On the critical set for discrete Laplacian parabolic equations with polynomial-type reactions

Abstract

In this paper, we study long-time behaviours of solutions to the discrete reaction-diffusion equations with nontrivial and nonnegative initial data, under the mixed boundary conditions. In particular, the function f satisfies for some constants and positive real numbers p and q. The purpose of this paper is to introduce a critical set depending on the nonnegative continuous function ψ. Also, we discuss about Fujita-type blow-up, which depends on the critical set .