Some results of the existence and behavior of solutions for p-Laplacian parabolic-type equations on networks

Abstract

In this paper, we consider a class of discrete p-Laplacian nonlinear parabolic equations with mixed boundary conditions. First, we establish the local existence and uniqueness of bounded solutions under suitable assumptions on the nonlinearity. We then prove a comparison principle and a strict comparison principle in the linear diffusion case, which provide key insights into the ordering of solutions. Furthermore, we show that solutions remain positive or vanish in finite time, depending on the parameters involved. Regarding blow-up behavior, we prove that solutions blow up in finite time for parameter values beyond a defined threshold, under a specific condition imposed on the nonlinearity. Finally, by removing this condition, we show that there is no finite-time blow-up of the solution in the linear diffusion case under some assumptions on the nonlinearity and the parameter range.