Life span of solutions to a semilinear parabolic equation on locally finite graphs

TL;DR

This paper investigates the lifespan of solutions to a semilinear heat equation on locally finite graphs, developing novel eigenvalue and maximum principle methods adapted to graph structures.

Contribution

It introduces the first eigenvalue method on graphs, extending classical PDE techniques to analyze solution lifespan on discrete structures.

Findings

Derived estimates for solution lifespan on graphs

Analyzed asymptotic behavior of solutions

Identified differences from continuous cases

Abstract

Let $G=(V,E)$ be a locally finite connected graph. We develop the first eigenvalue method on $G$ introduced in 1963 by Kaplan \cite{Kaplan} on Euclidean space, the discrete Phragm\'{e}n-Lindel\"{o}f principle of parabolic equations and upper and lower solutions method on $G$. Using these methods, we establish the estimates and asymptotic behaviour of the life span of solutions to a semilinear heat equation with initial data $\lambda\psi(x)$ for different scales of $\lambda$ on $G$ under some different conditions. Our results are different from the continuous case, which is related to the structure of the graph $G$.