A condition for blow-up solutions to discrete p-Laplacian parabolic equations under the mixed boundary conditions on networks

In this paper, we investigate the condition (C-p) alpha integral(u)(0) f(s) ds <= uf (u) + beta u(p) + gamma, u > 0 for some alpha > 2, gamma > 0, and 0 <= beta <= (alpha p)lambda(p,0)/p, where p > 1, and lambda(p,0) is the first eigenvalue of the discrete p-Laplacian Delta(p,omega). Using this condition, we obtain blow-up solutions to discrete p-Laplacian parabolic equations {u(t)(X, t) = Delta(p,omega)u(X, t) + f (u(X, t)), (X, t) is an element of S x (0,+infinity), mu(Z) partial derivative u/partial derivative(pn) (X, t) + sigma(Z)vertical bar u(X, t)vertical bar(p-2) 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, on a discrete network S, where partial derivative u/partial derivative(pn) denotes the discrete p-normal derivative. Here mu and sigma are nonnegative functions on the boundary partial derivative S of S with mu(z) + sigma(z) > 0, z is an element of partial derivative S. In fact, we will see that condition (C-p) improves the conditions known so far.