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

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  • 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.
    Mathematics Subject Classification: Primary: 39A14, 74G25; Secondary:35K58, 39A12.

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