DCDS
Existence and non-existence of global solutions for a discrete semilinear heat equation
Keisuke Matsuya Tetsuji Tokihiro
Discrete & Continuous Dynamical Systems - A 2011, 31(1): 209-220 doi: 10.3934/dcds.2011.31.209
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.
keywords: global solution. semilinear heat equation Discretization

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