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

Abstract

In this paper, we study blow up behavior of the semilinear parabolic inequality with $p$-Laplacian operator and nonlinear source $u_t-{\Delta }_{p}u\ge \sigma (x,t)\Phi (u)$ 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.