Blow-up solutions of parabolic $p$-Laplacian inequalities on locally finite graphs

TL;DR

This paper investigates the conditions under which solutions to a nonlinear parabolic inequality involving the p-Laplacian on graphs blow up, extending comparison principles and analyzing growth rate effects.

Contribution

It extends the comparison principle to graph settings and characterizes blow-up behavior based on the nonlinear source's growth rate.

Findings

Blow-up solutions exist when the nonlinear source grows faster than linearly.

Comparison principle is extended to the p-Laplacian on graphs.

Initial conditions influence the existence of blow-up solutions.

Abstract

In this paper, we study blow up behavior of the semilinear parabolic inequality with $p$-Laplacian operator and nonlinear source $u_t - \Delta_p u \geq \sigma(x, t)\Phi(u)$ on a locally finite connected weighted graph $G = (V, E)$. We extend the comparison principle and thereby establish the relationship between the initial value and the existence of blow-up solutions to the problem under different growth rates of $\Phi$. We prove that when the growth rate of $\Phi$ exceeds linear growth, blow-up solutions exist under appropriate initial conditions.