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

TL;DR

This paper investigates the conditions under which solutions to a discrete semilinear heat equation exist globally or blow up, revealing a critical parameter threshold analogous to the continuous case.

Contribution

It establishes the exact role of the nonlinearity parameter in the discrete setting, identifying the critical threshold for global solution existence.

Findings

Non-trivial global solutions exist for ter /d with small initial data.

No non-trivial global solutions for 0 <  /d.

The parameter  determines solution behavior similarly to the continuous case.

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 \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.