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.