Existence and non-existence of global solutions for a discrete semilinear heat equation

Existence of global solutions to initial value problems for a discrete analogue of a d-dimensional semilinear heat equation is investigated. We prove that a parameter \alpha in the partial difference equation plays exactly the same role as the parameter of nonlinearity does in the semilinear heat equation. That is, we prove non-existence of a non-trivial global solution for 0 < \alpha \le 2/d, and, for \alpha > 2/d, existence of non-trivial global solutions for sufficiently small initial data.