and the principal eigenpair of $-\Delta+a$ to delineate global existence vs blow-up regimes, and derives explicit blow-up time bounds and rates. The authors also connect PDE dynamics on graphs to complex dynamical networks, providing numerical demonstrations on a 25-node network that validate the theoretical criteria for convergence to zero or finite-time blow-up. Overall, the work extends classical diffusion and blow-up analysis from Euclidean spaces to graphs with potentials and offers a PDE-based lens for analyzing complex network synchronization and instability phenomena.

Abstract

For a nonlinear diffusion equation on graphs whose nonlinearity violates the Lipschitz condition, we prove short-time solution existence and characterize global well-posedness by establishing sufficient criteria for blow-up phenomena and quantifying blow-up rates. These theoretical results are then applied to model complex dynamical networks, with supporting numerical experiments. This work mainly makes two contributions: (i) generalization of existing results for diffusion equations on graphs to cases with nontrivial potentials, producing richer analytical results; (ii) a new PDE approach to model complex dynamical networks, with preliminary numerical experiments confirming its validity.

Figures (3)

• Figure 1: The network $G_{25}$ with $25$ nodes
• Figure 2: Dynamic curves of $G_{25}$
• Figure 3: Blow-up of $G_{25}$

Theorems & Definitions (16)

• Remark 1.1
• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Theorem 2.4
• Theorem 2.5
• Lemma 3.1
• proof
• Corollary 3.2
• Lemma 3.3