A complete characterization of the discrete p-Laplacian parabolic equations with q-nonlocal reaction with respect to the blow-up property.

Abstract In this paper, we consider the following discrete p -Laplacian nonlinear parabolic equations with q -nonlocal reaction { u t (x , t) = Δ p , ω u (x , t) + λ ∑ y ∈ S | u (y , t) | q − 1 u (y , t) , (x , t) ∈ S × (0 , ∞) , u (x , t) = 0 , (x , t) ∈ ∂ S × (0 , ∞) , u (x , 0) = u 0 ≥ 0 , x ∈ S ‾. Here, S is a network with boundary ∂ S. The goal of this paper is to characterize completely the parameters p > 1 and q > 0 to see when the solution blows up, vanishes, or exists globally. Indeed, the blow-up rates for the blow-up solutions are derived. [ABSTRACT FROM AUTHOR]