A new condition for blow-up solutions to discrete semilinear heat equations on networks

The purpose of this paper is to introduce a new condition (C) alpha integral(u)(0) f(s)ds <= uf (u) + beta u(2) + gamma, u > 0 for some alpha, beta, gamma > 0 with 0 < beta <= (alpha-2)lambda(0)/2, where lambda(0) is the first eigenvalue of discrete Laplacian Delta(omega) with which we obtain blow-up solutions to discrete semilinear heat equations {u(t) (x, t) = Delta(omega)u (x, t) +f (u(x, t)), (x, t) is an element of S x (0, +infinity), u (x, t) = 0, (x, t) is an element of partial derivative S x [0, +infinity), u (x, 0) = u(0) >= 0 (nontrivial), x is an element of<(S)over bar> on a discrete network S. In fact, it will be seen that the condition (C) improves the conditions known so far. (C) 2017 Elsevier Ltd. All rights reserved.