Nonexistence Results for a General Class of Parabolic Problems with a Potential on Weighted Graphs

TL;DR

This work studies nonexistence of nonnegative global solutions to the parabolic inequality $u_t \ge \Delta(F(u)) + v u^{\sigma}$ on infinite weighted graphs, with $0\le F(p)\le C p^{m}$ and $\sigma>1$, unifying the Porous Medium and Fast Diffusion Equations in a graph setting. It develops a space-time test-function method on annular regions $E_R$ and a cutoff $\phi_R$, deriving explicit growth conditions on the potential $v$ tied to the graph geometry via Assumption (A); under these conditions, any nonnegative global very weak solution must vanish. The results are illustrated on lattices, product graphs with finite groups, and homogeneous trees, and extended to finite graphs with a corresponding time-averaged potential condition, thereby broadening the nonexistence theory for parabolic diffusion on networks. Overall, the paper extends prior arXiv:2404.12058 results to a broad class of nonlinearities $F$ and graph geometries, yielding practical, geometry-aware nonexistence criteria with potential implications for diffusion processes on networks.

Abstract

We establish nonexistence conditions for nonnegative nontrivial solutions to a class of semilinear parabolic equations with a positive potential on weighted graphs, extending results in arXiv:2404.12058 [math.AP] to a broader setting that includes both the Porous Medium Equation and the Fast Diffusion Equation. We identify conditions related to the graph's geometry, the potential's behaviour at infinity, and bounds on the Laplacian of the distance function under which nonexistence holds. Using a test function argument, we derive explicit parameter ranges for nonexistence.

Theorems & Definitions (19)

• Definition 2.1
• Definition 2.2
• Remark 2.3
• Remark 2.4
• Theorem 2.5
• Corollary 2.6
• Corollary 2.7
• Corollary 2.8
• proof : Proof of Theorem \ref{['mainthm']}
• proof : Proof of Corollary \ref{['cor1']}