Blow-up of solutions to a Dirichlet problem for the discrete semi-linear   heat equation
  In this paper, the initial and boundary problem of the difference equation which is a discretization of the semi-linear heat equation. The difference equation derived by discretizing the semi-linear heat equation has solutions which show characteristics corresponding to the characteristics of the blow-up solutions for the original equation. The initial and boundary problem for the original equation has blow-up solutions when a certain condition is met. We prove that when a similar condition as that of the original solution is met in the initial and boundary problem for the difference equation, the difference equation has blow-up solutions having characteristics corresponding to the characteristics of the blow-up solutions for the original equation. 
Keisuke Matsuya
math.AP