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

In this paper, we study long-time behaviours of solutions to the discrete reaction-diffusion equations u(t) = Delta(omega)u + psi(t)f(u) with nontrivial and nonnegative initial data, under the mixed boundary conditions. In particular, the function f satisfies alpha (u(q) + u(p)) <= f(u) <= beta (u(q) + u(p)) for some constants 0 < alpha <= beta and positive real numbers p and q. The purpose of this paper is to introduce a critical set Lambda(psi) depending on the nonnegative continuous function psi. Also, we discuss about Fujita-type blow-up, which depends on the critical set Lambda(psi).