Abstract
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 αin 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 < α\le 2/d, and, for α> 2/d, existence of non-trivial global solutions for sufficiently small initial data.
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